If you are aware of an interesting new academic paper (that has been published in a peer-reviewed journal or has appeared on the arXiv), a conference talk (at an official professional scientific meeting), an external blog post (by a professional scientist) or a news item (in the mainstream news media), which you think might make an interesting topic for an FQXi blog post, then please contact us at forums@fqxi.org with a link to the original source and a sentence about why you think that the work is worthy of discussion. Please note that we receive many such suggestions and while we endeavour to respond to them, we may not be able to reply to all suggestions.

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Please also note that we do not accept unsolicited posts and we cannot review, or open new threads for, unsolicited articles or papers. Requests to review or post such materials will not be answered. If you have your own novel physics theory or model, which you would like to post for further discussion among then FQXi community, then please add them directly to the "Alternative Models of Reality" thread, or to the "Alternative Models of Cosmology" thread. Thank you.

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How can mindless mathematical laws give rise to aims and intention?

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RECENT POSTS IN THIS TOPIC

**Steve Dufourny**: *on* 9/5/15 at 10:16am UTC, wrote You are welcome, sincerilly

**Cristinel Stoica**: *on* 9/4/15 at 15:10pm UTC, wrote Dear Steve, Thank you very much for your kind and interesting comments. ...

**Steve Dufourny**: *on* 9/3/15 at 9:28am UTC, wrote Hello dear Mr Stoica, I read your essay, congratulations for your prize. I...

**Cristinel Stoica**: *on* 6/11/15 at 10:25am UTC, wrote Thanks, Christine, and congratulations to you too, for your beautiful...

**Christine Dantas**: *on* 6/11/15 at 10:12am UTC, wrote Congratulations! Kind regards, Christine

**Thomas Ray**: *on* 4/22/15 at 21:34pm UTC, wrote Ditto, re Petkov. :-)

**Cristinel Stoica**: *on* 4/22/15 at 21:16pm UTC, wrote Hi Jon, Thank you for the kind and interesting comments. I am not sure if...

**Cristinel Stoica**: *on* 4/22/15 at 21:08pm UTC, wrote Dear Tom, Thank you for noticing. Indeed, my Ph.D. Thesis appeared...

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RECENT ARTICLES

A tiny cooling device could help rewrite the thermodynamic rule book for quantum machines.

Philosopher Jenann Ismael invokes the thermodynamic arrow of time to explain how human intelligence emerged through culture.

Grounded physicists are exploring the use of online and virtual-reality conferencing, and AI-controlled experiments, to maintain social distancing. Post-pandemic, these positive innovations could make science more accessible and environmentally-friendly.

Untangling how the human perception of cause-and-effect might arise from quantum physics, may help us understand the limits and the potential of AI.

Physicists are using optogenetics techniques to make a rudimentary agent, from cellular components, which can convert measurements into actions using light.

FQXi FORUM

February 28, 2021

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: And the math will set you free by Cristinel Stoica [refresh]

TOPIC: And the math will set you free by Cristinel Stoica [refresh]

Can mathematics help us find our way through all the wonders and mysteries of the universe? When physicists describe the laws governing the physical world, mathematics is always involved. Is this due to the fact that the universe is, at least in part, mathematical? Or rather mathematics is merely a tool used by physicists to model phenomena? Is mathematics just a language to tell the story of our universe, a story which could be told with the same or even more effectiveness using another language? Or quite the opposite, the universe is just a mathematical structure?

Theoretical physicist. Research interests: foundations of physics, gauge theory, foundations of quantum mechanics, singularities in general relativity. Interested especially in the geometric aspects of the physical laws. ArXiv: http://arxiv.org/a/stoica_o_1 Blog: http://www.unitaryflow.com/ Scholar: https://scholar.google.com/citations?user=aleEOtsAAAAJ

ad Essay PDF File

Dear Cristi,

You say "*at least we know that there is room for free will*." You know that and I know that, though many dispute it. But free will, in any meaningful sense, is not deterministic, and, although one may draw an analogy with random numbers, I don't think one can draw it in any detail.

Thus I see no reason to contend that free will is isomorphic to math. Even if one...

view entire post

You say "

Thus I see no reason to contend that free will is isomorphic to math. Even if one...

view entire post

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Dear Edwin,

You say You say "at least we know that there is room for free will." and "Thus I see no reason to contend that free will is isomorphic to math.".

I wrote "at least we know that there is room for free will, whatever this may be", and also, when talking about neural correlates, I said "But by this I don't claim we can explain consciousness,...

view entire post

You say You say "at least we know that there is room for free will." and "Thus I see no reason to contend that free will is isomorphic to math.".

I wrote "at least we know that there is room for free will, whatever this may be", and also, when talking about neural correlates, I said "But by this I don't claim we can explain consciousness,...

view entire post

Dear Cristi,

I agree with both of your statements about free will and consciousness.

Your attitude toward Bell's theorem is reasonable. If you read my essay and understand it, but find it in error, there is no need to try to convince me. Yet I would appreciate it if you could give a hint of where you find the error.

Please understand that I do not claim his mathematical...

view entire post

I agree with both of your statements about free will and consciousness.

Your attitude toward Bell's theorem is reasonable. If you read my essay and understand it, but find it in error, there is no need to try to convince me. Yet I would appreciate it if you could give a hint of where you find the error.

Please understand that I do not claim his mathematical...

view entire post

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Dear Edwin,

I completely agree with you that we should question "The Gospel According to Bell" or anything we accept to be true, including classical locality. Bell himself was led to his results by questioning the results against hidden variables accepted in his time (so you are in good company). Reconsidering the foundations should be done systematically, just like one cleans our houses. I...

view entire post

I completely agree with you that we should question "The Gospel According to Bell" or anything we accept to be true, including classical locality. Bell himself was led to his results by questioning the results against hidden variables accepted in his time (so you are in good company). Reconsidering the foundations should be done systematically, just like one cleans our houses. I...

view entire post

"But can we find a property of time that can't possibly be described by mathematics? In fact, time was best understood due to mathematics, in relativity and thermodynamics."

This "understanding" is based on postulates that could be false. For instance, Einstein could have mistakenly "taken from the idea of light waves in the ether the one aspect that he needed" - then not only the...

view entire post

This "understanding" is based on postulates that could be false. For instance, Einstein could have mistakenly "taken from the idea of light waves in the ether the one aspect that he needed" - then not only the...

view entire post

report post as inappropriate

This is how science works, if a theory turns out to be wrong, we try to find a better one. I doubt that so far relativity and theoretical physics as a whole turned out to be so wrong, but I admit they don't answer everything yet, and they may be replaced by completely different theories someday.

Cristinel,

After Phipps found better Maxwell's equations, we don't even try and admit what is wrong with Poincaré's Lorentz transformation. This is how science doesn't work.

Everybody persistently uses the expression Michelson-Morley experiment although Morley did definitely not contribute to the concept of Michelson's experiments in Potsdam 1881 and in Cleveland...

view entire post

After Phipps found better Maxwell's equations, we don't even try and admit what is wrong with Poincaré's Lorentz transformation. This is how science doesn't work.

Everybody persistently uses the expression Michelson-Morley experiment although Morley did definitely not contribute to the concept of Michelson's experiments in Potsdam 1881 and in Cleveland...

view entire post

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Dear Eckard,

You said "Perhaps you cannot even calculate how many grandchildren your grandchildren will have."

Well, I frankly can't even calculate how many children I will have, let alone the grandchildren of my grandchildren :). But this is not what I said to Michel. I was talking about the power of mathematics to describe the objects we know, not to those we don't know, or which...

view entire post

You said "Perhaps you cannot even calculate how many grandchildren your grandchildren will have."

Well, I frankly can't even calculate how many children I will have, let alone the grandchildren of my grandchildren :). But this is not what I said to Michel. I was talking about the power of mathematics to describe the objects we know, not to those we don't know, or which...

view entire post

Dear Christi,

You try to go very far in the identification of the object and the words/mathematics that describe it. Your map of all sentences to numbers in the segment [0,1] is very illustrative. But you could as well have selected the formalism of QM where epistemological statements are the rule. I understand from your very good essay that you are a platonist, isn'it? What opinion do you have about the ontology of physics? Do you see a cutting edge betwwen the map/description and the physical object? Myself I am reluctant in accepting that physics is just maths and I am not sure that this reluctance can be formalized.

Michel

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You try to go very far in the identification of the object and the words/mathematics that describe it. Your map of all sentences to numbers in the segment [0,1] is very illustrative. But you could as well have selected the formalism of QM where epistemological statements are the rule. I understand from your very good essay that you are a platonist, isn'it? What opinion do you have about the ontology of physics? Do you see a cutting edge betwwen the map/description and the physical object? Myself I am reluctant in accepting that physics is just maths and I am not sure that this reluctance can be formalized.

Michel

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Dear Michel,

I am not sure I identify objects with mathematical descriptions. I just say it is possible to find for any object a mathematical description that captures everything that can be said about that object, precisely because mathematics is so versatile. And it is versatile even if our world is not really a mathematical structure, and that's why we can't really distinguish the two. The introductory discussion is just a toy to make a point. To make everything a point :)

I don't consider myself a Platonist. I don't care about abstract mathematical worlds, disconnected from our physical world. But I think that our world is indistinguishable from a mathematical structure, and if we rely on something non-mathematical, this is just provisional.

I think that we should admit*supermathematical** descriptions as final only if we are sure that we exhausted any hope for a *mathematical* description. And I don't think this is possible

Best wishes,

Cristi

_______________

**Supermathematical* is to *mathematical* what *supernatural* is to *natural*.

I am not sure I identify objects with mathematical descriptions. I just say it is possible to find for any object a mathematical description that captures everything that can be said about that object, precisely because mathematics is so versatile. And it is versatile even if our world is not really a mathematical structure, and that's why we can't really distinguish the two. The introductory discussion is just a toy to make a point. To make everything a point :)

I don't consider myself a Platonist. I don't care about abstract mathematical worlds, disconnected from our physical world. But I think that our world is indistinguishable from a mathematical structure, and if we rely on something non-mathematical, this is just provisional.

I think that we should admit

Best wishes,

Cristi

_______________

*

Dear Cristinel Stoica,

In your essay you wrote, "... it may be possible that we will be able to find the fundamental physical laws. The reason is that it seems that the universe seems to be very regular. The physical laws appear to be the same at any point and at any time." Does each universe in the multiverse have precisely the same physical laws? What, in your opinion, might be the explanation for the space roar?

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In your essay you wrote, "... it may be possible that we will be able to find the fundamental physical laws. The reason is that it seems that the universe seems to be very regular. The physical laws appear to be the same at any point and at any time." Does each universe in the multiverse have precisely the same physical laws? What, in your opinion, might be the explanation for the space roar?

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Dear David Brown,

You ask*Does each universe in the multiverse have precisely the same physical laws?*

I don't know what's outside our universe, assuming there is something. But if you refer to a particular multiverse theory, it comes with its own answer. For example, in the Many World Interpretation, as well in the multiverse predicted by some theories of inflation, the fundamental laws are the same. However, the laws that are not fundamental and depend on symmetry breaking may be different.

You ask*What, in your opinion, might be the explanation for the space roar?*

I didn't study the space roar and I don't have an opinion about it.

Cristi

You ask

I don't know what's outside our universe, assuming there is something. But if you refer to a particular multiverse theory, it comes with its own answer. For example, in the Many World Interpretation, as well in the multiverse predicted by some theories of inflation, the fundamental laws are the same. However, the laws that are not fundamental and depend on symmetry breaking may be different.

You ask

I didn't study the space roar and I don't have an opinion about it.

Cristi

Dear Cristi,

I always enjoy reading your essay, full of refreshing information and rational, logical and complete argument. I completely agree with you that there must be one coherent theory with one formula that produces only one Integer number normalized one "1" based on one principle derived from one source. I also completely in agreement with your hypothesis that "our universe has to be Turing complete". This is because the universe wants us to know its secret recipe not only for our own survival but also for the survival of Existence (Creator) itself. If not, it is not possible for mere human like us to find this theory and its formula. In other words, this theory of everything not only knows the mind of our Creator but it is the Creator itself, so that Existence, hence the Creator, would avoid its own extinction by allowing us, the avatar Creator, to save ourselves and our Creator. This is because Existence is precariously in the border of collapse and it does need fine tuning to stay on existing. Existence is living dangerously from extinction at any time, because Existence supposed not to exist. If I may plead that KQID might be such theory that would deliver the good.

Best wishes,

Leo KoGuan

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I always enjoy reading your essay, full of refreshing information and rational, logical and complete argument. I completely agree with you that there must be one coherent theory with one formula that produces only one Integer number normalized one "1" based on one principle derived from one source. I also completely in agreement with your hypothesis that "our universe has to be Turing complete". This is because the universe wants us to know its secret recipe not only for our own survival but also for the survival of Existence (Creator) itself. If not, it is not possible for mere human like us to find this theory and its formula. In other words, this theory of everything not only knows the mind of our Creator but it is the Creator itself, so that Existence, hence the Creator, would avoid its own extinction by allowing us, the avatar Creator, to save ourselves and our Creator. This is because Existence is precariously in the border of collapse and it does need fine tuning to stay on existing. Existence is living dangerously from extinction at any time, because Existence supposed not to exist. If I may plead that KQID might be such theory that would deliver the good.

Best wishes,

Leo KoGuan

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Dear Leo,

Thank you for the interesting comments. As you have noticed, I also tend to adopt the optimistic position that reason can lead us far, because the alternative seems to me that is limiting us, or even makes us focusing on excuses to avoid going deeper. By optimism I don't understand thinking that everything is understood, since this would be as unproductive as thinking we can't understand more.

Best wishes,

Cristi

Thank you for the interesting comments. As you have noticed, I also tend to adopt the optimistic position that reason can lead us far, because the alternative seems to me that is limiting us, or even makes us focusing on excuses to avoid going deeper. By optimism I don't understand thinking that everything is understood, since this would be as unproductive as thinking we can't understand more.

Best wishes,

Cristi

Dear Cristi,

On the contrary, it is like knowing the rule of the game of Chess, this enabling us to play Chess and the more we play the better we will become. More fun and exciting if we are playing against a grandmaster of Chess, losing and winning is part of the game. Yes, we are in the middle of mathematical game according to FQXI as the creature of Max Tegmark as the creator of this game. Let's continuing on playing as Newton pointed out that the Creator is assumed to help him doing the adjustment to fix the orbits correctly so that our solar system ( our universe at that time) are well behaving and not falling apart. My discovery points me that rather than our Creator is busy in doing so many adjustments in trillions trillions solar systems out there, we as the avatars of our Creator would be the ones doing the adjustments to save ourselves and be the Guardian of Existence.

No bad a job eh, being the guardians of our universe, we will be very busy like in the cartoon,

Leo KoGuan

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On the contrary, it is like knowing the rule of the game of Chess, this enabling us to play Chess and the more we play the better we will become. More fun and exciting if we are playing against a grandmaster of Chess, losing and winning is part of the game. Yes, we are in the middle of mathematical game according to FQXI as the creature of Max Tegmark as the creator of this game. Let's continuing on playing as Newton pointed out that the Creator is assumed to help him doing the adjustment to fix the orbits correctly so that our solar system ( our universe at that time) are well behaving and not falling apart. My discovery points me that rather than our Creator is busy in doing so many adjustments in trillions trillions solar systems out there, we as the avatars of our Creator would be the ones doing the adjustments to save ourselves and be the Guardian of Existence.

