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Current Essay Contest
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Media Partner: Scientific American
Previous Contests
Undecidability, Uncomputability, and Unpredictability Essay Contest
December 24, 2019 - April 24, 2020
Contest Partners: Fetzer Franklin Fund, and The Peter and Patricia Gruber Foundation
read/discuss • winners
What Is “Fundamental”
October 28, 2017 to January 22, 2018
Sponsored by the Fetzer Franklin Fund and The Peter & Patricia Gruber Foundation
read/discuss • winners
Wandering Towards a Goal
How can mindless mathematical laws give rise to aims and intention?
December 2, 2016 to March 3, 2017
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read/discuss • winners
Trick or Truth: The Mysterious Connection Between Physics and Mathematics
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Media Partner: Scientific American
read/discuss • winners
How Should Humanity Steer the Future?
January 9, 2014 - August 31, 2014
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It From Bit or Bit From It
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Questioning the Foundations
Which of Our Basic Physical Assumptions Are Wrong?
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Is Reality Digital or Analog?
November 2010 - February 2011
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RECENT POSTS IN THIS TOPIC
Giacomo D'Ariano: on 4/23/15 at 2:05am UTC, wrote Dear Noson, first let me say that I enjoyed very much reading you book...
Laurence Hitterdale: on 4/22/15 at 18:59pm UTC, wrote Dear Professor Yanofsky, Your example of James Bond leads to the obvious...
Alma Ionescu: on 4/22/15 at 12:12pm UTC, wrote Dear Noson, I read your essay a while ago but I realized I forgot to...
Noson Yanofsky: on 4/21/15 at 14:39pm UTC, wrote Thank you Cristi! Noson
Noson Yanofsky: on 4/21/15 at 14:37pm UTC, wrote Hi, MVV did not really address symmetry in mathematics. I have a hard...
Noson Yanofsky: on 4/21/15 at 13:41pm UTC, wrote Dear Armin, Thank you for the nice words. Applicability is the property...
Cristinel Stoica: on 4/21/15 at 10:54am UTC, wrote Dear Noson, I enjoyed very much your essay. One can't deny the success of...
Sylvain Poirier: on 4/21/15 at 7:11am UTC, wrote The idea of describing the mathematical character of physical laws in terms...
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Giacomo D'Ariano: on 4/23/15 at 2:05am UTC, wrote Dear Noson, first let me say that I enjoyed very much reading you book...
Laurence Hitterdale: on 4/22/15 at 18:59pm UTC, wrote Dear Professor Yanofsky, Your example of James Bond leads to the obvious...
Alma Ionescu: on 4/22/15 at 12:12pm UTC, wrote Dear Noson, I read your essay a while ago but I realized I forgot to...
Noson Yanofsky: on 4/21/15 at 14:39pm UTC, wrote Thank you Cristi! Noson
Noson Yanofsky: on 4/21/15 at 14:37pm UTC, wrote Hi, MVV did not really address symmetry in mathematics. I have a hard...
Noson Yanofsky: on 4/21/15 at 13:41pm UTC, wrote Dear Armin, Thank you for the nice words. Applicability is the property...
Cristinel Stoica: on 4/21/15 at 10:54am UTC, wrote Dear Noson, I enjoyed very much your essay. One can't deny the success of...
Sylvain Poirier: on 4/21/15 at 7:11am UTC, wrote The idea of describing the mathematical character of physical laws in terms...
RECENT FORUM POSTS
Foundational Questions Institute, FQXi: "On the latest edition of the FQXi podcast, quantum physicist Nicole Yunger..." in Quantum Steampunk -- FQXi...
Zeeya Merali: "İzzet Sakallı of Eastern Mediterranean University in Northern Cyprus and..." in Fingerprints of...
Balybin Urievich: "As for me, teaching the younger generation is one of the main tasks of..." in The Present State of...
John Cox: "Okay. In Topology, all points on a sphere are vector positions and..." in The Quantum Identity...
Vending Ways: "Vending Ways is a UAE-based leading distributor of Coin Operated Washing..." in MCQST2021 | The universe...
Steve Dufourny: "David Bohm developed a method for comcrete dialogues rather than debates..." in Global Collaboration
Steve Dufourny: "One of the problems is the difficulty to unite and convice. The majority..." in Global Collaboration
Thomas Ray: "Kobi, You write as if information accumulates like physical grains of sand..." in Mathematical Models of...
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click titles to read articles
The Math of Consciousness: Q&A with Kobi Kremnitzer
A meditating mathematician is developing a theory of conscious experience to help understand the boundary between the quantum and classical world.
Can We Feel What It’s Like to Be Quantum?
Underground experiments in the heart of the Italian mountains are testing the links between consciousness and collapse theories of quantum physics.
The Thermodynamic Limits of Intelligence: Q&A with David Wolpert
Calculating the energy needed to acquire and compute information could help explain the (in)efficiency of human brains and guide the search for extra-terrestrial intelligence.
Gambling Against the Second Law
Using precision thermometry to make mini heat engines, that might, momentarily, bust through the thermodynamic limit.
Mind and Machine: What Does It Mean to Be Sentient?
Using neural networks to test definitions of 'autonomy.'
FQXi FORUM
May 26, 2022
CATEGORY:
Trick or Truth Essay Contest (2015)
[back]
TOPIC: Why Mathematics Works So Well by Noson S. Yanofsky [refresh]
TOPIC: Why Mathematics Works So Well by Noson S. Yanofsky [refresh]
Essay Abstract
A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? We review the well-known fact that the symmetries of the laws of physics are their defining properties. We show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.
Author Bio
Noson S. Yanofsky has a PhD in mathematics (category theory). He is a professor of computer science in Brooklyn College. In addition to writing research papers he also co-authored “Quantum Computing for Computer Scientists”(Cambridge University Press, 2008) and “The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us” (MIT Press 2013). The second book is a popular science book that has been received very well both critically and popularly. He lives in Brooklyn with his wife and three children.
Download Essay PDF File
A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? We review the well-known fact that the symmetries of the laws of physics are their defining properties. We show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.
