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December 7, 2021

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Trick or Truth Essay Contest (2015)
[back]

TOPIC: Mathematics is Physics by Matthew Saul Leifer [refresh]

TOPIC: Mathematics is Physics by Matthew Saul Leifer [refresh]

In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theory-building based on the idea that human knowledge has the structure of a scale-free network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics.

Matt Leifer is a visiting researcher at Perimeter Institute for Theoretical Physics. His research interests include quantum foundations, quantum information, and particularly the intersection of the two. He is hoping to break the world record for the number of FQXi essay contests won by a single individual.

Your 2-page description of mathematics leaves out its most distinguishing feature. Mathematics has been axiomatized, and all mathematical knowledge is gained by proving theorems from axioms. That is what makes those abstractions work, and what gives math its autonomy. It is also what causes me to deviate from your naturalism, and I would say that my essay is more directly opposite Tegmark's view than yours.

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In Sec. 2.3, I said, "As the formalists suggest, mathematical theories are just abstract formal systems, but not all formal systems are mathematics." Because mathematical theories are formal systems I accept that they are defined by axioms and that theorems are proved from those axioms. If that is what you are referring to then I agree with you.

On the other hand, if you mean that all of mathematics can be axiomatized starting from set theory then I don't agree that this correctly captures the nature of mathematics. This sort of foundation is rather retrofitted to mathematics at a later date. The fact that there are several competing foundations for mathematics indicates that there is nothing unique about the foundations. I don't think there is anything wrong with foundational work of this kind. We can view it as creating a massive hub in the structure of mathematical knowledge that a lot of other theories can hang off. It is rather that I think the informal ideas and theories come first and they are slotted into the formal structure later.

On the other hand, if you mean that all of mathematics can be axiomatized starting from set theory then I don't agree that this correctly captures the nature of mathematics. This sort of foundation is rather retrofitted to mathematics at a later date. The fact that there are several competing foundations for mathematics indicates that there is nothing unique about the foundations. I don't think there is anything wrong with foundational work of this kind. We can view it as creating a massive hub in the structure of mathematical knowledge that a lot of other theories can hang off. It is rather that I think the informal ideas and theories come first and they are slotted into the formal structure later.

Axioms are not just for formalists; the Platonists use them also. And you call it "retrofitted", but the axiomatic method goes back 2000 years.

There are competing axiom systems for set theory. Probably the biggest difference is the axiom of choice, that you mention. However that does not undermine the axiomatization of mathematics. It only means that some mathematicians want to make dependence on that axiom explicit.

So when you say that "Mathematics is Physics", you are not talking about Mathematics has it has been understood for 2000 years. You are only talking about some narrow empirical subset of mathematics.

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There are competing axiom systems for set theory. Probably the biggest difference is the axiom of choice, that you mention. However that does not undermine the axiomatization of mathematics. It only means that some mathematicians want to make dependence on that axiom explicit.

So when you say that "Mathematics is Physics", you are not talking about Mathematics has it has been understood for 2000 years. You are only talking about some narrow empirical subset of mathematics.

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Yes, but why do mathematicians decide that some sets of axioms are more important than others? That is what I am really trying to get at.

Dear Matt Leifer,

Your essay describes your '*theory of mathematical theory building*' and contrasts it with (o.d.l.) Tegmark's MUH. As I agree completely with you, let me focus on a few specifics. First, you say "…*our fundamental laws of physics are formulated in terms of some of the most advanced branches of mathematics…*" I would probably have said that our fundamental...

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Your essay describes your '

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Let me first say that I do not generally respond to requests for me to read someone's essay made on this page, via email, or via any other method. I hope this does not offend you, or anyone else, but I get too many such requests and there are too many essays that I want to read anyway for it to be feasible for me to comply with all requests. Everyone who makes such a request claims that their...

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Dear Matt,

Congratulations on your thought provoking essay. As you would probably be entertaining other questions and have challenges on your time, I will have only one question for you:

Borrowing from different information sources, including cosmology,**can a Universe, either of the 'Physics is Mathematics' or 'Mathematics is Physics' variety perish?**

If the universe can perish, what is the possible implication for physics that mathematical/ physical objects are not eternally existing things but have a finite duration of existence?

Sorry, one question became two.

Regards,

Akinbo

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Congratulations on your thought provoking essay. As you would probably be entertaining other questions and have challenges on your time, I will have only one question for you:

Borrowing from different information sources, including cosmology,

If the universe can perish, what is the possible implication for physics that mathematical/ physical objects are not eternally existing things but have a finite duration of existence?

Sorry, one question became two.

Regards,

Akinbo

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I don't think my view has any implications for whether or not the universe will perish.

To answer your second question, I think it is helpful to first answer a related question, namely, in a universe with different laws of physics than our own, would mathematics be different?

According to my view, mathematical theories are just abstract formal systems, but only those formal systems...

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To answer your second question, I think it is helpful to first answer a related question, namely, in a universe with different laws of physics than our own, would mathematics be different?

According to my view, mathematical theories are just abstract formal systems, but only those formal systems...

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Dear Marc,

It was a real pleasure to read your essay, thank you.

In your essay, you state that physical theories will become increasingly abstract and mathematical. Who will be able to discover them if it gets even more complex? Do you believe we are close to hitting a knowledge wall?

Regards,

Christophe

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It was a real pleasure to read your essay, thank you.

In your essay, you state that physical theories will become increasingly abstract and mathematical. Who will be able to discover them if it gets even more complex? Do you believe we are close to hitting a knowledge wall?

Regards,

Christophe

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Abstract does not necessarily mean complex. In fact, I would argue that we introduce more abstraction precisely to reduce complexity, i.e. it reduces the number of direct connections between seemingly disparate parts of the knowledge network.

I don't think we are close to hitting a knowledge wall. One can work quite effectively on just a small portion of the knowledge network, i.e. just a few nodes clustered around a hub. The abstraction prevents an individual from having to know everything about everything in order to make any progress.

However, I would argue that in order to make progress we need to be good at identifying connections between nodes in the network, especially if those nodes are far away from one another. Given that any individual only knows a small part of the network, this means we require interdisciplinary collaboration to find the strong analogies that will be later abstracted into more powerful theories. We may well be hitting a knowledge wall for what a single individual can do on their own, but I believe we are just at the beginning of developing new modes of collaboration that can get around this.

I don't think we are close to hitting a knowledge wall. One can work quite effectively on just a small portion of the knowledge network, i.e. just a few nodes clustered around a hub. The abstraction prevents an individual from having to know everything about everything in order to make any progress.

However, I would argue that in order to make progress we need to be good at identifying connections between nodes in the network, especially if those nodes are far away from one another. Given that any individual only knows a small part of the network, this means we require interdisciplinary collaboration to find the strong analogies that will be later abstracted into more powerful theories. We may well be hitting a knowledge wall for what a single individual can do on their own, but I believe we are just at the beginning of developing new modes of collaboration that can get around this.

Greetings Matt

I enjoyed your essay. A network view of evolving mathematical knowledge does indeed exist. Even now, we are connecting more nodes, and physics seems to be giving a "garden of insights", fueling this process.

What is interesting about our current era, is how advanced physics has become. In an attempt formulate a theory of quantum gravity, it is clear Riemannian...

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I enjoyed your essay. A network view of evolving mathematical knowledge does indeed exist. Even now, we are connecting more nodes, and physics seems to be giving a "garden of insights", fueling this process.

