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FQXi FORUM

February 22, 2018

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

TOPIC: How not to factor a miracle. by Derek K Wise [refresh]

TOPIC: How not to factor a miracle. by Derek K Wise [refresh]

Wigner's famous and influential claim that mathematics is "unreasonably effective" in physics is founded on unreasonable assumptions about the nature of mathematics and its independence of physics. Here I argue that what is surprising is not the effectiveness of mathematics but the amenability of physics reductionist strategies. I also argue that while our luck may run out on the effectiveness of reduction, mathematics is still our best hope for surpassing this obstacle. While I agree that human understanding of the natural world in mathematical terms evinces a miracle, I see no way to factor out the human dimension of this miracle.

Derek Wise is a mathematical physicist at the University of Erlangen, and has been working at the interface of mathematics and physics since undergraduate studies. He was a Visiting Assistant Professor of Mathematics at UC Davis under the NSF VIGRE program until 2010, and has since been a Postdoctoral Fellow at the University of Erlangen, where he has held positions both in the Institute for Quantum Gravity and in the Department of Mathematics. He is dedicated to getting mathematicians and physicists to work together, and promoting mutual understanding.

Dear Derek Wise,

Thank you for your elegantly argued and written essay. Very enjoyable to read. Your point that mathematics is not nearly entirely separate from physics is well developed.

Bob Shour

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Thank you for your elegantly argued and written essay. Very enjoyable to read. Your point that mathematics is not nearly entirely separate from physics is well developed.

Bob Shour

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Dear Derek Wise

I really liked your essay for its depth,clarity and lucidity.But I liked the last paragraph of your essay the most wherein you raise a very important question and sum up your wisdom as well.My question is that how can we "bring in" the human element in the theories of physics and in mathematics as well?

If you look at my essay with Kauffmann we are discussing the same "Marvel" throughout our essay and there you will find that there is a way in which "human element" comes to the central place and the physics and mathematics follow from there.

Thanks for your essay.I would like to discuss more about it with you.

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I really liked your essay for its depth,clarity and lucidity.But I liked the last paragraph of your essay the most wherein you raise a very important question and sum up your wisdom as well.My question is that how can we "bring in" the human element in the theories of physics and in mathematics as well?

If you look at my essay with Kauffmann we are discussing the same "Marvel" throughout our essay and there you will find that there is a way in which "human element" comes to the central place and the physics and mathematics follow from there.

Thanks for your essay.I would like to discuss more about it with you.

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Dear Derek K. Wise,

In your essay you wrote: "Quantum gravity, the attempt to combine general relativity with quantum physics, faces a dilemma, since these theories involve orthogonal types of `ignoring.' Most approaches to the problem can be classified into one of two scenarios. In one scenario, we try to make gravity more like the rest of physics, for example by studying elementary particles gravity (`gravitons') or frameworks to generalize them. In the other scenario either we attempt to apply principles of quantum physics, which in fact we know entirely from the world of particles, to the picture of spacetime." It seems to me that the preceding is an excellent summary of some of the problems facing string theory. The string theorists need to put particle physics into geometry or to put geometry into particle physics — to somehow geometrize Feynman diagrams. I have suggested that the string theorists have underestimated Milgrom. Google "witten milgrom" for further information.

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In your essay you wrote: "Quantum gravity, the attempt to combine general relativity with quantum physics, faces a dilemma, since these theories involve orthogonal types of `ignoring.' Most approaches to the problem can be classified into one of two scenarios. In one scenario, we try to make gravity more like the rest of physics, for example by studying elementary particles gravity (`gravitons') or frameworks to generalize them. In the other scenario either we attempt to apply principles of quantum physics, which in fact we know entirely from the world of particles, to the picture of spacetime." It seems to me that the preceding is an excellent summary of some of the problems facing string theory. The string theorists need to put particle physics into geometry or to put geometry into particle physics — to somehow geometrize Feynman diagrams. I have suggested that the string theorists have underestimated Milgrom. Google "witten milgrom" for further information.

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Hi Derek,

You have presented honestly and robustly written essay. After careful reading I cannot find anything to criticize. You have described quantum gravity approaches as follows:

„Most approaches to the problem can be classified into one of two scenarios. In one scenario, we try to make gravity more like the rest of physics, for example by studying elementary particles gravity (`gravitons') or frameworks to generalize them. In the other scenario either we attempt to apply principles of quantum physics, which in fact we know entirely from the world of particles, to the picture of spacetime.”