Author Bio
Noson S. Yanofsky has a PhD in mathematics (category theory). He is a professor of computer science in Brooklyn College. In addition to writing research papers he also co-authored “Quantum Computing for Computer Scientists”(Cambridge University Press, 2008) and “The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us” (MIT Press 2013). The second book is a popular science book that has been received very well both critically and popularly. He lives in Brooklyn with his wife and three children.
Download Essay PDF File
Dear Prof. Yanofski,
Your essay is simply superb and sublime. You explain there is no mystery of the Siamese connection of math to physics because both derived from the symmetry of nature. However, I believe you have not answered why nature is symmetrical. Even more strange why do we even notice the symmetries, you simply pointed out of course we noticed them in fact because we are humans who are alive who want to to continue on living by noticing the vagaries of nature to stay alive. Why then we want to stay alive? You simply tautologically answer that if we are not we would not be alive to answer these questions, and so on. This is rather tautology, still superb answer but not satisfactory enough to answer Wigner's enlightens feeling of wonder why this connection exists? Something must be beyond our existential being. You have not dare to cross the taboo to go beyond your own church's dogma. I urge you to go beyond your church dogma and say that nature has its Creator. Who, what, how and why is it? This will answer our own question of wonder who, what, how and why I am?
However as a great mathematician with its tribe's culture and its religion and its church's dogma that as a member of the tribe you are obliged voluntarily obeyed its unspoken rule. If not you are sinned and you shall fall from your Eden by eating that forbidden fruit of knowledge "Apple". We are in the same conundrum when we started: we must follow the rule, following the rule we are only frogs in the well. We observe the sky through our own hole that we dig in ourselves.
I ask you to get out from your hole and cross the boundary and look for the Creator of this "Alice in Wonderland". Lewis Carroll wrote with his magic pen: 'But I don't want to go among mad people,' said Alice. 'Oh, you can't help that,' said the cat. 'We're all mad here.'
Sincerely,
Leo KoGuan
report post as inappropriate
Your essay is simply superb and sublime. You explain there is no mystery of the Siamese connection of math to physics because both derived from the symmetry of nature. However, I believe you have not answered why nature is symmetrical. Even more strange why do we even notice the symmetries, you simply pointed out of course we noticed them in fact because we are humans who are alive who want to to continue on living by noticing the vagaries of nature to stay alive. Why then we want to stay alive? You simply tautologically answer that if we are not we would not be alive to answer these questions, and so on. This is rather tautology, still superb answer but not satisfactory enough to answer Wigner's enlightens feeling of wonder why this connection exists? Something must be beyond our existential being. You have not dare to cross the taboo to go beyond your own church's dogma. I urge you to go beyond your church dogma and say that nature has its Creator. Who, what, how and why is it? This will answer our own question of wonder who, what, how and why I am?
However as a great mathematician with its tribe's culture and its religion and its church's dogma that as a member of the tribe you are obliged voluntarily obeyed its unspoken rule. If not you are sinned and you shall fall from your Eden by eating that forbidden fruit of knowledge "Apple". We are in the same conundrum when we started: we must follow the rule, following the rule we are only frogs in the well. We observe the sky through our own hole that we dig in ourselves.
I ask you to get out from your hole and cross the boundary and look for the Creator of this "Alice in Wonderland". Lewis Carroll wrote with his magic pen: 'But I don't want to go among mad people,' said Alice. 'Oh, you can't help that,' said the cat. 'We're all mad here.'
Sincerely,
Leo KoGuan
Dear Leo,
Thank you for the kind words.
As I wrote in the essay, saying there is a Creator would answer the question. But it raises many other deeper questions about the relationship between the Creator and the created.
Thank you for the interest.
All the best,
Noson
Thank you for the kind words.
As I wrote in the essay, saying there is a Creator would answer the question. But it raises many other deeper questions about the relationship between the Creator and the created.
Thank you for the interest.
All the best,
Noson
Dear Prof. Yanofsky,
My apology. I reread my comment, it sounds harsh. That is not what I want to convey. Like Moses before, I urge you to lead mathematician community to the promised land and search and define our Creator. Not the personal God, but Einstein's God of nature. I would add God as a Mathrmatician who infuses his creation with math. We are living in math world. Our Creator is living in our equations and breath fire to the equations.
I read yours with pleasure for your clarity of thought and logics. You have my vote.
Best wishes,
Leo KoGuan
report post as inappropriate
My apology. I reread my comment, it sounds harsh. That is not what I want to convey. Like Moses before, I urge you to lead mathematician community to the promised land and search and define our Creator. Not the personal God, but Einstein's God of nature. I would add God as a Mathrmatician who infuses his creation with math. We are living in math world. Our Creator is living in our equations and breath fire to the equations.
I read yours with pleasure for your clarity of thought and logics. You have my vote.
Best wishes,
Leo KoGuan
Dear Prof. Yanofski,
Your argument that the similarity in the symmetries shown by laws of physics and mathematics is the reason for the effectiveness of mathematics in physics. Is the similarity a chance coincidence? Or is there anything more fundamental?
I argue that the similarity is not a chance coincidence. Changes happen in the physical world entirely by way of motion. No motion implies no changes, and hence no laws. Motion is a space- time relation that follows mathematical laws. So all changes follow mathematical laws. In short, the physical world has only properties but no laws, of its own. The laws applicable to the world are mathematical. In any system, whether real or imaginary, the laws are invariably mathematical. This causes the similarity.
I invite you to read my essay: The physicalist interpretation of the relation between physics and mathematics".
report post as inappropriate
Your argument that the similarity in the symmetries shown by laws of physics and mathematics is the reason for the effectiveness of mathematics in physics. Is the similarity a chance coincidence? Or is there anything more fundamental?
I argue that the similarity is not a chance coincidence. Changes happen in the physical world entirely by way of motion. No motion implies no changes, and hence no laws. Motion is a space- time relation that follows mathematical laws. So all changes follow mathematical laws. In short, the physical world has only properties but no laws, of its own. The laws applicable to the world are mathematical. In any system, whether real or imaginary, the laws are invariably mathematical. This causes the similarity.
I invite you to read my essay: The physicalist interpretation of the relation between physics and mathematics".
Thank you for looking at my essay.
It seems to me that there are a lot of physical laws that are expressed with mathematics and have nothing to do with motion.
For me the similarity is not a chance coincidence.