No bad a job eh, being the guardians of our universe, we will be very busy like in the cartoon,

Leo KoGuan

report post as inappropriate

Dear Dr. Stoica:

Your essay focuses on the liberating aspect of mathematics: "And the Math Will Set You Free."

In contrast, my essay asserts that mathematics can enslave us and blind us from the truth. "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory" questions the orthodox Hilbert-Space Model of QM, and presents a simple realistic picture that makes directly testable experimental predictions, based on little more than Stern-Gerlach measurements. Remarkably, these simple experiments have never been done.

The accepted view of QM is that the physics (and mathematics) of the microworld are fundamentally different from those of the macroworld, which of course creates an inevitable boundary problem. I take the radical (and heretical) view that the fundamental organization is the same on both scales, so that the boundary problem immediately disappears. Quantum indeterminacy, superposition, and entanglement are artifacts of the inappropriate mathematical formalism. QM is not a universal theory of matter; it is rather a mechanism for distributed vector fields to self-organize into spin-quantized coherent domains similar to solitons. This requires nonlinear mathematics that is not present in the standard formalism.

So while mathematics provides essential insights into physics, an incorrect mathematical model that becomes established may be seen as virtually religious dogma which is not to be questioned. That prevents further progress.

Alan Kadin

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Your essay focuses on the liberating aspect of mathematics: "And the Math Will Set You Free."

In contrast, my essay asserts that mathematics can enslave us and blind us from the truth. "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory" questions the orthodox Hilbert-Space Model of QM, and presents a simple realistic picture that makes directly testable experimental predictions, based on little more than Stern-Gerlach measurements. Remarkably, these simple experiments have never been done.

The accepted view of QM is that the physics (and mathematics) of the microworld are fundamentally different from those of the macroworld, which of course creates an inevitable boundary problem. I take the radical (and heretical) view that the fundamental organization is the same on both scales, so that the boundary problem immediately disappears. Quantum indeterminacy, superposition, and entanglement are artifacts of the inappropriate mathematical formalism. QM is not a universal theory of matter; it is rather a mechanism for distributed vector fields to self-organize into spin-quantized coherent domains similar to solitons. This requires nonlinear mathematics that is not present in the standard formalism.

So while mathematics provides essential insights into physics, an incorrect mathematical model that becomes established may be seen as virtually religious dogma which is not to be questioned. That prevents further progress.

Alan Kadin

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Dear Dr. Kadin,

I completely agree with you that insisting on a mathematical model despite contrary evidence may blind us and lead us far from truth. But if someone hurts his finger with a hammer, it's not the fault of the hammer. This doesn't mean we can't make a better hammer. People can make anything into a dogma, and is their fault here, not of that thing. We should question anything.

Best regards,

Cristi

I completely agree with you that insisting on a mathematical model despite contrary evidence may blind us and lead us far from truth. But if someone hurts his finger with a hammer, it's not the fault of the hammer. This doesn't mean we can't make a better hammer. People can make anything into a dogma, and is their fault here, not of that thing. We should question anything.

Best regards,

Cristi

Dear Cristi Stoica,

Enjoyed your essay. Your comment "there must exist a mathematical structure which satisfies our observations about both the quantum world, and the general relativistic one. Maybe these theories are somehow limits of this theory. But the unified theory must exist, even if we don’t have it yet." was well argued and very believable.

I agree with your comment "The idea that the universe is nothing but a mathematical structure leads to many difficult and interesting questions".

My essay revolves around modelling the mathematical structures of the particles of the standard model. I hope you get a chance to have a look at the modelling I use for the particles of the standard model and look forward to any comments you may have.

Good luck on your essay, you deserve a good rating.

Regards,

Ed Unverricht

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Enjoyed your essay. Your comment "there must exist a mathematical structure which satisfies our observations about both the quantum world, and the general relativistic one. Maybe these theories are somehow limits of this theory. But the unified theory must exist, even if we don’t have it yet." was well argued and very believable.

I agree with your comment "The idea that the universe is nothing but a mathematical structure leads to many difficult and interesting questions".

My essay revolves around modelling the mathematical structures of the particles of the standard model. I hope you get a chance to have a look at the modelling I use for the particles of the standard model and look forward to any comments you may have.

Good luck on your essay, you deserve a good rating.

Regards,

Ed Unverricht

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Dear Cristi,

As always I greatly enjoyed you essay. If I understand you, the position you are taking is that we live in a mathematical structure, but can never express the full intricacy of that structure. The purpose of physics is to express the general laws that describe our universe:

“However, we are just looking for a theory describing the general laws, and not a complete description of this particular instance of the universe, which includes what every human thinks about the universe and themselves. This would not be feasible anyway for practical reasons.”

Is my interpretation of your idea correct? And if so, where would a mathematical structure that failed to match the universe’s general laws be said to exist? Or should we take it that one class of mathematical objects conforms to the real world and another are just humanly constructed?

Also, if you get a chance, please check out my essay where I imagine how a weak version of the MUH might be testable and let me know what you think along with your vote.

http://fqxi.org/community/forum/topic/2391

Best of luck in the contest!

Rick Searle

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As always I greatly enjoyed you essay. If I understand you, the position you are taking is that we live in a mathematical structure, but can never express the full intricacy of that structure. The purpose of physics is to express the general laws that describe our universe:

“However, we are just looking for a theory describing the general laws, and not a complete description of this particular instance of the universe, which includes what every human thinks about the universe and themselves. This would not be feasible anyway for practical reasons.”

Is my interpretation of your idea correct? And if so, where would a mathematical structure that failed to match the universe’s general laws be said to exist? Or should we take it that one class of mathematical objects conforms to the real world and another are just humanly constructed?

Also, if you get a chance, please check out my essay where I imagine how a weak version of the MUH might be testable and let me know what you think along with your vote.

http://fqxi.org/community/forum/topic/2391

Best of luck in the contest!

Rick Searle

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Dear Rick,

Thank you for the comments. Yes, I think that the universal laws can be known, but all details of the particular configuration can't be fully accessed by us, which are subsystems of the world, limited in time and space.

> where would a mathematical structure that failed to match the universe’s general laws be said to exist? Or should we take it that one class of mathematical objects conforms to the real world and another are just humanly constructed?

I find more compelling the idea that all mathematical structures have equal existence, but if our universe is a mathematical structure, then the other mathematical structures we know were obviously (re)constructed by us, since I find less likely that we gained some access to worlds outside ours. And if we will find the mathematical structure capturing completely the laws of our world, then we will do it by reason, hence by reconstruction too. However, what I argued is that if MUH is correct, we can't really prove it, not only because we are confined to only one of these structures. I reject Tegmark's attempt of proof on the grounds that if the universe, including us, is computational, then any universe with equal computational power offers the same benefits for the occurrence of intelligent life. But maybe other proofs can work, so I look forward to see your arguments for the testability of a weaker version of the MUH.

Best wishes,

Cristi

Thank you for the comments. Yes, I think that the universal laws can be known, but all details of the particular configuration can't be fully accessed by us, which are subsystems of the world, limited in time and space.

> where would a mathematical structure that failed to match the universe’s general laws be said to exist? Or should we take it that one class of mathematical objects conforms to the real world and another are just humanly constructed?

I find more compelling the idea that all mathematical structures have equal existence, but if our universe is a mathematical structure, then the other mathematical structures we know were obviously (re)constructed by us, since I find less likely that we gained some access to worlds outside ours. And if we will find the mathematical structure capturing completely the laws of our world, then we will do it by reason, hence by reconstruction too. However, what I argued is that if MUH is correct, we can't really prove it, not only because we are confined to only one of these structures. I reject Tegmark's attempt of proof on the grounds that if the universe, including us, is computational, then any universe with equal computational power offers the same benefits for the occurrence of intelligent life. But maybe other proofs can work, so I look forward to see your arguments for the testability of a weaker version of the MUH.

Best wishes,

Cristi

Cristi,

I will get to your essay as soon as possible. I have been on travel, but will be able to read it in a few days. Good luck with this.

Cheers LC

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I will get to your essay as soon as possible. I have been on travel, but will be able to read it in a few days. Good luck with this.

Cheers LC

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Do we really need math in physics? :)

Cool T-shirt! It would also work with "Mathematician" and "Physicist" reversed, which proves that math is physics and physics is math!

Marc

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Marc

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Dear Cristi,

I read with great interest your depth analytical essays. I totally agree: "So, if we can understand the universe, it is because this immense complexity can be reduced to a small number of laws. And this reduction is made possible by mathematics." The way is simple: Mathematics as constructive ontology should "grab" the dialectic of eidos and logos, the dialectic of absolute forms of existence of matter.

Kind regards,

Vladimir

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I read with great interest your depth analytical essays. I totally agree: "So, if we can understand the universe, it is because this immense complexity can be reduced to a small number of laws. And this reduction is made possible by mathematics." The way is simple: Mathematics as constructive ontology should "grab" the dialectic of eidos and logos, the dialectic of absolute forms of existence of matter.

Kind regards,

Vladimir

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Dear Vladimir,

Thank you for the kind comments. We seem to be lost in this complex diversity exhibited by the universe. So having some guiding order principles can help us making sense of the world. I look forward to read your essay.

Best wishes,

Cristi

Thank you for the kind comments. We seem to be lost in this complex diversity exhibited by the universe. So having some guiding order principles can help us making sense of the world. I look forward to read your essay.

Best wishes,

Cristi

You're right, Cristi. I invite you to see my analysis of the philosophical foundations of mathematics and physics, the method of ontological constructing of the primordial generating structure, "La Structure mère" as the ontological framework, carcass and foundation of knowledge, the core of which - **the ontological (structural, cosmic) memory**, and information - polyvalent phenomenon of the ontological (structural) memory of Universum as a whole. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld» (E.Husserl), as a picture of the world lyricists , poets and philosophers.

Kind regards,

Vladimir

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Kind regards,

Vladimir

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Dear Cristi,

In your excellent essay you ask crucial questions in current debate: what is mathematics? what is physics? and linking these: is mathematics merely a tool in physics? Referring also to Tegmark, I agree that everything is isomorphic to a mathematical structure and analogously to MUH I have coined Geometrical Universe Hypothesis. I propose to use the geometrization conjecture, proved by Perelman. We have the set of 8 Thurston geometries. We can treat them as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime… and get all interactions and matter.

GUH makes the testable prediction resulting from Thurston geometries.

If you are interested you can find details in my essay

http://fqxi.org/community/forum/topic/2452

I would appreciate your comments. Thank you.

Jacek

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In your excellent essay you ask crucial questions in current debate: what is mathematics? what is physics? and linking these: is mathematics merely a tool in physics? Referring also to Tegmark, I agree that everything is isomorphic to a mathematical structure and analogously to MUH I have coined Geometrical Universe Hypothesis. I propose to use the geometrization conjecture, proved by Perelman. We have the set of 8 Thurston geometries. We can treat them as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime… and get all interactions and matter.

GUH makes the testable prediction resulting from Thurston geometries.

If you are interested you can find details in my essay

http://fqxi.org/community/forum/topic/2452

I would appreciate your comments. Thank you.

Jacek

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Dear Dr. Stoica,

Could you please explain to me why you thought that my comment about the real Universe was inappropriate?

You are I hope aware that suppression of the truth is unethical.

Eagerly awaiting your answer,

Joe Fisher

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Could you please explain to me why you thought that my comment about the real Universe was inappropriate?

You are I hope aware that suppression of the truth is unethical.

Eagerly awaiting your answer,

Joe Fisher

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Dear Mr. Fisher,

I see that your comment, which was rather nice and interesting, along with my reply to it, are missing. I can only assume that someone considered them inappropriate, but it wasn't me that person, so I can't explain why they are missing. Have you tried to ask the administrator to allow them here again, or ask him what happened, rather than accusing me of "suppression of the truth"?

Best wishes,

Cristi

I see that your comment, which was rather nice and interesting, along with my reply to it, are missing. I can only assume that someone considered them inappropriate, but it wasn't me that person, so I can't explain why they are missing. Have you tried to ask the administrator to allow them here again, or ask him what happened, rather than accusing me of "suppression of the truth"?

Best wishes,

Cristi

I have asked Dr. Foster, the moderator of this competition and he claims that he did not remove it and for me to enquire of the person at the site. I posted a revised comment to your site, and that has been removed. I will repost for a third time.

Ruefully,

Joe Fisher

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Ruefully,

Joe Fisher

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Dear Dr. Stoica,

You wrote: “Some may hope that there are things in the universe which can't be described by mathematics. But can you name those things?” Thank you ever so much for asking. Please read and try to understand my definitive answer:

Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer...

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You wrote: “Some may hope that there are things in the universe which can't be described by mathematics. But can you name those things?” Thank you ever so much for asking. Please read and try to understand my definitive answer:

Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer...

view entire post

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Dear Cristi,

What a great essay! I enjoyed reading it very much. The topic of your essay echos that of mine . However, my collaborator and I take the view that mathematics is invented, and from that we question its accuracy in describing nature.

I particularly liked your arguments for a theory of everything. I agree with you that there must be a unified theory because the universe obeys one set of laws. I used to think Godel's theorem might prevent us from finding that theory, but you provided compelling reasons against that argument. Maybe we can find that theory, and maybe we cannot. Either way, we can only try.

Good luck in the contest.

Mohammed

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What a great essay! I enjoyed reading it very much. The topic of your essay echos that of mine . However, my collaborator and I take the view that mathematics is invented, and from that we question its accuracy in describing nature.

I particularly liked your arguments for a theory of everything. I agree with you that there must be a unified theory because the universe obeys one set of laws. I used to think Godel's theorem might prevent us from finding that theory, but you provided compelling reasons against that argument. Maybe we can find that theory, and maybe we cannot. Either way, we can only try.

Good luck in the contest.

Mohammed

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Dear Mohammed,

Thank you for the kind and interesting comments. We seem to agree that humans can at least hope to find the theory of everything. Maybe we have different opinions on what it will be, but we agree that we don't know it yet, so maybe it will be as one of us thinks it is, or totally different, who knows. Good luck in the contest!

Best wishes,

Cristi

Thank you for the kind and interesting comments. We seem to agree that humans can at least hope to find the theory of everything. Maybe we have different opinions on what it will be, but we agree that we don't know it yet, so maybe it will be as one of us thinks it is, or totally different, who knows. Good luck in the contest!

Best wishes,

Cristi

Dear Cristi,

now I had the chance to read your essay. Very interesting. In many parts, we both agreed (or as Pauli expressed it in a letter to Heisenberg: "boring agreement" ;-)

But at one point I went farther than: in my opinion at the center of modern math is structure (as you explained in one section). Math is not a simple theory to relate one number to another, it is rather a theory of structures. Even this structures will be also important in the future for other sciences. You mention model theory and even this point is crucial: one can change the model (including the logic), i.e. one can change the math completely and only the structure is left.

Maybe an agreement again?

Torsten

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now I had the chance to read your essay. Very interesting. In many parts, we both agreed (or as Pauli expressed it in a letter to Heisenberg: "boring agreement" ;-)

But at one point I went farther than: in my opinion at the center of modern math is structure (as you explained in one section). Math is not a simple theory to relate one number to another, it is rather a theory of structures. Even this structures will be also important in the future for other sciences. You mention model theory and even this point is crucial: one can change the model (including the logic), i.e. one can change the math completely and only the structure is left.