I have started my scenario from gravity, looking for a genuine correspondence rule and finished on using the geometrization conjecture, proved by Perelman. I have coined Geometrical Universe Hypothesis that can be broken down into:

- the correspondence rule that all interactions and matter are manifestations of spacetime geometry

- the empirical domain - gravitational, electromagnetic, strong nuclear and weak nuclear measurements and cosmological observations

- the geometric structure being a set of Thurston geometries with metrics and the wave transfer

If you are interested you can find details in my essay.

This view is developed in details by some physicists like Torsten Asselmeyer-Maluga.

I would appreciate your comments. Thank you.

Jacek

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You have presented honestly and robustly written essay. After careful reading I cannot find anything to criticize. You have described quantum gravity approaches as follows:

„Most approaches to the problem can be classified into one of two scenarios. In one scenario, we try to make gravity more like the rest of physics, for example by studying elementary particles gravity (`gravitons') or frameworks to generalize them. In the other scenario either we attempt to apply principles of quantum physics, which in fact we know entirely from the world of particles, to the picture of spacetime.”

I have started my scenario from gravity, looking for a genuine correspondence rule and finished on using the geometrization conjecture, proved by Perelman. I have coined Geometrical Universe Hypothesis that can be broken down into:

- the correspondence rule that all interactions and matter are manifestations of spacetime geometry

- the empirical domain - gravitational, electromagnetic, strong nuclear and weak nuclear measurements and cosmological observations

- the geometric structure being a set of Thurston geometries with metrics and the wave transfer

If you are interested you can find details in my essay.

This view is developed in details by some physicists like Torsten Asselmeyer-Maluga.

I would appreciate your comments. Thank you.

Jacek

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Hi Derek,

I enjoyed your essay. You wrote:

“However, it is worth pointing out that, for example, calculus and a large part of differential equations -still among the physicist's most important tools-were designed precisely for physical applications.”

I agree, but do you think that achieving an infinite is actually possible in the physical world? Would potential infinities be a more logical concept for physicists to employ? Or are you fine with physics only very closely approximating the real world?

“Moreover, as `quantum mathematicians' we would presumably not prove theorems according to the familiar

rules of logic, which are deeply tied to set theory, but according to rules of quantum logic which reflect the fuzzy, indistinct nature of propositions about fundamentally quantum mechanical systems.”

This is idea I was trying to get at in my question #4 which asks, “If quantum mechanics is a world where things can be both “yes” and “no” at the same time, should experimental results be analyzed with Zen Koans instead of logical inferences?”

I also like you questions regarding the choices of mathematical foundations, which I think is well... foundational.

Also, your talk of sub-objects and quotients of a set make me think that you might like to weigh in on a question I asked (#3): “Is there an analogy between the following relationships: a “class” vs. a “set” and “true” vs. “provable”?”

Please check out my Digital Physics movie essay if you have a chance.

Thanks,

Jon

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I enjoyed your essay. You wrote:

“However, it is worth pointing out that, for example, calculus and a large part of differential equations -still among the physicist's most important tools-were designed precisely for physical applications.”

I agree, but do you think that achieving an infinite is actually possible in the physical world? Would potential infinities be a more logical concept for physicists to employ? Or are you fine with physics only very closely approximating the real world?

“Moreover, as `quantum mathematicians' we would presumably not prove theorems according to the familiar

rules of logic, which are deeply tied to set theory, but according to rules of quantum logic which reflect the fuzzy, indistinct nature of propositions about fundamentally quantum mechanical systems.”

This is idea I was trying to get at in my question #4 which asks, “If quantum mechanics is a world where things can be both “yes” and “no” at the same time, should experimental results be analyzed with Zen Koans instead of logical inferences?”

I also like you questions regarding the choices of mathematical foundations, which I think is well... foundational.

Also, your talk of sub-objects and quotients of a set make me think that you might like to weigh in on a question I asked (#3): “Is there an analogy between the following relationships: a “class” vs. a “set” and “true” vs. “provable”?”

Please check out my Digital Physics movie essay if you have a chance.