All the best,
Noson
It seems to me that there are a lot of physical laws that are expressed with mathematics and have nothing to do with motion.
For me the similarity is not a chance coincidence.
All the best,
Noson
Dear Prof. Yanofsky
Thanks for such a beautiful essay considering symmetry as the primary connection between physics and mathematics.
We can imagine too such a primary symmetric rules in nature as if the symmetry in basic logical patterns that connect the hardware part (physics) and software part (mathematics) to unfold the same nature or universe including inseparable ourselves in it being the cognitive observers.
But such basic logical symmetries in nature also could be two types (instead only one) : deterministic or causal and probabilistic or broken causality. Therefore it seems that there would be a two types of basic logical symmetries as well - one like symmetries in causality and other symmetries in all broken-causality. Both are symmetries but could appear asymmetric from another.
Hence, there might be two connections, instead of one, in-between physics and mathematics e.g deterministic and probabilistic symmetries, because there are also two sets of respective physics and mathematics i.e. deterministic and probabilistic.
I also invite you to my submission "A tale of two logic".
Regards
Dipak Kumar Bhunia
report post as inappropriate
Thanks for such a beautiful essay considering symmetry as the primary connection between physics and mathematics.
We can imagine too such a primary symmetric rules in nature as if the symmetry in basic logical patterns that connect the hardware part (physics) and software part (mathematics) to unfold the same nature or universe including inseparable ourselves in it being the cognitive observers.
But such basic logical symmetries in nature also could be two types (instead only one) : deterministic or causal and probabilistic or broken causality. Therefore it seems that there would be a two types of basic logical symmetries as well - one like symmetries in causality and other symmetries in all broken-causality. Both are symmetries but could appear asymmetric from another.
Hence, there might be two connections, instead of one, in-between physics and mathematics e.g deterministic and probabilistic symmetries, because there are also two sets of respective physics and mathematics i.e. deterministic and probabilistic.
I also invite you to my submission "A tale of two logic".
Regards
Dipak Kumar Bhunia
Dear Prof. Yanofski,
I greatly enjoyed your essay tracing the efficacy of mathematics for physics to their shared features regarding symmetry.
Given your computer science background, I was curious as to what you thought of my understanding of Stephen Wolfram’s view at computation could serve as an alternative mathematics for physics? My take on that view is that Wolfram sees our current mathematics as “just one among many possible ones” and that mathematics success in physics is historically contingent and occurring bases on the types of problems mathematics has sought to address. Current physical laws are formulated in mathematical laws that are time symmetric- you can reverse them- in the same way a mathematical formula is reversible. But a physics based on computation may lack this symmetry. If you ran them a second time you would get an at least somewhat different answer.
Also, if you have the time, please read and vote on my own essay. While not as rigorous as yours it addresses many of the same issues.
http://fqxi.org/community/forum/topic/2391
Best of luck in the contest!
Rick Searle
report post as inappropriate
I greatly enjoyed your essay tracing the efficacy of mathematics for physics to their shared features regarding symmetry.
Given your computer science background, I was curious as to what you thought of my understanding of Stephen Wolfram’s view at computation could serve as an alternative mathematics for physics? My take on that view is that Wolfram sees our current mathematics as “just one among many possible ones” and that mathematics success in physics is historically contingent and occurring bases on the types of problems mathematics has sought to address. Current physical laws are formulated in mathematical laws that are time symmetric- you can reverse them- in the same way a mathematical formula is reversible. But a physics based on computation may lack this symmetry. If you ran them a second time you would get an at least somewhat different answer.
Also, if you have the time, please read and vote on my own essay. While not as rigorous as yours it addresses many of the same issues.
http://fqxi.org/community/forum/topic/2391
Best of luck in the contest!
Rick Searle
Noson -
Thank you for the superb essay. I enjoyed your book "The Outer Limits of Reason" but was a bit disappointed that it did not probe more deeply into the metaphysical implications of all those fascinating limits. This essay takes us there --- to the features of symmetry common to math and physics and to the question of why the world is the way it is. Well done.
I've taken a more metaphorical approach and hope you get a chance to comment on my essay "The Hole at the Center of Creation." My thesis poses a challenge you have not addressed, and the question may be asked this way: What is the ultimate symmetry from which all others emerge? While I use a different vocabulary in my essay, I would ask whether you agree that zero and infinity share an interesting quality: symmetry with infinite degrees of freedom.
With immense respect - George Gantz
I postulate that the Hole at the Center
report post as inappropriate
Thank you for the superb essay. I enjoyed your book "The Outer Limits of Reason" but was a bit disappointed that it did not probe more deeply into the metaphysical implications of all those fascinating limits. This essay takes us there --- to the features of symmetry common to math and physics and to the question of why the world is the way it is. Well done.
I've taken a more metaphorical approach and hope you get a chance to comment on my essay "The Hole at the Center of Creation." My thesis poses a challenge you have not addressed, and the question may be asked this way: What is the ultimate symmetry from which all others emerge? While I use a different vocabulary in my essay, I would ask whether you agree that zero and infinity share an interesting quality: symmetry with infinite degrees of freedom.
With immense respect - George Gantz
I postulate that the Hole at the Center
Dear Prof. Yanofski,
Your essay is a very enjoyable and thought provoking read. The concentration of symmetries for the definition of physics and then the extension of symmetries into the definition of mathematics provided a little different idea to the subject.
In your conclusion "In detail, for any physical law, symmetry of applicability states that the law can deal with swapping any appropriate object for any other appropriate object. If there is a mathematical statement that can describe this physical law, then we can substitute different values for the different objects that one is applying"
Your ideas of extending symmetry to help explain the relationship between math and physics is, in my opinion, very successful.
Good luck in the contest.
Regards,
Ed Unverricht
report post as inappropriate
Your essay is a very enjoyable and thought provoking read. The concentration of symmetries for the definition of physics and then the extension of symmetries into the definition of mathematics provided a little different idea to the subject.
In your conclusion "In detail, for any physical law, symmetry of applicability states that the law can deal with swapping any appropriate object for any other appropriate object. If there is a mathematical statement that can describe this physical law, then we can substitute different values for the different objects that one is applying"
Your ideas of extending symmetry to help explain the relationship between math and physics is, in my opinion, very successful.
