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**Armin Nikkhah Shirazi**: *on* 4/22/15 at 22:58pm UTC, wrote Dear Sophia, I had an interesting experience while I was reading your...

**Janko Kokosar**: *on* 4/22/15 at 15:25pm UTC, wrote Dear Sophia Your classification gives me to that I am between 1b and 2a,...

**James Hoover**: *on* 4/20/15 at 15:00pm UTC, wrote Sophia, Thanks for checking out my essay and for your kind words. Jim

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FQXi FORUM

May 17, 2021

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

TOPIC: Beyond Math by Sophia Magnusdottir [refresh]

TOPIC: Beyond Math by Sophia Magnusdottir [refresh]

In this essay I reflect on the use and usefulness of mathematics from the perspective of a pragmatic physicist. I first classify the different ways we presently think of the relation between our observations and mathematics. Then I explain how we can do physics without using math -- that we are in fact already doing it. In the end the pragmatic reader will know why math is reasonably effective, why we are all models, and how to go beyond math.

Sophia made a bachelors degree in mathematics before losing her way and ending up at the department of philosophy. She lives in Gothenburg with her partner, two sons, and three bunnies, and wishes she had studied physics.

Sophia,

First let me say it was refreshing to review an alternative point of view regarding this discussion and yes I read your entire essay twice :).

Assume you're in a dark room and you accidentally spill hot coffee-you would naturally anticipate the liquid falling to the floor because you have been conditioned to expect this observation through experience, but nothing is natural about this occurrence. In fact this assumption is merely a good bias approximate expectation based on your previous experiences. The day you spill the coffee and it floats in front of you or it pours out like molasses, you will reconsider what is truth and how can you define it to predict reoccurring observations. The relevance of mathematics is not only to describe nature accurately (we could use easier methods) but it is to predict accurately past and future events without a bias conclusion. We study this field to become better at predicting and accurately describing events we observe. Without mathematics supporting the laws of physics we would random exist...and Mr(s). Pragmatic would not find common sense with anything observed.

Best Regards,

D.C. Adams

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First let me say it was refreshing to review an alternative point of view regarding this discussion and yes I read your entire essay twice :).

Assume you're in a dark room and you accidentally spill hot coffee-you would naturally anticipate the liquid falling to the floor because you have been conditioned to expect this observation through experience, but nothing is natural about this occurrence. In fact this assumption is merely a good bias approximate expectation based on your previous experiences. The day you spill the coffee and it floats in front of you or it pours out like molasses, you will reconsider what is truth and how can you define it to predict reoccurring observations. The relevance of mathematics is not only to describe nature accurately (we could use easier methods) but it is to predict accurately past and future events without a bias conclusion. We study this field to become better at predicting and accurately describing events we observe. Without mathematics supporting the laws of physics we would random exist...and Mr(s). Pragmatic would not find common sense with anything observed.

Best Regards,

D.C. Adams

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Hi Demond,

Thanks for your thoughtful comment :) The point of my essay can be illustrated using your example. The fact that you (your brain) has an expectation about what is going to happen means that you have a (more or less accurate) model that you apply (and possibly revise). The relevant part is the model, not that you can formulate it in mathematical expressions. Yes, we can put it in mathematical terms for many systems, and we do it with success. These are the cases that physics focuses on today. That's a sufficient, but not a necessary requirement to do science.

-- Sophia

Thanks for your thoughtful comment :) The point of my essay can be illustrated using your example. The fact that you (your brain) has an expectation about what is going to happen means that you have a (more or less accurate) model that you apply (and possibly revise). The relevant part is the model, not that you can formulate it in mathematical expressions. Yes, we can put it in mathematical terms for many systems, and we do it with success. These are the cases that physics focuses on today. That's a sufficient, but not a necessary requirement to do science.

-- Sophia

The difference between a conceptual based physics and a math based physics is the starting point. The conceive first method then describe mathematically to form measurements appeals to me. The difficulty is the math first approach (restate an equation then go look for it in nature) such as in quantum mechanics and transformation of General relativity are not conceptual. Today’s physics is the latter. Perhaps the next great innovation will use the former method.

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Hi John,

Thanks for your comment. In spirit, that's what I'm trying to aim at in my essay, except that I don't think the math first approach is the difficult approach. It's the easy approach, which is the reason why it's the approach we're mostly using right now. The question I'm addressing in my essay is what if this does not always work?

-- Sophia

Thanks for your comment. In spirit, that's what I'm trying to aim at in my essay, except that I don't think the math first approach is the difficult approach. It's the easy approach, which is the reason why it's the approach we're mostly using right now. The question I'm addressing in my essay is what if this does not always work?

-- Sophia

Hi Sophia,

I agree that the math approach is the easy approach. That's why current theoretical physics developments are conducted almost exclusively on a mathematical basis. However as you pointed out the math fist approach may not work. The math of super string theories requires eleven dimensions of space and time and there is no way to confirm the existence of these extra dimensions experimentally.

I have chosen the physics first approach and I was able to come up with a physical model of our universe called Model Mechanics. The different aspects of Model Mechanics can be used to replace the various abstract math objects (such as field/virtual particles, curvature in space-time dark energy, dark matter....etc.) in our current math theories.

I invite to read my essay and give me your informed comments. Thanks.

Regards,

Ken Seto

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I agree that the math approach is the easy approach. That's why current theoretical physics developments are conducted almost exclusively on a mathematical basis. However as you pointed out the math fist approach may not work. The math of super string theories requires eleven dimensions of space and time and there is no way to confirm the existence of these extra dimensions experimentally.

I have chosen the physics first approach and I was able to come up with a physical model of our universe called Model Mechanics. The different aspects of Model Mechanics can be used to replace the various abstract math objects (such as field/virtual particles, curvature in space-time dark energy, dark matter....etc.) in our current math theories.

I invite to read my essay and give me your informed comments. Thanks.

Regards,

Ken Seto

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Hi Ken,

Thank you for drawing my attention to your essay, which I will be happy to have a look at. It is not true though that there is no way to confirm the existence of extra dimensions. If you have enough energy to excite string states, this would have observable consequences. The problem is that at least in the simplest models, the energy necessary for this is way too high for us to reach. There may be other signatures of the additional dimensions though, such as the dynamics in the early universe. You are probably aware of this, I am just saying it to remind you that in contrast to other aspects of the landscape (multiverse), some properties of string theory are testable - in principle.

-- Sophia

Thank you for drawing my attention to your essay, which I will be happy to have a look at. It is not true though that there is no way to confirm the existence of extra dimensions. If you have enough energy to excite string states, this would have observable consequences. The problem is that at least in the simplest models, the energy necessary for this is way too high for us to reach. There may be other signatures of the additional dimensions though, such as the dynamics in the early universe. You are probably aware of this, I am just saying it to remind you that in contrast to other aspects of the landscape (multiverse), some properties of string theory are testable - in principle.

-- Sophia

Dear Sophia,

What a refreshing essay. You have a delightful writing style that complements the clarity of your thinking.

I fully agree with your idea of analogs and duals that allow one to predict the behavior of the other. This is my model of consciousness, that is, our awareness of our brain modeling subsystems of the 'all'. Some, as you note, think of Friedmann's equations and their solution and they 'are' the model. I think of how classical spin works and then derive the theorem and the equations to describe the model. In this sense I think your thesis is right on target. I invite you to read my essay on the model I built first in my mind and then built a theory around. I believe modeling physics in your mind and then describing it mathematically is to be preferred to studying math and trying to guess what physics it describes. I believe that much math does not describe 'reality' in the same sense that much fantasy and fiction do not describe reality. Languages as maps are not restricted to real territory.

You've chosen a complex topic to write an essay on, as you note, almost blasphemy for some people. I think you did an excellent job presenting your ideas, and agree with them to the extent that I understand them correctly.

Thanks for contributing your essay, and good luck in the contest.

Edwin Eugene Klingman

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What a refreshing essay. You have a delightful writing style that complements the clarity of your thinking.

I fully agree with your idea of analogs and duals that allow one to predict the behavior of the other. This is my model of consciousness, that is, our awareness of our brain modeling subsystems of the 'all'. Some, as you note, think of Friedmann's equations and their solution and they 'are' the model. I think of how classical spin works and then derive the theorem and the equations to describe the model. In this sense I think your thesis is right on target. I invite you to read my essay on the model I built first in my mind and then built a theory around. I believe modeling physics in your mind and then describing it mathematically is to be preferred to studying math and trying to guess what physics it describes. I believe that much math does not describe 'reality' in the same sense that much fantasy and fiction do not describe reality. Languages as maps are not restricted to real territory.

You've chosen a complex topic to write an essay on, as you note, almost blasphemy for some people. I think you did an excellent job presenting your ideas, and agree with them to the extent that I understand them correctly.

Thanks for contributing your essay, and good luck in the contest.

Edwin Eugene Klingman

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Dear Edwin,

Thanks for your comment. I read your essay, and found it very inspiring. It seems your way to look at the questions posed in this contest is quite similar to mine, except that you refer to math as the map, not the a the image of the map, if I understand correctly? I too wish you good luck!

-- Sophia

Thanks for your comment. I read your essay, and found it very inspiring. It seems your way to look at the questions posed in this contest is quite similar to mine, except that you refer to math as the map, not the a the image of the map, if I understand correctly? I too wish you good luck!

-- Sophia

Dear Pragmatic Physicist,

I think that perhaps we are related. My name is Practical Engineer. I enjoyed your essay quite a bit. I especially liked it when the time traveler was introduced to the Fire God.