Maybe an agreement again?

Torsten

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Dear Torsten,

Thank you for the kind comments. Yes, the "boring agreement" between our viewpoints extends to your comments too :) I think math is the science of mathematical structures, and these are beyond models. I started reading your essay, and I will comment on your wall soon.

Best wishes,

Cristi

Thank you for the kind comments. Yes, the "boring agreement" between our viewpoints extends to your comments too :) I think math is the science of mathematical structures, and these are beyond models. I started reading your essay, and I will comment on your wall soon.

Best wishes,

Cristi

Dear Cristi,

thanks for your words and vote, which I will give back! Other essays concentrated on the close relation between math and physics to forecast experiments. But not yours and I like it (and also your scientific work).

Best

Torsten

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thanks for your words and vote, which I will give back! Other essays concentrated on the close relation between math and physics to forecast experiments. But not yours and I like it (and also your scientific work).

Best

Torsten

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I get a little worried when Goedel comes into the picture. This is not because I don’t think at some point this might rear its head, but because as you say this is misapplied. I think the real question is whether because we are a part of the universe that our attempt to completely codify the universe runs into this sort of incompleteness. We are in a way a piece of the universe that is working to encode its foundations, which is similar to a formal system listing or computing all its Goedel numbers. We may have an issue of this sort with the so called measurement problem and our inability to rectify quantum mechanics and measured outcomes with our macroscopic based thinking.

I am not sure one could say that 19th century physics was a unified theory as such. There were the three main areas of classical-Hamiltonian mechanics, thermodynamics, and electromagnetism. Thermodynamics ran into trouble with classical mechanics over the issue of time reversal invariance. Electromagnetic theory was not able to figure out how an arrangement of charges could be stable in a classical mechanical form. Then of course there was that annoying black body problem. I think physics then was filled with open questions that betrayed any idea of unification. I suppose in many at the time were confident that as Raleigh put it that all would be working once that black body problem is solved.

LC

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I am not sure one could say that 19th century physics was a unified theory as such. There were the three main areas of classical-Hamiltonian mechanics, thermodynamics, and electromagnetism. Thermodynamics ran into trouble with classical mechanics over the issue of time reversal invariance. Electromagnetic theory was not able to figure out how an arrangement of charges could be stable in a classical mechanical form. Then of course there was that annoying black body problem. I think physics then was filled with open questions that betrayed any idea of unification. I suppose in many at the time were confident that as Raleigh put it that all would be working once that black body problem is solved.

LC

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Dear Lawrence,

Thank you for the comments. I agree that us being part of the universe makes impossible to us to completely codify the universe. However, I don't think that trying to encode just the foundations is the same as trying to describe the whole, because the fundamental laws probably need a very small amount of memory to encode. Maybe a T-shirt, but even if it would be a book, would not be too much. I can imagine a computer containing a lot of information in it, including the complete specifications of the computer itself. Of course, it would not be possible to contain along with the specifications also a copy of the complete information contained in it, because it would have also to contain a copy of the copy and we regress to infinity. It is true that Maxwell's equations and the black body radiation require relativity and quantum mechanics, but what if these two tensions never appeared? A universe governed only by classical mechanics can easily contain a book of classical mechanics, or at least the page containing Newton's laws, without regression to infinity. I look forward to read your essay, to see your approach to the problem.

Best wishes,

Cristi

Thank you for the comments. I agree that us being part of the universe makes impossible to us to completely codify the universe. However, I don't think that trying to encode just the foundations is the same as trying to describe the whole, because the fundamental laws probably need a very small amount of memory to encode. Maybe a T-shirt, but even if it would be a book, would not be too much. I can imagine a computer containing a lot of information in it, including the complete specifications of the computer itself. Of course, it would not be possible to contain along with the specifications also a copy of the complete information contained in it, because it would have also to contain a copy of the copy and we regress to infinity. It is true that Maxwell's equations and the black body radiation require relativity and quantum mechanics, but what if these two tensions never appeared? A universe governed only by classical mechanics can easily contain a book of classical mechanics, or at least the page containing Newton's laws, without regression to infinity. I look forward to read your essay, to see your approach to the problem.

Best wishes,

Cristi

Dear Cristinel,

About "And the math will set you free", it may be but only after you have realized the truth - and only "The truth shall set you free".

Man is a created being absolute as he cannot create himself. The Creator is the God of man - the God who divides as well as the God who unites. Everything that we know of as well as things that have never ever crossed our mind are all...

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About "And the math will set you free", it may be but only after you have realized the truth - and only "The truth shall set you free".

Man is a created being absolute as he cannot create himself. The Creator is the God of man - the God who divides as well as the God who unites. Everything that we know of as well as things that have never ever crossed our mind are all...

view entire post

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Very nice essay, Cristi -- with the broad range of profound and interesting questions that one has come to expect from you. Well done.

A couple of things:

Being a student of Yaneer Bar-Yam (who by the way was just appointed as visiting professor to the MIT media lab) -- I appreciate your reference to his theory of multi-scale variety, which I regard as one of the most important theoretical advances in science today. It encompasses network-connected phenomena across every scale of physical self-organization, which brings me to the second thing:

Your claim that " ... we can’t prove the mathematical universe hypothesis by Tegmark’s method."

If one is a rationalist in the sense of Karl Popper, no scientific theory is ever proved. That is, no amount of empirical evidence that validates a theory, is proof that the theory is true -- only a falsifiable criterion makes a theory scientific at all. And Tegmark's criterion is very clear and unambiguously stated: if the universe is shown to be fundamentally based in randomness, the mathematical universe hypothesis is falsified.

There is also no ambiguity in Popper's position. Long ago, Popper made an important reversal of himself -- as important as Hawking's reversal in the case of black hole information loss -- that Darwin's theory of common ancestry is a falsifiable theory, after all. Not because we can refute common ancestry by a single experiment, but because biology is incoherent without the theory, and the auxiliary hypotheses that support the theory are clearly falsifiable.

In the same sense, I think that multi-scale variety can be seen as a unifying theory of complex systems, and Tegmark's MUH as a unifying theory of mathematical physics.

All best,

Tom

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A couple of things:

Being a student of Yaneer Bar-Yam (who by the way was just appointed as visiting professor to the MIT media lab) -- I appreciate your reference to his theory of multi-scale variety, which I regard as one of the most important theoretical advances in science today. It encompasses network-connected phenomena across every scale of physical self-organization, which brings me to the second thing:

Your claim that " ... we can’t prove the mathematical universe hypothesis by Tegmark’s method."

If one is a rationalist in the sense of Karl Popper, no scientific theory is ever proved. That is, no amount of empirical evidence that validates a theory, is proof that the theory is true -- only a falsifiable criterion makes a theory scientific at all. And Tegmark's criterion is very clear and unambiguously stated: if the universe is shown to be fundamentally based in randomness, the mathematical universe hypothesis is falsified.

There is also no ambiguity in Popper's position. Long ago, Popper made an important reversal of himself -- as important as Hawking's reversal in the case of black hole information loss -- that Darwin's theory of common ancestry is a falsifiable theory, after all. Not because we can refute common ancestry by a single experiment, but because biology is incoherent without the theory, and the auxiliary hypotheses that support the theory are clearly falsifiable.

In the same sense, I think that multi-scale variety can be seen as a unifying theory of complex systems, and Tegmark's MUH as a unifying theory of mathematical physics.

All best,

Tom

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Dear Tom,

Thank you for the interesting comments. Sure, Popper is right, "no scientific theory is ever proved". My argument regarding Tegmark's MUH is indeed that it can't be falsified by his criterion. You say that the criterion is "if the universe is shown to be fundamentally based in randomness, the mathematical universe hypothesis is falsified". I am not aware of Tegmark saying this, perhaps I don't get its meaning. I don't see why a mathematical structure can't be fundamentally based in randomness. I think Tegmark's idea is that there is a statistical way to show that our universe is a typical one supporting intelligent life, and this is what I disputed. My argument was that, since he also want them to be computable, most structures are computationally equivalent, for all practical purpose, with ours, so his criterion will fail. I am happy you were a student of Yaneer Bar-Yam, probably was a very interesting professor.

Best wishes,

Cristi

Thank you for the interesting comments. Sure, Popper is right, "no scientific theory is ever proved". My argument regarding Tegmark's MUH is indeed that it can't be falsified by his criterion. You say that the criterion is "if the universe is shown to be fundamentally based in randomness, the mathematical universe hypothesis is falsified". I am not aware of Tegmark saying this, perhaps I don't get its meaning. I don't see why a mathematical structure can't be fundamentally based in randomness. I think Tegmark's idea is that there is a statistical way to show that our universe is a typical one supporting intelligent life, and this is what I disputed. My argument was that, since he also want them to be computable, most structures are computationally equivalent, for all practical purpose, with ours, so his criterion will fail. I am happy you were a student of Yaneer Bar-Yam, probably was a very interesting professor.

Best wishes,

Cristi

Hi Cristi,

I didn't interpret Tegmark's falsifying criterion into existence. He states it explicitly in his book (p.356): "Looking toward the future, thee are two possibilities: If I'm wrong and the MUH is false, then physics will eventually hit an insurmountable roadblock beyond which no further progress is possible; there would be no further mathematical regularities to discover even though we still lacked a complete description of our physical reality. For example, a convincing demonstration that there's such a thing as fundamental randomness in the laws of nature (as opposed to deterministic observer cloning that merely *feels* random subjectively) would therefore refute the MUH. If I'm right, on the other hand, then there'll be no roadblock in our quest to understand reality, and we're limited only by our imagination!"

I explore this falsifying criterion in my essay.

The question of computability enters because Max approaches the hypothesis the same way that Wigner describes the role of mathematics in physics: discovery of physical regularities leads to discovery of mathematical regularities. I argue that if MUH is true, it has to work both backward and forward -- i.e., discovery of mathematical regularities leads to discovery of physical regularities. MUH is true if, and only if, the hypothetical final theory is mathematically complete.

(By student of Bar-Yam, I didn't mean a classroom student but rather a follower. He is a dozen years younger than I -- showing once again that there's little correlation between age and wisdom.)

Best,

Tom

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I didn't interpret Tegmark's falsifying criterion into existence. He states it explicitly in his book (p.356): "Looking toward the future, thee are two possibilities: If I'm wrong and the MUH is false, then physics will eventually hit an insurmountable roadblock beyond which no further progress is possible; there would be no further mathematical regularities to discover even though we still lacked a complete description of our physical reality. For example, a convincing demonstration that there's such a thing as fundamental randomness in the laws of nature (as opposed to deterministic observer cloning that merely *feels* random subjectively) would therefore refute the MUH. If I'm right, on the other hand, then there'll be no roadblock in our quest to understand reality, and we're limited only by our imagination!"

I explore this falsifying criterion in my essay.

The question of computability enters because Max approaches the hypothesis the same way that Wigner describes the role of mathematics in physics: discovery of physical regularities leads to discovery of mathematical regularities. I argue that if MUH is true, it has to work both backward and forward -- i.e., discovery of mathematical regularities leads to discovery of physical regularities. MUH is true if, and only if, the hypothetical final theory is mathematically complete.

(By student of Bar-Yam, I didn't mean a classroom student but rather a follower. He is a dozen years younger than I -- showing once again that there's little correlation between age and wisdom.)

Best,

Tom

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Hi Cristi,

As I told you in my FQXi page, I have read your pretty Essay. Here are some comments:

1) I find the title very intriguing.

2) I think the most brilliant text ever written is Einstein's 1915 paper on general relativity!

3) The idea that all human creation and knowledge is a point in the interval [0, 1] is upsetting!

4) I disagree with Smolin's statement that "there is no mathematical object which is isomorphic to the universe as a whole". I think we merely do not know if such a mathematical object exists or not.

5) Congrats for generalizing the Weyl curvature hypothesis, you know that I am an estimator of your research work.

6) I do not think that you are too reductionist in claiming that there is a mathematical structure which is isomorphic to the universe described by the extended list of propositions. In fact, I agree with you.

7) I am not completely sure of the real existence of a a theory of everything, but it should be a great pity if it does not exist! The same for black holes!

8) Your statement that "if we can understand the universe, it is because this immense complexity can be reduced to a small number of laws" is very intriguing. It I think should have been appreciated by Einstein who claimed that "The most unintelligible thing about the universe is that it is intelligible at all."

In any case, the reading of your intriguing Essay was interesting and enjoyable. It surely deserves the highest score that I am going to give you.

I wish you best luck in the Contest.

Cheers, Ch.

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As I told you in my FQXi page, I have read your pretty Essay. Here are some comments:

1) I find the title very intriguing.

2) I think the most brilliant text ever written is Einstein's 1915 paper on general relativity!

3) The idea that all human creation and knowledge is a point in the interval [0, 1] is upsetting!

4) I disagree with Smolin's statement that "there is no mathematical object which is isomorphic to the universe as a whole". I think we merely do not know if such a mathematical object exists or not.

5) Congrats for generalizing the Weyl curvature hypothesis, you know that I am an estimator of your research work.

6) I do not think that you are too reductionist in claiming that there is a mathematical structure which is isomorphic to the universe described by the extended list of propositions. In fact, I agree with you.

7) I am not completely sure of the real existence of a a theory of everything, but it should be a great pity if it does not exist! The same for black holes!

8) Your statement that "if we can understand the universe, it is because this immense complexity can be reduced to a small number of laws" is very intriguing. It I think should have been appreciated by Einstein who claimed that "The most unintelligible thing about the universe is that it is intelligible at all."

In any case, the reading of your intriguing Essay was interesting and enjoyable. It surely deserves the highest score that I am going to give you.

I wish you best luck in the Contest.

Cheers, Ch.

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Hi Christian,

Thank you for the comments and for the kind words! With 2), I think I very much agree with you, but I wanted to connect with most readers, and perhaps upset others. With 3), as well as 1), indeed I wanted to upset the reader. So connect and upset, to make sure what you say will be remembered one way or another :)). About the other points, I think you are right too. I was happy to read your excellent essay, and I hope this edition you will get a well-deserved prize!

Best wishes,

Cristi

Thank you for the comments and for the kind words! With 2), I think I very much agree with you, but I wanted to connect with most readers, and perhaps upset others. With 3), as well as 1), indeed I wanted to upset the reader. So connect and upset, to make sure what you say will be remembered one way or another :)). About the other points, I think you are right too. I was happy to read your excellent essay, and I hope this edition you will get a well-deserved prize!

Best wishes,

Cristi

Dear Cristi,

Thanks for your kind words, which honour me. I am happy to see that we are in perfect accord.

I wish you will get a well-deserved prize too! Let us cross our fingers!

Cheers, Ch.

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Thanks for your kind words, which honour me. I am happy to see that we are in perfect accord.

I wish you will get a well-deserved prize too! Let us cross our fingers!

Cheers, Ch.

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Dear Cristi,

The introduction to your essay is very clever. Most people do not appreciate that when going from the rationals to the real numbers, one enters a whole new ballgame, so to say. I did not appreciate it until I began to study some set theory, but your example brings this home very nicely.

I essentially agree with almost everything in the essay, but would like to add the...