Good luck in the contest.
Regards,
Ed Unverricht
Dear Ed Unverricht,
Thank you for the kind words.
You picked out the main sentence in the essay that explains the relationship. Thanks!
All the best,
Noson
Thank you for the kind words.
You picked out the main sentence in the essay that explains the relationship. Thanks!
All the best,
Noson
Dear Professor Yanofsky,
I agree with the appeal to the anthropic principle in the way that you present it. In order for human beings, or other similar entities, to ask and to answer questions about mathematics and physics, the beings asking the questions must exist. Furthermore, in order for these beings to exist, their environment must be sufficiently orderly and stable. As you say, “If the universe did not have some regularities, no life would be possible.” Yes, but this fact would seem to lead to the question whether the regularities must be mathematical rather than some other kind. According to your argument, the type of regularity prominent both in physics and in mathematics is the type of regularity congenial to human ways of thinking. You explicitly reject the view that mathematical order has a Platonic transmundane reality. Are mathematical regularities nonetheless objectively real in the physical world? Are these regularities independent of human cognition? Whichever way we answer, we would seem to be left with a further question. If we say that mathematical order is objectively real in nature, then the further question is why it is this kind of order rather than some non-mathematical type of regularity. At least we would need to understand why the order of nature is so thoroughly mathematical, with apparently no allowance for other kinds of order. On the other hand, if mathematical order is not inherent in nature, but is based on a human affinity for thinking in terms of symmetries, then we would confront two further questions. We would wonder why human beings happen to think this way, and we would wonder why this human thinking works so well in application to nature. So, the role of symmetries both in mathematics and in physics is important, but perhaps other explanatory factors are also needed.
Laurence Hitterdale
report post as inappropriate
I agree with the appeal to the anthropic principle in the way that you present it. In order for human beings, or other similar entities, to ask and to answer questions about mathematics and physics, the beings asking the questions must exist. Furthermore, in order for these beings to exist, their environment must be sufficiently orderly and stable. As you say, “If the universe did not have some regularities, no life would be possible.” Yes, but this fact would seem to lead to the question whether the regularities must be mathematical rather than some other kind. According to your argument, the type of regularity prominent both in physics and in mathematics is the type of regularity congenial to human ways of thinking. You explicitly reject the view that mathematical order has a Platonic transmundane reality. Are mathematical regularities nonetheless objectively real in the physical world? Are these regularities independent of human cognition? Whichever way we answer, we would seem to be left with a further question. If we say that mathematical order is objectively real in nature, then the further question is why it is this kind of order rather than some non-mathematical type of regularity. At least we would need to understand why the order of nature is so thoroughly mathematical, with apparently no allowance for other kinds of order. On the other hand, if mathematical order is not inherent in nature, but is based on a human affinity for thinking in terms of symmetries, then we would confront two further questions. We would wonder why human beings happen to think this way, and we would wonder why this human thinking works so well in application to nature. So, the role of symmetries both in mathematics and in physics is important, but perhaps other explanatory factors are also needed.
Laurence Hitterdale
Dear Laurence,
Thank you for taking such an interest in my paper.
You wrote: "... but this fact would seem to lead to the question whether the regularities must be mathematical rather than some other kind. " I think that we have to be clear about what we are talking about. I do not think the physical regularities are "mathematical" rather I think that the laws can be expressed in mathematical language because the mathematical language has the same regularities. As a simple case, if an experiment gives results here, then it will give the same results there. Mathematical language is true here and there also.
You write " Are mathematical regularities nonetheless objectively real in the physical world? Are these regularities independent of human cognition?" They are objective... That means we can all agree on it. But I am not sure that it really exists. We all agree that James Bond is a good guy and saves the world. But that does not mean he exists.
You ask very interesting questions. I have to think more about it.
All the best,
Noson
Thank you for taking such an interest in my paper.
You wrote: "... but this fact would seem to lead to the question whether the regularities must be mathematical rather than some other kind. " I think that we have to be clear about what we are talking about. I do not think the physical regularities are "mathematical" rather I think that the laws can be expressed in mathematical language because the mathematical language has the same regularities. As a simple case, if an experiment gives results here, then it will give the same results there. Mathematical language is true here and there also.
You write " Are mathematical regularities nonetheless objectively real in the physical world? Are these regularities independent of human cognition?" They are objective... That means we can all agree on it. But I am not sure that it really exists. We all agree that James Bond is a good guy and saves the world. But that does not mean he exists.
You ask very interesting questions. I have to think more about it.
All the best,
Noson
Dear Professor Yanofsky,
Your example of James Bond leads to the obvious question whether you are saying that mathematical regularities, like well-known fictional characters, are in the end items of inter-subjective human agreement. Many people would say that mathematical entities, such as pi or the integer eight, appear to have a definiteness which is independent of human perception and agreement. In this way, the mathematical entities seem unlike fictional characters. I think a similar dissimilarity holds in the case of mathematical truths as contrasted with truths about fictional entities. But, as you say, the issues here deserve more thought. In any event, thanks for your stimulating essay here and for your book, “The Outer Limits of Reason.”
Best wishes,
Laurence Hitterdale
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Your example of James Bond leads to the obvious question whether you are saying that mathematical regularities, like well-known fictional characters, are in the end items of inter-subjective human agreement. Many people would say that mathematical entities, such as pi or the integer eight, appear to have a definiteness which is independent of human perception and agreement. In this way, the mathematical entities seem unlike fictional characters. I think a similar dissimilarity holds in the case of mathematical truths as contrasted with truths about fictional entities. But, as you say, the issues here deserve more thought. In any event, thanks for your stimulating essay here and for your book, “The Outer Limits of Reason.”
Best wishes,
Laurence Hitterdale
Dear Noson,
A nice thought provoking essay. You are the expert, but if a non-expert may point out a few things, here are some;
"Galilean relativity demands that the laws of motion remain unchanged if a phenomenon is observed while stationary or moving at a uniform, constant velocity. Special relativity states that the laws of motion must remain the same even if the observers are...
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A nice thought provoking essay. You are the expert, but if a non-expert may point out a few things, here are some;
"Galilean relativity demands that the laws of motion remain unchanged if a phenomenon is observed while stationary or moving at a uniform, constant velocity. Special relativity states that the laws of motion must remain the same even if the observers are...