Myself and my fellow tribesmen have been doing what you describe for a long time although we did not realize that it had any formalisms. We know that certain problems in heat transfer and fluid mechanics and mass transfer can be represented by certain forms of differential equations. We also know that certain electrical devices can be described by the same types of differential equations. None of us are very good with math, but my cousin Sparky can build those electrical boxes really good. So what we do is simplify the mathematical equations into common forms that use dimensionless numbers. My favorite is the Reynolds number but there are others like Nusselt and Prandtl. So cousin Sparky builds the box. Then we measure the voltage and the current at the different points of interest. Then we use the dimensionless groups to convert from volts and amps into mass and pressure or whatever we need to know.

The only problem is that it is a little expensive and slow. And if cousin Sparky makes a mistake then we have to start over.

There is another tribe on the other side of the mountain. We are on friendly terms with them. They aren't very good with math either. They made this thing they call a wind-tunnel. They claim to be able to fly. We think they're crazy.

Best Regards and Good Luck,

Gary Simpson

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I think that perhaps we are related. My name is Practical Engineer. I enjoyed your essay quite a bit. I especially liked it when the time traveler was introduced to the Fire God.

Myself and my fellow tribesmen have been doing what you describe for a long time although we did not realize that it had any formalisms. We know that certain problems in heat transfer and fluid mechanics and mass transfer can be represented by certain forms of differential equations. We also know that certain electrical devices can be described by the same types of differential equations. None of us are very good with math, but my cousin Sparky can build those electrical boxes really good. So what we do is simplify the mathematical equations into common forms that use dimensionless numbers. My favorite is the Reynolds number but there are others like Nusselt and Prandtl. So cousin Sparky builds the box. Then we measure the voltage and the current at the different points of interest. Then we use the dimensionless groups to convert from volts and amps into mass and pressure or whatever we need to know.

The only problem is that it is a little expensive and slow. And if cousin Sparky makes a mistake then we have to start over.

There is another tribe on the other side of the mountain. We are on friendly terms with them. They aren't very good with math either. They made this thing they call a wind-tunnel. They claim to be able to fly. We think they're crazy.

Best Regards and Good Luck,

Gary Simpson

report post as inappropriate

Dear Sophia,

Your essay is near the top in quality of those that have appeared so far. Your topic sort of overlaps the last part of mine

http://fqxi.org/community/forum/topic/2320

I discuss largely the nature of mathematics I see shaping up that will be used by physics.

The reason that mathematics is used in physics is that we measure things and express that according to numbers. This of course goes way back with measuring weights, or distances and so forth. Physics of course has taken this to considerable extremes with measuring quantum eigenvalues and using measurements of scattering cross sections to back out quantum amplitudes and strengths of gauge interactions.

There are of course sciences that do not as heavily rely upon mathematics, such as biology. We have subjects such as psychology that have some small overlap with science, and this tends to reach a limit with sociology. Even these have some mathematics enter into their practice.

At the end you mention the linking of brains or minds, which will be facilitated with cybernetics. That might happen, and that will pave the way towards humanity becoming something other than it has been. We may become something completely different in the long run. This is of course assuming that we can survive far into the future. Under those conditions we can’t know now what sorts of mental structures we will be thinking, or maybe groking or mind-melding, about.

Much of these alternatives you discuss are I think not demonstrable, or at least I see nothing either empirically possible or logically provable in order to support of verify one of these.

LC

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Your essay is near the top in quality of those that have appeared so far. Your topic sort of overlaps the last part of mine

http://fqxi.org/community/forum/topic/2320

I discuss largely the nature of mathematics I see shaping up that will be used by physics.

The reason that mathematics is used in physics is that we measure things and express that according to numbers. This of course goes way back with measuring weights, or distances and so forth. Physics of course has taken this to considerable extremes with measuring quantum eigenvalues and using measurements of scattering cross sections to back out quantum amplitudes and strengths of gauge interactions.

There are of course sciences that do not as heavily rely upon mathematics, such as biology. We have subjects such as psychology that have some small overlap with science, and this tends to reach a limit with sociology. Even these have some mathematics enter into their practice.

At the end you mention the linking of brains or minds, which will be facilitated with cybernetics. That might happen, and that will pave the way towards humanity becoming something other than it has been. We may become something completely different in the long run. This is of course assuming that we can survive far into the future. Under those conditions we can’t know now what sorts of mental structures we will be thinking, or maybe groking or mind-melding, about.

Much of these alternatives you discuss are I think not demonstrable, or at least I see nothing either empirically possible or logically provable in order to support of verify one of these.

LC

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Dear Lawrence,

Thanks for the kind words :) I will go and check out your essay! I have been travelling and I am somewhat behind reading.

Yes, I share your opinion that we may become something completely different in the long run. It's one of the reasons I don't buy into the idea that artificial intelligence is a danger to mankind. I don't think we will actually get to see a split in...

view entire post

Thanks for the kind words :) I will go and check out your essay! I have been travelling and I am somewhat behind reading.

Yes, I share your opinion that we may become something completely different in the long run. It's one of the reasons I don't buy into the idea that artificial intelligence is a danger to mankind. I don't think we will actually get to see a split in...

view entire post

The threat of AI is not something that keeps me awake at night as much as other things. Our species has been very good at exploiting the environment to engineer positive feedbacks for us. We are now converting the planetary biosphere into a garbage heap. We are demolishing our life support system. These things keep me far more concerned than the problem of AI.

The problem of AI taking over is worth keeping in mind though. I think we are seeing the integration of digital technology into us. Now we have wearable tech and Microsoft has its holographic view system, even though it has nothing to do with real holograms. I do see a real prospect for more direct brain computer interfaces (BCI) in the next few decades. We may see this intrude into the inner aspects of our consciousness, and we may in many ways end up in a sort of cyber mind-meld. Star Trek had the BORG, and I would not be too surprised if we end up in that state in the second half of this century.

The question of AI “taking over,” assuming there is much left here worth taking over, may come if we have adaptive and learning AI systems hooked to human brains. They might adapt and learn how to become more like the brain, which could in the end supplant the brain. These might form the basis for von Neumann probes that migrate out into the solar system and after hundreds of millions of years migrate across the galaxy. I see some prospect for this sort of thing happening.

Cheers LC

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The problem of AI taking over is worth keeping in mind though. I think we are seeing the integration of digital technology into us. Now we have wearable tech and Microsoft has its holographic view system, even though it has nothing to do with real holograms. I do see a real prospect for more direct brain computer interfaces (BCI) in the next few decades. We may see this intrude into the inner aspects of our consciousness, and we may in many ways end up in a sort of cyber mind-meld. Star Trek had the BORG, and I would not be too surprised if we end up in that state in the second half of this century.

The question of AI “taking over,” assuming there is much left here worth taking over, may come if we have adaptive and learning AI systems hooked to human brains. They might adapt and learn how to become more like the brain, which could in the end supplant the brain. These might form the basis for von Neumann probes that migrate out into the solar system and after hundreds of millions of years migrate across the galaxy. I see some prospect for this sort of thing happening.

Cheers LC

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Thanks for your essay Sophia. It is interesting and well written.

In the chapter "How to do science without mathematics", you explain that a computer might simply compare two subsets to find solutions without the need for a mathematical model. I might be wrong, but comparing two subsets is simple maths but it is still mathematics?

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In the chapter "How to do science without mathematics", you explain that a computer might simply compare two subsets to find solutions without the need for a mathematical model. I might be wrong, but comparing two subsets is simple maths but it is still mathematics?

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Hi Christophe,

I am glad that you liked my essay! It is a topic very to my heart. I am not sure what you refer to exactly, maybe it is a misunderstanding? In the examples that I mentioned, the computer is one of the subsystems, it does not do a comparison. The comparison would be a test of the hypothesis that the computer is a good model. Ie, you'd make some measurement and see if your model (the one subsystem) is good to describe (or predict) the other subsystem. Or were you referring to something else?

-- Sophia

I am glad that you liked my essay! It is a topic very to my heart. I am not sure what you refer to exactly, maybe it is a misunderstanding? In the examples that I mentioned, the computer is one of the subsystems, it does not do a comparison. The comparison would be a test of the hypothesis that the computer is a good model. Ie, you'd make some measurement and see if your model (the one subsystem) is good to describe (or predict) the other subsystem. Or were you referring to something else?

-- Sophia

I agree, the pragmatic physicist does not have the right to risk knowledge advancement on ethical restraint, on social conventions, nor on philosophical disputation. Predictability is the dominant reality. This is not unlike Machiavelli for the world of true science. Use what works.

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John,

I'm not sure what you mean with "does not have the right". Who says what these rights are? My perspective on this is more one based on the mechanism of selection and adaption. That what works is what will bring us progress. It's somewhat tautological. The question whether scientists should conform to ethical codes or engage in philosophical disputation isn't that easy to answer because ethics also "work" towards something, so does philosophical discussion. They just work towards something different than describing nature. That's why I have been very careful with explaining what I mean with "it works".

-- Sophia

I'm not sure what you mean with "does not have the right". Who says what these rights are? My perspective on this is more one based on the mechanism of selection and adaption. That what works is what will bring us progress. It's somewhat tautological. The question whether scientists should conform to ethical codes or engage in philosophical disputation isn't that easy to answer because ethics also "work" towards something, so does philosophical discussion. They just work towards something different than describing nature. That's why I have been very careful with explaining what I mean with "it works".