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The introduction to your essay is very clever. Most people do not appreciate that when going from the rationals to the real numbers, one enters a whole new ballgame, so to say. I did not appreciate it until I began to study some set theory, but your example brings this home very nicely.

I essentially agree with almost everything in the essay, but would like to add the...

view entire post

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Dear Armin,

Thank you for reading my essay and for the comments. You raised interesting points, and I will address them.

1. As you know, in mathematics the term "exists" is the same as in logic. For example, "there exists a field which extends the field of real numbers and is algebraically closed". This doesn't have the meaning of physical existence, but rather of logical...

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Thank you for reading my essay and for the comments. You raised interesting points, and I will address them.

1. As you know, in mathematics the term "exists" is the same as in logic. For example, "there exists a field which extends the field of real numbers and is algebraically closed". This doesn't have the meaning of physical existence, but rather of logical...

view entire post

Dear Christinel,

"When physicists describe the laws governing the physical world, mathematics is always involved." This sentence of your abstract is the only one without question mark.

Let me ask you to provide something indispensable but possibly neglected: What about causality and about the border between past and future?

Regards,

Eckard

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"When physicists describe the laws governing the physical world, mathematics is always involved." This sentence of your abstract is the only one without question mark.

Let me ask you to provide something indispensable but possibly neglected: What about causality and about the border between past and future?

Regards,

Eckard

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Dear Eckard,

Thank you for the questions, but I would like to ask you to be more specific. Could you please describe the specific features or instances of causality and present which you refer to, and why you consider that math is not involved in them? Or if this wasn't what you meant, could you please explain me the question?

Best regards,

Cristi

Thank you for the questions, but I would like to ask you to be more specific. Could you please describe the specific features or instances of causality and present which you refer to, and why you consider that math is not involved in them? Or if this wasn't what you meant, could you please explain me the question?

Best regards,

Cristi

Dear Cristi,

Being a fan of the good old Budeanu, I hope you as a new voice of Bucuresti might forgive me my misspelled Cristinel and decide without prejudice which variant is most appealing to you:

1) The late Einstein called the distinction between past and future an illusion. Nonetheless admitting that the now worries him seriously he considered it something outside science. In principle, this is the accepted "spacetime" view of modern physics.

2) Spencer Scoular instead argues for a qualitative theory of physics.

3) Tim Maudlin suggests a notion of number that provides the arrow of time. Spencer and Tim further discussed their views in Maudlin's thread.

4) I gave already in Fig. 1 of an earlier essay of mine an alternative explanation: Only elapsed time is measurable. Only future processes can be influenced. I agree with Tim Maudlin on the good old Euclidean notion of number as a measure, not a pebble.

5) You might have your own idea.

With best hopes and wishes,

Eckard

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Being a fan of the good old Budeanu, I hope you as a new voice of Bucuresti might forgive me my misspelled Cristinel and decide without prejudice which variant is most appealing to you:

1) The late Einstein called the distinction between past and future an illusion. Nonetheless admitting that the now worries him seriously he considered it something outside science. In principle, this is the accepted "spacetime" view of modern physics.

2) Spencer Scoular instead argues for a qualitative theory of physics.

3) Tim Maudlin suggests a notion of number that provides the arrow of time. Spencer and Tim further discussed their views in Maudlin's thread.

4) I gave already in Fig. 1 of an earlier essay of mine an alternative explanation: Only elapsed time is measurable. Only future processes can be influenced. I agree with Tim Maudlin on the good old Euclidean notion of number as a measure, not a pebble.

5) You might have your own idea.

With best hopes and wishes,

Eckard

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Dear Eckard,

Thank you for the clarifications.

I explained how I think time may be both appear to flow and be frozen here. The point is that whatever one may feel in the present, it has neural correlates and is recorded in the state of the universe at that point. Each instant contains the feelings that we have at that time, including the feeling of "now". And there is no need to be a highlight of one moment of time, since each of them contains the feeling that it is highlighted in its own present. I see no problem here. I agree that some may disagree, and this is why you can find as many references as you want. But no matter what property of time one may consider that it is characteristic to the now, it is part of the instant, of the slice of spacetime at that time, and each instant contains such thoughts. Yesterday you considered that that time was now, tomorrow you will say the same. Today you may say that yesterday isn't now, but yesterday you didn't. So I don't think it is so easy to find something that distinguishes now from other instants, except the fact that our instances in that moment call it now. Regarding endowing time with an arrow, this is done by the thermodynamical arrow. We don't know why the universe started with such a low entropy, but we do know that solving the problem for the big bang solves it for the other times too. Adding by hand an arrow, like Tim does, doesn't solve the issue, since to each of his structures there is a dual structure in which the directions are reversed, and there is no way to distinguish one from its dual. You can ask him, I am sure that this is what he will answer too.

Best regards,

Cristi

Thank you for the clarifications.

I explained how I think time may be both appear to flow and be frozen here. The point is that whatever one may feel in the present, it has neural correlates and is recorded in the state of the universe at that point. Each instant contains the feelings that we have at that time, including the feeling of "now". And there is no need to be a highlight of one moment of time, since each of them contains the feeling that it is highlighted in its own present. I see no problem here. I agree that some may disagree, and this is why you can find as many references as you want. But no matter what property of time one may consider that it is characteristic to the now, it is part of the instant, of the slice of spacetime at that time, and each instant contains such thoughts. Yesterday you considered that that time was now, tomorrow you will say the same. Today you may say that yesterday isn't now, but yesterday you didn't. So I don't think it is so easy to find something that distinguishes now from other instants, except the fact that our instances in that moment call it now. Regarding endowing time with an arrow, this is done by the thermodynamical arrow. We don't know why the universe started with such a low entropy, but we do know that solving the problem for the big bang solves it for the other times too. Adding by hand an arrow, like Tim does, doesn't solve the issue, since to each of his structures there is a dual structure in which the directions are reversed, and there is no way to distinguish one from its dual. You can ask him, I am sure that this is what he will answer too.

Best regards,

Cristi

Dear Cristi,

Thank you for your comments on my essay. I discussed your comment about the paradoxical nature of immortality in the context of a multi/maxiverse in a reply on my page.

Congratulations once again on an outstanding entry to this FQXi contest! I found myself highlighting a lot of your statements that I wholeheartedly agree with. In particular, in the section “Is there...

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Thank you for your comments on my essay. I discussed your comment about the paradoxical nature of immortality in the context of a multi/maxiverse in a reply on my page.

Congratulations once again on an outstanding entry to this FQXi contest! I found myself highlighting a lot of your statements that I wholeheartedly agree with. In particular, in the section “Is there...

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Dear Marc,

Thank you for the kind and interesting comments. You said "I found myself highlighting a lot of your statements that I wholeheartedly agree with." I had the same feeling while reading yours! There is a perfect harmony and complementarity between our essays :)

You wrote "I have some questions concerning your affirmation 'at least we know that there is room for free will, whatever this may be'." Yes, I gave some citations to previous essays and other works. I think a place to start are these two, Flowing with a Frozen River (pages 4-7) and Modern physics, determinism and free-will, both used in Aaronson's The Ghost in the Quantum Turing Machine. My arguments come from quantum mechanics, and the conclusion about free-will is very close to Hoefer's Freedom from the Inside Out. Also in "The Tao of It from Bit" I discuss a bit the issue. Please let me know what you think, or if you have questions. The bottom line is that I think free will is compatible with both determinism and indeterminism (indeterminism alone anyway doesn't guarantee it, because If quantum randomness would equal free will, then any Geiger counter would have free will.. But I only say there is room from free will, I don't know what it is :)

Best wishes,

Cristi

Thank you for the kind and interesting comments. You said "I found myself highlighting a lot of your statements that I wholeheartedly agree with." I had the same feeling while reading yours! There is a perfect harmony and complementarity between our essays :)

You wrote "I have some questions concerning your affirmation 'at least we know that there is room for free will, whatever this may be'." Yes, I gave some citations to previous essays and other works. I think a place to start are these two, Flowing with a Frozen River (pages 4-7) and Modern physics, determinism and free-will, both used in Aaronson's The Ghost in the Quantum Turing Machine. My arguments come from quantum mechanics, and the conclusion about free-will is very close to Hoefer's Freedom from the Inside Out. Also in "The Tao of It from Bit" I discuss a bit the issue. Please let me know what you think, or if you have questions. The bottom line is that I think free will is compatible with both determinism and indeterminism (indeterminism alone anyway doesn't guarantee it, because If quantum randomness would equal free will, then any Geiger counter would have free will.. But I only say there is room from free will, I don't know what it is :)

Best wishes,

Cristi

Dear Christinel,

You are convinced that maths and physics are much related, as in Tegmark's thesis. I suggest you read Leifer's essay and in an another direction the multiverse essay of Laura Mersini-Houghton. As you worked in cosmology and QM, I would be glad to have your view about the multiverse as a possible way to connect these two separate fields. Myself I am quite innnocent on this subject. I am working at this essay by Laura.

I am also rating your essay now.

Thanks in advance.

Michel

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You are convinced that maths and physics are much related, as in Tegmark's thesis. I suggest you read Leifer's essay and in an another direction the multiverse essay of Laura Mersini-Houghton. As you worked in cosmology and QM, I would be glad to have your view about the multiverse as a possible way to connect these two separate fields. Myself I am quite innnocent on this subject. I am working at this essay by Laura.

I am also rating your essay now.

Thanks in advance.

Michel

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Dear Michel,

Thank you for the comments. Indeed, I tried to bring some arguments supporting/explaining the connection between math and physics. The essays by Matt and Laura are on my to do list, I hope to get there soon. Regarding the connection between cosmology and QM, if you refer to the connection between inflation and QM, as advocated by Sean Carroll, I am not sure what to say about this. If you refer to Quantum Gravity and Quantum Cosmology, I think that it is premature the standard view that perturbative methods fail for Quantum Gravity. I have a paper on the connection between singularities and dimensional reduction in perturbative quantum gravity (Metric dimensional reduction at singularities with implications to Quantum Gravity, Annals of Physics 347C (2014), pp. 74-91). Many researchers found that various dimensional reduction effects may help quantum gravity. I argue that we don't need to put by hand these various dimensional reductions, since they occur already due to singularities in GR. But I also hope there is a better, nonperturbative way to quantize gravity, yet to be found.

Best wishes,

Cristi

Thank you for the comments. Indeed, I tried to bring some arguments supporting/explaining the connection between math and physics. The essays by Matt and Laura are on my to do list, I hope to get there soon. Regarding the connection between cosmology and QM, if you refer to the connection between inflation and QM, as advocated by Sean Carroll, I am not sure what to say about this. If you refer to Quantum Gravity and Quantum Cosmology, I think that it is premature the standard view that perturbative methods fail for Quantum Gravity. I have a paper on the connection between singularities and dimensional reduction in perturbative quantum gravity (Metric dimensional reduction at singularities with implications to Quantum Gravity, Annals of Physics 347C (2014), pp. 74-91). Many researchers found that various dimensional reduction effects may help quantum gravity. I argue that we don't need to put by hand these various dimensional reductions, since they occur already due to singularities in GR. But I also hope there is a better, nonperturbative way to quantize gravity, yet to be found.

Best wishes,

Cristi

Dear Christi,

Thank you for the references. I am interested by the quantum gravity subject but it may take a while before I am able to produce a good idea. These days I was exploring F-theory also because it involves modular group concepts.

Michel

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Thank you for the references. I am interested by the quantum gravity subject but it may take a while before I am able to produce a good idea. These days I was exploring F-theory also because it involves modular group concepts.

Michel

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Dear Michel,

I too tend to focus more where the previous research leads me. I was not interested in perturbative Quantum Gravity, but the singularities which I studied turned out to have dimensional reduction effects, so this is how I become interested. To make a parallel, I move from one field of interest to another rather by analytical continuation, than by jumps :) So I guess for you is more natural to explore F-theory, given that it involves modular group concepts.

Best wishes,

Cristi

I too tend to focus more where the previous research leads me. I was not interested in perturbative Quantum Gravity, but the singularities which I studied turned out to have dimensional reduction effects, so this is how I become interested. To make a parallel, I move from one field of interest to another rather by analytical continuation, than by jumps :) So I guess for you is more natural to explore F-theory, given that it involves modular group concepts.

Best wishes,

Cristi

Hi Christi,

This is a very interesting essay indeed! I have to object on one point though. Just because you can list certain properties about a system doesn't mean you know that there must exist a mathematical structure describing what the system does. It is trivially true that if you collect certain quantities that describe some system, then that is some kind of table, which you can call a "mathematical structure" if you wish, but this is just data collecting. The point of science is to come up with a useful mathematical model for this. In my essay I explain what I mean with this in more detail. I enjoyed reading your essay, also because it is well structured.

-- Sophia

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This is a very interesting essay indeed! I have to object on one point though. Just because you can list certain properties about a system doesn't mean you know that there must exist a mathematical structure describing what the system does. It is trivially true that if you collect certain quantities that describe some system, then that is some kind of table, which you can call a "mathematical structure" if you wish, but this is just data collecting. The point of science is to come up with a useful mathematical model for this. In my essay I explain what I mean with this in more detail. I enjoyed reading your essay, also because it is well structured.

-- Sophia

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Dear Sophia,

"Just because you can list certain properties about a system doesn't mean you know that there must exist a mathematical structure describing what the system does. It is trivially true that if you collect certain quantities that describe some system, then that is some kind of table, which you can call a "mathematical structure" if you wish, but this is just data collecting."

Well, I didn't refer to data collecting. I meant that if you have**all** the propositions that are true for a system, then there is a mathematical model which makes these propositions true. You have an axiomatic system (the collection of true statements), and there is a mathematical model of it (in the sense of model theory). It would be ridiculous to think that collecting some numbers that follow from a series of measurements would give a mathematical model. So I agree with you that "The point of science is to come up with a useful mathematical model for this.". I was not talking about collecting data from experiments, but about the existence of a mathematical model of a system which can be described by propositions and is free of logical inconsistencies.

Best wishes,

Cristi

"Just because you can list certain properties about a system doesn't mean you know that there must exist a mathematical structure describing what the system does. It is trivially true that if you collect certain quantities that describe some system, then that is some kind of table, which you can call a "mathematical structure" if you wish, but this is just data collecting."

Well, I didn't refer to data collecting. I meant that if you have

Best wishes,

Cristi

Dear Cristinel,

In your essay you discuss many serious and intriguing topics. I have a question about one matter in particular.

Toward the middle of the essay you ask whether everything is isomorphic to a mathematical structure. I believe that your answer to this question is “Yes.” I wonder about the converse question: Is every mathematical structure isomorphic to some physical structure. From your discussion of the mathematical universe hypothesis, I am not quite clear how you would answer this question. I understand that you deny that the mathematical universe hypothesis has been established as true; in other words, for all we know, some mathematical structures might have no application to physical existence. What, then, is the ontological status of the physically irrelevant mathematical structures? Do they have an abstract platonic reality? Are they totally unreal? Do they exist only as mental constructions in human minds? Or something else?

I appreciate also your example of the real number line between 0 and 1, and your discussion of Godel’s incompleteness theorem, but I have no specific questions or comments on those parts of your essay.

Best wishes,

Laurence Hitterdale

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In your essay you discuss many serious and intriguing topics. I have a question about one matter in particular.