What is interesting about our current era, is how advanced physics has become. In an attempt formulate a theory of quantum gravity, it is clear Riemannian...

view entire post

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Dear Matt,

I'm not sure I understand the sense in which mathematics is supposed to be "about the physical world" as you understand it. In one sense, the truth value of any claim about the physical world depends on how the physical world is, that is, it is physically contingent. Had the physical world been different, the truth value of the claim would be different. Now take a claim about the integers, such as Goldbach's conjecture. Do you mean to say that the truth or falsity of Goldbach's conjecture depends on the physical world: if the physical world is one way then it is true and if it is another way it is false? What feature of the physical world could the truth or falsity of the conjecture possibly depend on? Do you think the conjecture could fail to have a truth value at all? The formalists tried to reduce mathematical truth to theoremhood, but Gödel proved that won't work. We know that if Goldbach's conjecture is false then (in some sense) it is provably false by direct calculation (although the calculation might take more steps than there are elementary particles). But if it is true, it may not be provable from any acceptable axioms. It is very hard to make any sense of what we all believe about a case like this (the conjecture is either true or false) without, in some sense, being Platonistic. I can see you don't like Platonism, but I can't see how you deal with questions like these about mathematical truth. Could you provide an example of a purely mathematical claim whose truth depends on the physical world, and point to the feature of the physical world it depends on?

Cheers,

Tim

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I'm not sure I understand the sense in which mathematics is supposed to be "about the physical world" as you understand it. In one sense, the truth value of any claim about the physical world depends on how the physical world is, that is, it is physically contingent. Had the physical world been different, the truth value of the claim would be different. Now take a claim about the integers, such as Goldbach's conjecture. Do you mean to say that the truth or falsity of Goldbach's conjecture depends on the physical world: if the physical world is one way then it is true and if it is another way it is false? What feature of the physical world could the truth or falsity of the conjecture possibly depend on? Do you think the conjecture could fail to have a truth value at all? The formalists tried to reduce mathematical truth to theoremhood, but Gödel proved that won't work. We know that if Goldbach's conjecture is false then (in some sense) it is provably false by direct calculation (although the calculation might take more steps than there are elementary particles). But if it is true, it may not be provable from any acceptable axioms. It is very hard to make any sense of what we all believe about a case like this (the conjecture is either true or false) without, in some sense, being Platonistic. I can see you don't like Platonism, but I can't see how you deal with questions like these about mathematical truth. Could you provide an example of a purely mathematical claim whose truth depends on the physical world, and point to the feature of the physical world it depends on?

Cheers,

Tim

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"Could you provide an example of a purely mathematical claim whose truth depends on the physical world, and point to the feature of the physical world it depends on?"

No, because I don't think there is one. Mathematical theories are still formal systems in my view, so truth or falsehood is supposed to be decided by the usual methods of proof.

Titling my essay "Mathematics is...

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No, because I don't think there is one. Mathematical theories are still formal systems in my view, so truth or falsehood is supposed to be decided by the usual methods of proof.

Titling my essay "Mathematics is...

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OK, but really this is a Platonistic answer. Of course, which mathematical systems can be usefully used to describe the world depends on how the world is! That is not in dispute. But your claim that settling on a axiom system settles Goldbach's conjecture (and does so independently of the world) is Platonistic. The notion of a "theorem" (i.e. something that follows, by application of rules, from a set of axions) is also Platonistic: you don't think that what the theorems are depends on the world. And this does not address the status of claims that are not theorems and whose negations are not theorems.

Regular arithmetic and modular arithmetic are different mathematical structures. Which is useful for describing things is of course contingent on those things, but the purely mathematical structure of the structures does not. What naturalistic fact determines whether or not Goldbach's conjecture follows from the Peano axioms, and what naturalistic fact determines its truth if neither it nor it's negation follow?

I recognize that this is just philosophy of math—and not really central to your essay, which can be read as about which mathematical systems we discover (or invent) and decide to deploy in physics and why. But a thoroughly non-Platonistic account of mathematics is both very hard to formulate (since one thinks of the axiom/theorem relation in a completely Platonistic way) and inconsistent with normal beliefs about mathematics (since truth cannot be reduced to theoremhood in any case).

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Regular arithmetic and modular arithmetic are different mathematical structures. Which is useful for describing things is of course contingent on those things, but the purely mathematical structure of the structures does not. What naturalistic fact determines whether or not Goldbach's conjecture follows from the Peano axioms, and what naturalistic fact determines its truth if neither it nor it's negation follow?

I recognize that this is just philosophy of math—and not really central to your essay, which can be read as about which mathematical systems we discover (or invent) and decide to deploy in physics and why. But a thoroughly non-Platonistic account of mathematics is both very hard to formulate (since one thinks of the axiom/theorem relation in a completely Platonistic way) and inconsistent with normal beliefs about mathematics (since truth cannot be reduced to theoremhood in any case).

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You are being too modest to call it "just" philosophy of math. It's important to get the background right and I am relatively naive in this area, so I appreciate your attempts to pin me down.

I am disturbed by your suggestion that I am a platonist. To me, platonism suggests the existence of an abstract world, independent of the physical world, to which we somehow have access through...

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I am disturbed by your suggestion that I am a platonist. To me, platonism suggests the existence of an abstract world, independent of the physical world, to which we somehow have access through...

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Dear Dr. Leifer,

I very much appreciated your essay on how abstract mathematics still possesses an empirical tether to the natural world (and I like that word, ``tether’’). Your figure showing mesh connections going to ``star’’ really helped in understanding your point. We don’t want our networks to have too many links. The abstractions of mathematics are easier for us than having way too many links. [``we introduce more abstraction precisely to reduce complexity’’ (your note of 2/27)]. And then we continue to do that for abstractions of abstractions.

Just a few little comments: The dividing line between natural versus ``spiritual’’ foundations for the Forms is slightly obscured by the possibility that the basic fundamental physical Forms are the quantum fields of the Vacuum – which are of course ``natural’’ but in a highly unusual non-classical and somewhat intangible way. This also makes Tegmark’s view a little more plausible in that these basic fields are highly mathematical and ``ethereal.’’ If the ancient platonic view was brought more up-to-date and modified for current relevance, then the math and physics Forms become more ``natural,’’ and the ``where’’ becomes various but numerous intelligences in the universe viewing nature produced from the ``Vacuum’’ (everything in quotes because our old words don’t quite fit).

Thanks again, Dave.

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I very much appreciated your essay on how abstract mathematics still possesses an empirical tether to the natural world (and I like that word, ``tether’’). Your figure showing mesh connections going to ``star’’ really helped in understanding your point. We don’t want our networks to have too many links. The abstractions of mathematics are easier for us than having way too many links. [``we introduce more abstraction precisely to reduce complexity’’ (your note of 2/27)]. And then we continue to do that for abstractions of abstractions.

Just a few little comments: The dividing line between natural versus ``spiritual’’ foundations for the Forms is slightly obscured by the possibility that the basic fundamental physical Forms are the quantum fields of the Vacuum – which are of course ``natural’’ but in a highly unusual non-classical and somewhat intangible way. This also makes Tegmark’s view a little more plausible in that these basic fields are highly mathematical and ``ethereal.’’ If the ancient platonic view was brought more up-to-date and modified for current relevance, then the math and physics Forms become more ``natural,’’ and the ``where’’ becomes various but numerous intelligences in the universe viewing nature produced from the ``Vacuum’’ (everything in quotes because our old words don’t quite fit).

Thanks again, Dave.

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Thanks for your comments. Regarding quantum fields, of course there is widespread disagreement about quantum theory and what can be said to exist in the quantum world. I like to say that the biggest problem we have with quantum theory is the problem of quantum jumps, i.e. quantum physicists are always jumping to conclusions.

Dear M. S. Leifer,

It is very interesting your figure 2. It is very applicable to my article.

I would like to you to fill your hopes

Best Regards,

Branko Zivlak

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It is very interesting your figure 2. It is very applicable to my article.

I would like to you to fill your hopes

Best Regards,

Branko Zivlak

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"Since mathematical theories are derived from the natural world..."

No. Just like physical theories, they are derived from initial assumptions (axioms, postulates) that could be arbitrary and false. In Big Brother's world, a new arithmetic theory has been derived from Big Brother's postulate "2+2=5":

"In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable what then?"