-- Sophia

``The pragmatic physicist – first name Pragmatic, last name Physicist – wants to describe observations and only bothers to think if thinking seems useful for this description.’’

I didn’t notice any ethical restraint in this description. Ethical restraint, social conventions, and philosophical disputation have had a tendency to inhibit knowledge advancement.

Survival is the only moral goal of life suggests ``rights’’ come from the ability to survive. Those technologically advanced societies overcome lesser societies. Humanists would like to think Kant’s view would win. But history has shown Hobbes’ view of the leviathan is what nature allows over a long term. Kautilya’s ``Arthashastra’’ and Machiavelli’s ``The Prince’’ are good descriptions of international relations.

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I didn’t notice any ethical restraint in this description. Ethical restraint, social conventions, and philosophical disputation have had a tendency to inhibit knowledge advancement.

Survival is the only moral goal of life suggests ``rights’’ come from the ability to survive. Those technologically advanced societies overcome lesser societies. Humanists would like to think Kant’s view would win. But history has shown Hobbes’ view of the leviathan is what nature allows over a long term. Kautilya’s ``Arthashastra’’ and Machiavelli’s ``The Prince’’ are good descriptions of international relations.

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Hi. Interesting essay. Here are some thoughts about it:

You pointed out that when we need to approach or make predictions about processes from the world (subsystems of our universe), solutions ("models") can be developed not only in the form of mathematical theories but also as other processes from the world that somehow "behave the same". My opinion is that almost all processes in the world are so relatively complex and depending on many parameters (both physical constants and initial states of systems), that it would be extremely unlikely to discover 2 processes that behave the same (give the same result), except for a few very simple phenomena, or if they are made of the exact same stuff, or if one of them has been precisely well-designed for the purpose of matching the other. And I think only 2 kinds of systems (that you mentioned) are generally able to receive that needed amount of design for this purpose: well-programmed computers, and intelligent minds understanding a subject.

I mean, I don't believe in large usefulness for the other kind of "model" that you mentioned with the example of Analogue Gravity : they might provide vague similarities for specific processes but no full similarity, and cannot approach any accuracy I guess (sorry I did not check the article you gave as reference). For example, space-time outside the horizon of a black hole remains perfectly time-symmetric (if not distorted by a falling big mass); the time asymmetry only concerns what crosses the horizon. The acoustic model with a moving fluid does not show this fact, or at least does not make it intuitive. It can suggest some aspects but cannot be accurate because, well, moving fluids remain in a Galilean space-time and just cannot have a faithful correspondence with the curved space-time of General Relativity. Generally, digital computers can reproduce quite well any results that analogue models could provide (as any kind of analogue physical process can be described by known laws of physics, thus analyzed by these laws with methods of numerical analysis, except for quantum phenomena where classical computation would be inefficient, in which case making a quantum model, either "different stuff" quantum analog or by quantum computation, would be a hard problem anyway), with the advantage that they can be programmed to apply the exactly right law, while analog processes being subject to the specific laws ruling the stuff they are made of, are unlikely to behave the same as what we want.

Strictly speaking, computer simulations are mathematical stuff, even if they can look quite different from what is usually presented as "mathematics" at school. However, computer solutions can be considered largely non-mathematical when they heavily depend on the input of big data, which has a non-mathematical origin (see Jaron Lanier's talk on the myth of AI describing the situation of automatic translation systems).

Intelligent minds can also have both mathematical and non-mathematical abilities, that can be developed depending on needs. The advantage of mathematical abilities being their capacity of perfection to match a given rule, in case it would happen to be the right one; but I consider that only a mind can understand another mind, which is an ability beyond maths.

Now about your attempt at classifying views with sentences such as "Observations can be math", "Observations are described by math but are not math". I don't think such sentences make much sense. You don't seem to take them very seriously in the rest of your essay, but why did you develop them in the first place ? In particular, I doubt anyone would claim an observation "is math". An observation may be said to be "described by math" if some mathematical structures can be found there, in the sense that the observation is shown to not be entirely random. However I do not see it as a binary question "can be described or cannot be described by math", but rather a roughly continuous, quantitative question "how much can it be mathematically explained" in the sense of "compressed by some algorithm". So it is another sort of mathematical way of describing how mathematical something is, but it takes more subtle mathematical concepts than such basic set theory concepts as you did. See more explanations in my essay.

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You pointed out that when we need to approach or make predictions about processes from the world (subsystems of our universe), solutions ("models") can be developed not only in the form of mathematical theories but also as other processes from the world that somehow "behave the same". My opinion is that almost all processes in the world are so relatively complex and depending on many parameters (both physical constants and initial states of systems), that it would be extremely unlikely to discover 2 processes that behave the same (give the same result), except for a few very simple phenomena, or if they are made of the exact same stuff, or if one of them has been precisely well-designed for the purpose of matching the other. And I think only 2 kinds of systems (that you mentioned) are generally able to receive that needed amount of design for this purpose: well-programmed computers, and intelligent minds understanding a subject.

I mean, I don't believe in large usefulness for the other kind of "model" that you mentioned with the example of Analogue Gravity : they might provide vague similarities for specific processes but no full similarity, and cannot approach any accuracy I guess (sorry I did not check the article you gave as reference). For example, space-time outside the horizon of a black hole remains perfectly time-symmetric (if not distorted by a falling big mass); the time asymmetry only concerns what crosses the horizon. The acoustic model with a moving fluid does not show this fact, or at least does not make it intuitive. It can suggest some aspects but cannot be accurate because, well, moving fluids remain in a Galilean space-time and just cannot have a faithful correspondence with the curved space-time of General Relativity. Generally, digital computers can reproduce quite well any results that analogue models could provide (as any kind of analogue physical process can be described by known laws of physics, thus analyzed by these laws with methods of numerical analysis, except for quantum phenomena where classical computation would be inefficient, in which case making a quantum model, either "different stuff" quantum analog or by quantum computation, would be a hard problem anyway), with the advantage that they can be programmed to apply the exactly right law, while analog processes being subject to the specific laws ruling the stuff they are made of, are unlikely to behave the same as what we want.

Strictly speaking, computer simulations are mathematical stuff, even if they can look quite different from what is usually presented as "mathematics" at school. However, computer solutions can be considered largely non-mathematical when they heavily depend on the input of big data, which has a non-mathematical origin (see Jaron Lanier's talk on the myth of AI describing the situation of automatic translation systems).

Intelligent minds can also have both mathematical and non-mathematical abilities, that can be developed depending on needs. The advantage of mathematical abilities being their capacity of perfection to match a given rule, in case it would happen to be the right one; but I consider that only a mind can understand another mind, which is an ability beyond maths.

Now about your attempt at classifying views with sentences such as "Observations can be math", "Observations are described by math but are not math". I don't think such sentences make much sense. You don't seem to take them very seriously in the rest of your essay, but why did you develop them in the first place ? In particular, I doubt anyone would claim an observation "is math". An observation may be said to be "described by math" if some mathematical structures can be found there, in the sense that the observation is shown to not be entirely random. However I do not see it as a binary question "can be described or cannot be described by math", but rather a roughly continuous, quantitative question "how much can it be mathematically explained" in the sense of "compressed by some algorithm". So it is another sort of mathematical way of describing how mathematical something is, but it takes more subtle mathematical concepts than such basic set theory concepts as you did. See more explanations in my essay.

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Hi Sylvain:

Thanks for your thoughtful comment. I am not sure why you say that "it would be extremely unlikely to discover 2 processes that behave the same" because that's what all of science presently works like! And the fact that it does then speaks against your opinion "that almost all processes in the world are so relatively complex and depending on many parameters". Or maybe I am misunderstanding what you are saying? As a matter of fact most systems that are presently described in physics are dramatically simple and depend on very few parameters. Think harmonic oscillator.

The reason I have distinguished between the options that observations are math or can be described by math is that both are represented in the literature, and it seems a very fitting topic for the question of this essay contest. Y

ou write "I doubt anyone would claim an observation "is math"." Then you haven't read Tegmark's paper, and apparently you also missed the explanation of it in my essay. If you do not think that observations are math, then what it is that makes an observation different from math? It can't be describable by math, can it?

In any case, the whole point of my essay is to point out that while we may differ on the philosophical underpinning of science, in practice it doesn't matter.

-- Sophia

Thanks for your thoughtful comment. I am not sure why you say that "it would be extremely unlikely to discover 2 processes that behave the same" because that's what all of science presently works like! And the fact that it does then speaks against your opinion "that almost all processes in the world are so relatively complex and depending on many parameters". Or maybe I am misunderstanding what you are saying? As a matter of fact most systems that are presently described in physics are dramatically simple and depend on very few parameters. Think harmonic oscillator.

The reason I have distinguished between the options that observations are math or can be described by math is that both are represented in the literature, and it seems a very fitting topic for the question of this essay contest. Y

ou write "I doubt anyone would claim an observation "is math"." Then you haven't read Tegmark's paper, and apparently you also missed the explanation of it in my essay. If you do not think that observations are math, then what it is that makes an observation different from math? It can't be describable by math, can it?

In any case, the whole point of my essay is to point out that while we may differ on the philosophical underpinning of science, in practice it doesn't matter.

-- Sophia

Hi Sophia,

I did appreciate your point in the conclusions section that there is no particular reason why models must be mathematical to be useful.

I agree with this and I think there is a great value in spending time to get the descriptive model correct. It should be obvious to everyone that having a quantum theory in which there is no agreement on interpretation is missing something.