Toward the middle of the essay you ask whether everything is isomorphic to a mathematical structure. I believe that your answer to this question is “Yes.” I wonder about the converse question: Is every mathematical structure isomorphic to some physical structure. From your discussion of the mathematical universe hypothesis, I am not quite clear how you would answer this question. I understand that you deny that the mathematical universe hypothesis has been established as true; in other words, for all we know, some mathematical structures might have no application to physical existence. What, then, is the ontological status of the physically irrelevant mathematical structures? Do they have an abstract platonic reality? Are they totally unreal? Do they exist only as mental constructions in human minds? Or something else?

I appreciate also your example of the real number line between 0 and 1, and your discussion of Godel’s incompleteness theorem, but I have no specific questions or comments on those parts of your essay.

Best wishes,

Laurence Hitterdale

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Dear Laurence,

Thank you for the comments, your raise an excellent question: " Is every mathematical structure isomorphic to some physical structure?". It may be, but I think that not all mathematical structures are isomorphic to physical structures from our universe. For any mathematical structure A, it is possible to find another mathematical structure B which is not isomorphic to a substructure of A. Hence, if there is a mathematical structure isomorphic to our entire universe (including its past and future), there have to be mathematical structures not isomorphic to physical structures from our universe. But maybe there are other universes with which they are isomorphic. I find appealing MUH, precisely because it removes the distinction between mathematical structures that have physical counterpart, and those that don't. Simply, according to MUH, each mathematical structure is a physical structure, and this is the physical structure isomorphic to it. But, as I argued in the essay, this maybe is impossible to prove, and is not falsifiable.

I look forward to read your essay, and I wish you good luck in the contest!

Best wishes,

Cristi

Thank you for the comments, your raise an excellent question: " Is every mathematical structure isomorphic to some physical structure?". It may be, but I think that not all mathematical structures are isomorphic to physical structures from our universe. For any mathematical structure A, it is possible to find another mathematical structure B which is not isomorphic to a substructure of A. Hence, if there is a mathematical structure isomorphic to our entire universe (including its past and future), there have to be mathematical structures not isomorphic to physical structures from our universe. But maybe there are other universes with which they are isomorphic. I find appealing MUH, precisely because it removes the distinction between mathematical structures that have physical counterpart, and those that don't. Simply, according to MUH, each mathematical structure is a physical structure, and this is the physical structure isomorphic to it. But, as I argued in the essay, this maybe is impossible to prove, and is not falsifiable.

I look forward to read your essay, and I wish you good luck in the contest!

Best wishes,

Cristi

Christinel,

Thanks for the positive assessment of my paper. I gave your paper a pretty high score a few weeks ago. I did this while I was on travel and I don't think I had time to write a post on your blog page. I will try to write a comment, which will probably require rereading your paper.

There is a paper by Schreiber on directly applying HOTT to physics. This is a difficult and...

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Thanks for the positive assessment of my paper. I gave your paper a pretty high score a few weeks ago. I did this while I was on travel and I don't think I had time to write a post on your blog page. I will try to write a comment, which will probably require rereading your paper.

There is a paper by Schreiber on directly applying HOTT to physics. This is a difficult and...

view entire post

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Dear Cristi Stoica

Your essay is clearly written and you included all the essential notions, which are important at explanation, for instance, consciousness. I also look on physics from reductionical view. But, I disagree that neural correlates are enough to explain consciousness. Not only me, Tononi and Koch also claim for panpsychism. Thus neural correlates only explain consciousness in...

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Your essay is clearly written and you included all the essential notions, which are important at explanation, for instance, consciousness. I also look on physics from reductionical view. But, I disagree that neural correlates are enough to explain consciousness. Not only me, Tononi and Koch also claim for panpsychism. Thus neural correlates only explain consciousness in...

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Dear Yanko,

Thank you for reading and commenting on my essay. I would take this opportunity to make some clarifications, and after that, you will probably feel that we agree more than you thought.

You say "I disagree that neural correlates are enough to explain consciousness." Well, this makes two of us, since I wrote "I don't claim we can explain consciousness, with or without mathematics. My only claim is that its physical manifestations are describable by mathematics, at least in principle."

Also, when I mentioned the duck principle, or "Leinbiz's identity of indiscernibles principle", I did it to delimit the domain of physics and objective science, and not as a claim that this explains also the subjective.

Related to what you said about the relation between quantum and free will, may I point you to some references: Flowing with a Frozen River (pages 4-7) and Modern physics, determinism and free-will, both used in Aaronson's The Ghost in the Quantum Turing Machine. My arguments come from quantum mechanics, and the conclusion about free-will is very close to Hoefer's Freedom from the Inside Out. Also in The Tao of It from Bit I discuss a bit the issue. Related to what you said about the Turing test, I wrote something that you might like.

About gravity, I agree with Petkov that it is not a force, at least in general relativity is just inertia on curved spacetime. Make it just a force doesn't seem to me either to be the right way. However, I think that to some extent can still be treated in a quantum manner, given that the other forces are geometric in nature too, and in this case maybe my own approach to quantum gravity, that singularities help removing the infinities in perturbative QG, may help, even though essentially it is not a force.

I agree very much with your words: "Because foundations of physics should be simple, the task of math is to describe quantum gravity on a t-shirt."

Best wishes,

Cristi

Thank you for reading and commenting on my essay. I would take this opportunity to make some clarifications, and after that, you will probably feel that we agree more than you thought.

You say "I disagree that neural correlates are enough to explain consciousness." Well, this makes two of us, since I wrote "I don't claim we can explain consciousness, with or without mathematics. My only claim is that its physical manifestations are describable by mathematics, at least in principle."

Also, when I mentioned the duck principle, or "Leinbiz's identity of indiscernibles principle", I did it to delimit the domain of physics and objective science, and not as a claim that this explains also the subjective.

Related to what you said about the relation between quantum and free will, may I point you to some references: Flowing with a Frozen River (pages 4-7) and Modern physics, determinism and free-will, both used in Aaronson's The Ghost in the Quantum Turing Machine. My arguments come from quantum mechanics, and the conclusion about free-will is very close to Hoefer's Freedom from the Inside Out. Also in The Tao of It from Bit I discuss a bit the issue. Related to what you said about the Turing test, I wrote something that you might like.

About gravity, I agree with Petkov that it is not a force, at least in general relativity is just inertia on curved spacetime. Make it just a force doesn't seem to me either to be the right way. However, I think that to some extent can still be treated in a quantum manner, given that the other forces are geometric in nature too, and in this case maybe my own approach to quantum gravity, that singularities help removing the infinities in perturbative QG, may help, even though essentially it is not a force.

I agree very much with your words: "Because foundations of physics should be simple, the task of math is to describe quantum gravity on a t-shirt."

Best wishes,

Cristi

Dear Cristi,

About interrelations among consciousness, math and physics, we are close, we only need to see details. But you do not mentioned a word ''Panpsychism'' and not many about ''quantum consciousness''. Thus maybe you can give some words about this. I looked two your links about free will, but here are clear differences.

If you wish you can read my essay. The main speculation in my quantum consciosness is that quantum randomness is free will, thus that the human mind MAYBE sometimes changes randomness of quantum phenomena.

Best regards

Janko Kokosar

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About interrelations among consciousness, math and physics, we are close, we only need to see details. But you do not mentioned a word ''Panpsychism'' and not many about ''quantum consciousness''. Thus maybe you can give some words about this. I looked two your links about free will, but here are clear differences.

If you wish you can read my essay. The main speculation in my quantum consciosness is that quantum randomness is free will, thus that the human mind MAYBE sometimes changes randomness of quantum phenomena.

Best regards

Janko Kokosar

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Dear Janko,

You said "The main speculation in my quantum consciosness is that quantum randomness is free will, thus that the human mind MAYBE sometimes changes randomness of quantum phenomena." In the first link I gave you I actually propose an experiment to test whether mind influences the randomness. The experiment is perhaps impossible to perform today, but in the future, who knows. I look forward to read your essay.

Best regards,

Cristi

You said "The main speculation in my quantum consciosness is that quantum randomness is free will, thus that the human mind MAYBE sometimes changes randomness of quantum phenomena." In the first link I gave you I actually propose an experiment to test whether mind influences the randomness. The experiment is perhaps impossible to perform today, but in the future, who knows. I look forward to read your essay.

Best regards,

Cristi

Very boring essay. Such a filling of the maximum allowed length for so little... maybe I'm just already too familiar with the fact that mathematics is the study of structures to have any interest reading it again, but...

Despite your try to develop a visual metaphor to illustrate the question of discovered vs. invented, you did not even come up with any decent answer to this question. Your...

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Despite your try to develop a visual metaphor to illustrate the question of discovered vs. invented, you did not even come up with any decent answer to this question. Your...

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Dear Sylvain,

Thank you for reading, and for letting me know that you were bored (my readers seem to be polarized between boring agreement and boring disagreement, I think you are at this latter pole). When I first saw your essay-sized comment, I thought it is just a revenge for me being boring to you. But I wasn't bored by your comments at all, and I am grateful for them. The argument with...

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Thank you for reading, and for letting me know that you were bored (my readers seem to be polarized between boring agreement and boring disagreement, I think you are at this latter pole). When I first saw your essay-sized comment, I thought it is just a revenge for me being boring to you. But I wasn't bored by your comments at all, and I am grateful for them. The argument with...

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Sorry for the ambiguity of some of my words that could be perceived more negatively than I meant. I did not mean that you had any intention to twist something. About this, I only meant I saw your interpretation of quantum physics as a sort of more complicated equivalent to other concepts I know. I am aware that when some ideas are discovered, they usually first appear in forms which are not the...

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Dear Sylvain,

OK, so you now no longer say that my interpretation is a more complicated version of the standard one, but of some others you know.

Related to the line, I don't think I get what you mean, and how bringing atoms into discussion supported your claim.

About NDEs, you are missing/ignoring my point, which was about distinguishing between remembering and imagination based on neural correlates. Discussing anecdotal evidence will not change this, even if it is true.

About your statement "If you don't assume that the laws of physics should be able to address arithmetic then all comments on the consequences of the incompleteness theorem on the laws of physics should be dismissed as inapplicable", it doesn't make sense. It is like saying that if you don't assume the existence of tooth fairy it doesn't make sense to criticize a viewpoint which assumes the existence of tooth fairy.

Branching time in the context of indeterministic dynamical systems makes perfect sense, because I addressed any possible indeterministic theory, to show that indeterminism doesn't guarantee free-will. Rejecting them because the only one you want to consider is quantum indeterminism is unjustified. And, even if we talk about branching in the context of quantum mechanics doesn't mean MWI, since I was talking about branching in the space of states, the branching of the history into two possible histories, and not into two real worlds.

Best wishes,

Cristi

OK, so you now no longer say that my interpretation is a more complicated version of the standard one, but of some others you know.

Related to the line, I don't think I get what you mean, and how bringing atoms into discussion supported your claim.

About NDEs, you are missing/ignoring my point, which was about distinguishing between remembering and imagination based on neural correlates. Discussing anecdotal evidence will not change this, even if it is true.

About your statement "If you don't assume that the laws of physics should be able to address arithmetic then all comments on the consequences of the incompleteness theorem on the laws of physics should be dismissed as inapplicable", it doesn't make sense. It is like saying that if you don't assume the existence of tooth fairy it doesn't make sense to criticize a viewpoint which assumes the existence of tooth fairy.

Branching time in the context of indeterministic dynamical systems makes perfect sense, because I addressed any possible indeterministic theory, to show that indeterminism doesn't guarantee free-will. Rejecting them because the only one you want to consider is quantum indeterminism is unjustified. And, even if we talk about branching in the context of quantum mechanics doesn't mean MWI, since I was talking about branching in the space of states, the branching of the history into two possible histories, and not into two real worlds.

Best wishes,

Cristi

Dear Sir, I was not convinced. Your argument that mathematics is not a human invention was not convincing. I am more interested in the problem of why scientists believe in Einstein relativity when the mathematics involved in that theory is completely false. We are supposed to believe it because experiments validate it. So we believe in a false mathematical theory because the experiments say it is correct. Pretty funny.

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Dear Sir,

Thank you for the comment, and for warning me of the fact that "scientists believe in Einstein relativity when the mathematics involved in that theory is completely false". I have in plan to write someday an essay in which I will discuss some misconceptions viz. special and general relativity, but I don't know when I will find time to do it.

Best regards,

Cristi

Thank you for the comment, and for warning me of the fact that "scientists believe in Einstein relativity when the mathematics involved in that theory is completely false". I have in plan to write someday an essay in which I will discuss some misconceptions viz. special and general relativity, but I don't know when I will find time to do it.

Best regards,

Cristi

Dear Cristinel, conclusion of your essay "So we can’t prove the mathematical universe

hypothesis by Tegmark’s method" provokes me to let you know that in our essay this hypothesis of Tegmark is refuted.

Best regards,

Alexey Burov.

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hypothesis by Tegmark’s method" provokes me to let you know that in our essay this hypothesis of Tegmark is refuted.

Best regards,

Alexey Burov.

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Dear Christinel,

in your essay you outline clearly the 'isomorphism-paradigm' of the relation between physics and mathematics. I quote from you essay :1. 'If the universe is describable by a list of propositions, then there is a mathematical structure describable by the same propositions'.

And a bit later :2'....the unified theory must exists...because we can not live simultaneously in two disconnected worlds.'

I would like to point out that you make two strong metaphysical assumptions, based upon the ontology of classical mechanics :

1. That it is possible to come up with a complete list of properties of the entire! universe.

This assumption is clearly wrong!Why? Because you leave out the agent compiling the list.According to quantum mechanics it is simply not possible to auto-describe a state to a system.A quantum mechanical state ( even of the 'universe') is always assign by an observer outside the system. In other words, your 'complete' list always has a blind spot.

2.You make the assumption that there is only 'the-one-reality', with all of its parts in mutual 'connection'.

But it is again quantum mechanics which tells us otherwise. The parts of the reality 'to be connected' is not only a logical operation ( like the 'AND'-operator), but also a physical interaction between a system and an observer (another system).During the interaction the system will in general undergo a random change of state.

That means that your 'complete' list of properties not only has a blind spot, but it also undergoes random fluctuations.

As a conclusion, the notion of 'the-one-whole-reality' ( can be traced back to Parmenides) is exposed as a fairy tale.

We should learn to live and do physics without it.

In my essay I describe how physics is better understood as Darwinian process, instead of a static description of 'the -one-reality'.

regards

Frank

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in your essay you outline clearly the 'isomorphism-paradigm' of the relation between physics and mathematics. I quote from you essay :1. 'If the universe is describable by a list of propositions, then there is a mathematical structure describable by the same propositions'.

And a bit later :2'....the unified theory must exists...because we can not live simultaneously in two disconnected worlds.'

I would like to point out that you make two strong metaphysical assumptions, based upon the ontology of classical mechanics :

1. That it is possible to come up with a complete list of properties of the entire! universe.

This assumption is clearly wrong!Why? Because you leave out the agent compiling the list.According to quantum mechanics it is simply not possible to auto-describe a state to a system.A quantum mechanical state ( even of the 'universe') is always assign by an observer outside the system. In other words, your 'complete' list always has a blind spot.

2.You make the assumption that there is only 'the-one-reality', with all of its parts in mutual 'connection'.