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Dear Akinbo,
Thank you for taking such an interest in my paper.
While the 1887MM experiment might show that the motion of the observer "is not even in the equation", I think what I wrote about modern special relativity is correct.
I agree with you about limiting the domain of discourse for the statement 9+4=13. But I have much simpler counterexamples. If I have heaps of sand it does not work. If you add one heap of sand to another, you get one heap: 1+1=1. Also my wife once sent me out to get size 4 diapers. The store did not have size 4 so I purchased two size 2 packages. Needless to say, my wife was not happy. We can conclude from this that 2+2 =/= 4. So we have to limit the domain
of discourse to be discrete objects that add in the appropriate way.
As for your last point, I do not know. I will have to look at your paper carefully. To me, there is no physical circle whose diameter and circumference have the ratio of pi. This is something true only in mathematics. If there is a Plank's length, then the infinite precision of pi is wrong. But even without that, human beings cannot deal with infinite precision. So the statement is metaphysical. We cannot deal with it.
Again, thank you for taking an interest in my paper.
I will look at your paper.
All the best,
Noson
Thank you for taking such an interest in my paper.
While the 1887MM experiment might show that the motion of the observer "is not even in the equation", I think what I wrote about modern special relativity is correct.
I agree with you about limiting the domain of discourse for the statement 9+4=13. But I have much simpler counterexamples. If I have heaps of sand it does not work. If you add one heap of sand to another, you get one heap: 1+1=1. Also my wife once sent me out to get size 4 diapers. The store did not have size 4 so I purchased two size 2 packages. Needless to say, my wife was not happy. We can conclude from this that 2+2 =/= 4. So we have to limit the domain
of discourse to be discrete objects that add in the appropriate way.
As for your last point, I do not know. I will have to look at your paper carefully. To me, there is no physical circle whose diameter and circumference have the ratio of pi. This is something true only in mathematics. If there is a Plank's length, then the infinite precision of pi is wrong. But even without that, human beings cannot deal with infinite precision. So the statement is metaphysical. We cannot deal with it.
Again, thank you for taking an interest in my paper.
I will look at your paper.
All the best,
Noson
Dear Noson,
I know experts do not like amateurs who say things differently from what the authority have proclaimed so I hope this comment does not put you off before reading my essay.
When you say, "While the 1887 MM experiment might show that the motion of the observer "is not even in the equation"...
In explaining the null result using Special relativity, there is a v in the length contraction and time dilation equations of SR and that v is supposed to represent the velocity of the observer.
The length contraction equation of the Lorentz transformation is
L' = L √(1 - v2/c2)
While the time dilation equation is
t' = t √(1 - v2/c2)
Also in concluding the paper, Michelson remarked that the relative velocity between any possible stationary ether and the earth was certainly less than one sixth of the earth’s (observer's) velocity.
Best regards,
Akinbo
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I know experts do not like amateurs who say things differently from what the authority have proclaimed so I hope this comment does not put you off before reading my essay.
When you say, "While the 1887 MM experiment might show that the motion of the observer "is not even in the equation"...
In explaining the null result using Special relativity, there is a v in the length contraction and time dilation equations of SR and that v is supposed to represent the velocity of the observer.
The length contraction equation of the Lorentz transformation is
L' = L √(1 - v2/c2)
While the time dilation equation is
t' = t √(1 - v2/c2)
Also in concluding the paper, Michelson remarked that the relative velocity between any possible stationary ether and the earth was certainly less than one sixth of the earth’s (observer's) velocity.
Best regards,
Akinbo
Noson,
Your diaper story reminded me of one about Sierpinski. He and his wife were waiting at the train station, and he was upset because one of the bags seemed to be missing. His puzzled wife said, "What's bothering you? I see that all six bags are here." "No!" replied Sierpinski, "I've counted them several times -- zero, 1, 2, 3, 4, 5!"
Tom
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Your diaper story reminded me of one about Sierpinski. He and his wife were waiting at the train station, and he was upset because one of the bags seemed to be missing. His puzzled wife said, "What's bothering you? I see that all six bags are here." "No!" replied Sierpinski, "I've counted them several times -- zero, 1, 2, 3, 4, 5!"
Tom
Noson,
Our world views are so similar that I'm at a loss to think of what to say. I loved the essay of course, so I hope you can visit my forum and we can hopefully engage in a discourse of the rational idealism that motivates us.
Thanks, and please accept my best wishes and highest mark!
Tom
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Our world views are so similar that I'm at a loss to think of what to say. I loved the essay of course, so I hope you can visit my forum and we can hopefully engage in a discourse of the rational idealism that motivates us.
Thanks, and please accept my best wishes and highest mark!
Tom
Dear Tom,
Thank you. I will look at your paper. I hope we agree.
All the best,
Noson
Thank you. I will look at your paper. I hope we agree.
All the best,
Noson
Dear Noson,
You introduce the idea of 'symmetry of applicability' for characterizing physics and that of 'symmetry of semantics' for maths. As for the first concept, do you consider it distinct from what we call 'universality'? As for the second concept, is not the 'universality' of mathematical concepts that allow us to apply them in different contexts? But that may just be a rewording of what you are writing.
I like your advanced examples about the changes of semantics of mathematical statements (the Hilbert's Nullstellentsatz) that capture well what is sometimes also called the tautology in maths. In your paper [YanZel2] you explain how the language of category theory formalize these facts.
And for the relation of maths to physics you write "Rather the regularities of phenomena and thoughts are seen and chosen by human beings in the same way" and later you explain that the adaptation of the human being to his environnement forces him to perceive and organize the regularities. Myself I am not a platonist and I tend to consider that phys and maths are just two different cognitive processes that are constrained by the world external to us.
My essay is of a different taste but does not contadict you. I hope you can find time to read it, as you are a mathematician you can understand many parts of it.
Michel
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You introduce the idea of 'symmetry of applicability' for characterizing physics and that of 'symmetry of semantics' for maths. As for the first concept, do you consider it distinct from what we call 'universality'? As for the second concept, is not the 'universality' of mathematical concepts that allow us to apply them in different contexts? But that may just be a rewording of what you are writing.