I do however, feel that a model which has a clear top level description regarding properties that can be measured and related to a mathematical model is the most convincing and useful type of model.

Thank you for reading my essay titled Solving the Mystery and I hope I answered your question on realism to your satisfaction.

With best regards

Richard

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I did appreciate your point in the conclusions section that there is no particular reason why models must be mathematical to be useful.

I agree with this and I think there is a great value in spending time to get the descriptive model correct. It should be obvious to everyone that having a quantum theory in which there is no agreement on interpretation is missing something.

I do however, feel that a model which has a clear top level description regarding properties that can be measured and related to a mathematical model is the most convincing and useful type of model.

Thank you for reading my essay titled Solving the Mystery and I hope I answered your question on realism to your satisfaction.

With best regards

Richard

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Dear Madam,

Your essay is poetry written in prose format. We wish you could have used a gender neutral format like ‘it’ because there is no bare charge or bare mass. Every perceivable information / object is composite with positive (male) charge in the center (central like bone) confined by negative charge (female exterior like flesh) with both protecting each other differently. Their...

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Your essay is poetry written in prose format. We wish you could have used a gender neutral format like ‘it’ because there is no bare charge or bare mass. Every perceivable information / object is composite with positive (male) charge in the center (central like bone) confined by negative charge (female exterior like flesh) with both protecting each other differently. Their...

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Dear Sophia,

I find the quality of many of the essays in this competition below expectation. Yours is a refreshing exception, although I would have wanted to challenge Mr Pragmatic Physicist with some questions about what is real and what is not. I also find refreshing the subtle mix of humour in a serious discussion. I think the essay is destined to do well in my opinion at least.

Since in matters of physics and mathematics, you seem to have a pragmatic and practical mind all rolled into one, I might want to ask you a few questions on what is real, if it is pragmatically correct to ask it here or will do so at my thread, whichever is your preference.

Regards,

Akinbo

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I find the quality of many of the essays in this competition below expectation. Yours is a refreshing exception, although I would have wanted to challenge Mr Pragmatic Physicist with some questions about what is real and what is not. I also find refreshing the subtle mix of humour in a serious discussion. I think the essay is destined to do well in my opinion at least.

Since in matters of physics and mathematics, you seem to have a pragmatic and practical mind all rolled into one, I might want to ask you a few questions on what is real, if it is pragmatically correct to ask it here or will do so at my thread, whichever is your preference.

Regards,

Akinbo

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Dear Akinbo,

I agree on what the quality of the essays is concerned... :/ what bothers me even more though is this weird rating procedure in which participants rate each other's essays. To make matters worse, they can even see the average rating from other's ratings. I certainly wish that the responsible FQXi folks would read Surowiecki's "Wisdom of Crowds", esp the part about information cascades, then they'd see that nothing sensible can come out of this.

In any case, it seems rather futile to complain about this. It must be difficult for FQXi, one the one hand to be open-minded, on the other hand to not be overrun by low quality entries, and then find a fast way to sort out the trash.

I don't know what is real, or let me say Pragmatist doesn't know, just so we can get over somebody's personal (your or mine) experience. The only way Pragmatist can make sense of it is as a relative measure. You could say then that something "is as real as" something else. The question then is whether mathematics is "as real as", say, you. This is a question which seems well-defined. I don't know what to make of an an absolute "reality", I cannot find any basis for it. I myself have an unfortunate tendency towards solipsism, based on some personal experience, but Pragmatist would discard this as useless. I will check out your thread.

-- Sophia

I agree on what the quality of the essays is concerned... :/ what bothers me even more though is this weird rating procedure in which participants rate each other's essays. To make matters worse, they can even see the average rating from other's ratings. I certainly wish that the responsible FQXi folks would read Surowiecki's "Wisdom of Crowds", esp the part about information cascades, then they'd see that nothing sensible can come out of this.

In any case, it seems rather futile to complain about this. It must be difficult for FQXi, one the one hand to be open-minded, on the other hand to not be overrun by low quality entries, and then find a fast way to sort out the trash.

I don't know what is real, or let me say Pragmatist doesn't know, just so we can get over somebody's personal (your or mine) experience. The only way Pragmatist can make sense of it is as a relative measure. You could say then that something "is as real as" something else. The question then is whether mathematics is "as real as", say, you. This is a question which seems well-defined. I don't know what to make of an an absolute "reality", I cannot find any basis for it. I myself have an unfortunate tendency towards solipsism, based on some personal experience, but Pragmatist would discard this as useless. I will check out your thread.

-- Sophia

Dear Pragmatic,

very pleasant read indeed. A message that builds up during reading but also, it seems, during writing! Nice narrative idea and style.

Perhaps my favourite passage is the one in which you admit to ignore why the All displays recurrent, reproducible, often self-similar subsystems, but identify in these features the key for an effective coupling with the mathematical...

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very pleasant read indeed. A message that builds up during reading but also, it seems, during writing! Nice narrative idea and style.

Perhaps my favourite passage is the one in which you admit to ignore why the All displays recurrent, reproducible, often self-similar subsystems, but identify in these features the key for an effective coupling with the mathematical...

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Dear Tommaso,

You have picked out a very important point indeed, one that I had to gloss over due to lack of space. The real mystery is, in fact, why do we find ourselves in an environment that has so many self-similarities - both over space and over time?

I don't think that mathematics is really necessary to extract them, but it is definitely useful, and I think that this is essentially the reason why we find mathematics useful, 'unreasonably useful' even.

One may suspect - and I apologize in advance for the anthropic smell ;) - that these self-similarities are necessary somehow for the evolution of life, or for that life to be able to start recognizing any regularities at all, which is essential for evolution. See, if nature wasn't so reproducible and, in a sense, reliable, life would never adapt and could never evolve.

Somewhere in the multiverse there is a Computationalist in Pragmatic's family tree...

-- Sophia

You have picked out a very important point indeed, one that I had to gloss over due to lack of space. The real mystery is, in fact, why do we find ourselves in an environment that has so many self-similarities - both over space and over time?

I don't think that mathematics is really necessary to extract them, but it is definitely useful, and I think that this is essentially the reason why we find mathematics useful, 'unreasonably useful' even.

One may suspect - and I apologize in advance for the anthropic smell ;) - that these self-similarities are necessary somehow for the evolution of life, or for that life to be able to start recognizing any regularities at all, which is essential for evolution. See, if nature wasn't so reproducible and, in a sense, reliable, life would never adapt and could never evolve.

Somewhere in the multiverse there is a Computationalist in Pragmatic's family tree...

-- Sophia

Dear Ms. Magnusdottir,

I enjoyed your story of Pragmatic Physicist.

I am a pragmatic physicist with a vivid pictorial imagination. In my mind's eye, I see real waves and particles propagating through space and interacting with each other. I believe that this physical intuition provides a better insight into physical theory than mathematical formalism. Indeed, humans have a natural propensity for visual image processing; mathematics is more difficult. However, theoretical physicists are taught to reject physical intuition since it is unreliable, relying only on mathematical formalism. I believe that this is a mistake.

In my essay, "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory", I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What’s more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

Alan Kadin

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I enjoyed your story of Pragmatic Physicist.

I am a pragmatic physicist with a vivid pictorial imagination. In my mind's eye, I see real waves and particles propagating through space and interacting with each other. I believe that this physical intuition provides a better insight into physical theory than mathematical formalism. Indeed, humans have a natural propensity for visual image processing; mathematics is more difficult. However, theoretical physicists are taught to reject physical intuition since it is unreliable, relying only on mathematical formalism. I believe that this is a mistake.

In my essay, "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory", I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What’s more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

Alan Kadin

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Dear Alan,

I agree with you that humans have a natural propensity for visual image processing. In fact, the human eye-brain team is still vastly better at analyzing visual information than any computer. I think that the relevance of data vizualisation for human understanding and, ultimately, scientific modeling, is often underestimated. Alas, I see no reason why not this should eventually be possible to do by a computer.

I am vary of the idea though that intuition is better than mathematical formalism. Human intuition did not develop to explain phenomena that we have no physical perception of. I will look at your essay and read it with interest.

-- Sophia

I agree with you that humans have a natural propensity for visual image processing. In fact, the human eye-brain team is still vastly better at analyzing visual information than any computer. I think that the relevance of data vizualisation for human understanding and, ultimately, scientific modeling, is often underestimated. Alas, I see no reason why not this should eventually be possible to do by a computer.

I am vary of the idea though that intuition is better than mathematical formalism. Human intuition did not develop to explain phenomena that we have no physical perception of. I will look at your essay and read it with interest.

-- Sophia

Physics without math appears impossible because we need numbers when we make measurements. Numbers are part of math. I do see see how we can do physics without math. Probably someone who knows no math can conceive a physical theory but physics goes beyond conception and requires measurements. Measurements are impossible without math. Can you offer a compelling argument about the possibility of measurements without any math? That would be really interesting.

There is also another alternative: Math progresses so much that humans are no longer necessary, they can described mathematically and saved on a chip along with their world.

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There is also another alternative: Math progresses so much that humans are no longer necessary, they can described mathematically and saved on a chip along with their world.

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Numbers are an intermediary. They are handy, but you don't need them. You can determine for example whether the height of a quicksilver column is as high as the height of a sandpile without ever writing down the height of either. What you need to make predictions is not the number, you need to know what to do in reaction - you need a model system, but that system doesn't have to be numerical in any sense.