But it is again quantum mechanics which tells us otherwise. The parts of the reality 'to be connected' is not only a logical operation ( like the 'AND'-operator), but also a physical interaction between a system and an observer (another system).During the interaction the system will in general undergo a random change of state.

That means that your 'complete' list of properties not only has a blind spot, but it also undergoes random fluctuations.

As a conclusion, the notion of 'the-one-whole-reality' ( can be traced back to Parmenides) is exposed as a fairy tale.

We should learn to live and do physics without it.

In my essay I describe how physics is better understood as Darwinian process, instead of a static description of 'the -one-reality'.

regards

Frank

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Dear Frank,

Thank you for reading my essay and for the comments.

1. I agree that a being inside the universe can't get a complete description of the universe. And I agreed also when I wrote the essay: "However, we are just looking for a theory describing the general laws, and not a complete description of this particular instance of the universe, which includes what every human thinks about the universe and themselves. This would not be feasible anyway for practical reasons"

2. I don't see why "it is again quantum mechanics which tells us otherwise", that is, that there is not one connected mathematical structure describing it, given that quantum mechanics is already a mathematical theory about one such mathematical structure, and it includes fluctuations, entanglement, contextuality and all that.

Thank you for pointing me to your essay, which I look forward to read.

Best regards,

Cristi

Thank you for reading my essay and for the comments.

1. I agree that a being inside the universe can't get a complete description of the universe. And I agreed also when I wrote the essay: "However, we are just looking for a theory describing the general laws, and not a complete description of this particular instance of the universe, which includes what every human thinks about the universe and themselves. This would not be feasible anyway for practical reasons"

2. I don't see why "it is again quantum mechanics which tells us otherwise", that is, that there is not one connected mathematical structure describing it, given that quantum mechanics is already a mathematical theory about one such mathematical structure, and it includes fluctuations, entanglement, contextuality and all that.

Thank you for pointing me to your essay, which I look forward to read.

Best regards,

Cristi

Dear Cristinel,

Lots of sweeping statements in this essay and many attempts to refute other arguments without sufficient logic or evidence and more of of attempts to persuade with a general sweeping comment (where is your evidence for "Even if we manage to extend the list with new truths about the universe, there is a mathematical structure which is isomorphic to the universe described by...

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Lots of sweeping statements in this essay and many attempts to refute other arguments without sufficient logic or evidence and more of of attempts to persuade with a general sweeping comment (where is your evidence for "Even if we manage to extend the list with new truths about the universe, there is a mathematical structure which is isomorphic to the universe described by...

view entire post

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Dear Donald,

I read your comments and realized that you did not read with care my essay. Of course our**mathematical representations** of the universe evolved together with physics. And these representations change, and I even wrote "We know laws that were never contradicted by our observations, and their validity seems to be eternal, but this doesn't mean they will not be contradicted by experiment someday, or that there are no corners of the universe where they don't apply."

I was talking about the possibility that the physical world is isomorphic with a mathematical structure which**we are still searching**. So your argument is not against what I said, but what you think I said.

Also, you call Godel in your defense, but I also discuss this in the essay. Your statement that mathematics is not complete refers to mathematics which include arithmetic, and the incompleteness refers to the incompleteness of a finite set of axioms, in the sense that finite length proofs can't reach any truth. Drawing from this the conclusion that math is some broken toy that can't be fixed is forced. The conclusion, by contrary, is that math is much beyond the limited capacity of humans to express it in finite length proofs based on a finite number of axioms.

I don't understand your last argument, the one that the universe has contradictions and all evidence show this. You mean logical contradictions?

Thanks for your comments,

Cristi

I read your comments and realized that you did not read with care my essay. Of course our

I was talking about the possibility that the physical world is isomorphic with a mathematical structure which

Also, you call Godel in your defense, but I also discuss this in the essay. Your statement that mathematics is not complete refers to mathematics which include arithmetic, and the incompleteness refers to the incompleteness of a finite set of axioms, in the sense that finite length proofs can't reach any truth. Drawing from this the conclusion that math is some broken toy that can't be fixed is forced. The conclusion, by contrary, is that math is much beyond the limited capacity of humans to express it in finite length proofs based on a finite number of axioms.

I don't understand your last argument, the one that the universe has contradictions and all evidence show this. You mean logical contradictions?

Thanks for your comments,

Cristi

Now I more closely examined your article "Modern physics, Determinism and free-will", for properly replying to what you actually put there (sorry I was not careful enough last time). Indeed, I confused between your "interpretation" that you support, and your (incorrect) description of the wave function collapse with the role of free will in wavefunction collapse (that you criticize). I see now...

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Dear Sylvain,

Thank you for keeping your promise that you will "read and refute" my article, and to continue our "dialogue". Unfortunately, you are again missing the point and misinterpreting what I write :)

Let me explain what I am doing in the article you discuss, and in others: I am exploring the consequences of a modification of the standard interpretation of quantum mechanics,...

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Thank you for keeping your promise that you will "read and refute" my article, and to continue our "dialogue". Unfortunately, you are again missing the point and misinterpreting what I write :)

Let me explain what I am doing in the article you discuss, and in others: I am exploring the consequences of a modification of the standard interpretation of quantum mechanics,...

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"*to make a choice, you need to know what that choice means, some of its consequences. Otherwise what kind of a choice is that? The alternative is letting the randomness choose for you.*"

You are talking about fantasy, not about real facts. Of course it would be nice if we were always correctly informed about the consequences of our choices before making them, but this is not how...

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You are talking about fantasy, not about real facts. Of course it would be nice if we were always correctly informed about the consequences of our choices before making them, but this is not how...

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"You are talking about fantasy, not about real facts.". Then free will is a fantasy, and that's all. Note that I was not talking about knowing that your choice of lottery numbers is winning or that of you choose to save someone's life, this will not turn out to be a new Hitler. I was talking about being able to perform elementary actions. For example, choosing between going left or right. If the choice is made at the branching, is made by the branching. But randomness is not freedom. If this would be so, then any Geiger counter would have free will.

Dear Cristinel,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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Dear Cristi

I really enjoyed reading your beautiful essay.

Here some considerations.

(1)" A universe in a dot". This is a lovely peace of writing. It reinforces my conviction that the principle of finite information density must hold, as Feynman himself believed (see his 1982 Int. J. Th. Phys.) The continuum is paradoxical: you can write the whole history of the universe on...

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I really enjoyed reading your beautiful essay.

Here some considerations.

(1)" A universe in a dot". This is a lovely peace of writing. It reinforces my conviction that the principle of finite information density must hold, as Feynman himself believed (see his 1982 Int. J. Th. Phys.) The continuum is paradoxical: you can write the whole history of the universe on...

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Dear Mauro,

Thank you very much for reading the essay and for the interesting comments. I find myself in agreement with what you say. The continuum seems to allow much more information than it actually uses, so seems to be a waste :). So perhaps the continuum is rather an ordering principle for discrete information. With respect to Smolin's recent philosophy, I think you and I agree again, and your points 2a and 2b are right on the spot. And yes, you are definitely right about a theory being an efficient way to compress information. Indeed, we would have a lot to discuss, and I look forward to see you again at a conference, to discuss in more detail.

Best wishes,

Cristi

Thank you very much for reading the essay and for the interesting comments. I find myself in agreement with what you say. The continuum seems to allow much more information than it actually uses, so seems to be a waste :). So perhaps the continuum is rather an ordering principle for discrete information. With respect to Smolin's recent philosophy, I think you and I agree again, and your points 2a and 2b are right on the spot. And yes, you are definitely right about a theory being an efficient way to compress information. Indeed, we would have a lot to discuss, and I look forward to see you again at a conference, to discuss in more detail.

Best wishes,

Cristi

Daer Cristinel

I find your essay very interesting, indeed, I consider mine an extension to it. I agree on almost everything.

I believe, that the universe is "an hyper-equation". "A mathematical structure inside an other, and depends on another external"

"Mathematics and physics are two sides of the same thing. numbers and physical entities in motion”.

Now, I think...

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I find your essay very interesting, indeed, I consider mine an extension to it. I agree on almost everything.

I believe, that the universe is "an hyper-equation". "A mathematical structure inside an other, and depends on another external"

"Mathematics and physics are two sides of the same thing. numbers and physical entities in motion”.

Now, I think...

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attachments: Theorem_1.jpg, Theorem_345.jpg

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Christi,

Congratulations on another fine essay, and on hitting the top. Not only do I agree with most of your propositions (I axiomise that maths CAN describe all) but more importantly you set them out and describe them succinctly. I was also pleased to read you agree that a TOE IS possible and that Godel does NOT preclude a mathematical description. Do you agree that recursive higher orders should be able to model that formalism asymptotically?

One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible? I assign this 'false proof' issue to much anomaly and paradox in physics and identify an important example in my own essay, confounding logic in QM. My suggestion is that if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time (which I agree is 'physical'). I hope you'll read and give your views.

You don't suggest a direction for any new formalism pointing towards a TOE, which is not a criticism as very few do, but again I hope you'll analyse the hierarchical 'rules of brackets' which i suggest does so consistently with your analysis (as far as I can see, but your view?)

I'm not surprised you're in the top group again and hope my score will help keep you there.

Very best of luck in the final judgements.

Peter

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Congratulations on another fine essay, and on hitting the top. Not only do I agree with most of your propositions (I axiomise that maths CAN describe all) but more importantly you set them out and describe them succinctly. I was also pleased to read you agree that a TOE IS possible and that Godel does NOT preclude a mathematical description. Do you agree that recursive higher orders should be able to model that formalism asymptotically?

One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible? I assign this 'false proof' issue to much anomaly and paradox in physics and identify an important example in my own essay, confounding logic in QM. My suggestion is that if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time (which I agree is 'physical'). I hope you'll read and give your views.

You don't suggest a direction for any new formalism pointing towards a TOE, which is not a criticism as very few do, but again I hope you'll analyse the hierarchical 'rules of brackets' which i suggest does so consistently with your analysis (as far as I can see, but your view?)

I'm not surprised you're in the top group again and hope my score will help keep you there.

Very best of luck in the final judgements.

Peter

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Peter,

Thank you for the interesting remarks. Let me congratulate you as well for your place. Regarding your question "Do you agree that recursive higher orders should be able to model that formalism asymptotically?", I would say maybe.

> "One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible?"

I think I addressed the question of models that are not "true" in a section of my essay named "Is mathematics merely a tool in physics?". If you refer to models that give all the correct outcomes, and can't be distinguished by other models by no matter what experiment, I think it is possible. But I think that if these can be related to anomalies and paradoxes, then they can be distinguished and falsified. Regarding "if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time", well, correct maths is right, but being right doesn't make it true about our physical world.

"You don't suggest a direction for any new formalism pointing towards a TOE". I think it is too early for such a suggestion.

I look forward to read your essay, which is on my todo list.

Best wishes,

Cristi

Thank you for the interesting remarks. Let me congratulate you as well for your place. Regarding your question "Do you agree that recursive higher orders should be able to model that formalism asymptotically?", I would say maybe.

> "One thing perhaps 'missing' was any reminder that a mathematical structure may not necessarily precisely model any physical process for which it's developed, though giving apparently correct outcomes. Do you agree this is possible?"

I think I addressed the question of models that are not "true" in a section of my essay named "Is mathematics merely a tool in physics?". If you refer to models that give all the correct outcomes, and can't be distinguished by other models by no matter what experiment, I think it is possible. But I think that if these can be related to anomalies and paradoxes, then they can be distinguished and falsified. Regarding "if we say ALL maths must ALL be right if the bottom line proves right then we can be misguided about how nature actually evolves in time", well, correct maths is right, but being right doesn't make it true about our physical world.

"You don't suggest a direction for any new formalism pointing towards a TOE". I think it is too early for such a suggestion.

I look forward to read your essay, which is on my todo list.

Best wishes,

Cristi

Hi Cristinel,

Your excellent essay makes a compelling case for mapping the universe to mathematical structure, and you presented it in an upbeat and enjoyable narrative. Your demonstration of how pivotal topics from relativity and incompleteness do not limit this isomorphism, provides a very persuasive argument. I like that you explained there is no conflict between mathematics and causality, and your discussion on physical explanation and mathematical description was very pertinent. My essay also discusses the strong connection between physics and mathematics, and for example how changing the mathematical representation affects physical explanation. I also show what was considered a limitation on computation due to (Gödel) incompleteness, can in fact be an expansion when attempting to physically model the mathematical structure of indecidability. This may address or at least support points raised in your essay.

Also, I appreciated the wide variety of examples encompassed in your essay, and am motivated to check out your work on the Weyl curvature hypothesis, singularities, and free will.

A very interesting contribution that thoroughly addresses this topic, I give it the highest rating.

Please take a moment to read my essay and rate it as well. Thanks again,

Steve Sax

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Your excellent essay makes a compelling case for mapping the universe to mathematical structure, and you presented it in an upbeat and enjoyable narrative. Your demonstration of how pivotal topics from relativity and incompleteness do not limit this isomorphism, provides a very persuasive argument. I like that you explained there is no conflict between mathematics and causality, and your discussion on physical explanation and mathematical description was very pertinent. My essay also discusses the strong connection between physics and mathematics, and for example how changing the mathematical representation affects physical explanation. I also show what was considered a limitation on computation due to (Gödel) incompleteness, can in fact be an expansion when attempting to physically model the mathematical structure of indecidability. This may address or at least support points raised in your essay.

Also, I appreciated the wide variety of examples encompassed in your essay, and am motivated to check out your work on the Weyl curvature hypothesis, singularities, and free will.

A very interesting contribution that thoroughly addresses this topic, I give it the highest rating.

Please take a moment to read my essay and rate it as well. Thanks again,

Steve Sax

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Hi Steve,

Thanks for the interesting comments and kind remarks. Indeed, some may qualify my essay as (too) optimistic, and I don't really think there are limitations that should stop us from trying to learn more. By the same criteria, any attempt to understand the universe has to be optimistic. In the same time, I advocate an agnostic and skeptical position regarding the belief in the absolute validity of our theories. And as you said, I indeed see no conflict between mathematics and causality. I think you have a keen eye and understood well my points. Your remarks about how changing the mathematical representation affects physical explanation, and the idea to take advantage from Gödel's incompleteness theorem seem to me very interesting. I look forward to read your essay. Good luck in the contest!

Best regards,

Cristi

Thanks for the interesting comments and kind remarks. Indeed, some may qualify my essay as (too) optimistic, and I don't really think there are limitations that should stop us from trying to learn more. By the same criteria, any attempt to understand the universe has to be optimistic. In the same time, I advocate an agnostic and skeptical position regarding the belief in the absolute validity of our theories. And as you said, I indeed see no conflict between mathematics and causality. I think you have a keen eye and understood well my points. Your remarks about how changing the mathematical representation affects physical explanation, and the idea to take advantage from Gödel's incompleteness theorem seem to me very interesting. I look forward to read your essay. Good luck in the contest!

Best regards,

Cristi

Cristi,

Thanks for being engaged in my essay and for your kind words.

Jim

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Thanks for being engaged in my essay and for your kind words.

Jim

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Christi,

I don't know how I missed your essay.

Your essay is delightful, like you have the essence of Socrates on your shoulder, distilling with questions and answers his method of inquiry and discussion. I think the potion you create works. It helps to clarify the mind-numbing. You also throw in helpful progressions of studies, GR beyond Newton, singularities.