There are paradoxes in the new arithmetic theory. Here is one of them (it can be juxtaposed with the twin paradox in Einstein's theory of relativity):

3(2 + 2) = 3x5 = 15

3(2 + 2) = 3x2 + 3x2 = 6 + 6 = 12

Pentcho Valev

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No. Just like physical theories, they are derived from initial assumptions (axioms, postulates) that could be arbitrary and false. In Big Brother's world, a new arithmetic theory has been derived from Big Brother's postulate "2+2=5":

"In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable what then?"

There are paradoxes in the new arithmetic theory. Here is one of them (it can be juxtaposed with the twin paradox in Einstein's theory of relativity):

3(2 + 2) = 3x5 = 15

3(2 + 2) = 3x2 + 3x2 = 6 + 6 = 12

Pentcho Valev

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If you don't think physical theories are derived from the natural world, then I don't think we have much to talk about. Of course, other factors go into the construction of our physical theories, but if they are not designed to account for the empirical facts then what is the point of them?

I think physical theories are derived from ASSUMPTIONS about the natural world that could be false. For instance, recently the assumption that light always travels at the same speed in a vacuum has been refuted:

"A team of Scottish scientists has made light rays travel slower than the speed of light."

"Spatially structured photons that travel in free space slower than the speed of light"

Pentcho Valev

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"A team of Scottish scientists has made light rays travel slower than the speed of light."

"Spatially structured photons that travel in free space slower than the speed of light"

Pentcho Valev

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Your essay view of math follows mine nearly exactly.

Mine is a little stronger on the in the natural world because it is apart of the natural world like gravity. I also add the idea of fractals (self similarity) rather than hubs. Thus math developed out of our human scale and applies to other scales because the universe is fractal.

There are some problems with the abstractions that deviate with the observation of math such as irrational numbers, division, and infinity. These things are not observed in our scale and are part of the human introduced postulates that are false. Indeed, the current study of math allows the introduction of postulates and the reasoning from those postulates. It is called ``pure’’ but it is really only unjustified abstraction. This is not necessarily physics. For example, the introduction of non-Euclidean geometry is unreal - its use in cosmology is problematical because the universe has been measured to be flat (Euclidean).

My view allows the idea of using a mathematical structure that is observed such as by statistics or group theory to be considered real. For example, the periodic table was developed first by noting common characteristics of elements. A few holes were filled (predicted) by where the hole was in the classification scheme. Later, the causal underlying structure of atoms explained the periodic table. Indeed, the position of an element indicated something about the atomic structure. The same type of classification is true for the particle group models. Holes in the group model have been used to predict particles that were found. Can this be used to imply an underlying structure of particles? My model says yes.

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Mine is a little stronger on the in the natural world because it is apart of the natural world like gravity. I also add the idea of fractals (self similarity) rather than hubs. Thus math developed out of our human scale and applies to other scales because the universe is fractal.

There are some problems with the abstractions that deviate with the observation of math such as irrational numbers, division, and infinity. These things are not observed in our scale and are part of the human introduced postulates that are false. Indeed, the current study of math allows the introduction of postulates and the reasoning from those postulates. It is called ``pure’’ but it is really only unjustified abstraction. This is not necessarily physics. For example, the introduction of non-Euclidean geometry is unreal - its use in cosmology is problematical because the universe has been measured to be flat (Euclidean).

My view allows the idea of using a mathematical structure that is observed such as by statistics or group theory to be considered real. For example, the periodic table was developed first by noting common characteristics of elements. A few holes were filled (predicted) by where the hole was in the classification scheme. Later, the causal underlying structure of atoms explained the periodic table. Indeed, the position of an element indicated something about the atomic structure. The same type of classification is true for the particle group models. Holes in the group model have been used to predict particles that were found. Can this be used to imply an underlying structure of particles? My model says yes.

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Matt,

Though I couldn't disagree with you more, I really enjoyed your essay.

We will have much to debate -- my view agrees with Max's, and my own upcoming essay deals with the issues of Godel and Goldbach that Tim raised.

Two things for the time being:

1. "There is no 'adding zeroes and ones to the end of binary strings' research group in any mathematics department. " Sure there is. Chaitin's number is maximally unknowable, and its algorithm cannot predict the next binary digit of the value. What's more, the value is dependent on the language in which the algorithm is written.

2. Hierarchical knowledge? What if knowledge is laterally distributed on multiple scales in the hub-connected complex network? No hierarchy -- which was Bar-Yam's solution to the problem of bounded rationality (Herbert Simon).

Tom

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Though I couldn't disagree with you more, I really enjoyed your essay.

We will have much to debate -- my view agrees with Max's, and my own upcoming essay deals with the issues of Godel and Goldbach that Tim raised.

Two things for the time being:

1. "There is no 'adding zeroes and ones to the end of binary strings' research group in any mathematics department. " Sure there is. Chaitin's number is maximally unknowable, and its algorithm cannot predict the next binary digit of the value. What's more, the value is dependent on the language in which the algorithm is written.

2. Hierarchical knowledge? What if knowledge is laterally distributed on multiple scales in the hub-connected complex network? No hierarchy -- which was Bar-Yam's solution to the problem of bounded rationality (Herbert Simon).

Tom

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1. I was referring to the specific (very boring) formal system that I had introduced earlier. There is no mathematical research group studying that. I initially thought to include a footnote that of course there are people studying the general structure of formal systems, which would cover the kind of thing you are talking about here.

2. I am not sure I understand all of the technical terms you are using here, but indeed I do expect that knowledge is distributed on multiple scales. I am not familiar with the literature on bounded rationality, but it sounds like something I should look into.

2. I am not sure I understand all of the technical terms you are using here, but indeed I do expect that knowledge is distributed on multiple scales. I am not familiar with the literature on bounded rationality, but it sounds like something I should look into.

Hi Matt,

I delayed replying until my essay entry was posted.

You should be able to find all you want to know about multi-scale variety, on the home page of the New England Complex Systems Institute.

Bounded rationality was formally developed by Herbert Simon, as a heuristic approach to decision problems. Because Bayes' theorem is also motivated by decision problems -- I...

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I delayed replying until my essay entry was posted.

You should be able to find all you want to know about multi-scale variety, on the home page of the New England Complex Systems Institute.

Bounded rationality was formally developed by Herbert Simon, as a heuristic approach to decision problems. Because Bayes' theorem is also motivated by decision problems -- I...

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Dear Matthew Leifer,

At certain times I take a stance similar to this. You might by way of comparison look at Peterson’s paper for a different perspective. In my paper I am primarily concerned with what I call mathematics that has “meat” or “body,” by which I mean things that are computed in some rather explicit way and that have reference to physical properties. I look informally at decidability issues, by treating this in a somewhat physical way, and make arguments with respect to the complexity of numbers.

There is what might be called the “soul” of mathematics, which is all of that Platonist stuff. I am not committed to saying this exists or does not exist. This is in some way connected to mind or consciousness, but connected in way that I don’t understand and I don’t think anyone else does either. Whether one want to argue for the existence of this “soul” is a matter of choice or almost what might be called faith. I don’t think there ever will be some decidable criterion whereby we can say Platonia exists or not. I will put on the hat of Platonism at times and at other times not wear it. In my essay I largely keep it off.

“On a dark night in a city that knows how to keep its secrets; on the tenth floor of the atlas building one man searches for answers to life’s persistent questions, Guy Noir private eye.” Garrison Keillor “Prairie Home Companion. That about states where the deep question about the relationship between mathematics and physics lies.

LC

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At certain times I take a stance similar to this. You might by way of comparison look at Peterson’s paper for a different perspective. In my paper I am primarily concerned with what I call mathematics that has “meat” or “body,” by which I mean things that are computed in some rather explicit way and that have reference to physical properties. I look informally at decidability issues, by treating this in a somewhat physical way, and make arguments with respect to the complexity of numbers.