Besides, the point isn't that we should do science entirely without math, but that math might not be sufficient, and that's no reason to give up on doing science all together. Ie, use numbers where useful, but what do you do when they're not useful? That's the point I addressed in my essay.

Besides, the point isn't that we should do science entirely without math, but that math might not be sufficient, and that's no reason to give up on doing science all together. Ie, use numbers where useful, but what do you do when they're not useful? That's the point I addressed in my essay.

"Numbers are an intermediary. They are handy, but you don't need them. You can determine for example whether the height of a quicksilver column is as high as the height of a sandpile without ever writing down the height of either."

What about determining the volume of a hypersphere? Is there anything to compare it to? I do not only disagree with what you say but I also argue that as long as we establish a comparison we have essentially established a number system: we can call the standard element "1" and start from there, out standard meter for example.

",,, but that math might not be sufficient,"

I agree with that. Obviously math is a tool but not what reveals the truth, if truth exists anyway. Thanks.

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What about determining the volume of a hypersphere? Is there anything to compare it to? I do not only disagree with what you say but I also argue that as long as we establish a comparison we have essentially established a number system: we can call the standard element "1" and start from there, out standard meter for example.

",,, but that math might not be sufficient,"

I agree with that. Obviously math is a tool but not what reveals the truth, if truth exists anyway. Thanks.

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Hi Sophia,

I certainly agree with you that regularities and self-similarities are very useful, perhaps necessary, for the appearance and support of life, and for us to have some chances to make sense of this universe; and I certainly accept your apologies for the anthropic smell of the idea :-}

But you can’t imagine how cheap and frequent it is to obtain periodic and selfsimilar structures from randomly chosen algorithms that run on simple or random inputs - as your remote computationalist relative could confirm.

When I first bumped into self-similarity (I bought Mandelbrot’s 1977 book on Fractals in 1978) I thought these were cute but rather abstract and abstruse forms. In fact, it is easy to see that they are simply another form of periodicity. And, for example, out of 256 elementary cellular automata (with elementary initial conditions), over 20 develop self-similar patterns. Fractals are the A-B-C of the infant computational universe.

Ciao

Tommaso

PS - I replied to your kind comments in my page.

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I certainly agree with you that regularities and self-similarities are very useful, perhaps necessary, for the appearance and support of life, and for us to have some chances to make sense of this universe; and I certainly accept your apologies for the anthropic smell of the idea :-}

But you can’t imagine how cheap and frequent it is to obtain periodic and selfsimilar structures from randomly chosen algorithms that run on simple or random inputs - as your remote computationalist relative could confirm.

When I first bumped into self-similarity (I bought Mandelbrot’s 1977 book on Fractals in 1978) I thought these were cute but rather abstract and abstruse forms. In fact, it is easy to see that they are simply another form of periodicity. And, for example, out of 256 elementary cellular automata (with elementary initial conditions), over 20 develop self-similar patterns. Fractals are the A-B-C of the infant computational universe.

Ciao

Tommaso

PS - I replied to your kind comments in my page.

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Hi Sophia,

your "mathological" classification is really cute. Introducing the word "ALL" for those wishing a multiverse with 1e500 universes is very instructive. All in German means really all, i.e. everything including infinite in space and time.

I read your essay from the first line to the last and liked it.

Best

Lutz

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your "mathological" classification is really cute. Introducing the word "ALL" for those wishing a multiverse with 1e500 universes is very instructive. All in German means really all, i.e. everything including infinite in space and time.

I read your essay from the first line to the last and liked it.

Best

Lutz

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How can a non-physicist take the perspective of a pragmatic physicist? Is it beyond grasp.

I think this essay is the result of a lack of understanding of what physicists do.

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I think this essay is the result of a lack of understanding of what physicists do.

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Dear Sophia,

Your narrative is beautiful. Words speak volumes that numbers cannot begin to represent. You don’t need to be a physicist to think; indeed one cannot think without words. You are correct in pointing out that 'It is shortsighted to just dismiss philosophy.' We do not need to be reminded of Plato's perception that philosophy is the 'spectator of all time and all existence'...

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Your narrative is beautiful. Words speak volumes that numbers cannot begin to represent. You don’t need to be a physicist to think; indeed one cannot think without words. You are correct in pointing out that 'It is shortsighted to just dismiss philosophy.' We do not need to be reminded of Plato's perception that philosophy is the 'spectator of all time and all existence'...

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Dear Gary,

Thanks for the kind words. I think you express what many people's intuition tells then. I think one shouldn't dismiss such intuition, but as a scientist one also has to find a way to state it more precisely, which is what I have attempted. I am not at all sure, for example, that not all observations can be described by math, which you say you are certain of. Clearly, we cannot right now describe all observations by math, but is there a fundamental limit to what we can do? We might never find out. But the thing is, as I have pointed out in my essay, that we can pragmatically ignore this and still do science, with or without math, though certainly not without love :)

-- Sophia

Thanks for the kind words. I think you express what many people's intuition tells then. I think one shouldn't dismiss such intuition, but as a scientist one also has to find a way to state it more precisely, which is what I have attempted. I am not at all sure, for example, that not all observations can be described by math, which you say you are certain of. Clearly, we cannot right now describe all observations by math, but is there a fundamental limit to what we can do? We might never find out. But the thing is, as I have pointed out in my essay, that we can pragmatically ignore this and still do science, with or without math, though certainly not without love :)

-- Sophia

Dear Pragmatic Physicist -

Thank you for such a delightful essay! Such a practical and commonsense approach, I admire it greatly - Hakuna Matata!. I was thinking you had given me a great new way of thinking about mathematics, physics and the world - one that really made sense. But then you asked whether M is in M, and I must admit I've been spinning my wheels ever since. It also made me wonder - I observe myself, so I am in O, but since O is that which I observe, O must be in me, and then I think this statement must be false. Oh dear, I just can't keep this all straight, and I just can't see how taking the math our\t of physics is going to help......

Hoping you can enlighten me! With sincere regards - Musing Metaphysician (aka George Gantz)

.

I must beall that I observe, so I must be outside of O. isn't O in me or outside of O since I make observations in O, and one observation is that I observe myself making observations in O.

n O are about mysel making statements so am I in O?

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Thank you for such a delightful essay! Such a practical and commonsense approach, I admire it greatly - Hakuna Matata!. I was thinking you had given me a great new way of thinking about mathematics, physics and the world - one that really made sense. But then you asked whether M is in M, and I must admit I've been spinning my wheels ever since. It also made me wonder - I observe myself, so I am in O, but since O is that which I observe, O must be in me, and then I think this statement must be false. Oh dear, I just can't keep this all straight, and I just can't see how taking the math our\t of physics is going to help......

Hoping you can enlighten me! With sincere regards - Musing Metaphysician (aka George Gantz)

.

I must beall that I observe, so I must be outside of O. isn't O in me or outside of O since I make observations in O, and one observation is that I observe myself making observations in O.

n O are about mysel making statements so am I in O?

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APOLOGIES FOR THE TYPOS! Let me try again:

Dear Pragmatic Physicist -

Thank you for such a delightful essay! Such a practical and commonsense approach, I admire it greatly - Hakuna Matata!. I was thinking you had given me a great new way of thinking about mathematics, physics and the world - one that really made sense. But then you asked whether M is in M, and I must admit I've been spinning my wheels ever since. It also made me wonder - I observe myself, so I am in O, but since O is that which I observe, O must be in me, and then I think this statement must be false. Oh dear, I just can't keep this all straight, and I just can't see how taking the math out of physics is going to help......

Hoping you can enlighten me! With sincere regards - Musing Metaphysician (aka George Gantz)

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Dear Pragmatic Physicist -

Thank you for such a delightful essay! Such a practical and commonsense approach, I admire it greatly - Hakuna Matata!. I was thinking you had given me a great new way of thinking about mathematics, physics and the world - one that really made sense. But then you asked whether M is in M, and I must admit I've been spinning my wheels ever since. It also made me wonder - I observe myself, so I am in O, but since O is that which I observe, O must be in me, and then I think this statement must be false. Oh dear, I just can't keep this all straight, and I just can't see how taking the math out of physics is going to help......

Hoping you can enlighten me! With sincere regards - Musing Metaphysician (aka George Gantz)

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Dear George,

Thanks for your comment :) I am sorry for confusing you by adding such an admittedly involved question in the passing. I just didn't want to leave it out, but then there wasn't enough space to discuss it further. It's a well-known problem within any axiomatic mathematical theory (of sufficient complexity) that there are questions that cannot be answered. One such question is for example: Does the set of all sets that don't contain themselves contain itself? Well, if it doesn't contain itself, then it does, and if it does contain itself, then it doesn't contain itself. Headache now? The thing is that you run into these problems by creating meta-statements (about sets that contain sets) in a lower-level language. A similar, more popular phrasing is the Barber paradox:

http://en.wikipedia.org/wiki/Barber_paradox

So what I was saying is that the question whether M contains itself, while not in and by itself paradoxical, is also such a meta-question that one can't properly answer within set theory. But then the whole point of my essay was to say that mathematics might not be all there is anyway, and that using math to find out if there is something more than math can only be a first step anyway, so there is no need to get a headache over it :)

-- Sophia

Thanks for your comment :) I am sorry for confusing you by adding such an admittedly involved question in the passing. I just didn't want to leave it out, but then there wasn't enough space to discuss it further. It's a well-known problem within any axiomatic mathematical theory (of sufficient complexity) that there are questions that cannot be answered. One such question is for example: Does the set of all sets that don't contain themselves contain itself? Well, if it doesn't contain itself, then it does, and if it does contain itself, then it doesn't contain itself. Headache now? The thing is that you run into these problems by creating meta-statements (about sets that contain sets) in a lower-level language. A similar, more popular phrasing is the Barber paradox:

http://en.wikipedia.org/wiki/Barber_paradox

So what I was saying is that the question whether M contains itself, while not in and by itself paradoxical, is also such a meta-question that one can't properly answer within set theory. But then the whole point of my essay was to say that mathematics might not be all there is anyway, and that using math to find out if there is something more than math can only be a first step anyway, so there is no need to get a headache over it :)

-- Sophia

Hi Sophia –

It was a relief to find your delightful and intelligent essay back in January, when the contest was otherwise looking pretty bleak. It’s still the best-written of the bunch. And I entirely agree with your viewpoint, nicely expressed in your comment above – “The relevant part is the model, not that you can formulate it in mathematical expressions.” If we were...