Thanks,

Jon

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Hi Derek, I enjoyed your essay and found it educational. Re. in your conclusion -"It is one thing for the universe to be sensible in some precise way, and quite another for some entity within the universe to make sense of it."Here's a thought - Have we made sense of it or have we fabricated a sense ( or a number of different types of sense) that we then presume the universe to correspond to. Only because it correlates with the data received, though the sense made is not necessarily the causal reality?" That could apply to personal misinterpretation of events or sense as widely accepted as the space-time continuum.

Good Luck, Georgina

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Good Luck, Georgina

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Derek,

Having re read your essay, I would like to add that I really like the imagining being a quantum scale human and the description of how that would alter development of our mathematics.I found that an ingenious way to demonstrate the differences between the scales and why we have the mathematics that we have.

Regards, Georgina

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Having re read your essay, I would like to add that I really like the imagining being a quantum scale human and the description of how that would alter development of our mathematics.I found that an ingenious way to demonstrate the differences between the scales and why we have the mathematics that we have.

Regards, Georgina

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Nice essay Derek, and I've given it an appropriate community rating. I do hope you find something in this forum that helps you resolve our place in the scheme of things,

Cheers,

Rowan

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Cheers,

Rowan

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Derek,

Like your surname, your essay is pithy yet profound. I too believe that you cannot factor out the human element, mentioning the connections of math, mind, and physics, as well as the quantum with the classical, but eschewing reductionism.

The book "Outlier" documents how mastery of any endeavor takes years of dedication. Having a primitive understanding of mathematics, I can certainly agree with math requiring an added level of effort.

Great job.

Jim

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Like your surname, your essay is pithy yet profound. I too believe that you cannot factor out the human element, mentioning the connections of math, mind, and physics, as well as the quantum with the classical, but eschewing reductionism.

The book "Outlier" documents how mastery of any endeavor takes years of dedication. Having a primitive understanding of mathematics, I can certainly agree with math requiring an added level of effort.

Great job.

Jim

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Derek,

Time grows short, so I am revisiting essays I’ve read to assure I’ve rated them. I find that I rated yours on 3/21, rating it as one I could immediately relate to. 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 to assure I’ve rated them. I find that I rated yours on 3/21, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Dear Derek,

I really like some of the points you make in this very nicely written essay.

I think your rebuttal of Wigner's arguments are persuasive. His characterization of mathematics is clearly very poor. You are also surely right to comment that that large parts of mathematics (the differential and integral calculus) were invented precisely to describe physical effects, so it is hardly surprising that they do work well in that field.

I would like to try to add to your comments a further reason why mathematics has proved to be so effective in physics. It relates to your remark "that we can find useful mathematical abstractions of the simplest (you italicize this word) objects in our experience" and Einstein's claim that the comprehensibility of the world is "eternally incomprehensible". I think he discounted the fact that the mind is certainly not independent of the physical processes that take place in our brains and bodies. Indeed, much indicates that it and all our thoughts are strongly and perhaps totally dependent on them.

My comments derive in part from a book about cognitive science with the somewhat daunting title Philosophy in the Flesh with subtitle The Embodied Mind and its Challenge to Western Thought. The authors are George Lakoff and Mark Johnson. Lakoff has also written a book called Where Mathematics Comes From. I read about the first quarter of the first of these books a couple of years ago and found it illuminating. The book's thrust is well caught in these passages from the opening pages:

"Reason is not disembodied ... but arises from the nature of our brains, bodies, and bodily experiences. ... the very structure of reason itself comes from the details of our embodiment. The same neural and cognitive mechanisms that allow us to perceive and move around also create our conceptual systems and modes of reason."

Although I do not recall the authors arguing exactly as I do now, I parse these sentences as follows: we are each a self-aware physical system that forms key concepts based precisely on the most important physical facts that enable us to function and maintain our more or less autonomous existence. We have a direct 'feel' for the physical world precisely because we are part of it. Darwinian evolution means that some of the most important facts about how we function in the world as physical systems (and what aspects of the world are crucial to our survival) are hard wired into our neural makeup. Some of these deep truths are revealed to us in consciousness, above all the properties of the integers, the notion of continuity and basic geometrical notions such as point, line, angle and distance. As physical systems, we are finely attuned to vitally important simple verities that are simultaneously physical and mathematical in nature. We could not have survived as a species if that were not so.