Impressive essay.

Jim

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I don't know how I missed your essay.

Your essay is delightful, like you have the essence of Socrates on your shoulder, distilling with questions and answers his method of inquiry and discussion. I think the potion you create works. It helps to clarify the mind-numbing. You also throw in helpful progressions of studies, GR beyond Newton, singularities.

Impressive essay.

Jim

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Dear Cristi,

Since this comment relates to both yours and our essays, I am reproducing it here, in your blog, with a slight abbreviation.

Thank you so much for your compliments; it is a true pleasure to be highly appreciated by one of the experts!

You underline that your main take away from our essay "is the uniqueness of the laws.” Maybe, our laws of nature are not exactly unique, but they definitely belong to a very special and narrow set of mathematical structures, much more narrow than Tegmark’s multiverse suggests. In other words, our laws are truly beautiful in that deep meaning of mathematical beauty which was professed by Pythagoreans of all times, from Pythagoras and Euclid to Kepler and Newton and to Einstein and Dirac.

In that light the statement of your essay that,

“Mathematics is already there, eternal and unchanging. What we invent is the discovery of mathematics,”

is revealed as having an even deeper meaning than it may at first seem.

Cheers and good luck!

Alexey and Lev

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Since this comment relates to both yours and our essays, I am reproducing it here, in your blog, with a slight abbreviation.

Thank you so much for your compliments; it is a true pleasure to be highly appreciated by one of the experts!

You underline that your main take away from our essay "is the uniqueness of the laws.” Maybe, our laws of nature are not exactly unique, but they definitely belong to a very special and narrow set of mathematical structures, much more narrow than Tegmark’s multiverse suggests. In other words, our laws are truly beautiful in that deep meaning of mathematical beauty which was professed by Pythagoreans of all times, from Pythagoras and Euclid to Kepler and Newton and to Einstein and Dirac.

In that light the statement of your essay that,

“Mathematics is already there, eternal and unchanging. What we invent is the discovery of mathematics,”

is revealed as having an even deeper meaning than it may at first seem.

Cheers and good luck!

Alexey and Lev

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I enjoyed your exploratory approach to the essay question, Cristi.

You parse it into many questions. But I get the impression that you are not dogmatically attached to any of your answers. Rather you enjoy the dialectic struggle between opposing ideas.

I would enjoy a joust over a beer with you sometime!

......David

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You parse it into many questions. But I get the impression that you are not dogmatically attached to any of your answers. Rather you enjoy the dialectic struggle between opposing ideas.

I would enjoy a joust over a beer with you sometime!

......David

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Dear Christi Stoica,

I enjoyed reading your essay. In particular, I liked your approach to include many questions. Most of the competitors (including me) are too eager to give answers, but I think it is an important goal to get the reader to wonder about these questions first.

In addition, I would like to discuss two points.

The idea of illustrating how you can code various texts as numbers in the [0,1] interval seems well chosen for the intended audience. On the other hand, I worry that you may have stretched it a bit too far when you write: "That line contains your entire life". After all, my life is not a text. ;-) And even if it would be narrated, a lot would depend on the wording - in particular: the first word! (If the narrater always starts with "Well,..." then all our lives end up close to 1.)

You write at some that "the fact that mathematics is useful doesn't mean that the universe is mathematical". At a later point, you say that a full isomorphism (unknown so far) between the universe and a mathematical structure would mean that there is no difference between the two. This second part is not so clear to me: could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure. This makes all the difference: ascribing a structure to the universe leaves a lot of degrees of freedom, but more importantly, it would leave out the 'substance' of the universe, and it would avoid the conclusion that the universe is mathematical.

Still, my general impression of your essay is a positive one. And the fact that it leaves topics for discussion is a sign of its quality, not a criticism.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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I enjoyed reading your essay. In particular, I liked your approach to include many questions. Most of the competitors (including me) are too eager to give answers, but I think it is an important goal to get the reader to wonder about these questions first.

In addition, I would like to discuss two points.

The idea of illustrating how you can code various texts as numbers in the [0,1] interval seems well chosen for the intended audience. On the other hand, I worry that you may have stretched it a bit too far when you write: "That line contains your entire life". After all, my life is not a text. ;-) And even if it would be narrated, a lot would depend on the wording - in particular: the first word! (If the narrater always starts with "Well,..." then all our lives end up close to 1.)

You write at some that "the fact that mathematics is useful doesn't mean that the universe is mathematical". At a later point, you say that a full isomorphism (unknown so far) between the universe and a mathematical structure would mean that there is no difference between the two. This second part is not so clear to me: could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure. This makes all the difference: ascribing a structure to the universe leaves a lot of degrees of freedom, but more importantly, it would leave out the 'substance' of the universe, and it would avoid the conclusion that the universe is mathematical.

Still, my general impression of your essay is a positive one. And the fact that it leaves topics for discussion is a sign of its quality, not a criticism.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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Dear Sylvia,

Thank you for the comments. You raise interesting questions.

One point you make is about my words "That line contains your entire life". You thoughtfully reply "After all, my life is not a text". Well, could you please tell me the part of your life which is not a text? I don't intend to be too curious, rather to ask you a trick question, since if you can tell this, you will make it into a text :) A quick reading of my essay may leave the reader with the feeling that I choose to ignore consciousness for example, but as I wrote, "I don't claim we can explain consciousness, with or without mathematics." What I wrote about this may clarify what I mean: "However, any feeling we may have, there are neural correlates associated to it, and hence, physical correlates. And these physical correlates are in the domain of known physics, which is strongly mathematized." About your excellent remark that you can choose to narrate the same life differently, and make it closer to one number or another, I fully agree, but I can't see why would this be a problem. As you know, there is a bijection between the points on a segment and those in a square, or the entire space, but that bijection can't be continuous. I understand that your remark reveals that representing everything on a segment is counterintuitive, and I like it. I just wanted to make a point regarding whether math is discovered or invented.

Another point you make is "could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure." I agree, isomorphism is between structures. For example, there are more isomorphisms between the set of real numbers and other structures. The real line is isomorphic with a square, if we refer to the category of sets, with a line if we refer to the category of topological spaces, with a totally ordered set if we refer to the order, with a vector space, with a metric space, with a group, semigroup, ring, field, etc., it all depends on the structure we are interested in. As I explained in the essay, the structure is captured in the relations, all relations that can be described by propositions. So there's nothing that can be left outside the structure, if we take into account all the true propositions about the world. Saying "ascribing a structure to the universe leaves a lot of degrees of freedom" is not necessarily true, I mean, of course it is true*if* we leave outside some of the truths. About the substance, I don't know what you mean by this. Is it something that has effects? Then its properties are captured in the structure. If you think that there are properties of the substance that don't have effects to the structure, then I have nothing to say about it, and anything we would say would be out of our possibilities of verifications. I think the text in a book is what makes a book, and not the paper or the electronic memory used to keep a copy of that text.

It was a deep pleasure to talk with you, and I wish your essay will do well, since I loved it.

Best wishes,

Cristi

Thank you for the comments. You raise interesting questions.

One point you make is about my words "That line contains your entire life". You thoughtfully reply "After all, my life is not a text". Well, could you please tell me the part of your life which is not a text? I don't intend to be too curious, rather to ask you a trick question, since if you can tell this, you will make it into a text :) A quick reading of my essay may leave the reader with the feeling that I choose to ignore consciousness for example, but as I wrote, "I don't claim we can explain consciousness, with or without mathematics." What I wrote about this may clarify what I mean: "However, any feeling we may have, there are neural correlates associated to it, and hence, physical correlates. And these physical correlates are in the domain of known physics, which is strongly mathematized." About your excellent remark that you can choose to narrate the same life differently, and make it closer to one number or another, I fully agree, but I can't see why would this be a problem. As you know, there is a bijection between the points on a segment and those in a square, or the entire space, but that bijection can't be continuous. I understand that your remark reveals that representing everything on a segment is counterintuitive, and I like it. I just wanted to make a point regarding whether math is discovered or invented.

Another point you make is "could there really be an isomorphism between the universe and a mathematical structure? In my opinion, at best we can find an isomorphism between _the structure of_ the universe and a mathematical structure." I agree, isomorphism is between structures. For example, there are more isomorphisms between the set of real numbers and other structures. The real line is isomorphic with a square, if we refer to the category of sets, with a line if we refer to the category of topological spaces, with a totally ordered set if we refer to the order, with a vector space, with a metric space, with a group, semigroup, ring, field, etc., it all depends on the structure we are interested in. As I explained in the essay, the structure is captured in the relations, all relations that can be described by propositions. So there's nothing that can be left outside the structure, if we take into account all the true propositions about the world. Saying "ascribing a structure to the universe leaves a lot of degrees of freedom" is not necessarily true, I mean, of course it is true

It was a deep pleasure to talk with you, and I wish your essay will do well, since I loved it.

Best wishes,

Cristi

Dear Christi,

Thank you for your (fast!) response.

If I tell you an episode of my life, my life itself will not turn into words. To keep it simpler, let's talk about colours: mentioning a colour does not produce that colour (at best, it may trigger a memory of it). Even if I would be able to say everything about a particular colour (not just 'red', but the spectrum, possibly partial translucence, etc.) I would still not have reproduced the colour. Sure, I could use the information, in computer graphics for instance, to reproduce it. But I would need some hardware to run it on (part of the universe). With the universe as a whole, I don't see how a full description (for simplicity, let's rely on an outdated materialistic view: some mass here and some mass there) is enough (in the materialistic example: you would still need to get some mass in addition to a mere description thereof).

On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?

Best wishes,

Sylvia

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Thank you for your (fast!) response.

If I tell you an episode of my life, my life itself will not turn into words. To keep it simpler, let's talk about colours: mentioning a colour does not produce that colour (at best, it may trigger a memory of it). Even if I would be able to say everything about a particular colour (not just 'red', but the spectrum, possibly partial translucence, etc.) I would still not have reproduced the colour. Sure, I could use the information, in computer graphics for instance, to reproduce it. But I would need some hardware to run it on (part of the universe). With the universe as a whole, I don't see how a full description (for simplicity, let's rely on an outdated materialistic view: some mass here and some mass there) is enough (in the materialistic example: you would still need to get some mass in addition to a mere description thereof).

On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?

Best wishes,

Sylvia

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Dear Sylvia,

Thank you for the answer. I think I understand what you mean, but I don't think I made you understand what I mean.

"If I tell you an episode of my life, my life itself will not turn into words." Sure it will not. And even if you give me the complete list of episodes, your life will not turn into words. What I mean is that its structure will be captured in those words. So, going back to my words "That line contains your entire life", I should have said perhaps "That line contains a complete description of your entire life". The main subject of my essay was about the relation between mathematics and physics, and I was careful to mention that I don't aim to explain unphysical stuff, perhaps consciousness being an example.

You give a good example, that of a color. But you answer it yourself, you can reproduce it on a computer (and that information can be expressed if we wish as words, or as a point on a segment). Your worry that there will be not enough room for hardware in the universe to simulate the color is a new element you add. But the proof of principle remains: the possibility to describe that particular color exists. I am talking about the possibility of a description, not a real implementation of it, not of a hardware. I don't think that, in order to prove it, one should effectively build that description. For instance, the number π exists even if its decimals are nowhere written completely, and even if the entire universe is not enough to write it.

"On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?" If you destroy all writings containing Pythagora's thereom, you will not destroy it, you will destroy the information about it that we have. It will be soon rediscovered. But if you destroy all books by Shakespeare, I don't expect that we will rewrite them soon. They will still be in that segment, but we will lose the address to retrieve them. Think for example at a computer. When you normally delete a file, you delete a reference to it, but the information remains on the hard drive, and can be recovered by special software, until you overwrite it with other information. When you lose the address, the file is not lost. A home is not lost when you lose the address. But you lose your access to it. In the case of a book, the address is the book.

But when we talk about life, the things are different... I don't think that a complete description of your life values at least 0.000...01% of your life. You may think that this is a contradiction: on the one hand I claim that everything is isomorphic to a mathematical structure, on the other hand, I don't reject the possibility that consciousness or life is more than this. Maybe this is not obvious in my essay of this year (where I focused on the relation between mathematics and the physical world), but perhaps the one for last year is more on this topic.

Best wishes,

Cristi

Thank you for the answer. I think I understand what you mean, but I don't think I made you understand what I mean.

"If I tell you an episode of my life, my life itself will not turn into words." Sure it will not. And even if you give me the complete list of episodes, your life will not turn into words. What I mean is that its structure will be captured in those words. So, going back to my words "That line contains your entire life", I should have said perhaps "That line contains a complete description of your entire life". The main subject of my essay was about the relation between mathematics and physics, and I was careful to mention that I don't aim to explain unphysical stuff, perhaps consciousness being an example.

You give a good example, that of a color. But you answer it yourself, you can reproduce it on a computer (and that information can be expressed if we wish as words, or as a point on a segment). Your worry that there will be not enough room for hardware in the universe to simulate the color is a new element you add. But the proof of principle remains: the possibility to describe that particular color exists. I am talking about the possibility of a description, not a real implementation of it, not of a hardware. I don't think that, in order to prove it, one should effectively build that description. For instance, the number π exists even if its decimals are nowhere written completely, and even if the entire universe is not enough to write it.

"On your view, can you destroy all copies of a book (including our memories of reading it etc.) without destroying the text?" If you destroy all writings containing Pythagora's thereom, you will not destroy it, you will destroy the information about it that we have. It will be soon rediscovered. But if you destroy all books by Shakespeare, I don't expect that we will rewrite them soon. They will still be in that segment, but we will lose the address to retrieve them. Think for example at a computer. When you normally delete a file, you delete a reference to it, but the information remains on the hard drive, and can be recovered by special software, until you overwrite it with other information. When you lose the address, the file is not lost. A home is not lost when you lose the address. But you lose your access to it. In the case of a book, the address is the book.

But when we talk about life, the things are different... I don't think that a complete description of your life values at least 0.000...01% of your life. You may think that this is a contradiction: on the one hand I claim that everything is isomorphic to a mathematical structure, on the other hand, I don't reject the possibility that consciousness or life is more than this. Maybe this is not obvious in my essay of this year (where I focused on the relation between mathematics and the physical world), but perhaps the one for last year is more on this topic.

Best wishes,

Cristi

Dear Christi,

Yes, it is easier for me to agree with the new formulation "That line contains a complete description of your entire life".

Just a clarification: my point with the color example was not that there isn't enough room in the universe for the hardware. Simply that we need hardware, which presupposes that we already have the universe at our disposal* - and exactly this does not work when it is the entire universe we want to describe. If we have that mathematical description and delete the physical universe, it does not become reality. Our description will at best be isomorphic to the structure of the universe, but still something is missing to be able to bridge the gap from a mere description to reality.

Thanks for your reply to my question about the book text! It was really nice to have this exchange with you.

Best wishes and good luck,

Sylvia

*: It's a bit like the Sagan quote: "If you wish to make an apple pie from scratch, you must first invent the universe" ;-)

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Yes, it is easier for me to agree with the new formulation "That line contains a complete description of your entire life".

Just a clarification: my point with the color example was not that there isn't enough room in the universe for the hardware. Simply that we need hardware, which presupposes that we already have the universe at our disposal* - and exactly this does not work when it is the entire universe we want to describe. If we have that mathematical description and delete the physical universe, it does not become reality. Our description will at best be isomorphic to the structure of the universe, but still something is missing to be able to bridge the gap from a mere description to reality.