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It was a relief to find your delightful and intelligent essay back in January, when the contest was otherwise looking pretty bleak. It’s still the best-written of the bunch. And I entirely agree with your viewpoint, nicely expressed in your comment above – “The relevant part is the model, not that you can formulate it in mathematical expressions.” If we were...

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Hi Conrad,

I find myself agreeing on most of your comment. I will make sure to check out your essay :) It is a point that is often not appreciated that physics is ultimately about relations between physical things. (I'm not sure parameter is a word I would have chosen - it has a distinct meaning in many theories that I don't think you refer to, but I think I know what you mean.) In a sense, that is also what my essay aims at expressing.

You are right about the context-dependence, in principle. In practice it is believed of course that much of the context doesn't matter. One can question whether this is indeed so. Especially when it comes to complex systems, it is far from clear that there is even *any* situation in which one can neglect the context. But it arguably works in many cases (spherical cows etc).

The one point you raise that I don't quite agree on is that electromagnetism, the SM, and and gravity, have nothing in common. They have quite a lot in common actually. To begin with, and to state the obvious, they're all field theories. They are also all local theories. They are defined on differentiable manifolds. They all can be formulated as geometric theories. They all have a notion of parallel transport. It is exactly these similarities that makes so many people believe that there probably is some underlying unifying theory. (I have no strong opinion on this. I'll believe it when I see it ;) ). Quantum mechanics and general relativity have less in common because quantum mechanics is only an approximation. You shouldn't compare quantum mechanics to general relativity, but to the equation of motion for particles in general relativity (not a field theory any more).

Finally, let me say, that I was happy to see that the quality of essays has considerably risen since January :)

-- Sophia

I find myself agreeing on most of your comment. I will make sure to check out your essay :) It is a point that is often not appreciated that physics is ultimately about relations between physical things. (I'm not sure parameter is a word I would have chosen - it has a distinct meaning in many theories that I don't think you refer to, but I think I know what you mean.) In a sense, that is also what my essay aims at expressing.

You are right about the context-dependence, in principle. In practice it is believed of course that much of the context doesn't matter. One can question whether this is indeed so. Especially when it comes to complex systems, it is far from clear that there is even *any* situation in which one can neglect the context. But it arguably works in many cases (spherical cows etc).

The one point you raise that I don't quite agree on is that electromagnetism, the SM, and and gravity, have nothing in common. They have quite a lot in common actually. To begin with, and to state the obvious, they're all field theories. They are also all local theories. They are defined on differentiable manifolds. They all can be formulated as geometric theories. They all have a notion of parallel transport. It is exactly these similarities that makes so many people believe that there probably is some underlying unifying theory. (I have no strong opinion on this. I'll believe it when I see it ;) ). Quantum mechanics and general relativity have less in common because quantum mechanics is only an approximation. You shouldn't compare quantum mechanics to general relativity, but to the equation of motion for particles in general relativity (not a field theory any more).

Finally, let me say, that I was happy to see that the quality of essays has considerably risen since January :)

-- Sophia

Dear Sophia,

This was a very enjoyable read and I liked the rigor that you apply and the clarity that you show when outlining the mathological classification.

I also think that the paper does a very good job when describing simulations and their (actual and potential) use. This made me consider the next question. Since models based on overall classification lack in the direction of a clear decomposition by parts, how large a risk is that a black box understanding of model similarities (eg analog gravity to turbulence) would lead to a ritualization of scientific endeavor in the long run? As a philosopher, I think this is a question you might enjoy analyzing.

That being said, I'll add that I'd love to hear your opinion on my essay.

Warm regards,

Alma

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This was a very enjoyable read and I liked the rigor that you apply and the clarity that you show when outlining the mathological classification.

I also think that the paper does a very good job when describing simulations and their (actual and potential) use. This made me consider the next question. Since models based on overall classification lack in the direction of a clear decomposition by parts, how large a risk is that a black box understanding of model similarities (eg analog gravity to turbulence) would lead to a ritualization of scientific endeavor in the long run? As a philosopher, I think this is a question you might enjoy analyzing.

That being said, I'll add that I'd love to hear your opinion on my essay.

Warm regards,

Alma

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Dear Alma,

I will have a look at your essay :) Don't you think that mathematics too is a ritualization?

-- Sophia

I will have a look at your essay :) Don't you think that mathematics too is a ritualization?

-- Sophia

Dear Sophia,

To show what I had in mind, I can give some examples as to why I think math should not be a ritualization, except for the part where they drink lots of coffee to produce theorems. I am sure you are familiar with the cases, still I should outline them here in detail. The four color theorem was one of the first computer proofs; the computer assistance was necessary because they...

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To show what I had in mind, I can give some examples as to why I think math should not be a ritualization, except for the part where they drink lots of coffee to produce theorems. I am sure you are familiar with the cases, still I should outline them here in detail. The four color theorem was one of the first computer proofs; the computer assistance was necessary because they...

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Dear Sophia,

Thank you for the excellent essay. It was a pleasure to read the entire one, not only the abstract and conclusions. I love your detailed description of Pragmatic Physicist and the other Pragmatic persons and the honest confession referred to essays’ reading. There is more than 100 essays in the contest so the decision what to read and comment is difficult and needs a...

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Thank you for the excellent essay. It was a pleasure to read the entire one, not only the abstract and conclusions. I love your detailed description of Pragmatic Physicist and the other Pragmatic persons and the honest confession referred to essays’ reading. There is more than 100 essays in the contest so the decision what to read and comment is difficult and needs a...

view entire post

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Hi Sophia,

Great essay! It is well-written and well-argued. In my essay, we argue that mathematics provides models about nature, and does not tell us the reality of nature. Maybe someday we will be able to describe nature in a language other than mathematics. I would be glad to take your opinion about my essay.

Best regards,

Mohammed

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Great essay! It is well-written and well-argued. In my essay, we argue that mathematics provides models about nature, and does not tell us the reality of nature. Maybe someday we will be able to describe nature in a language other than mathematics. I would be glad to take your opinion about my essay.

Best regards,

Mohammed

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Dear Sophia,

You have a creative essay here. I understand you saying basically that science works by modeling, just as language does. You agree thus that ALL communication is at last mediated and the relevant "medium" I suppose to be what you mean by a MODEL.

But my worry is that physics agrees that there can be "physical" communication without physical medium (such as in quantum entanglement and action at a distance or even the so-called "fields"; see Graneau's essay). Isn't this then communication without a model?

In this sense imagery of any kind and hence language actually fails yet mathematics works in that math/physics actually can assign "nothing" a quantity by the name of a constant (as in a "conservation law" or "energy" or "entropy" or the "quantum" state).

Otherwise really how would you actively model "nothing" without maths? It seems that which ever**imagery** one may adopt of "nothing" leads to a conflict for this state insists on being WITHOUT an observable trait.

And I think that this exactly is the argument that both Godel and Heisenberg formalize (in mathematics and physics respectively) namely: we should learn to accept NOTHING as also a legitimate trait (indeed the most fundamental trait) that nature can have.

I think therefore of "Nothing" as the state that models itself; the set that contains itself; perhaps what Steven P. Sax calls in his essay the self referential state.

It is what I have called in my essay the observer/initial condition. And O yes, It is by definition a conflict both in logic and imagery which assertion only brings us back to Godel and Heisenberg namely: it is nevertheless legitimate nature.

Would like to have your comment on my essay.

Regards,

Chidi

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You have a creative essay here. I understand you saying basically that science works by modeling, just as language does. You agree thus that ALL communication is at last mediated and the relevant "medium" I suppose to be what you mean by a MODEL.

But my worry is that physics agrees that there can be "physical" communication without physical medium (such as in quantum entanglement and action at a distance or even the so-called "fields"; see Graneau's essay). Isn't this then communication without a model?

In this sense imagery of any kind and hence language actually fails yet mathematics works in that math/physics actually can assign "nothing" a quantity by the name of a constant (as in a "conservation law" or "energy" or "entropy" or the "quantum" state).

Otherwise really how would you actively model "nothing" without maths? It seems that which ever

And I think that this exactly is the argument that both Godel and Heisenberg formalize (in mathematics and physics respectively) namely: we should learn to accept NOTHING as also a legitimate trait (indeed the most fundamental trait) that nature can have.

I think therefore of "Nothing" as the state that models itself; the set that contains itself; perhaps what Steven P. Sax calls in his essay the self referential state.