There is what might be called the “soul” of mathematics, which is all of that Platonist stuff. I am not committed to saying this exists or does not exist. This is in some way connected to mind or consciousness, but connected in way that I don’t understand and I don’t think anyone else does either. Whether one want to argue for the existence of this “soul” is a matter of choice or almost what might be called faith. I don’t think there ever will be some decidable criterion whereby we can say Platonia exists or not. I will put on the hat of Platonism at times and at other times not wear it. In my essay I largely keep it off.

“On a dark night in a city that knows how to keep its secrets; on the tenth floor of the atlas building one man searches for answers to life’s persistent questions, Guy Noir private eye.” Garrison Keillor “Prairie Home Companion. That about states where the deep question about the relationship between mathematics and physics lies.

LC

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Hello. This is the second essay I read after that of Lee Smolin, that tries to give a naturalistic philosophy of mathematics, as opposed to a Platonistic one. As I commented there, I have yet to see a coherent formulation of naturalism. The Stanford encyclopedia article you refer to on this point, admits it directly : "The term ‘naturalism’ has no very precise meaning in contemporary...

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There is a lot of food for thought in your comments, and I don't have time to answer all of them in one go. This is just the first of several replies and I will address the rest of the issues you have raised in due course.

For now though, I just want to comment on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If...

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For now though, I just want to comment on "naturalism". I think it is unfair to criticize the term on the grounds that it does not have a clear and unique meaning. If...

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Matt, you write, " ... I want a metaphysics that does not posit entities beyond those of physics, or at least one that does so only minimally."

I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not.

" ... when I call something 'real', what I really mean is that it is useful, indispensable, etc. to the entities who practice science."

What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?

Tom

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I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not.

" ... when I call something 'real', what I really mean is that it is useful, indispensable, etc. to the entities who practice science."

What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?

Tom

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"I think this could not be a clearer statement of an anti-rationalist viewpoint. Those entities that are metaphysically real -- such as the moon when no one is looking at it -- is not just minimally real. It is real or it is not."

"Minimally real" is not a terminology I have introduced or at all relevant to what I am saying. The atoms and molecules that make up the moon are real. There is a particular arrangement of those atoms and molecules that we call "the moon". That is also real.

"What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?"

The pragmatist criterion of "usefulness" is very often misunderstood. It is intended in a very broad sense. Theories of the solar system that say that the moon is real are more coherent and tell a more consistent story than those that do not. A scientist who goes around thinking that the moon is real will have a far easier time reasoning about what goes on in the solar system than one who does not. It passes the pragmatic test of "usefulness".

In fact, I would argue that realist theories are pragmatically preferred in general, as they provide a better explanatory framework than anti-realist theories.

"Minimally real" is not a terminology I have introduced or at all relevant to what I am saying. The atoms and molecules that make up the moon are real. There is a particular arrangement of those atoms and molecules that we call "the moon". That is also real.

"What is useful or indispensable about the moon when no one is looking at it, to a scientist or to anyone else?"

The pragmatist criterion of "usefulness" is very often misunderstood. It is intended in a very broad sense. Theories of the solar system that say that the moon is real are more coherent and tell a more consistent story than those that do not. A scientist who goes around thinking that the moon is real will have a far easier time reasoning about what goes on in the solar system than one who does not. It passes the pragmatic test of "usefulness".

In fact, I would argue that realist theories are pragmatically preferred in general, as they provide a better explanatory framework than anti-realist theories.

Dear Matt,

Good to hear you're a naturalist, not a pragmatist. Or maybe both? :)

Anyway, I take your argument to be more about mathematical physics than mathematics per se. But even if we restrict ourselves to mathematical physics (which is a branch of mathematics, of course), what about the problem of multiplicity? To follow your argument, the objective world of mathematics is just the physical world, but we know that the physical world admits competing mathematical descriptions (e.g., different formalisms of the same quantum theory: operators, path integrals, geometric approaches, etc.). Why would this multiplicity arise if the mathematical world and the physical world were one?

Cheers,

Alexei

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Good to hear you're a naturalist, not a pragmatist. Or maybe both? :)

Anyway, I take your argument to be more about mathematical physics than mathematics per se. But even if we restrict ourselves to mathematical physics (which is a branch of mathematics, of course), what about the problem of multiplicity? To follow your argument, the objective world of mathematics is just the physical world, but we know that the physical world admits competing mathematical descriptions (e.g., different formalisms of the same quantum theory: operators, path integrals, geometric approaches, etc.). Why would this multiplicity arise if the mathematical world and the physical world were one?

Cheers,

Alexei

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I am both a naturalist and a pragmatist, and I think that is important for answering your questions.

Our physical and mathematical theories are both highly constrained by the natural world, but they are not completely determined by them. There is also the constraint that our knowledge derives from a social process and must be represented in a form that is useful to that society (this is the pragmatism part). The fact that there may be several different such representations is therefore not much of a problem for me. I am not saying that the mathematics literally is the physical world.

I also think you misunderstand me if you think I am talking just about mathematical physics. It could be read that way for sure, but I really intend it as a theory of all of mathematics. I recognize that this is a harder thesis to defend, but I wish to defend it.

Our physical and mathematical theories are both highly constrained by the natural world, but they are not completely determined by them. There is also the constraint that our knowledge derives from a social process and must be represented in a form that is useful to that society (this is the pragmatism part). The fact that there may be several different such representations is therefore not much of a problem for me. I am not saying that the mathematics literally is the physical world.

I also think you misunderstand me if you think I am talking just about mathematical physics. It could be read that way for sure, but I really intend it as a theory of all of mathematics. I recognize that this is a harder thesis to defend, but I wish to defend it.

Would you say that another way to say 'regularities within regularities' is 'math theory X surrounds math theory Y'?

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Dear Matt Leifer,

Is it 1. "*our universe is nothing but a mathematical structure and that all possible mathematical structures exist in the same sense as our universe.*"

or 2. "*mathematics is a natural science—just like physics, chemistry, or biology—albeit ... fundamentally a theory about our physical universe and, as such, it should come as no surprise that our fundamental theories of the universe are formulated in terms of mathematics.*

Very interesting argument. I guess one idea that would support your side is that it is hard to imagine people developing things as simple as numbers and counting, if there was nothing to count or organize into classes and groups in the real world, ie. you need something to count to invent the concept of counting...

Enjoyed your essay, thanks.

Regards,

Ed Unverricht

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Is it 1. "

or 2. "

Very interesting argument. I guess one idea that would support your side is that it is hard to imagine people developing things as simple as numbers and counting, if there was nothing to count or organize into classes and groups in the real world, ie. you need something to count to invent the concept of counting...

Enjoyed your essay, thanks.

Regards,

Ed Unverricht

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Hi Matt,

Enjoyed reading your essay, thanks for posting it! I find the network idea appealing and hope you're pursuing it further.

I must say though that the essays I've read so far already show that the phrase "the only" in your Conclusions is quite untenable.

"I have argued that viewing mathematics as a natural science is the only reasonable way of understanding why mathematics plays such a central role in physics."

All the best,

Tapio

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Enjoyed reading your essay, thanks for posting it! I find the network idea appealing and hope you're pursuing it further.

I must say though that the essays I've read so far already show that the phrase "the only" in your Conclusions is quite untenable.

"I have argued that viewing mathematics as a natural science is the only reasonable way of understanding why mathematics plays such a central role in physics."

All the best,

Tapio

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Dear Saul,

very simple and elegant idea, very convincingly expressed. For me, the text has appealed to visual intuition even more than the pictures. Another strong plus is that your essay is one of the few that hits the central question of the Contest right on the head.

One observation. You stress in various ways (e.g. with your first image) that your approach is opposite to...