Thanks for your reply to my question about the book text! It was really nice to have this exchange with you.

Best wishes and good luck,

Sylvia

*: It's a bit like the Sagan quote: "If you wish to make an apple pie from scratch, you must first invent the universe" ;-)

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Dear Cristi,

A very enjoyable and easy to read essay, with interesting thoughts and observations. Especially noteworthy for us were your remarks on Godel's theorem.

The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?

And if pushed, what stance would you take: Platonism, or non-Platonism? :-)

Our best wishes,

Anshu, Tejinder

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A very enjoyable and easy to read essay, with interesting thoughts and observations. Especially noteworthy for us were your remarks on Godel's theorem.

The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?

And if pushed, what stance would you take: Platonism, or non-Platonism? :-)

Our best wishes,

Anshu, Tejinder

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Dear Anshu and Tejinder,

Thank you for the kind comments.

"The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?"

I don't really know what to answer, since I really know neither the nature of mathematics nor that of material substance.

If we call material substance the thing from which things are made, what is that thing? Things are made of atoms, which seem hollow, and made of protons and neutrons, which seem hollow, which are made of quarks. Which are what? and why can't they have independent existence? Are particles waves of some stuff, or waves of probability? Are they fields of what? Can we know about these things more than mathematical abstractions? Are they more than clicks in detectors? You see, I am very confused about all this, and I must admit I don't know what is the substance making all this stuff. I don't want to sound like an ancient oriental wise pondering about the illusion of everything but nothingness, nor like a post modernist cubist surrealist, I just really don't know. What we know are operations from which we infer relations. We know propositions of the form "if we do this we obtain that outcome, with that probability". And we know that in the world there are some regularities, which look like the regularities we find in mathematical structures and nowhere else. So maybe these two things, matter and mathematics, are just one thing, the thread that connects these regularities. Call it the substance making the world, call it mathematical structure, in both cases is just a thread connecting the regularities. So if we say that mathematics lives on a material support, or that what we call material substance is in fact a collection of relations, or of propositions which are true about those relations, or mathematics, would it be a difference? And if there is a substance upon which the regularities are imposed in a way which looks like mathematics, then how can two so different things be so intimatelly connected, if they are not one and the same?

So I agree that "a description of the duck is different from the duck", but we don't have ducks, we only have descriptions of them. We are talking here about two descriptions of ducks. If the descriptions are identical and there is no way to check how the real ducks are (or how real the ducks are), can they be different?

"And if pushed, what stance would you take: Platonism, or non-Platonism? :-)"

None. Should I pick one? If the Platonist view distinguishes between the ideal world and our world, which is just an approximate pale shadow of the former, I clearly am not satisfied with it. Why have an ideal world of shapes and live in the cave? What use would be for that ideal world? And, as you said, how can we know about it, other than by extra-sensorial perceptions, which I don't even know what can be? If non-Platonism means that mathematics is just a secretion of the thinking matter, then again I am not satisfied, because the very source of that secretion is subject to mathematical laws. To avoid circularity, I feel forced to admit identity between the two.

But if I admit the identity of the two, this looks like a mathematical universe hypothesis, or mathematical monism. In this case, what brings the mathematical structure into existence? I don't know, but whatever we would consider to be the reality (including material substance), we face the same question: what brings it into existence? I don't have an answer for this, but I would rather have a single answer about the existence of the two ducks which are one, than two answers about what brings into existence two so different ducks, and another answer about what makes them so similar. Maybe all that there is is just mind, which contemplates an infinite diversity of propositions, which all arise from the principle of explosion (this essay page 9), and selects from these subworlds which are logically consistent, creating by this all mathematical structures, hence all possible worlds. Maybe. But I don't know :)

Best wishes,

Cristi

Thank you for the kind comments.

"The duck: for us, a description of the duck is different from the duck; thus we said that mathematics lacks in material substance. Would you agree?"

I don't really know what to answer, since I really know neither the nature of mathematics nor that of material substance.

If we call material substance the thing from which things are made, what is that thing? Things are made of atoms, which seem hollow, and made of protons and neutrons, which seem hollow, which are made of quarks. Which are what? and why can't they have independent existence? Are particles waves of some stuff, or waves of probability? Are they fields of what? Can we know about these things more than mathematical abstractions? Are they more than clicks in detectors? You see, I am very confused about all this, and I must admit I don't know what is the substance making all this stuff. I don't want to sound like an ancient oriental wise pondering about the illusion of everything but nothingness, nor like a post modernist cubist surrealist, I just really don't know. What we know are operations from which we infer relations. We know propositions of the form "if we do this we obtain that outcome, with that probability". And we know that in the world there are some regularities, which look like the regularities we find in mathematical structures and nowhere else. So maybe these two things, matter and mathematics, are just one thing, the thread that connects these regularities. Call it the substance making the world, call it mathematical structure, in both cases is just a thread connecting the regularities. So if we say that mathematics lives on a material support, or that what we call material substance is in fact a collection of relations, or of propositions which are true about those relations, or mathematics, would it be a difference? And if there is a substance upon which the regularities are imposed in a way which looks like mathematics, then how can two so different things be so intimatelly connected, if they are not one and the same?

So I agree that "a description of the duck is different from the duck", but we don't have ducks, we only have descriptions of them. We are talking here about two descriptions of ducks. If the descriptions are identical and there is no way to check how the real ducks are (or how real the ducks are), can they be different?

"And if pushed, what stance would you take: Platonism, or non-Platonism? :-)"

None. Should I pick one? If the Platonist view distinguishes between the ideal world and our world, which is just an approximate pale shadow of the former, I clearly am not satisfied with it. Why have an ideal world of shapes and live in the cave? What use would be for that ideal world? And, as you said, how can we know about it, other than by extra-sensorial perceptions, which I don't even know what can be? If non-Platonism means that mathematics is just a secretion of the thinking matter, then again I am not satisfied, because the very source of that secretion is subject to mathematical laws. To avoid circularity, I feel forced to admit identity between the two.

But if I admit the identity of the two, this looks like a mathematical universe hypothesis, or mathematical monism. In this case, what brings the mathematical structure into existence? I don't know, but whatever we would consider to be the reality (including material substance), we face the same question: what brings it into existence? I don't have an answer for this, but I would rather have a single answer about the existence of the two ducks which are one, than two answers about what brings into existence two so different ducks, and another answer about what makes them so similar. Maybe all that there is is just mind, which contemplates an infinite diversity of propositions, which all arise from the principle of explosion (this essay page 9), and selects from these subworlds which are logically consistent, creating by this all mathematical structures, hence all possible worlds. Maybe. But I don't know :)

Best wishes,

Cristi

Hi Cristi--

I loved your essay! Incredibly well-written, well-structured, and thought-provoking. Your analysis of the applicability of Gödel’s theorems to physics is spot on. Also, I think that you did an excellent job of rebutting Smolin’s objections regarding time and particularities.

Nonetheless, I do not share your view that “the universe is isomorphic to mathematics”. Nor do I believe that “the universe is nothing but a mathematical structure”. Let me ask just one “quick question” to sketch out my objection.

If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics? You know what I’m talking about. We are forced to rely on analytical simplifications, numerical approximations, linearizations, and perturbations (to name just a few techniques), every day and in every way, to make progress in physics. Just how isomorphic can mathematics be to the physical world if physicists must typically rely on such mathematical techniques to get the job done? Put differently, if “A Supreme Something” had ordered me to design a physical world—and to do so in way isomorphic to mathematics—I’d like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in! And you say?

I firmly believe that constructive criticism, including disagreement, is the engine that drives progress in physics. I may not agree with your overall position, but you did a great job setting it out and getting me to think about our points of agreement and disagreement. Accordingly, I have given you a high rating. (Not that you’ll notice, as you ratings are already very high!). Congratulations.

Very best regards,

Bill.

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I loved your essay! Incredibly well-written, well-structured, and thought-provoking. Your analysis of the applicability of Gödel’s theorems to physics is spot on. Also, I think that you did an excellent job of rebutting Smolin’s objections regarding time and particularities.

Nonetheless, I do not share your view that “the universe is isomorphic to mathematics”. Nor do I believe that “the universe is nothing but a mathematical structure”. Let me ask just one “quick question” to sketch out my objection.

If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics? You know what I’m talking about. We are forced to rely on analytical simplifications, numerical approximations, linearizations, and perturbations (to name just a few techniques), every day and in every way, to make progress in physics. Just how isomorphic can mathematics be to the physical world if physicists must typically rely on such mathematical techniques to get the job done? Put differently, if “A Supreme Something” had ordered me to design a physical world—and to do so in way isomorphic to mathematics—I’d like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in! And you say?

I firmly believe that constructive criticism, including disagreement, is the engine that drives progress in physics. I may not agree with your overall position, but you did a great job setting it out and getting me to think about our points of agreement and disagreement. Accordingly, I have given you a high rating. (Not that you’ll notice, as you ratings are already very high!). Congratulations.

Very best regards,

Bill.

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Hi Bill,

Thank you for the kind comments.

You ask "If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics?"

Why would we be able to analytically solve any of our PDEs in physics? You seem to state it as if it would be an inevitable consequence of the hypothesis that the universe is isomorphic to a mathematical structure.

"if “A Supreme Something” had ordered me to design a physical world—and to do so in way isomorphic to mathematics—I’d like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in!"

Why would you do it computationally efficacious? And could you do it like this, and in the same time allow the complexity we observe and we need to exist?

Anyway, our current mathematical models of the physical world are very good approximations. Put it conversely, the universe seems to be able to approximate efficiently our mathematical models, which are indeed not so computationally efficacious. So even if the universe would not be isomorphic to math, it seems to be doing so well the job of a mathematical structure, including these computations.

Thanks again for these interesting questions, and good luck in the contest!

Best wishes,

Cristi

Thank you for the kind comments.

You ask "If the universe is isomorphic to math, then why we are unable to analytically solve any of our PDEs in physics?"

Why would we be able to analytically solve any of our PDEs in physics? You seem to state it as if it would be an inevitable consequence of the hypothesis that the universe is isomorphic to a mathematical structure.

"if “A Supreme Something” had ordered me to design a physical world—and to do so in way isomorphic to mathematics—I’d like to think that I could have concocted a physical setup far more computationally efficacious than the one we now find ourselves in!"

Why would you do it computationally efficacious? And could you do it like this, and in the same time allow the complexity we observe and we need to exist?

Anyway, our current mathematical models of the physical world are very good approximations. Put it conversely, the universe seems to be able to approximate efficiently our mathematical models, which are indeed not so computationally efficacious. So even if the universe would not be isomorphic to math, it seems to be doing so well the job of a mathematical structure, including these computations.

Thanks again for these interesting questions, and good luck in the contest!

Best wishes,

Cristi

Hi Cristinel,

Turing Machines, Game of Life, Free Will, Godel, Rule 110, etc.?! I think you'd like my Digital Physics essay, and the actual movie even more so.

Are you fine with using all of mathematics to explain our universe? When using math to describe physical phenomenon, are you fine with incoroporating axioms that are merely known to be independent assumptions or would you prefer axioms to be self-evident? I think the concept of actual infinity, and its different guises (e.g. Axiom of Choice, Continuum, etc.) are the root of many paradoxes in both mathematics and physics. Have mathematicians and physicists forgot that a reductio ad absurdum means we should re-examine our assumptions?

Please check out my essay if you get the chance.

Jon

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Turing Machines, Game of Life, Free Will, Godel, Rule 110, etc.?! I think you'd like my Digital Physics essay, and the actual movie even more so.

Are you fine with using all of mathematics to explain our universe? When using math to describe physical phenomenon, are you fine with incoroporating axioms that are merely known to be independent assumptions or would you prefer axioms to be self-evident? I think the concept of actual infinity, and its different guises (e.g. Axiom of Choice, Continuum, etc.) are the root of many paradoxes in both mathematics and physics. Have mathematicians and physicists forgot that a reductio ad absurdum means we should re-examine our assumptions?

Please check out my essay if you get the chance.

Jon

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Hi Jon,

Thank you for the kind and interesting comments. I am not sure if all mathematics can be used to explain the universe. Maybe there are parts that don't have correspondent in the universe, although they may have in other universes. Also, some mathematical theories are based on opposite axioms. For example, Euclidean and non-Euclidean geometries contradict one another when it comes about parallel lines. But there is a way to incorporate contradiction and use it as a fecundity principle to create mathematical universes, including ours (see this essay page 9). I look forward to read your essay, in this brief time that remains.

Best wishes,

Cristi

Thank you for the kind and interesting comments. I am not sure if all mathematics can be used to explain the universe. Maybe there are parts that don't have correspondent in the universe, although they may have in other universes. Also, some mathematical theories are based on opposite axioms. For example, Euclidean and non-Euclidean geometries contradict one another when it comes about parallel lines. But there is a way to incorporate contradiction and use it as a fecundity principle to create mathematical universes, including ours (see this essay page 9). I look forward to read your essay, in this brief time that remains.

Best wishes,

Cristi

Cristi,

I haven't seen you announce it, though since I have Email confirmation from the Minkowski Institute Press that your PhD thesis has been published, please allow me to extend my congratulations. I note that your acknowledgments include David Finkelstein, one of the Minkowski Institute's founding members (along with Abhay Asktekar whom you also acknowledge, among other highly distinguished academics) -- Finkelstein is one of my favorite scientists.

All best,

Tom

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I haven't seen you announce it, though since I have Email confirmation from the Minkowski Institute Press that your PhD thesis has been published, please allow me to extend my congratulations. I note that your acknowledgments include David Finkelstein, one of the Minkowski Institute's founding members (along with Abhay Asktekar whom you also acknowledge, among other highly distinguished academics) -- Finkelstein is one of my favorite scientists.

All best,

Tom

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Dear Tom,

Thank you for noticing. Indeed, my Ph.D. Thesis appeared yesterday at Amazon. It was published by Minkowski Institute Press. I wish to thank for this to Vesselin Petkov, the author of this excellent essay.

Best wishes,

Cristi

Thank you for noticing. Indeed, my Ph.D. Thesis appeared yesterday at Amazon. It was published by Minkowski Institute Press. I wish to thank for this to Vesselin Petkov, the author of this excellent essay.

Best wishes,

Cristi

Hello dear Mr Stoica,

I read your essay, congratulations for your prize. I recognize your analyze of maths.

It is relevant considering the natural automata like the turing machine. That said,if God has inserted mathematical Tools and foundamental laws, so can we utilize the extrapolations without limits. It is important for the prédictions of the evolution.The principle of uniquity is so important considering the entropy and its uniqueness. Of course maths are relevant but can we superimpose all what we want , like we want.I am not sure.

In all case , your essay is interesting in a whole point of vue.

Best Regards

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I read your essay, congratulations for your prize. I recognize your analyze of maths.

It is relevant considering the natural automata like the turing machine. That said,if God has inserted mathematical Tools and foundamental laws, so can we utilize the extrapolations without limits. It is important for the prédictions of the evolution.The principle of uniquity is so important considering the entropy and its uniqueness. Of course maths are relevant but can we superimpose all what we want , like we want.I am not sure.

In all case , your essay is interesting in a whole point of vue.

Best Regards

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