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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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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 laws of physics can be formulated in this way. The necessity to do so is not obvious to me.

Second, you say "unless the platonist can give us an account of where the abstract realm actually is in physical reality, and how our brains interact with it…" My understanding is generally that the ideal realm they claim is outside of physical reality, but then, as you say, how can our brains interact with it?

I also agree with you that "the search for a theory of everything is not fruitless; I just do not expect it to ever terminate." You conclude that "theories are just convenient representations of the regularities… [iterated] of the physical world."

My essay begins with an approach to discovering these regularities and to deriving formal representations. This approach to a "theory of theory" is discussed in my first pages and in my endnotes. I believe it to be fully compatible with your ideas.

The meat of my essay however concerns Bell's mathematical formulation of a simple model of physics and his assumptions that are not clearly stated. In fact, after specifying quite clearly that the "hidden variables" can take almost any form, he introduces his basic assumption with one word, the conjunction 'and', followed by his equation (1).

As I agree with you on both the nature of math and on an approach to theorizing about theory building [as I understand you], and as I quote you in my endnotes to define the current state of quantum mechanics, I hope you will find the time to read my essay and to provide me with feedback on it.

My very best regards,

Edwin Eugene Klingman

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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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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 geometry must give way to a new geometrical structure. What this new structure is, will become one of the ultimate examples of the unity of mathematics and physics. For at present it seems current mathematics, in all its complexity, is not sufficient to describe the microscopic structure of spacetime.

There a different approaches to quantum gravity, such as loop quantum gravity, Connes' noncommutative geometry, and string/M-theory. And although there has been a backlash in public opinion towards string/M-theory (our most promising theory of unified physics), one must remember it has its origins in hadronic physics, not quantum gravity. Moreover, pure bosonic string theory is only consistent in 26-dimensions, a fact which is very mysterious from a mathematical viewpoint.

If string/M-theory, in its complete nonperturbative form, proves to be the unique theory of quantum gravity, not only for our universe, but for all other universe in a multiverse system, it is a testament to the usefulness of exceptional mathematics, instead of general mathematical structures. General mathematical structures, hold no preference for a given n-dimensional space, with fixed value of n. Special values of n occur in the study of sporadic groups, exceptional Lie groups, and other "odd ball" mathematical constructions.

What seems to be occuring, at present time, is the unifying of these exceptional mathematical structures, using insights from string/M-theory, loop quantum gravity and noncommutative geometry. For example, it has been argued by Horowitz and Susskind that it is difficult to relate bosonic string theory in 26-dimensions to M-theory in 11-dimensions because there should be a 27-dimensional bosonic M-theory that reduces to bosonic string theory, just as M-theory reduces to the 10-dimensional superstring theories by compactification. In other words, at strong coupling, the extra dimension "opens up" and there should be a stable ground state as the full extra dimension is restored.

Smolin (a father of loop quantum gravity), years ago, introduced a Chern-Simons type matrix model in 27-dimensions as a candidate for bosonic M-theory, that effectively unites loop quantum gravity with M-theory. The matrix model makes use of an algebra of observables, that was discovered as an exceptional case by Jordan and Von Neumann in 1934. The algebra makes of 8-dimensional variables called the octonions (found in 1843), which themselves are a maximal case of the so-called (finite dimensional) division algebras. The real, complex and quaternion algebras are the only three algebras of this type, existing in 1, 2 and 4-dimensions.

There is mounting evidence that bosonic M-theory may indeed be formulated as a matrix model in 27-dimensions. If this is the case, the dimension with value n=27, is a result of the maximality of construction of an algebra of observables over the octonions. Such an algebra, permits only 3x3 matrices, at most, which together close into a hermitian matrix algebra that leads to a sensible quantum mechanics. This hermitian matrix algebra, the exceptional Jordan algebra, is the self-adjoint part of a unique Jordan C*-algebra. This C*-algebra allows a noncommutative geometrical interpretation.

So here, a historical series of accidental mathematical and physics discoveries, could lead to a theory of unified physics, that unites all promising approaches to quantum gravity. Moreover, it seems likely sporadic groups, exceptional Lie groups, L-functions, Shimura varieties and noncommutative geometry can all be connected in the process. This is quite an embarrassment of riches considering the haphazard process by which it stemmed. This is not to say, ultimate reality is necessarily mathematical. For as Hawking once asked, “What is it that breathes fire into the equations and makes a universe for them to describe?” What is the structure of the ancient Pythagorean's apeiron? Is there a pre-mathematical description of reality? Is Nature, at its core, acausal? Hopefully, in the coming years, we can begin to address such questions in a sensible manner.