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very simple and elegant idea, very convincingly expressed. For me, the text has appealed to visual intuition even more than the pictures. Another strong plus is that your essay is one of the few that hits the central question of the Contest right on the head.

One observation. You stress in various ways (e.g. with your first image) that your approach is opposite to...

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If math is a natural science then the correct logic to reason with should be quantum logic.

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I see a lot in common between your scale-free network and my view of universality. It is all about the things that are in common between different topics that are the most interesting. These form the subject areas that mathematicians like to study.

It is curious that emergent, self-organised structures have this scale-free, self-similar, fractal form. You mention how this is related to category theory and that is how I see it too.

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It is curious that emergent, self-organised structures have this scale-free, self-similar, fractal form. You mention how this is related to category theory and that is how I see it too.

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Hi Matt,

it is an interesting idea, but I don't think it's very well defined. I don't know for example what you mean with "knowledge" or a "theory" to begin with. Besides this, as you probably know, none of the real-world networks that you list are truly scale-free. They are just approximately scale-free over some orders of magnitude. I am not even sure that knowledge is fundamentally a discrete thing. We arguably use a discretization in reality (chunks of papers and websites and so on).

In a nutshell what you seem to be saying is that one can try to understand knowledge discovery with a mathematical model as well. I agree that one can do this, though we can debate whether the one you propose is correct. But that doesn't explain why many of the observations that we have lend themselves to mathematical description. Why do we find ourselves in a universe that does have so many regularities? (And regularities within regularities?) That really is the puzzling aspect of the "efficiency of mathematics in the natural science". I don't see that you address it at all.

I don't think that consciousness and the nature of now will remain outside physics for much longer, but then that's just my opinion. There may be aspects of our observations that will remain outside of our possibility to describe them with math though, I could agree on that.

I quite like your essay because you're a good example for the pragmatic physicist of my essay. Maybe you like to pick a philosophy from the categories in my essay? :)

-- Sophia

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it is an interesting idea, but I don't think it's very well defined. I don't know for example what you mean with "knowledge" or a "theory" to begin with. Besides this, as you probably know, none of the real-world networks that you list are truly scale-free. They are just approximately scale-free over some orders of magnitude. I am not even sure that knowledge is fundamentally a discrete thing. We arguably use a discretization in reality (chunks of papers and websites and so on).

In a nutshell what you seem to be saying is that one can try to understand knowledge discovery with a mathematical model as well. I agree that one can do this, though we can debate whether the one you propose is correct. But that doesn't explain why many of the observations that we have lend themselves to mathematical description. Why do we find ourselves in a universe that does have so many regularities? (And regularities within regularities?) That really is the puzzling aspect of the "efficiency of mathematics in the natural science". I don't see that you address it at all.

I don't think that consciousness and the nature of now will remain outside physics for much longer, but then that's just my opinion. There may be aspects of our observations that will remain outside of our possibility to describe them with math though, I could agree on that.

I quite like your essay because you're a good example for the pragmatic physicist of my essay. Maybe you like to pick a philosophy from the categories in my essay? :)

-- Sophia

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If you read my writings on the foundations of quantum theory, you will see that I am not Pragmatic Physicist. I am somewhat of a pragmatist in the philosophical sense, but I define "usefulness" more broadly than you do in your essay. A concept or idea that has explanatory power is useful even if it is not, strictly speaking, needed in order to predict the observations. In other words, a...

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Dear Dr. Leifer,

I posted a comment at your site that was unnecessarily contemptuous and devoid of the civility all contributors are entitled to. I deeply regret having done so, and I do hope that you can forgive my slurring of your fully deserved reputation.

I suspect that I may be suffering a relapse of Asperger’s Disorder. While this might explain my distasteful action, it cannot in any way justify it.

Respectfully,

Joe Fisher

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I posted a comment at your site that was unnecessarily contemptuous and devoid of the civility all contributors are entitled to. I deeply regret having done so, and I do hope that you can forgive my slurring of your fully deserved reputation.

I suspect that I may be suffering a relapse of Asperger’s Disorder. While this might explain my distasteful action, it cannot in any way justify it.

Respectfully,

Joe Fisher

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Hey Matt,

I thoroughly enjoyed this essay and find I agree with most of your points. I am not entirely convinced that human knowledge is devoid of hierarchy, though. Take your comment about sociologists believing that knowledge is a social construct, for example. It would seem to me that the successes of modern science and the fact that many discoveries are independently and often unknowingly verified by different people in entirely different social settings, directly counteracts that argument. Perhaps even stronger evidence might be some of the basic mathematical and physical concepts that can be independently grasped and indeed "discovered" by other species.

Anyway, my point is that if there truly is an objective reality out there (which I personally believe there must be), then it would seem that there ought to be at least some, albeit rough, hierarchy to our knowledge of it. We can abstract away from that objective reality in any number of ways, but all have the commonality that they are looking for either regularities within regularities within regularities, or regularities on TOP of regularities on top of regularities. Either way, objective reality is the starting point and it would seem to me that certain fields are closer to that objective reality than others.

I have one other minor quibble, though it is not necessarily with you. I know that it is traditional to view logic, set theory, category theory, etc. as competing theories (on an essentially equal footing) for the foundations of mathematics. But it seems to me that there is an undercurrent of what I might call "intuitionist" logic (not quite "informal" logic, which is an actual field) common to all of them. I mean, think for example about the very process of "creating" category theory or set theory: start with some basic premises and a few axioms, and reason from there. That in and of itself is reason enough to think that there is some deeper, singular foundational "truth" to mathematics that underlies everything.

Cheers,

Ian

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I thoroughly enjoyed this essay and find I agree with most of your points. I am not entirely convinced that human knowledge is devoid of hierarchy, though. Take your comment about sociologists believing that knowledge is a social construct, for example. It would seem to me that the successes of modern science and the fact that many discoveries are independently and often unknowingly verified by different people in entirely different social settings, directly counteracts that argument. Perhaps even stronger evidence might be some of the basic mathematical and physical concepts that can be independently grasped and indeed "discovered" by other species.

Anyway, my point is that if there truly is an objective reality out there (which I personally believe there must be), then it would seem that there ought to be at least some, albeit rough, hierarchy to our knowledge of it. We can abstract away from that objective reality in any number of ways, but all have the commonality that they are looking for either regularities within regularities within regularities, or regularities on TOP of regularities on top of regularities. Either way, objective reality is the starting point and it would seem to me that certain fields are closer to that objective reality than others.

I have one other minor quibble, though it is not necessarily with you. I know that it is traditional to view logic, set theory, category theory, etc. as competing theories (on an essentially equal footing) for the foundations of mathematics. But it seems to me that there is an undercurrent of what I might call "intuitionist" logic (not quite "informal" logic, which is an actual field) common to all of them. I mean, think for example about the very process of "creating" category theory or set theory: start with some basic premises and a few axioms, and reason from there. That in and of itself is reason enough to think that there is some deeper, singular foundational "truth" to mathematics that underlies everything.

Cheers,

Ian

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Let me reply to your second point first, as that is the easier of the two. I agree that there is a core of what we might call "informal logic" that is common to all foundations of mathematics. That core is what most mathematicians actually use in their daily work of proving theorems, and indeed it is what we all use when we try to make rational arguments. This informal logic is a massive hub in...

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Dear Matt,

Seeing the whole human knowledge as a scale-free network (like the WWW, Internet, cellular and ecological networks: your ref. [13]) seemed to me first counterintuitive but its scientific soundness got into me gradually. I realize how much the network of subfields I met in my career had an impact in my today research.

Putnam's solgan: 'meanings' just ain't in the head, that he develops in his twin earth thought experiment, also gave me something to think about and I now start to understand why you see mathematics as a natural science, not just as subfield of cognitive science.