It is moreover striking how much of the advanced mathematics essential for theoretical physics grew out of relatively simple and even obvious generalization of notions we acquire as children. I will mention only the transformation of Euclidean geometry, the understanding of which at both neural and conscious level was essential for our survival as hunter gatherers, by the concept of curvature by first Gauss and then Riemann. Curvature is something we literally see and the attempt to quantify it is very natural.

Thus, it seems to me that Wigner could hardly be further from the truth in defining mathematics as "the science of skillful operations with concepts invented just for this purpose". The brain's very development means that it operates on the basis of deep but simple truths and has a highly honed sense for what is important. Survival has built skill into our mental capacity. It is not used to play games but to sift the essential from the inessential. Since profound truths are nearly always simple, you are right to suggest it is "not surprising that we can find useful mathematical abstractions of the simplest objects in our experience."

I even think Einstein got it wrong. It is not eternally incomprehensible that the world is comprehensible. The world is manifestly aware of itself, for we are part of the world and are self-aware. The workings of the brain must reflect the workings of the larger world. Otherwise it would not survive and function. We would not be here if sensory experience were not organized in our minds. That is what thinking is. The eternally hard problem is the existence of consciousness in a world that does not seem to need it.

Perhaps the puzzlement felt by Einstein and Wigner arose because they died before cognitive science got into its stride.

Julian Barbour.

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I really like some of the points you make in this very nicely written essay.

I think your rebuttal of Wigner's arguments are persuasive. His characterization of mathematics is clearly very poor. You are also surely right to comment that that large parts of mathematics (the differential and integral calculus) were invented precisely to describe physical effects, so it is hardly surprising that they do work well in that field.

I would like to try to add to your comments a further reason why mathematics has proved to be so effective in physics. It relates to your remark "that we can find useful mathematical abstractions of the simplest (you italicize this word) objects in our experience" and Einstein's claim that the comprehensibility of the world is "eternally incomprehensible". I think he discounted the fact that the mind is certainly not independent of the physical processes that take place in our brains and bodies. Indeed, much indicates that it and all our thoughts are strongly and perhaps totally dependent on them.

My comments derive in part from a book about cognitive science with the somewhat daunting title Philosophy in the Flesh with subtitle The Embodied Mind and its Challenge to Western Thought. The authors are George Lakoff and Mark Johnson. Lakoff has also written a book called Where Mathematics Comes From. I read about the first quarter of the first of these books a couple of years ago and found it illuminating. The book's thrust is well caught in these passages from the opening pages:

"Reason is not disembodied ... but arises from the nature of our brains, bodies, and bodily experiences. ... the very structure of reason itself comes from the details of our embodiment. The same neural and cognitive mechanisms that allow us to perceive and move around also create our conceptual systems and modes of reason."

Although I do not recall the authors arguing exactly as I do now, I parse these sentences as follows: we are each a self-aware physical system that forms key concepts based precisely on the most important physical facts that enable us to function and maintain our more or less autonomous existence. We have a direct 'feel' for the physical world precisely because we are part of it. Darwinian evolution means that some of the most important facts about how we function in the world as physical systems (and what aspects of the world are crucial to our survival) are hard wired into our neural makeup. Some of these deep truths are revealed to us in consciousness, above all the properties of the integers, the notion of continuity and basic geometrical notions such as point, line, angle and distance. As physical systems, we are finely attuned to vitally important simple verities that are simultaneously physical and mathematical in nature. We could not have survived as a species if that were not so.

It is moreover striking how much of the advanced mathematics essential for theoretical physics grew out of relatively simple and even obvious generalization of notions we acquire as children. I will mention only the transformation of Euclidean geometry, the understanding of which at both neural and conscious level was essential for our survival as hunter gatherers, by the concept of curvature by first Gauss and then Riemann. Curvature is something we literally see and the attempt to quantify it is very natural.

Thus, it seems to me that Wigner could hardly be further from the truth in defining mathematics as "the science of skillful operations with concepts invented just for this purpose". The brain's very development means that it operates on the basis of deep but simple truths and has a highly honed sense for what is important. Survival has built skill into our mental capacity. It is not used to play games but to sift the essential from the inessential. Since profound truths are nearly always simple, you are right to suggest it is "not surprising that we can find useful mathematical abstractions of the simplest objects in our experience."

I even think Einstein got it wrong. It is not eternally incomprehensible that the world is comprehensible. The world is manifestly aware of itself, for we are part of the world and are self-aware. The workings of the brain must reflect the workings of the larger world. Otherwise it would not survive and function. We would not be here if sensory experience were not organized in our minds. That is what thinking is. The eternally hard problem is the existence of consciousness in a world that does not seem to need it.