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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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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 philosophy." It is remarkable to see that still after quite a long time that a majority of physicists and philosophers who care about metaphysics are trying to defend and develop naturalism, no clear formulation of its actual meaning could even reach a status of notability without being also loaded with big troubles (such as those of Bohmian mechanics); while I would explain the lack of well-known coherent formulation of idealism by the lack of serious tries by competent physicists and philosophers, a gap I care to fill by my essay.

The main difference I see between yours and Lee's exposition, is that Lee entered in specific details, especially about where he draws the line of existence between things. His claims are totally incoherent, both logically and with respect to existing knowledge in math and physics as I explained in my many comments there, however he has at least this merit, of taking that risk of making specific claims and thus expose himself to refutation. It seems you found a pretty good way to minimize the amount of criticism that you will get, by minimizing the quantity of claims contained in your essay. Indeed, most of the ideas there are already given in the abstract. You managed to approach the standard size of 9 pages not by adding up many ideas but rather by diluting the few you had. Anyway, by lack of effective matter to discuss from your essay, and also because I don't like to repeat myself, I invite you to read my comments to Lee's article (I'll still have more to write there), for the general remarks I made about this common topic between yours and his essay, of naturalism vs. Platonism, and for the specific analysis of his tries to effectively specify the claims of naturalism (I'm curious how would your options differ from his).

Now for your few specific claims :

You wrote: "Naturalism is the position that everything arises from natural properties and causes, i.e. supernatural or spiritual explanations are excluded. In particular, natural science is our best guide to what exists, so natural science should guide our theorizing about the nature of mathematical objects."

It is true that sciences of "natural" things (physics, biology) were extremely successful and these aspects of the world are a privileged field for scientific inquiry. However, like so many naturalist authors, you fail to distinguish between the scientific method and materialistic metaphysical prejudices, by using the ambiguous expression of "natural science" that may actually mean either the scientific method or the body of discovered knowledge in physics and biology, but where the adjective "natural" it contains can be irrationally played on in philosophical discussions to make it look as if the scientific method required to adopt such metaphysical prejudices. It doesn't.

"A naturalistic theory has no place for a dualistic mind that is independent of the structure of our brains. Therefore, if we have intuitive access to an abstract realm, our physical brains must interact with it in some way. Our best scientific theories contain no such interaction."

Our best scientific theories do not contain any explanation of anything psychological : what it may mean for a mind to understand something in general. Thus, there is no wonder why they do not explain either how we (I mean, some people) can understand mathematics in particular.

"The only external reality that our brains interact with is physical reality, via our sensory experience. Therefore, unless the platonist can give us an account of where the abstract mathematical realm actually is in physical reality, and how our brains interact with it, platonism falls afoul of naturalism."

The content of this argument looks like an absolute argument against platonism, to which I replied in comment to Lee's article. Strangely, for no reason I could see, the last phrase "platonism falls afoul of naturalism" looks as if it was only an argument relative to naturalism (thus void outside it). But then in this case, it implicitly seems to assume that a remark that "Naturalism contradicts X" should be taken as an argument against X, thus taking for granted that naturalism should be true. However I still beg for a rational argument for naturalism, in the name of which it would make sense to consider a view as being made less plausible just by its conflict with naturalism.

"The various attempts to reduce all of mathematics to logic or arithmetic reflect a desire [to] view mathematical knowledge as hanging hierarchically from a common foundation. However, the fact that mathematics now has multiple competing foundations, in terms of logic, set theory or category theory, indicates that something is wrong with this view." ; and in your comments: "What I mean is rather that which formal systems we decide to call mathematics, out of all the myriad of arbitrary axiom systems we might choose to lay down, is a matter of physics."

Well no. There is, for example, a clear absolute sense of which rules can form a valid and complete system of proofs for first-order logic, and satisfy the completeness theorem. It is clearly independent of physics. Now for the choice of an axiom system to best serve as the foundation of mathematics, I do not see such a competition between possibilities as you assume. Instead, as I explained in my work on the foundations of maths which I developed showing its intrinsic necessities independently of physics, I rather see a coherent whole of complementary parts of the foundations, with a dynamic articulation between them. Where there are alternative possibilities in competition for the same purpose (such as axiomatizations of set theory) I see a situation where in some aspects there is a preferred kind of formalization, and in other aspects such as the continuum hypothesis there is a remaining real diversity of acceptable systems corresponding to a real diversity of acceptable realities, that can often be understood as due to the real fact of ambiguity of the powerset of infinite sets.