Going back to the power law of a scale-free network, the words of Henri Poincaré in Science and hypothesis, came to my mind

"We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment. If that be so, the rough data of experiment, which are our sensations, could be measured. We might, indeed, be tempted to believe that this is so, for in recent times there has been an attempt to measure them, and a law has even been formulated, known as Fechner’s law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine the experiments by which the endeavour has been made to establish this law, we shall be led to a diametrically opposite conclusion."

It may be that, as in Poincaré's quote, if you zoom into the netwoork, you get a different structure like the resonant bubbles in Hamitonian chaos (e.g. resonances between subfields of maths and physics, of chemistry and life sciences, philosophy and language and so on that would create voids in the network). But even if the network is not scale free it is quite interesting to see the human knowledge as a complex and entangled system.

Best,

Michel

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Seeing the whole human knowledge as a scale-free network (like the WWW, Internet, cellular and ecological networks: your ref. [13]) seemed to me first counterintuitive but its scientific soundness got into me gradually. I realize how much the network of subfields I met in my career had an impact in my today research.

Putnam's solgan: 'meanings' just ain't in the head, that he develops in his twin earth thought experiment, also gave me something to think about and I now start to understand why you see mathematics as a natural science, not just as subfield of cognitive science.

Going back to the power law of a scale-free network, the words of Henri Poincaré in Science and hypothesis, came to my mind

"We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment. If that be so, the rough data of experiment, which are our sensations, could be measured. We might, indeed, be tempted to believe that this is so, for in recent times there has been an attempt to measure them, and a law has even been formulated, known as Fechner’s law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine the experiments by which the endeavour has been made to establish this law, we shall be led to a diametrically opposite conclusion."

It may be that, as in Poincaré's quote, if you zoom into the netwoork, you get a different structure like the resonant bubbles in Hamitonian chaos (e.g. resonances between subfields of maths and physics, of chemistry and life sciences, philosophy and language and so on that would create voids in the network). But even if the network is not scale free it is quite interesting to see the human knowledge as a complex and entangled system.

Best,

Michel

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The hypothesis that knowledge forms a scale-free network should certainly be put to empirical test, as it could certainly have some other type of structure. However, I am led to the scale-free hypothesis not just from the structure of web links and citations, which can be viewed as very rough approximations to the knowledge network, but also because of several toy models for how knowledge might...

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Matt,

I also believe that math "connects to the physical world via our direct empirical observations."

You do not expect the search for a theory of everything to ever end, but is not fruitless? In 1000 or 2000 years when we enter the realm of a type 2 civilization, do you think that perspective might change? Is "one new hub" necessary for a theory of everything?

My "connections" speaks of some of the same ideas but less eloquently.

Jim

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I also believe that math "connects to the physical world via our direct empirical observations."

You do not expect the search for a theory of everything to ever end, but is not fruitless? In 1000 or 2000 years when we enter the realm of a type 2 civilization, do you think that perspective might change? Is "one new hub" necessary for a theory of everything?

My "connections" speaks of some of the same ideas but less eloquently.

Jim

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I think that the search for a theory of everything is a search for connections at ever deeper levels within the knowledge network. I don't expect it to end because the network itself is always evolving.

I don't think the scale of energy available to a civilization has anything to do with the fundamental theory of how knowledge grows. So long as we are talking about a society of finite beings, our knowledge will reflect that structure. The only thing I can imagine that might change things is if we evolve to a borg-like entity with a single consciousness, but even then we are still talking about a network of finite entities interacting with one another, so maybe this would just speed the process up without changing its overall structure.

I don't quite know what you mean by "one new hub", but a theory of everything would be a single hub to which everything is connected heirarchically, i.e. in technical terms the knowledge graph would be a tree. It is not impossible for a scale-free network to have this structure, but I just don't think this is the structure of our actual knowledge network.

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I don't think the scale of energy available to a civilization has anything to do with the fundamental theory of how knowledge grows. So long as we are talking about a society of finite beings, our knowledge will reflect that structure. The only thing I can imagine that might change things is if we evolve to a borg-like entity with a single consciousness, but even then we are still talking about a network of finite entities interacting with one another, so maybe this would just speed the process up without changing its overall structure.

I don't quite know what you mean by "one new hub", but a theory of everything would be a single hub to which everything is connected heirarchically, i.e. in technical terms the knowledge graph would be a tree. It is not impossible for a scale-free network to have this structure, but I just don't think this is the structure of our actual knowledge network.

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Hi Matt,

My system is a counterexample to your thesis. If you don't have the time just read the electron mass section and run the program (click "program link" at the end of the section) , it will execute in less than a minute.

Essay

Thanks and good luck.

P.S. That was a nice one you pulled on Lubos, some people actually believed it.

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My system is a counterexample to your thesis. If you don't have the time just read the electron mass section and run the program (click "program link" at the end of the section) , it will execute in less than a minute.

Essay

Thanks and good luck.

P.S. That was a nice one you pulled on Lubos, some people actually believed it.

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Dear Saul,

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

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

Joe Fisher

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

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

Joe Fisher

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Dear Matthew Saul Leifer,

I read and enjoyed your essay. You presented your ideas well analytically and logically, I thought.

I did not think of my essay contribution as advancing the point of view of mathematics as naturalistic, but it seems to me that would be a fair characterization. In fact, your remark in your conclusion that "Mathematics is constructed out of the physical world" accords with the idea in my essay that the rate at which it is constructed can be measured and quantified (using C log (n) ).

On your point at page 2 that human knowledge has the structure of scale-free network, that may be so if we can say that energy distribution in the universe has the structure of a scale-free network. To utilize energy that is distributed like a scale free network recipient systems would have to model themselves the same way. At least one might argue that. In light of your remarks about networks, you might find some of the references in my essay of interest.

Thank you for having taken the time to write your interesting essay.

Best wishes,

Bob Shour

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I read and enjoyed your essay. You presented your ideas well analytically and logically, I thought.

I did not think of my essay contribution as advancing the point of view of mathematics as naturalistic, but it seems to me that would be a fair characterization. In fact, your remark in your conclusion that "Mathematics is constructed out of the physical world" accords with the idea in my essay that the rate at which it is constructed can be measured and quantified (using C log (n) ).

On your point at page 2 that human knowledge has the structure of scale-free network, that may be so if we can say that energy distribution in the universe has the structure of a scale-free network. To utilize energy that is distributed like a scale free network recipient systems would have to model themselves the same way. At least one might argue that. In light of your remarks about networks, you might find some of the references in my essay of interest.

Thank you for having taken the time to write your interesting essay.

Best wishes,

Bob Shour

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Hi Matt,

I really enjoyed your network perspective, and it's even changed my worldview about math somewhat. I think you're absolutely right that some new mathematical "nodes" are developed for exactly the reasons you describe here. (Strong analogies between existing nodes make it more natural to build a new node, leading to a more-efficient common structure, etc.)

But is this the...

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I really enjoyed your network perspective, and it's even changed my worldview about math somewhat. I think you're absolutely right that some new mathematical "nodes" are developed for exactly the reasons you describe here. (Strong analogies between existing nodes make it more natural to build a new node, leading to a more-efficient common structure, etc.)

But is this the...

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I do not think that much "pure" exploring actually goes on in mathematics. If that's what mathematicians were really doing then why would they not just explore arbitrary axiom systems? Mathematicians typically identify two strands of mathematical work: theory-building and problem solving. In reality, these are not completely separate from each other. A theory may need to be built in the...

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Matt,

You write "I do not think that much 'pure' exploring actually goes on in mathematics. If that's what mathematicians were really doing then why would they not just explore arbitrary axiom systems?"

Because axiomatics -- the logically deductive framework for mathematics -- is only a small part of the discipline. As Godel explained.

" ... the main thing we need to explain is why differential geometry showed up in GR in the first place, and that is the type of thing I am trying to account for."

