A draft of an essay I’ve written, with plans to submit it to the FQXI essay contest, is available here. Constructive comments welcome…
People who have a take on the subject that has nothing to do with what I’m writing about are encouraged to submit their own essays to FQXI, but not to post them here.
Update: Thanks to all commenters for often helpful comments. I’ve revised the essay a bit, mostly by adding some material at the end, material that to some extent addresses important issues raised by some commenters.
Update: The essay has been submitted and is posted here.
Nice essay. But I don’t see what is so geometrical about the CKM mixing, and suspect that it is Perfectly Reasonable Effectiveness of Mathematics.
I don’t know, but I always felt that saying “it seems unreasonable that mathematics, which is a creation of the human mind, should be effective in understanding the world” is kind of the same as a bird saying “it seems unreasonable that my wings, which were created by my body, should be effective in helping me navigate through space”
I’ve never understood why anyone thinks mathematics being effective is surprising. I mean, basic math is the real world. Animals count. When my dog is chasing a rabbit, it’s doing all sorts of fancy computations, but it’s just nature. Same as when I’m catching a baseball. No one says “wow, how surprising, English (or fill in your favorite other language) is so surprisingly effective at describing the real world.” Math is a language. It’s got more uncompromising rules, but still.
Peter,
I like the way you showed how the development of mathematics and physics since Wigner has continued develop new and more powerful links between them – but I doubt that Wigner would have been surprised. I’m not sure though, that the aspired to unification is any closer than it was in Newton’s day.
Mathematics continues to proliferate worlds in in endless profusion, but physics (I hope) still seems to be pretty singular, despite the MW movement.
PS – Since you referenced it, I looked at your QT, Groups, and Representations book. The part I read looks very promising, and a nicely chosen level. Publication date?
That’s an excellent piece. It could, though, use a minor copyedit–mostly some minor wording adjustments and a sprinkling of commas. You might also add to the abstract the point you are making that not only has mathematics been “surprisingly” useful in describing nature, but that advances in our description of nature have been “surprisingly” leading to new and deep mathematics. If I were to add something else, it would be that the success of mathematics, and specifically the ubiquity of Lie groups, representation theory, and differential geometry, in our description of nature strongly suggests that our world is, fundamentally, some rich and beautiful mathematical structure, come to life.
Thanks Garrett!
CIP,
Thanks! I’m working on the last half of the book this semester while teaching a course based on this, hope is to have something fairly complete by the end of the semester in May. For a completely finished product, sometime next year…
@Anderson and Jeff M
The mathematics at hand is far from basic mathematics. For example, Einstein’s theory of General Relativity makes use of Riemann’s mathematical creation, differential geometry. I’m pretty certain in 1853, when Gauss asked Riemann to prepare a Habilitationsschrift thesis on non-Euclidean geometry, nobody could foresee higher dimensional differential geometry being applied in 1907 by Einstein to describe gravity as curvature in a four-dimensional spacetime. Moreover, the theory of Riemann surfaces is central to the study of worldsheet dynamics in string theory, a theory of quantum gravity. And let’s not get started on Riemann’s celebrated zeta function, as its generalization, L-functions play a pivotal role in the theory of Motives, which arises in the Langland’s program.
I would argue that the “intriguing pattern of U(1) hypercharges” (§4.2) is to a large extent understood: Instead of formulating the Standard Model with the group U(1) x SU(2) x SU(3), one should — essentially equivalently — use S(U(2) x U(3)), which has U(1) x SU(2) x SU(3) as a sixfold covering. This reveals that the Standard Model fermion representation has a rather simple form: It is the direct sum of two parts, each of which is the restriction of a representation of the identity component of Spin(3,1) x U(2) x U(3). The first part is the tensor product of [A] the right-handed Dirac representation of the identity component of Spin(3,1); [B] the representation Lambda^even of U(2) on the exterior product Lambda^even(W), where W = C^2 is a 2-dimensional unitary complex vector space corresponding to weak interactions; [C] the representation Lambda^odd of U(3) on the exterior product Lambda^odd(S), where S = C^3 is a 3-dimensional unitary complex vector space corresponding to strong interactions; and [D] the trivial representation on a 3-dimensional unitary complex vector space C^3 corresponding to particle generations. The second part is the same up to replacing the right-handed with the left-handed Dirac representation and replacing the representation Lambda^even of U(2) with Lambda^odd.