"Firstly, in network language, the concept of a “theory of everything” corresponds to a network with one enormous hub, from which all other human knowledge hangs via links that mean “can be derived from”. This represents a hierarchical view of knowledge, which seems unlikely to be true if the structure of human knowledge is generated by a social process. It is not impossible for a scale-free network to have a hierarchical structure like a branching tree, but it seems unlikely that the process of knowledge growth would lead uniquely to such a structure. It seems more likely that we will always have several competing large hubs..."

If things went as you describe here, we would not have got the amazing success of mathematics in physics we had, with an amazingly universal agreement on what is the right fundamental theory of gravitation on the one hand, of microscopic physics on the other hand, how amazingly well, each on their side, they indirectly explain so many things on almost all physical processes that could be tested. So I do not see your philosophy coherent with the state of science that could be observed.

"...and that some aspects of human experience, such as consciousness and why we experience a unique present moment of time, will be forever outside the scope of physics." What a nicely non-naturalistic claim ! ;-)

"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." Only under the assumption of naturalism. But, if, as I hold, naturalism is irrational, and the only reasonable way of understanding science is an idealistic one (more precisely a mind/mathematics dualism), then your conclusion fails.

Now for the general ideas : your account of mathematics and its success is much too vague, and fails to relate to the effective contents of how successful is mathematics for physics. You present an abstraction of explanation for an abstraction of a problem. But the difference between mathematical abstractions with their success in science (as abstractions of real problems), and the abstraction of your approach to the problem of the success of mathematics in physics, is that the success of mathematical abstractions depends on their care and success to keep effective logical articulations (as you say, "tether") with the real problems they are abstractions of, but your abstraction of approach to the issue of usefulness of mathematics, is a non-mathematical abstraction that fails at keeping a tether to the reality of this usefulness. If the success of mathematics in physics could be explained as simply as what you describe, there would have been no amazement at this success in the first place.

Now, of course, it would make little sense to only express this objection in the abstract: to justify it, I need to enter the specifics about how your approach fails to account for the specific success of mathematics. So here are a few specific details of how you fail.

First, you explain mathematics as "the study of regularities, within regularities, within ..., within regularities of the natural world". In this case, we have a hierarchy of different levels of abstractions, where some concepts are more abstract than others, as they are not directly natural objects but abstract generalities about wide ranges of natural objects, or generalities about generalities about natural objects. When abstract generalities are developed as abstract generalities, they are not themselves the natural objects. That structure of knowledge you describe can only be expected to be useful as a path of reasoning through which we might discover new natural objects by first going up to the abstraction and then down again from the abstraction to concepts of new objects that would be particular objects inside classes described by these abstractions: the new fundamental objects we discover should display this same dependence to mathematical abstract generalities as the objects of our experience do.

But this is not what we observe. Instead, what we observe is that what we discover as fundamental objects of physical reality are some very abstract mathematical objects themselves. And what I mean here as "abstract mathematical objects", does not look like any generality like a general description of regularities among a range of many particular objects in a naturalistic sense. You may make this confusion because some of the most famous mathematical concepts, such as category theory, precisely look like generalities describing a wide range of particular cases (it is a generality of generalities). But this is not the case for all highly abstract mathematical concepts, and this is not how it goes for the success of mathematical concepts used in physics. In short, "high abstraction" and "generality" are not synonyms.

And to explain how the concepts of "abstraction" and "generality" differ, I need to take a specific example. I would like you to consider the case of the Dirac equation. This is a particular case of equation of a particular object (electrons or other spin 1/2 massive particles), but nevertheless a very abstract one.

This equation describes the field of electronic presence as taking values in the space of bispinors of space-time. To say roughly, we can define this space as the sum of 2 spinor spaces (with conjugate types), where a spinor space is a 2-dimensional complex space E such that the space of hermitian forms on E is identified with the tangent space to this point of space-time. Namely, tangent vectors (x,y,z,t) to this point are identified with Hermitian forms on E with matrix

(t+x y−iz)

(y+iz t−x)

since the determinant of this matrix coincides with the relativistic invariant (t²−x²−y²−z²).