It is what I have called in my essay the observer/initial condition. And O yes, It is by definition a conflict both in logic and imagery which assertion only brings us back to Godel and Heisenberg namely: it is nevertheless legitimate nature.

Would like to have your comment on my essay.

Regards,

Chidi

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I enjoyed your essay. Essentially my essay, "Modeling Reality with Mathematics" was an attempt to present thoughts similar to yours with a few stories instead of logic. I was afraid I would be blown out of the saddle with my opinions. Several here have offered a similar point of view about modeling and math. I feel better. However, I have noticed that a few that seem to agree with you do not act on those agreements. In my opinion, the roots of the beliefs in math as reality is far deeper than many would like to believe.

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Dear Al,

Yes, I agree with you, there is much more believe underlying physics than many physicists would want to admit. This is more apparent though in unconfirmed (actually questionable) hypothesis that are rarely questioned, rarely even recognized as such, such as naturalness and simplicity. That math can be used is also a hypothesis - arguably one that is very well confirmed, but that might have its limits too. I think it is important to keep in mind that it is just that, a hypothesis, and those who believe that there is a deep connection between math and reality, that they might even be the same, are just expressing exactly this: a belief. I will have a look at your essay :)

-- Sophia

Yes, I agree with you, there is much more believe underlying physics than many physicists would want to admit. This is more apparent though in unconfirmed (actually questionable) hypothesis that are rarely questioned, rarely even recognized as such, such as naturalness and simplicity. That math can be used is also a hypothesis - arguably one that is very well confirmed, but that might have its limits too. I think it is important to keep in mind that it is just that, a hypothesis, and those who believe that there is a deep connection between math and reality, that they might even be the same, are just expressing exactly this: a belief. I will have a look at your essay :)

-- Sophia

Sophia,

I like your good humor and insightful psychology! -- indeed, I jumped to the conclusion, after having rejected the idea that anything such as a Pragmatic Physicist even exists.

At least, not a mathematical physicist.

Thing is, though -- in principle, everything that is described by mathematical symbols can in fact be translated into natural language. To avoid loss of precision and self-consistency, however, more than the most simple results would be exceedingly labored and tedious. Even the logician's little piece of foundational mathematics, such as Russell and Whitehead undertook in *Principia Mathematica* filled 300 some odd pages of dense and mind-numbing symbols to prove 1 + 1 = 2. And then along came Godel ...

It would be folly to think that a natural language description of significant physical phenomena would be less labored, less tedious, would it not? If one wants to do away with representational formalism entirely, that's beyond pragmatism -- it's the radical empiricism science rejected over 300 years ago.

I disagree with what you said, but I loved the way you said it. :-) (Your naming of "Pragmatic Physicist" reminded me that when my daughter was a little girl she wrote and illustrated a story starring a character named "Binomial Nomenclature.")

All best,

Tom

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I like your good humor and insightful psychology! -- indeed, I jumped to the conclusion, after having rejected the idea that anything such as a Pragmatic Physicist even exists.

At least, not a mathematical physicist.

Thing is, though -- in principle, everything that is described by mathematical symbols can in fact be translated into natural language. To avoid loss of precision and self-consistency, however, more than the most simple results would be exceedingly labored and tedious. Even the logician's little piece of foundational mathematics, such as Russell and Whitehead undertook in *Principia Mathematica* filled 300 some odd pages of dense and mind-numbing symbols to prove 1 + 1 = 2. And then along came Godel ...

It would be folly to think that a natural language description of significant physical phenomena would be less labored, less tedious, would it not? If one wants to do away with representational formalism entirely, that's beyond pragmatism -- it's the radical empiricism science rejected over 300 years ago.

I disagree with what you said, but I loved the way you said it. :-) (Your naming of "Pragmatic Physicist" reminded me that when my daughter was a little girl she wrote and illustrated a story starring a character named "Binomial Nomenclature.")

All best,

Tom

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Wow, Sophia your essay was great! And I loved the graphics. You really captured some ideas I was groping towards when I tried to lay out Lem's position in my own essay.

Please check mine out, tell me what you think, and give me your vote:

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

Best of luck in the contest!

Rick Searle

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Please check mine out, tell me what you think, and give me your vote:

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

Best of luck in the contest!

Rick Searle

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Sophia,

As the pragmatic physicist you will use the math and modeling that works, realizing that it is only as good as your inputs of Instrumathism. As you say, Math is constant. The person between math and physics is not. I speak of the pragmatic approach as well, demonstrating the connections of math, physics and the human brain in modeling the classical and the quantum worlds, coming up with new concepts explaining nature: quantum biology, LHC, and DNA.

Your essay has a Ben Franklin common sense flow that makes perfect sense.

Jim

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As the pragmatic physicist you will use the math and modeling that works, realizing that it is only as good as your inputs of Instrumathism. As you say, Math is constant. The person between math and physics is not. I speak of the pragmatic approach as well, demonstrating the connections of math, physics and the human brain in modeling the classical and the quantum worlds, coming up with new concepts explaining nature: quantum biology, LHC, and DNA.

Your essay has a Ben Franklin common sense flow that makes perfect sense.

Jim

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Hi Sophia,

I enjoyed your essay a lot, although I suspect you will not find too many Pragmatic Physicists in the FQXi community, as we are an impractical lot who like to debate the nature of reality.

I liked your idea that a simulation of one system by another could be considered as a generalization of the idea of a mathematical model. Your case could be bolstered by noting that when this simulation takes the form of a quantum computation, it could easily become impractical for a mathematician to verify its conclusions via conventional means due to the exponential complexity blowup. Therefore, we could regard quantum computation as already providing a concrete example of a geneneralization of mathematical modelling.

Finally, please could you supply some further references on the history of 16th century Hip-Hop?

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I enjoyed your essay a lot, although I suspect you will not find too many Pragmatic Physicists in the FQXi community, as we are an impractical lot who like to debate the nature of reality.

I liked your idea that a simulation of one system by another could be considered as a generalization of the idea of a mathematical model. Your case could be bolstered by noting that when this simulation takes the form of a quantum computation, it could easily become impractical for a mathematician to verify its conclusions via conventional means due to the exponential complexity blowup. Therefore, we could regard quantum computation as already providing a concrete example of a geneneralization of mathematical modelling.

Finally, please could you supply some further references on the history of 16th century Hip-Hop?

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Hi Matt,

You are making a good point, one that I should clearly take into account if I should ever expand on the topic, which is the problem of verification that, as one can see from prominent recent examples, is becoming a very real problem in mathematics. Thanks for that!

Regarding 16th century Hip Hop, its existence can be proved by any of the methods listed here

http://staffhome.ecm.uwa.edu.au/~00043886/humour/invalid

.proofs.html

make your pick ;)

-- Sophia

You are making a good point, one that I should clearly take into account if I should ever expand on the topic, which is the problem of verification that, as one can see from prominent recent examples, is becoming a very real problem in mathematics. Thanks for that!

Regarding 16th century Hip Hop, its existence can be proved by any of the methods listed here

http://staffhome.ecm.uwa.edu.au/~00043886/humour/invalid

.proofs.html

make your pick ;)

-- Sophia

Dear Sophia,

I very much enjoyed your essay and liked a lot the idea of "circumventing" math in terms of mapping certain systems to equivalent ones. This is surely an interesting approach and actually the ultimate goal of universal quantum simulators.

Your paper will be among the winners in my view. It is very well-written and clearly argued - congratulations!

However, I dare to differ in one key point: in my humble opinion, it is not possible to replace math - simply because it is the very tool we need to explain and/or define the physical systems!

Of course I can for instance model one system via an appropriate quantum simulator, but for that to be the case I would need the mathematical description (Hamiltonian) of the respective system to implement it. Without it, how would I know what system my quantum simulation matches with? What obervables would I measure? How would I do the comparison/matching?

We have to resort to reductionism and take the system apart mathematically to know what exactly it is that we are matching to another system. Otherwise it is just a "black box" that happens to behave in an equivalent way to another box without us having any true understanding of the two boxes.

The mathematical description is indispensable and in my little opera "Map = Territory" I even argue for a possible merger of the description and the described in fundamental physics.

Wish you all the best and great success!

With deep respect,

Martin

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I very much enjoyed your essay and liked a lot the idea of "circumventing" math in terms of mapping certain systems to equivalent ones. This is surely an interesting approach and actually the ultimate goal of universal quantum simulators.

Your paper will be among the winners in my view. It is very well-written and clearly argued - congratulations!

However, I dare to differ in one key point: in my humble opinion, it is not possible to replace math - simply because it is the very tool we need to explain and/or define the physical systems!

Of course I can for instance model one system via an appropriate quantum simulator, but for that to be the case I would need the mathematical description (Hamiltonian) of the respective system to implement it. Without it, how would I know what system my quantum simulation matches with? What obervables would I measure? How would I do the comparison/matching?

We have to resort to reductionism and take the system apart mathematically to know what exactly it is that we are matching to another system. Otherwise it is just a "black box" that happens to behave in an equivalent way to another box without us having any true understanding of the two boxes.

The mathematical description is indispensable and in my little opera "Map = Territory" I even argue for a possible merger of the description and the described in fundamental physics.

Wish you all the best and great success!

With deep respect,

Martin

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Dear Martin,

Thank you for your thoughtful comment and the kind words. I will have a look at your essay :) You are right of course in that if you program a system, you are putting in a mathematical description already. The point of the examples in my paper is to explain that this isn't strictly speaking necessary. The only thing you ultimately need to do is to compare two systems and see how they behave. How do you do the comparison - well that's the task of science. You already do this now by using math. How do you know what is the momentum, if not because we've calculated certain behaviors and have noticed they do describe observations? You can do the same thing by constructing (with your hands - or maybe nano-tweezers) some system and measuring its properties, then comparing it to another system.