I agree with you -- so did Einstein -- because general relativity is true only *up to diffeomorphism.* It could not be a final theory of gravity, then, and Einstein explicitly didn't intend it to be. What I am at a loss to understand, is why you invest so much interest in problem solving ("All life is problem solving," said Popper, a sentiment which I much appreciate), and yet don't attempt to differentiate the problem and solution:

"Mathematicians typically identify two strands of mathematical work: theory-building and problem solving. In reality, these are not completely separate from each other."

For course they are. Were they not, one could not differentiate theory from result in any rigorous way; no theorem could be a true, logically closed judgment. By such reasoning, how would one *ever* account for the role of differential geometry in Einstein's theory? -- the fact that space and time (Minkowski space) are not physically real except in a "union of the two," is the principle that motivates physical applications of differential geometry (the physics of continuous functions). Vesselin Petkov explains it as eloquently as anyone I've ever read.

Best,

Tom

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You write "I do not think that much 'pure' exploring actually goes on in mathematics. If that's what mathematicians were really doing then why would they not just explore arbitrary axiom systems?"

Because axiomatics -- the logically deductive framework for mathematics -- is only a small part of the discipline. As Godel explained.

" ... the main thing we need to explain is why differential geometry showed up in GR in the first place, and that is the type of thing I am trying to account for."

I agree with you -- so did Einstein -- because general relativity is true only *up to diffeomorphism.* It could not be a final theory of gravity, then, and Einstein explicitly didn't intend it to be. What I am at a loss to understand, is why you invest so much interest in problem solving ("All life is problem solving," said Popper, a sentiment which I much appreciate), and yet don't attempt to differentiate the problem and solution:

"Mathematicians typically identify two strands of mathematical work: theory-building and problem solving. In reality, these are not completely separate from each other."

For course they are. Were they not, one could not differentiate theory from result in any rigorous way; no theorem could be a true, logically closed judgment. By such reasoning, how would one *ever* account for the role of differential geometry in Einstein's theory? -- the fact that space and time (Minkowski space) are not physically real except in a "union of the two," is the principle that motivates physical applications of differential geometry (the physics of continuous functions). Vesselin Petkov explains it as eloquently as anyone I've ever read.

Best,

Tom

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I am not saying that theories are not distinct from problems. Of course they are. All I am saying is that the two different activities that mathematicians consider themselves to be engaged in are not entirely separate. One may need to build a new theory in order to solve a problem and one has to solve problems in the course of theory-building.

Dear Matthew,

You are suggesting an argumentation as to why it is natural to expect mathematics being suitable to physics. Essentially your point is that mathematics is a natural science, so a compatibility between the two natural sciences is also natural. I would not argue against this logic, but I do not see how it is an answer to Wigner's point of wonder.

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. This makes a difference between your paper and one by Tegmark.

Regards,

Alexey Burov.

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You are suggesting an argumentation as to why it is natural to expect mathematics being suitable to physics. Essentially your point is that mathematics is a natural science, so a compatibility between the two natural sciences is also natural. I would not argue against this logic, but I do not see how it is an answer to Wigner's point of wonder.

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. This makes a difference between your paper and one by Tegmark.

Regards,

Alexey Burov.

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I was meaning to discuss elegance in my essay, but I removed that section for lack of space, and because I thought it best to focus on the main argument. It is hard to pin down exactly what one means by an aesthetic notion like "elegance" in the context of mathematics and physics. My own feeling is that it has to do with economy and compactness of representation. That is, if we find a compact...

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Dear Matt,

Your well written and well argued essay looks like it's close to breaking that record mentioned atop of the page and with good reason. That math is derived from the observation of the physical universe is quite a compelling idea and your witty sense of humor can only benefit the exposition. It's a very good observation that you underscore at the end, about how your point of view is different from the Tegmarkverse, where the universe varieties can only be weakly interacting. I found the idea of regularities within regularities being quite striking as I considered it in my essay as well, only just naming it modularity instead. I also enjoyed the idea of knowledge as a scale free network. I am wondering what are the consequences of such a treatment and if a part of this network can undergo a change when the number of points in the region crosses a certain threshold - what I have in mind is creativity or the generation of new ideas that complete the pattern.

Thank you for a very good read and wish you best of luck in the contest! Should you have enough time and the curiosity to read my essay, I'd appreciate your comments.

Warm regards,

Alma

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Your well written and well argued essay looks like it's close to breaking that record mentioned atop of the page and with good reason. That math is derived from the observation of the physical universe is quite a compelling idea and your witty sense of humor can only benefit the exposition. It's a very good observation that you underscore at the end, about how your point of view is different from the Tegmarkverse, where the universe varieties can only be weakly interacting. I found the idea of regularities within regularities being quite striking as I considered it in my essay as well, only just naming it modularity instead. I also enjoyed the idea of knowledge as a scale free network. I am wondering what are the consequences of such a treatment and if a part of this network can undergo a change when the number of points in the region crosses a certain threshold - what I have in mind is creativity or the generation of new ideas that complete the pattern.

Thank you for a very good read and wish you best of luck in the contest! Should you have enough time and the curiosity to read my essay, I'd appreciate your comments.

Warm regards,

Alma

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I think it is difficult to see what the consequences of knowledge as a scale-free network are right now. This is because the idea is only a sketch at the moment. Work is needed to determine whether this really is a good representation of human knowledge and what the processes are by which it grows and changes. Nonetheless, if all this can be established rigorously then there is certainly scope for all sorts of statistical mechanics phenomena such as phase transitions to occur within the network. The idea that a "paradigm change" might be represented by a phase transition in the network is quite appealing.

I was thinking either of a paradigm change (knowledge pertaining to society) or a model for how brains generate ideas. The former is maybe more difficult to asses as two individuals with the same number of connections may have a different general impact whereas the latter is perhaps something on the lines of the minimum number of information points needed to deduce a new piece of the puzzle, as related to the complexity dimension of the concept; as in how much information did you need so the idea of a knowledge network can pop into existence :) Anyway excellent idea!

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Sorry, I just realized I forgot to rate your essay so I fixed that now. Good luck!

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I've probably misunderstood something, but in virtue of which physical processes is the mathematical equation 2 + 3 = 5 true?

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Matt,

As time grows short, so I am revisiting essays I’ve read (3/26) to assure I’ve rated them. I find that I did not rate yours, so I am rectifying that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345

Jim

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As time grows short, so I am revisiting essays I’ve read (3/26) to assure I’ve rated them. I find that I did not rate yours, so I am rectifying that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345

Jim

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Dear Matthew Saul Leifer

Your essay has some positive aspects (+) and some where I disagree (-).

(+) Your thoughts are shown very concise, especially your figure 4.

(+) I never liked to be said that ''math is a thing of axioms, not a thing of intuition.'' For instance, the question, which number of hairs is a border between baldness and non-baldness is a thing of intuition....

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Your essay has some positive aspects (+) and some where I disagree (-).

(+) Your thoughts are shown very concise, especially your figure 4.

(+) I never liked to be said that ''math is a thing of axioms, not a thing of intuition.'' For instance, the question, which number of hairs is a border between baldness and non-baldness is a thing of intuition....

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Dear Matt Leifer

I should to add, that modeling with consciousness is very similar to physicalism. But otherwise physicalism is in contradiction with panspchism. This also does not mean that I do not defend scienfic approach to this question.

My essay

Best regards

Janko Kokosar

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I should to add, that modeling with consciousness is very similar to physicalism. But otherwise physicalism is in contradiction with panspchism. This also does not mean that I do not defend scienfic approach to this question.

My essay

Best regards

Janko Kokosar

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Dear Prof. Matthew,

Wonderful essay! I enjoyed reading it, and we seem to agree in many points, as my essay reflects, especially that mathematics is a study of regularities in nature. I would be glad to take your opinion in my essay.