This is very abstract, but not any kind of "generality of things" like what category theory does by describing regularity classes of particular systems that may go down to objects that look "natural" in a naturalistic sense. Now what is amazing with the success of mathematics is that this spinor space E was found to be "what electrons (and other fermions) are actually made of". Yet its connection to space-time, as described above, is... quite abstract. Spinors are not "made of" space-time connections, since any spinor (element of E) would correspond to a light-like direction of space-time but any physical description by such a direction would fail to fix the phase of this spinor. Concretely, a big problem with a spinor is that its phase is reversed when you apply to it a rotation with angle 2pi.

So, unless you provide a naturalistic explanation of how an object can be reversed when applying a rotation with angle 2pi, and how such an amazing thing as the Dirac equation can be relevant to physics, I must consider that naturalism fails to account for the mathematical aspects of the physical reality as we observed.

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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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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 Tegmark’s. In my opinion, they can still coexist. If reality is ultimately a complex mathematical structure (let’s not worry about the multiverse aspects), what’s wrong with imagining homo-sapiens building the knowledge network for describing that external reality just as you indicated? Your meta-theory works independent of the origin or status of that reality.

Another point. I notice your prudence in envisaging the eventual formation of a final, mega-hub at the top. Indeed, there are phenomena and driving forces in nature, as found in our evolving biosphere, that seem to escape a precise mathematical formulation, and to resist the math-based game of regularity finding/aggregation/abstraction. Hence, the scenario implied by your meta-theory is one in which multiple separate disciplines - physics being one - will keep existing and developing, forever separated from one another by their degree of math-friendliness.

Is some stronger unification possible/desirable?

Perhaps it is, by looking not only at the top of knowledge but also at the bottom of reality. Let me explain. Your essay is extremely effective in covering the evolution of human knowledge (the observer side), and suggests a unification process going upwards, looking at the top of knowledge, while the observed object - the universe - remains passive and static. But if we viewed it as dynamic, and managed to find the seed at the bottom of reality, and that seed turned out to be a simple algorithm (as some crazy people dare suggesting), from which everything would emerge - fields, matter, but also biospheres - then we would have a more unified scenario: that seed would certainly deserve a very special place in the network of human knowledge. Perhaps the top, although it seems to me that it would not be a hub exactly as you conceive them...

Best regards

Tommaso

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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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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 *only* way that new mathematical "nodes" are developed? It seems to me that an equally-important process is simple exploration/extrapolation from existing nodes, tweaking the premises that lead to one bit of math, and seeing what new bits might result. Sure, this isn't going to necessarily lead to the efficient network structure that you describe, but it would supply a bunch of "raw material" which your efficiency-driven process could then organize.

One big distinction between the nodes generated by an "exploration process" (rather than your "efficiency process") is that there's generally no requirement that they hold together self-consistently. This is what I had in mind when I talked about different "caves" in my own essay, even though you're right that eventually your efficiency-driven process tends to merge the caves together into much larger networks... maybe even to the point where you can claim that there is one "pragmatically-true" type of math when it comes to things like the axiom of choice.

If you're with me so far, then I think your comment on my essay reflects that we're trying to answer two somewhat different interesting questions. You are addressing the mystery of why the most-efficient network groups of math tend to map to physics; I'm trying to address the mystery of why so many separate explorative-nodes tend to find a use in physics.

I think there are some aspects that make your task harder than mine, and some that make them easier. On the easier front (which I think you hinted at), mathematicians know about physics when they are looking for efficiency-nodes. When faced with mathematical choices about the details of differential geometry, say, I'd be a bit surprised if mathematicians refused to think about GR. But while GR could potentially inform part of a subsequent efficiency-process, it certainly didn't inform the original development of non-Euclidean geometry; that's basically the mystery that I was trying to address.

On the other hand (as you noted on my page), given a collection of unrelated but internally self-consistent axiomatic systems, it shouldn't be terribly surprising that *some* of them find a use in physics. (Indeed, that's a key part of my argument.) But if mathematicians truly think that some of these network-systems are more "correct" than others, then it is quite interesting that the deemed "correct" systems are more useful in physics than the equally-logically-valid but "incorrect" ones (as judged by the overall mathematics community). I guess the big question here is just how much influence higher physics has had in framing the judgment of mathematicians (as per my GR example), or if it really is all built up from counting apples, and very basic stuff like that. Do you see higher-level physics as impacting higher-level math, or is it all one-way-influence at that level?

Thanks again for a thought-provoking essay!

Ken

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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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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.

(+) I like naturalism, because I read at Smolin that formulae exist in time, not in timeless environment.

(+) At least, you mentioned consciousness, because ignorance of it is not good.