I like your advanced examples about the changes of semantics of mathematical statements (the Hilbert's Nullstellentsatz) that capture well what is sometimes also called the tautology in maths. In your paper [YanZel2] you explain how the language of category theory formalize these facts.
And for the relation of maths to physics you write "Rather the regularities of phenomena and thoughts are seen and chosen by human beings in the same way" and later you explain that the adaptation of the human being to his environnement forces him to perceive and organize the regularities. Myself I am not a platonist and I tend to consider that phys and maths are just two different cognitive processes that are constrained by the world external to us.
My essay is of a different taste but does not contadict you. I hope you can find time to read it, as you are a mathematician you can understand many parts of it.
Michel
Dear Michel,
For me "universality" is something about unifying two seemingly different domains. So for example in physics, Newton unified celestial mechanics and terrestrial mechanics. In math, universal algebra unifies many branches of algebra. Category theory shows that many tools in many different parts of mathematics, theoretical computer science and theoretical physics are the same and so category theory unifies all these different areas.
I will most definitely look at your paper.
All the best,
Noson
For me "universality" is something about unifying two seemingly different domains. So for example in physics, Newton unified celestial mechanics and terrestrial mechanics. In math, universal algebra unifies many branches of algebra. Category theory shows that many tools in many different parts of mathematics, theoretical computer science and theoretical physics are the same and so category theory unifies all these different areas.
I will most definitely look at your paper.
All the best,
Noson
(I am copying my reply on your forum as a notice. I also made some observation above on the possibly touchy subject of the velocity of the observer in special relativity. Probably, you decided to ignore this for the moment, which is okay)
Dear Noson,
Thanks for finding the time to comment on my essay.
As regards, your first query why the real number system works so well in...
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Dear Noson,
Thanks for finding the time to comment on my essay.
As regards, your first query why the real number system works so well in...
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Dear Noson,
I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
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I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.
All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.
Joe Fisher
Dear Noson,
You are suggesting an explanation why "any existing structure in our perceived physical universe is naturally expressed in the language of mathematics". Essentially your point is that since both physics and mathematics are about symmetries, a compatibility between the two sciences is reasonable.
However, Wigner's wonder about the relation of physics and mathematics is not just abut the fact that there are some mathematical forms describing laws of nature. He is fascinated by something more: that these forms are both elegant, while covering a wide range of parameters, and extremely precise. I do not see anything in your paper which relates to that amazing and highly important fact about the relation of physics and mathematics.
Regards,
Alexey Burov.
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You are suggesting an explanation why "any existing structure in our perceived physical universe is naturally expressed in the language of mathematics". Essentially your point is that since both physics and mathematics are about symmetries, a compatibility between the two sciences is reasonable.
However, Wigner's wonder about the relation of physics and mathematics is not just abut the fact that there are some mathematical forms describing laws of nature. He is fascinated by something more: that these forms are both elegant, while covering a wide range of parameters, and extremely precise. I do not see anything in your paper which relates to that amazing and highly important fact about the relation of physics and mathematics.
Regards,
Alexey Burov.
Dear Alexey,
Thank you for taking an interest in my paper.
Its not that they share symmetry. Its that with since they both have symmetry, they are chosen the same way.
I do not think Wigner mentions "elegant" in his paper.Perhaps you mean "beauty". Either way, I don't think that beauty plays a role in either physics or mathematics. Its a subjective feeling that different people have about different subjects. Usually when you learn about something new in the context of something else that you already know you have the feeling of the new thing being beautiful. But there is no reason why the world or mathematics should be beautiful or elegant. Einstein is quoted as saying, “If you are out to describe the truth, leave elegance to the tailor.” (Something similar was said earlier by Ludwig Boltzmann.)
As for precision, I write this in my essay: "The fact that symmetry of semantics does not permit any counterexamples within the domain of discourse implies a certain precision of thought and language which people associate with mathematics."
Again, thank you for taking an interest in my paper.
All the best,
Noson
Thank you for taking an interest in my paper.
Its not that they share symmetry. Its that with since they both have symmetry, they are chosen the same way.
I do not think Wigner mentions "elegant" in his paper.Perhaps you mean "beauty". Either way, I don't think that beauty plays a role in either physics or mathematics. Its a subjective feeling that different people have about different subjects. Usually when you learn about something new in the context of something else that you already know you have the feeling of the new thing being beautiful. But there is no reason why the world or mathematics should be beautiful or elegant. Einstein is quoted as saying, “If you are out to describe the truth, leave elegance to the tailor.” (Something similar was said earlier by Ludwig Boltzmann.)
As for precision, I write this in my essay: "The fact that symmetry of semantics does not permit any counterexamples within the domain of discourse implies a certain precision of thought and language which people associate with mathematics."
Again, thank you for taking an interest in my paper.
All the best,
Noson
Dear Noson,
Your denial of "that beauty plays a role in either physics or mathematics" strongly contradicts to the history of fundamental science, from Kepler and Newton to Einstein and Dirac. Your quotation of Einstein is an example of an extreme misinterpretation by taking out of context. The only reliable source where I know Einstein mentions this 'tailor' comparison is his preface to...
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Your denial of "that beauty plays a role in either physics or mathematics" strongly contradicts to the history of fundamental science, from Kepler and Newton to Einstein and Dirac. Your quotation of Einstein is an example of an extreme misinterpretation by taking out of context. The only reliable source where I know Einstein mentions this 'tailor' comparison is his preface to...
view entire post
Dear Alexei,
Thank you. I never knew the origin of the Einstein quote and I agree it is indeed out of context.
But you cannot make your point by an "appeal to authority". Please give a definition of beauty/elegance/unity of ideas etc? Are such concepts quantifiable? Can a computer determine when a physical or mathematical idea is beautiful? (Can a computer tell when a painting is beautiful? ) If yes, please tell. If not, then we are talking about some vague wishey washy feeling that many great physicists had. That is fine. But it does not really say anything about the physical universe.
This is interesting stuff. Thanks!
All the best,
Noson
Thank you. I never knew the origin of the Einstein quote and I agree it is indeed out of context.
But you cannot make your point by an "appeal to authority". Please give a definition of beauty/elegance/unity of ideas etc? Are such concepts quantifiable? Can a computer determine when a physical or mathematical idea is beautiful? (Can a computer tell when a painting is beautiful? ) If yes, please tell. If not, then we are talking about some vague wishey washy feeling that many great physicists had. That is fine. But it does not really say anything about the physical universe.