Best regards,

Mohammed

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Wonderful essay! I enjoyed reading it, and we seem to agree in many points, as my essay reflects, especially that mathematics is a study of regularities in nature. I would be glad to take your opinion in my essay.

Best regards,

Mohammed

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Matt,

You argued hard for your interesting and original hypothesis but I was left with a few obvious questions.

Your 'main proof' (that we've been getting more mathematical) is wholly circumstantial, so based on other assumptions which were unsupported, i.e. that physics is not getting ever more mathematical purely due to being confounded by logical analysis. And that may perhaps be the WRONG direction for improving understanding. That may be right or wrong but seems equally possible.

You don't seem to consider the case of genius and advancement by those who never even learned maths, or perhaps anything beyond basic arithmetic, and haven't used it in their achievements. It seems most of the greatest achievers in history a fall into that category! Physics is after all only a small slice of humanity and it may be argued that slice has made less not more progress in recent times!

You seem to accept all maths as 'correct' per se and don't highlight or even seem to refer to cases where mathematics we employ does NOT model natures mechanisms and can mislead us (which I address in my own essay). Do you not think we should take better care of HOW we employ mathematics?

I look forward to your responses, but a well written and presented essay with an original hypothesis.

Best of luck hitting you record number of FQXi wins.

Peter

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You argued hard for your interesting and original hypothesis but I was left with a few obvious questions.

Your 'main proof' (that we've been getting more mathematical) is wholly circumstantial, so based on other assumptions which were unsupported, i.e. that physics is not getting ever more mathematical purely due to being confounded by logical analysis. And that may perhaps be the WRONG direction for improving understanding. That may be right or wrong but seems equally possible.

You don't seem to consider the case of genius and advancement by those who never even learned maths, or perhaps anything beyond basic arithmetic, and haven't used it in their achievements. It seems most of the greatest achievers in history a fall into that category! Physics is after all only a small slice of humanity and it may be argued that slice has made less not more progress in recent times!

You seem to accept all maths as 'correct' per se and don't highlight or even seem to refer to cases where mathematics we employ does NOT model natures mechanisms and can mislead us (which I address in my own essay). Do you not think we should take better care of HOW we employ mathematics?

I look forward to your responses, but a well written and presented essay with an original hypothesis.

Best of luck hitting you record number of FQXi wins.

Peter

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Dear Matt S. Leifer,

Is this is correct summary of your main thesis (in section 4)? : "First, humans studied many aspects the world, gathering knowledge. At some point, it made sense to start studying the structure of that knowledge. (And further iterations.) This is called mathematics."

Although I find this idea appealing (and I share your general preference for a naturalistic...

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Is this is correct summary of your main thesis (in section 4)? : "First, humans studied many aspects the world, gathering knowledge. At some point, it made sense to start studying the structure of that knowledge. (And further iterations.) This is called mathematics."

Although I find this idea appealing (and I share your general preference for a naturalistic...

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Apologies for not replying sooner.

I certainly did not intend my essay to provide a comprehensive analysis of all forms of mathematical enquiry. I was just trying to make it plausible that there is a process of mathematical abstraction that keeps enough of a tether to empirical reality to explain the later use of those theories in physics. I agree that there are other processes going on...

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I certainly did not intend my essay to provide a comprehensive analysis of all forms of mathematical enquiry. I was just trying to make it plausible that there is a process of mathematical abstraction that keeps enough of a tether to empirical reality to explain the later use of those theories in physics. I agree that there are other processes going on...

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Dear Matt,

Thank you for the essay. Mathematics is not a science because there are no physical experiments that will change a proof. Mathematics is not a lesser thing because it is not a physical science, some would say it is more because it is more "pure". There are many examples of outdated or just wrong physics were the mathematical systems are still valid (equations in classical physics easily have particle going faster than light and not following the rules of quantum mechanics). Any gravitational interaction between more than three particles does not have a true solution in current mathematics. The relationship between math and physics is a wonderful and useful intersection, not the creation of either.

All the best,

Jeff

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Thank you for the essay. Mathematics is not a science because there are no physical experiments that will change a proof. Mathematics is not a lesser thing because it is not a physical science, some would say it is more because it is more "pure". There are many examples of outdated or just wrong physics were the mathematical systems are still valid (equations in classical physics easily have particle going faster than light and not following the rules of quantum mechanics). Any gravitational interaction between more than three particles does not have a true solution in current mathematics. The relationship between math and physics is a wonderful and useful intersection, not the creation of either.

All the best,

Jeff

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Dear Matt,

I had read your essay while the contest was underway but never got to comment on it. Better late than never!

In my essay, I side with Tegmark’s view that can summarized as “Physics is Math”, so I was naturally intrigued by your claim that the opposite is the case. After reading your essay, I think it all comes down to the different way we define mathematics. In your...

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I had read your essay while the contest was underway but never got to comment on it. Better late than never!

In my essay, I side with Tegmark’s view that can summarized as “Physics is Math”, so I was naturally intrigued by your claim that the opposite is the case. After reading your essay, I think it all comes down to the different way we define mathematics. In your...

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I don't disagree too strongly with your characterization of my position, but I would modify it somewhat:

(1) It doesn't have to be physics per se unless one takes the position that all of natural science and all of human knowledge can ultimately be derived from physics. I don't take that position. It has to be derived from generalizations of observations of the natural world. Many of...

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(1) It doesn't have to be physics per se unless one takes the position that all of natural science and all of human knowledge can ultimately be derived from physics. I don't take that position. It has to be derived from generalizations of observations of the natural world. Many of...

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Dear Matt:

I would find this meal more substantial if the distinguishing characteristics of mathematics as a formal system were mapped to what we know about physical reality. Playfully, thinking about realities in which mathematics would not apply might be revelatory. Let's consider a reality in which you had as many fingers as toes when you went to bed and woke up with half as many of the former. Or where a sheep arrived overnight (like flies on meat in the theory of spontaneous generation) - except that as you looked at it, it seemed to be more like a llama.

This guides me to a suggestion that the existence of an irreducible scale (quantization), fermion number conservation, and locality (charge cancellation and finite velocity in the propagation of effects) might all be critical to mapping mathematics to physics.

But I do like you idea that mathematics is an abstraction of physical reality. I see an explanation, in fact, of the observation that physics becomes more and more like mathematics every day. Physicists are reasoning about things that they cannot see. Their grasp of reality has become wholly Platonic. Therefore abstractions upon abstractions is all that they are left with. Rather like theologians arguing about angels dancing on the head of a pin, they have grasped the prop of the legitimacy of classical dynamics, extended it into the world of the unseen through quantum field theory, and then just gone off on a lark with group theory, discarding entirely the proposition that the complexity we observe might be explained by positing additional structure (as it always was in the past).

Brian

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I would find this meal more substantial if the distinguishing characteristics of mathematics as a formal system were mapped to what we know about physical reality. Playfully, thinking about realities in which mathematics would not apply might be revelatory. Let's consider a reality in which you had as many fingers as toes when you went to bed and woke up with half as many of the former. Or where a sheep arrived overnight (like flies on meat in the theory of spontaneous generation) - except that as you looked at it, it seemed to be more like a llama.

This guides me to a suggestion that the existence of an irreducible scale (quantization), fermion number conservation, and locality (charge cancellation and finite velocity in the propagation of effects) might all be critical to mapping mathematics to physics.

But I do like you idea that mathematics is an abstraction of physical reality. I see an explanation, in fact, of the observation that physics becomes more and more like mathematics every day. Physicists are reasoning about things that they cannot see. Their grasp of reality has become wholly Platonic. Therefore abstractions upon abstractions is all that they are left with. Rather like theologians arguing about angels dancing on the head of a pin, they have grasped the prop of the legitimacy of classical dynamics, extended it into the world of the unseen through quantum field theory, and then just gone off on a lark with group theory, discarding entirely the proposition that the complexity we observe might be explained by positing additional structure (as it always was in the past).

Brian

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