Perhaps the puzzlement felt by Einstein and Wigner arose because they died before cognitive science got into its stride.

Julian Barbour.

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I feel there's a conjugation of all the applied theories of succession of this pertained Mystery.

- With regards,

Miss. Sujatha Jagannathan

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- With regards,

Miss. Sujatha Jagannathan

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Dear Sir, your essay started out with a lot of promise, but alas ended with not much of a resolution. There were some good ideas at the start but they were not carried through to a conclusion that I found to be acceptable. In my opinion there is a simple answer to this problem. Mathematics is effective in physics because we humans want it to be so. That is we invent the world to make it comprehensible to us. If magic were fashionable we would still have a world of magic. In some respects the world of physics is magic and that is why mathematics is so effective. We invent a magic of mathematics to make the world behave as we desire it to be in our imagination, and if the math doesn't work out as it should we make the world fit the mathematics. So it is inevitable that mathematics is unreasonably effective in physics, because we humans require it to be so. I wish academics would see the world as it is rather than trying to make the world into what is the politically correct current cultural opinion of the right fashionable thing to belief.

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Dear Derek,

Your essay reads very well and clear, so it invites many comments. In addition, I feel very close to many of your ideas.

1. You seem to depart from Wigner's definition of mathematics, and to avoid giving one, till you finally wonder how we can factor out the human element. What do you think of Grothendieck's testimony, in Récoltes et semailles? He describes...

view entire post

Your essay reads very well and clear, so it invites many comments. In addition, I feel very close to many of your ideas.

1. You seem to depart from Wigner's definition of mathematics, and to avoid giving one, till you finally wonder how we can factor out the human element. What do you think of Grothendieck's testimony, in Récoltes et semailles? He describes...

view entire post

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Dear Derek,

I liked your short essay very much and I keep

"The real miracle is the level of complexity of the world relative to our own intelligence".

As I am a mathematical physicist, I am interested by your opinion that "quantum groupoid" is a right language, I follow you with groupoids and I am not too far with the G-set structure of dessins d'enfants, I am less convinced by the quantum groups whose role in quantum information (and more generally physics) is questionable although topological quantum error correction is a possible instance.

I tend not to put too much abstraction and I favour constructive mathematics.

Thanks for a pleasant reading.

Best regards,

Michel

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I liked your short essay very much and I keep

"The real miracle is the level of complexity of the world relative to our own intelligence".

As I am a mathematical physicist, I am interested by your opinion that "quantum groupoid" is a right language, I follow you with groupoids and I am not too far with the G-set structure of dessins d'enfants, I am less convinced by the quantum groups whose role in quantum information (and more generally physics) is questionable although topological quantum error correction is a possible instance.

I tend not to put too much abstraction and I favour constructive mathematics.

Thanks for a pleasant reading.

Best regards,

Michel

report post as inappropriate

Dear Derek,

I liked your distinctions in the way of ignoring, which I think is not sufficiently appreciated. The terminology "co-reductionism" may help towards mending this, but I don't know how widely it is used. My suspicion is that most people would lump both together as examples of a broader conception of reductionism, but this would obscure that important differences between them, as you rightly point out.

If I may make one critical comment, given the title you chose for your essay, I felt that there was more that could have been said in the last section of the essay, particularly 1) why it is so amazing that the universe is sensible to us 2) why you see no sensible way to factor it out?

Overall your essay was very clearly written, and your argument that how we conceive the interplay between mathematics and physics affects how we study their interface resonates with me.

Best wishes,

Armin

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I liked your distinctions in the way of ignoring, which I think is not sufficiently appreciated. The terminology "co-reductionism" may help towards mending this, but I don't know how widely it is used. My suspicion is that most people would lump both together as examples of a broader conception of reductionism, but this would obscure that important differences between them, as you rightly point out.

If I may make one critical comment, given the title you chose for your essay, I felt that there was more that could have been said in the last section of the essay, particularly 1) why it is so amazing that the universe is sensible to us 2) why you see no sensible way to factor it out?

Overall your essay was very clearly written, and your argument that how we conceive the interplay between mathematics and physics affects how we study their interface resonates with me.

Best wishes,

Armin

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