This description encodes in particular the U(1) hypercharges. It is not mysterious that Lambda^odd(S) occurs here instead of Lambda^even(S): switching between these two exchanges particles with antiparticles. But the fact that the representation Lambda^* of U(2) — which has no obvious geometric origin — is coupled with the — clearly geometric — Dirac representation hints at a geometric origin at least of the weak interaction. In this sense I agree that we are indeed “missing some piece of geometrical structure”, as you (PW) write. But while we do not yet have a good explanation of the Standard Model fermion representation, we should not make things even more mysterious than they are by expressing them in terms of U(1) x SU(2) x SU(3) instead of S(U(2) x U(3)).
By the way: “Wigners” -> “Wigner’s”, “solutons” -> “solutions”, “or details” -> “for details”.
great essay! Minor typo: in second bullet of Sec 8, it should be “for details” (not “or details”).
Also “underly” and “solutons”.
Ivor Grattan-Guinness (who passed away last december) wrote in 2008 a paper that I would like to see quoted just as often as Wigner’s essay:
Grattan-Guinness, I., “Solving Wigner’s mystery: the reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences.” The mathematical intelligencer 30.3 (2008): 7-17.
how similar is wigner’s thought concerning the “unreasonable effectiveness of mathematics” to putnam’s no-miracles argument for scientifc realism? putnam thought that if theories weren’t “real,” confirmation of their novel predictions was miraclulous.
While this article is far more coherent and persuasive than Lee Smolin’s, it has one glaring weakness in common with Smolin’s – it doesn’t anywhere say what mathematics is. I don’t think it’s tenable to argue for a “grand unified theory of mathematics and physics” unless you answer this question.
For background, I’m a mathematician with a recreational interest in the philosophy of the subject. I read Wigner’s article some time ago and wasn’t particularly convinced by it, not least because the same wonder he feels about applications of mathematics to physics is often felt by mathematicians about applications of one field of mathematics to another – but this sense of wonder doesn’t stop Platonism from being an obsolete philosophical framework.
As a specific comment, I think the quote you give from Frenkel about the Langlands programme is hyperbole. (However, I see the book can be downloaded online, so perhaps I’ll be convinced by it…)
Einstein’s starting point for the general theory was the equivalence principle:
http://en.wikipedia.org/wiki/Equivalence_principle
heavy mass is inertial mass.
Einstein thus tried to formulate a theory in which this distinction is not made anymore. Only after that did the mathematics come into view.
Another thing is that physical models are only effective, and thus not everyting derived mathematically from a useful model is necessarily true. See vacuum fluctuations. The idea that the vector potential A can be thought of as being a connection field only came after the Maxwell equations had been formulated, and these derived from the older descriptions of electricity and magnetism. Maxwell’s idea to include the displacement current D was a physical insight.
The idea that mathematics can lift the veil of Nature in a post-empiricist manner is a form of Idealism.
One last typo (because the devil is in the details): ” (…) it’s difficult to impossible to predict (…)”. Difficult or impossible? 😉
Also “it’s” -> “it is”
Thanks for sharing. Good luck.
I liked your essay, it is a bold suggestion that String Theory could be replaced one day by mathematical structures coming from ideas in the Langlands program.
I wonder how you decide what type of mathematics is going to be useful for describing nature – for example, is a rigorous measure theoretic Hilbert Space formulation of QM required? If not, why is measure theory not as applicable as, for example, number theory?
Thanks to all who pointed out typos, those at least are fixed.
JG (and others),
I’m not making an argument that one can just look to mathematics to decide how to describe nature. One needs to first look at our empirically most successful theories (e.g. the SM and general relativity), and then try and understand, amongst the many ways of mathematically formalizing them, which of these ways relates them to the deepest known mathematical structures. If there really is a unity there, the known connections between successful physical ideas and successful mathematical ideas will suggest other such connections worth looking into. This may lead to progress, on either the mathematical or physics end.