Keep in mind though that I never said that we should do without math altogether. In practice, you'd certainly still use math for some aspects. But maybe not for all. See, even if you program your computer with some mathematical input, that computer is a physical thing. It might be doing something else besides your programming. Now presently this is something we are trying to avoid at all costs because executing the math is the modelling that we want to do. But maybe for some purposes we'd be better of not trying to control the system too much, and see if that helps us with other things.

-- Sophia

Thank you for your thoughtful comment and the kind words. I will have a look at your essay :) You are right of course in that if you program a system, you are putting in a mathematical description already. The point of the examples in my paper is to explain that this isn't strictly speaking necessary. The only thing you ultimately need to do is to compare two systems and see how they behave. How do you do the comparison - well that's the task of science. You already do this now by using math. How do you know what is the momentum, if not because we've calculated certain behaviors and have noticed they do describe observations? You can do the same thing by constructing (with your hands - or maybe nano-tweezers) some system and measuring its properties, then comparing it to another system.

Keep in mind though that I never said that we should do without math altogether. In practice, you'd certainly still use math for some aspects. But maybe not for all. See, even if you program your computer with some mathematical input, that computer is a physical thing. It might be doing something else besides your programming. Now presently this is something we are trying to avoid at all costs because executing the math is the modelling that we want to do. But maybe for some purposes we'd be better of not trying to control the system too much, and see if that helps us with other things.

-- Sophia

Dear Sophia,

Thank you for a very interesting essay.

There is a philosophy book that came out in 1980 that I think is in agreement with your main argument. The book is titled "Science Without Numbers: A Defence of Nominalism" by Hartry H. Field.

Please take a look at my essay.

All the best,

Noson Yanofsky

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Thank you for a very interesting essay.

There is a philosophy book that came out in 1980 that I think is in agreement with your main argument. The book is titled "Science Without Numbers: A Defence of Nominalism" by Hartry H. Field.

Please take a look at my essay.

All the best,

Noson Yanofsky

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Hi Sophia,

I followed a fair amount of your argument, but take away your main message that mathematics is at the moment the best language we have to communicate physics with fellow scientists. In this regard it is to be highly treasured, but not to be given a status beyond the physical events to which it bears relation.

Your essay is very nicely written and has given the best description of mathematics and its role that I have seen so far in this collection.

Regards

Neal

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I followed a fair amount of your argument, but take away your main message that mathematics is at the moment the best language we have to communicate physics with fellow scientists. In this regard it is to be highly treasured, but not to be given a status beyond the physical events to which it bears relation.

Your essay is very nicely written and has given the best description of mathematics and its role that I have seen so far in this collection.

Regards

Neal

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Dear Sophia,

I am pragmatic meteorologist. For me "universe" mean all there is. “Pragmatic meteorologist’s view on science is shaped by what he learned as a student”. That is mostly thermodynamic and mathematics. At this moment, the most powerful computers in the world are processed numerous observation data using mathematical formulas to get a good weather forecast. The problem is not in the atmosphere even in mathematics. The problem is the accuracy of the measured data and approximation formula used. The same is true for everything, even for the universe. We cannot blame nature and mathematics for our ignorance of the exact relationships a whole and parts.

Giants of natural philosophy in the past knew how to choose the right math to come up with results that we now use. I hope that I am on their way and that I chose the right mathematical approach to describe the relationship a whole (of the Universe) and parts in my essay. If you find an error in my calculations acknowledge them.

Your philosophical approach is for me understandable and useful. What I cannot accept is when physicists try to be philosophers but stop halfway. In short, there are plenty of essays that are violent towards elementary mathematical rules or improper use mathematics.

Regards,

Branko

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I am pragmatic meteorologist. For me "universe" mean all there is. “Pragmatic meteorologist’s view on science is shaped by what he learned as a student”. That is mostly thermodynamic and mathematics. At this moment, the most powerful computers in the world are processed numerous observation data using mathematical formulas to get a good weather forecast. The problem is not in the atmosphere even in mathematics. The problem is the accuracy of the measured data and approximation formula used. The same is true for everything, even for the universe. We cannot blame nature and mathematics for our ignorance of the exact relationships a whole and parts.

Giants of natural philosophy in the past knew how to choose the right math to come up with results that we now use. I hope that I am on their way and that I chose the right mathematical approach to describe the relationship a whole (of the Universe) and parts in my essay. If you find an error in my calculations acknowledge them.

Your philosophical approach is for me understandable and useful. What I cannot accept is when physicists try to be philosophers but stop halfway. In short, there are plenty of essays that are violent towards elementary mathematical rules or improper use mathematics.

Regards,

Branko

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Sophia,

Time grows short, so I am revisiting essays I’ve read (3/26) to assure I’ve rated them. I find that I haven't rated yours, so I will rectify that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Time grows short, so I am revisiting essays I’ve read (3/26) to assure I’ve rated them. I find that I haven't rated yours, so I will rectify that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Dear Sophia

Your classification gives me to that I am between 1b and 2a, or more 1b. Namely, I think that elementary math, which is necessary for physics is only math of quantum gravity theory and math of distinctions of qualia. Thus this model does not need Tegmark's multiverse. But, I am from street. I think that math cannot describe everything, we need also primitive consciousness. By consequences, this model is very similar to physicalism, only physics is replaced by consciousness. In my opinion, graphical classificaton is in right direction, but it needs to be improved.

Your non-mathematical approach seems to me similar, as Penrose's in one of his books? As a pragmatic person, you shold still give some opinion about consciousness? Because when reading your section 6, I cannot say if your non-math can include non-math part of consciousness.

Maybe all graphs do not describe precisely your points?

I think that you think to write in the first paragraph in section 3 that ''T maps the observations to models.'' Otherwise expressions T(O) are not correct.

I agree that it is not very important if people agree in philosohical view, of course maybe this can be verified with measurements.

I like your word pragmatic, because it can means to be open to other approaches, and to find essential things. It is like positivism, but they are too much negativistic :-) to other approaches, for instance consciousness and interpretation of quantum mechanics.

My essay

Best regards

Janko Kokosar

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Your classification gives me to that I am between 1b and 2a, or more 1b. Namely, I think that elementary math, which is necessary for physics is only math of quantum gravity theory and math of distinctions of qualia. Thus this model does not need Tegmark's multiverse. But, I am from street. I think that math cannot describe everything, we need also primitive consciousness. By consequences, this model is very similar to physicalism, only physics is replaced by consciousness. In my opinion, graphical classificaton is in right direction, but it needs to be improved.

Your non-mathematical approach seems to me similar, as Penrose's in one of his books? As a pragmatic person, you shold still give some opinion about consciousness? Because when reading your section 6, I cannot say if your non-math can include non-math part of consciousness.

Maybe all graphs do not describe precisely your points?

I think that you think to write in the first paragraph in section 3 that ''T maps the observations to models.'' Otherwise expressions T(O) are not correct.

I agree that it is not very important if people agree in philosohical view, of course maybe this can be verified with measurements.

I like your word pragmatic, because it can means to be open to other approaches, and to find essential things. It is like positivism, but they are too much negativistic :-) to other approaches, for instance consciousness and interpretation of quantum mechanics.

My essay

Best regards

Janko Kokosar

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Dear Sophia,

I had an interesting experience while I was reading your essay.

While reading the first half, I thought that, well, your "mathological" classification system may certainly be interesting to people who seriously consider the MUH, but for others (like me) it essentially boils down to a system of distinctions without a difference. In short, my reaction was, frankly, "Meh".

Then I read the second half of your essay, and the idea that the model of reality is not the mathematical structure itself but the outcome of the process of engaging with it struck me as truly profound. I had never conceptualized "modeling reality" in this way. And I think you are right, it does broaden our perspective to consider mathematics as only one (but highly effective) means for modeling our observations. Your "black box" example makes this point very nicely.

I think this is very important idea that deserves to be more widely considered. I hope you explore it further in your future work.

As a result of being exposed to an idea that has shifted my view of an aspect of the world that is important to me, my view of your essay underwent a dramatic shift toward the positive, something I had not experienced in reviewing any other FQXi essay.

Let me close by saying that I also really enjoyed your writing style with its understated humor.

Best wishes,

Armin

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I had an interesting experience while I was reading your essay.

While reading the first half, I thought that, well, your "mathological" classification system may certainly be interesting to people who seriously consider the MUH, but for others (like me) it essentially boils down to a system of distinctions without a difference. In short, my reaction was, frankly, "Meh".

Then I read the second half of your essay, and the idea that the model of reality is not the mathematical structure itself but the outcome of the process of engaging with it struck me as truly profound. I had never conceptualized "modeling reality" in this way. And I think you are right, it does broaden our perspective to consider mathematics as only one (but highly effective) means for modeling our observations. Your "black box" example makes this point very nicely.

I think this is very important idea that deserves to be more widely considered. I hope you explore it further in your future work.

As a result of being exposed to an idea that has shifted my view of an aspect of the world that is important to me, my view of your essay underwent a dramatic shift toward the positive, something I had not experienced in reviewing any other FQXi essay.

Let me close by saying that I also really enjoyed your writing style with its understated humor.

Best wishes,

Armin

report post as inappropriate

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