This is interesting stuff. Thanks!
All the best,
Noson
Noson,
Is it a paradox of symmetry that our presence is asymmetrical in a seemingly ordered universe? Curiously does our presence give it order through our observations?
Such questions are mind-boggling. In contrast,your essay is straightforward and logical.
The mathematician Hermann Weyl gave a succinct definition of symmetry:
“A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.”
Some say we are an asymmetrical lump in that symmetry.
My essay shows my cowardice in avoiding such questions. I am straightforward in showing the connections of math, physics and the human mind.
Jim
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Is it a paradox of symmetry that our presence is asymmetrical in a seemingly ordered universe? Curiously does our presence give it order through our observations?
Such questions are mind-boggling. In contrast,your essay is straightforward and logical.
The mathematician Hermann Weyl gave a succinct definition of symmetry:
“A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.”
Some say we are an asymmetrical lump in that symmetry.
My essay shows my cowardice in avoiding such questions. I am straightforward in showing the connections of math, physics and the human mind.
Jim
Dear Jim,
Thank you for taking an interest in my essay.
Following Weyl's definition, I would say:
A true statement is mathematical if you can change its semantics (what it refers to) for another and get a similar true statement.
All the best,
Noson
Thank you for taking an interest in my essay.
Following Weyl's definition, I would say:
A true statement is mathematical if you can change its semantics (what it refers to) for another and get a similar true statement.
All the best,
Noson
Dear Noson,
Your Essay is one more argument as to why math is effective in the more philosophical conceptual sense, among many here and elsewhere. That is good, even those who are not real Platonist justify the effectiveness of math beautifully. Ironically the whole debate is an indication of the power of math looking at it from multitudes of angle. And that is precisely why people like Wolfram, Conway, Tegmark and others came to the natural conclusion that Mathematics has some very deep connection with reality.
Although I did not know about Wolfram and others at the time, just from basic interest in physics I took a similar guess as to math having a very intimate connection with reality. And so I took a guess as to the nature of such connection, but I was very lucky with my guess with everything turning out to be just right. And indeed my theory which shows the origin of the design of reality has a hint of your idea about the symmetry by semantics, and even at one point I so some connection with category theory, among many other connection(some in the essay). I will explain more once you get interested in it.
Essay
Thanks and good luck
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Your Essay is one more argument as to why math is effective in the more philosophical conceptual sense, among many here and elsewhere. That is good, even those who are not real Platonist justify the effectiveness of math beautifully. Ironically the whole debate is an indication of the power of math looking at it from multitudes of angle. And that is precisely why people like Wolfram, Conway, Tegmark and others came to the natural conclusion that Mathematics has some very deep connection with reality.
Although I did not know about Wolfram and others at the time, just from basic interest in physics I took a similar guess as to math having a very intimate connection with reality. And so I took a guess as to the nature of such connection, but I was very lucky with my guess with everything turning out to be just right. And indeed my theory which shows the origin of the design of reality has a hint of your idea about the symmetry by semantics, and even at one point I so some connection with category theory, among many other connection(some in the essay). I will explain more once you get interested in it.
Essay
Thanks and good luck
Dear Prof. Yanofski,
Great essay! You offered a precise definitions of physics and mathematics and the relation between them. We seem to agree in many points, as my essay reflects, especially the importance of symmetry, and how regularities in nature allow mathematics to be so effective. I would be glad to take your opinion in my essay.
Best regards,
Mohammed
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Great essay! You offered a precise definitions of physics and mathematics and the relation between them. We seem to agree in many points, as my essay reflects, especially the importance of symmetry, and how regularities in nature allow mathematics to be so effective. I would be glad to take your opinion in my essay.
Best regards,
Mohammed
Dear Noson,
I enjoyed your crisply and clearly written essay very much. The fact that symmetries are lurking everywhere if we only look for them is a lesson that everyone who wants to understand the nature of reality should take deeply to hear. I had not previously thought of logical validity as a symmetry, or, for that matter, of the symmetry of applicability. While it is clear that the conserved "thing" in the former is truth, I am wondering what it is for the latter?
If I my give just one little piece of constructive criticism, I found ending the essay with an appeal to the anthropic principle a little bit of a downer. In its uncontroversial form it is a tautology, and if one wants to draw stronger conclusions from it, then it invites a host of questions the answers to which based on the strong versions stretch credulity.
However, you did mention at the very end that the anthropic principle does not answer all the deep questions.
On final note, I came across some reviews of your book "The outer limits of reason" which seems like a very interesting book and which I will add to my "to read" list.
Best wishes,
Armin
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I enjoyed your crisply and clearly written essay very much. The fact that symmetries are lurking everywhere if we only look for them is a lesson that everyone who wants to understand the nature of reality should take deeply to hear. I had not previously thought of logical validity as a symmetry, or, for that matter, of the symmetry of applicability. While it is clear that the conserved "thing" in the former is truth, I am wondering what it is for the latter?
If I my give just one little piece of constructive criticism, I found ending the essay with an appeal to the anthropic principle a little bit of a downer. In its uncontroversial form it is a tautology, and if one wants to draw stronger conclusions from it, then it invites a host of questions the answers to which based on the strong versions stretch credulity.
However, you did mention at the very end that the anthropic principle does not answer all the deep questions.
On final note, I came across some reviews of your book "The outer limits of reason" which seems like a very interesting book and which I will add to my "to read" list.
Best wishes,
Armin
Dear Armin,
Thank you for the nice words.
Applicability is the property that is preserved by symmetry of applicability. Let me explain. If a certain rule works with an object of a certain type and you swap that object for another object of the same type then the rule will still be applicable for the new object. Agree?
I agree with you that the anthropic principle is true but there is a feeling of cheating with it. As you say, I pointed out its shortcomings.
Please let me know what you think of my book when you read it.
Sincerely,
Noson
Thank you for the nice words.
Applicability is the property that is preserved by symmetry of applicability. Let me explain. If a certain rule works with an object of a certain type and you swap that object for another object of the same type then the rule will still be applicable for the new object. Agree?