(-) About consciousness, I have a similar standpoint as Poirer and still many people in this contest. Consciousness causes movements, thus it is part physics. If one philosophy of physics does not include consciousness, is not good.

(-) You said to Poirer that panpsychism disagrees with physicalism. This is not true. My model includes physicalism and reductionism. It does not need supernatural and spiritual in the first intoduction. Even, panpsychism is defended also by Koch and Tononi. But emergentism has not yet answered anything.

(?) In my essay I gave also speculation about Pythagora theorem, that it is a consequence of energy law, that Euclidean law is a consequence of physics. What is your opinion?

(?) I gave also an example where physics adjusts to math, this is dimensionless nature of Planck spacetime. What is your opinion?

(?) One important question is simplicity of fundamental physics. Smolin does not believe in simplicity, but what about you?

(?) What do you think about positivism, like this of Roger Schlafly? I support it, but 100% positivism is not correct, by my opinion.

My opinion is that merging of fundamental physics causes that the number of axioms is reduced. Thus math is foundation of physics, but not on the same way as Platonic math.

Maluga wrote, that the essence of math is to describe physics more simply, because our brains have not capacity to think on all parameters. My addition is that the goal of math is to show that physics is simple. Here I also obtained answer, why Economy is not simple. Answer: Economy is a part of physics, thus simplicity is home in physics.

My essay

Best regards

Janko Kokosar

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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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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 approach), it is not obvious to me that this captures all (or even the majority) of mathematical theories. In mathematicians, we can take anything as a source of inspiration (including the world, our the structure of our knowledge thereof), but we are not restricted to studying it in that form: for instance, we may deliberately negate one of the properties in the structure that was the original inspiration, simply because we have a hunch that doing so may lead to interesting mathematics. Or do you see this differently?

On the other hand, the picture you describe does ring true with my subjective experience that learning supposedly 'difficult' theories never really turns out to be that difficult when you actually try: somehow, you keep dealing with combining a limited number of 'things' according to specific rules. To arrive at the relevant 'things', however, may require a long chain of explanation and abstraction. Your idea of replacing connections with a new hub seems to be related to the steps in that chain. So, I do think that this ability of 'going meta' is crucial to mathematics - not just to philosophy. ;-)

But as I indicated before, it is not clear to me that this is sufficient to consider mathematics as an empirical science. Or maybe it is possible, if you consider a large part of our knowledge to be suppositional knowledge?

I also like the conjecture of section 3 that human knowledge may be a scale-free network, which goes against the hierarchical view but still explains how it may appear to us like that. In this picture, I guess the old ideal of a universal human would translate to focussing on multiple hubs: still a great way to 'keep in touch' with the actual network structure of knowledge.

In addition, I think your essay is accessible for a general audience. And you actually thought of adding pictures. :-)

Great job. My vote is 9/10.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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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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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 essay, you write:

“ [F]or me, abstract mathematical objects can be called real insofar as they are useful for our scientific reasoning”

and

“[M]athematical theories are just abstract formal systems, but not all formal systems are mathematics. Instead, mathematical theories are those formal systems that maintain a tether to empirical reality through a process of abstraction and generalization from more empirically grounded theories, aimed at achieving a pragmatically useful representation of regularities that exist in nature.”

If I read you correctly, you essentially define mathematics so that,

(1) if it not ultimately derived from the generalization of physics, it is not math ;

(2) if it not even remotely useful for reasoning about physics, it is not math.

If mathematics is defined in this way, I fully agree with you that mathematics is (a subset of) physics.

On the other hand, if we define Mathematics in a wider sense that encompasses all abstract formal systems, including those that are too big, too complex or too irregular to be grasped and studied by human-level minds, I think it is quite possible that “All-of-Physics” (in the sense of all possible physical realities, observable or not), is “generated” by “All-of-Math”. With this larger definition, it is Physics that is (a subset of) Mathematics!

That said, I really like the way you explained how abstract mathematical theories can arise:

“The main idea is that when a sufficiently large number of strong analogies are discovered between existing nodes in the knowledge network, it makes sense to develop a formal theory of their common structure, and replace the direct connections with a new hub, which encodes the same knowledge more efficiently. […] In this way, mathematics can become increasingly abstract, and develop its own independent structure, whilst maintaining a tether to the empirical world.”

This essay contest cast a very wide net by asking about the relationship between math and physics, and it is fascinating to see the diversity of answers that were proposed. Thank you for contributing a very interesting essay, and all the best!

Marc

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