I agree with you that the anthropic principle is true but there is a feeling of cheating with it. As you say, I pointed out its shortcomings.
Please let me know what you think of my book when you read it.
Sincerely,
Noson
The idea of describing the mathematical character of physical laws in terms of the abundance of symmetries, was also present in the essay by Milen Velchev Velev. See my reply there. See also the references of essays with effective descriptions of "what is remarkable about the success of mathematics in physics" I collected in my review of this contest, which your essay does not account for.
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Hi,
MVV did not really address symmetry in mathematics.
I have a hard time seeing where my essay sits in your idiological oppositions chart.
Noson
MVV did not really address symmetry in mathematics.
I have a hard time seeing where my essay sits in your idiological oppositions chart.
Noson
Dear Noson,
I enjoyed very much your essay. One can't deny the success of mathematics, as you illustrated by predictions like those by Le Verrier, Maxwell, Dirac, Kepler, etc. You criticize well the positions of theologians and Plantonists, and especially the standard response, which presents mathematics as relative to the human experience. Indeed, the role played by symmetry must be fundamental, both in physics and mathematics, and may explain the connection between the two of them. I plan to read your references on the symmetry in math. I am particularly interested since some years ago I identified a symmetry in mathematics that connects many different structures under one umbrella, but I never find time to finish that paper (I think I thought that not many would consider it to be very useful to worth the effort). Of course, the idea is different than yours. I like your definition "a statement is mathematics if we can swap what it refers to and remain true" and "anything that satisfies symmetry of semantics, is mathematics". I agree with "Mathematics is invariant with respect to symmetry of semantics. We are claiming that this type of symmetry is as fundamental to mathematics as symmetry is to physics." Congrats for a so well written essay, which truly addresses the question of effectiveness of mathematics in physics.
Best wishes,
Cristi Stoica
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I enjoyed very much your essay. One can't deny the success of mathematics, as you illustrated by predictions like those by Le Verrier, Maxwell, Dirac, Kepler, etc. You criticize well the positions of theologians and Plantonists, and especially the standard response, which presents mathematics as relative to the human experience. Indeed, the role played by symmetry must be fundamental, both in physics and mathematics, and may explain the connection between the two of them. I plan to read your references on the symmetry in math. I am particularly interested since some years ago I identified a symmetry in mathematics that connects many different structures under one umbrella, but I never find time to finish that paper (I think I thought that not many would consider it to be very useful to worth the effort). Of course, the idea is different than yours. I like your definition "a statement is mathematics if we can swap what it refers to and remain true" and "anything that satisfies symmetry of semantics, is mathematics". I agree with "Mathematics is invariant with respect to symmetry of semantics. We are claiming that this type of symmetry is as fundamental to mathematics as symmetry is to physics." Congrats for a so well written essay, which truly addresses the question of effectiveness of mathematics in physics.
Best wishes,
Cristi Stoica
Thank you Cristi!
Noson
Noson
Dear Noson,
I read your essay a while ago but I realized I forgot to comment. My sincerest apologies for this, because I enjoyed your writing very much! I found that there is much similarity of thought between us.
This is an extraordinarily precise and accurate analysis that you are making. You are saying that the problem only exists if “one considers both physics and mathematics to each be perfectly formed, objective and independent of human observers”. This is absolutely true! Unless one starts with the assumption that they are unrelated, the problem disappears. You are going to the heart of the problem when you bring into discussion the symmetry and conservation laws of physics, because it is this reducibility to invariant quantities that is characteristic to both math and physics that makes them work together. Can the symmetry of syntax that makes math be a good support for physics be called a conservation law as well? I am thinking here on the lines of the internal logical consistency of math.
Wish you the best of luck in the contest!
Alma
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I read your essay a while ago but I realized I forgot to comment. My sincerest apologies for this, because I enjoyed your writing very much! I found that there is much similarity of thought between us.
This is an extraordinarily precise and accurate analysis that you are making. You are saying that the problem only exists if “one considers both physics and mathematics to each be perfectly formed, objective and independent of human observers”. This is absolutely true! Unless one starts with the assumption that they are unrelated, the problem disappears. You are going to the heart of the problem when you bring into discussion the symmetry and conservation laws of physics, because it is this reducibility to invariant quantities that is characteristic to both math and physics that makes them work together. Can the symmetry of syntax that makes math be a good support for physics be called a conservation law as well? I am thinking here on the lines of the internal logical consistency of math.
Wish you the best of luck in the contest!
Alma
Dear Noson,
first let me say that I enjoyed very much reading you book "The Outer Limits of Reason". It is one of the few books that I read more than 60% scratching my personal notes on it and highlighting the most interesting parts. The reason is that you penetrate the epistemology of physics and the structure of a scientific theory with focus and magnetizing style. In your nice essay I can recognize your style of the book.
I completely share the idea that the key to the solution of why mathematics works so well in physics is group theory and the notion of symmetry, as I also emphasized in my essay. I came to the conclusion that a lot of physics can be derived in terms of group representations. For example, we recently gave a definition of reference frame and boost that is purely group theoretical (without space, time, kinematics, and mechanics!) corresponding to a general invariance that only in a limiting situation (the so-called relativistic limit) leads to Lorentz invariance of QFT, but more generally is a nonlinear version of Lorentz group. This allowed us to extend the notion of boost to a discrete Planck scale.
My best wishes
Mauro
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first let me say that I enjoyed very much reading you book "The Outer Limits of Reason". It is one of the few books that I read more than 60% scratching my personal notes on it and highlighting the most interesting parts. The reason is that you penetrate the epistemology of physics and the structure of a scientific theory with focus and magnetizing style. In your nice essay I can recognize your style of the book.
I completely share the idea that the key to the solution of why mathematics works so well in physics is group theory and the notion of symmetry, as I also emphasized in my essay. I came to the conclusion that a lot of physics can be derived in terms of group representations. For example, we recently gave a definition of reference frame and boost that is purely group theoretical (without space, time, kinematics, and mechanics!) corresponding to a general invariance that only in a limiting situation (the so-called relativistic limit) leads to Lorentz invariance of QFT, but more generally is a nonlinear version of Lorentz group. This allowed us to extend the notion of boost to a discrete Planck scale.
My best wishes
Mauro
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