Paul Levy,
Yes, the unreasonable and unexpected effectiveness of one field of mathematics in another is common and I’d argue sometimes evidence for unity in mathematics that we don’t yet understand. The argument for unity with fundamental physics is an argument that there is something going on here at the deepest levels that we don’t understand, and from that point of view we likely don’t yet know the correct best definition of “mathematics”. There are lots of definitions one can now come up with for “mathematics”, from looking at what mathematicians do and coming up with some generality that covers it all, but I don’t think that leads to anything particularly interesting or truly gets to the heart of what the deepest mathematical ideas are telling us about.
A bit funny, but totally unsurprising I guess, how you write about all these nice subjects and ideas and totally avoid mentioning the words string theory even once, despite it being so intimately connected both to the geometric Langlands and the (2,0) theory. It’s ok I guess, but somehow feels a bit bordering on intellectual dishonesty: string theory is really quite an important part of how people came to these ideas, so my advice would be to add a bit about how the Langlands duality is really a S-duality of string theory, how (2,0) is the world-volume theory of M5 branes and so on. Of course that would go totally against the narrative this blog is trying to sell, so as I said, it’s really not surprising.
hopffiber,
I don’t see how adding discussion of the complicated story of often heavily over-hyped claims about string theory and the Langlands program would add anything worthwhile to the essay. If you want to discuss the ideas that are in the essay, that’s fine, but if you or others just want to argue about “string theory”, you’ll have to find another posting where it’s relevant.
I think the essay has lots of problems and it has no chance to win the contest:
– It’s obviously written in a great hurry
– Very low-profile, but if that’s your style that’s OK
– There is no effort in being pedagogical, soon you go into rings and Lie groups, bundles and whatever. A basic understanding of what kind of physics is going on is completely missing.
My suggestion is, if you are serious about this, you should try to tell a story that people smart enough and with a background in physics, but not your own field, should be able to follow. Nevertheless, I do agree on the general ideas. For some quite different thoughts on the connection between mathematics and physics, you might want to take a look at the words of V.I. Arnold,
http://pauli.uni-muenster.de/~munsteg/arnold.html
Typos: “underly”!!!, “Milss”
tomate,
Thanks. Actually I did put quite a bit of time into thinking about the essay, wrote a couple initial versions that I discarded as being unworkable. It is true that I put very little time into proof-reading, thanks to you and others for helping me there.
The main problem with the initial versions is that I started out trying to explain in detail more of the mathematics and physics, but then realized that I was starting to write something that would be at least 90 pages, not 9. To really explain Weil’s use of the Heisenberg algebra and the adelic point of view on theta functions, what an automorphic representation is, topological quantum field theory, etc., etc. is a major project. The rules of the contest say you can add a page or two of technical material, but that wouldn’t help significantly. So, I finally decided the best I could do was just lay out in the space given an outline of the ideas I’m referring to. Some experts will recognize what I’m talking about, others I hope will find something intriguing that encourages them to go out and learn more (I should try and find some more, better expository references). Trying to write about these topics without referring to rings or Lie groups, or that level of basic tools in mathematical physics I don’t think is really possible without producing something content-free.
In any case, trying to win a contest is not why I decided to do this. This was just a good opportunity to put together these thoughts and get something written that perhaps others will find of interest.
In the bibliography, the publication date for Clifford Algebras and Lie Theory by Meinrenken is 2013 not 2003.
Refs. 1,2: bold type goes too far.
Refs. 2,4,8: “Birkhauser” -> “Birkhäuser”.
Ref. 5: “A., and” -> “A. and”, comma after “Program” missing.
Ref. 14: “Grassmanians” -> “Grassmannians”, “curves.” -> “curves,”, “113(1982)” -> “113 (1988)”.
Ref. 15: “3(1988)” -> “117 (1988)”.
Ref. 16: Comma after “E.” missing.
Maybe add a ref for Atiyah and Singer? and some discrimination of your conclusion from the MUH? You cover a lot of ground…
One comment:
I think the relationship between mathematics and physics is more
of an N to 1 correspondence than “unity”. It is true that the laws of
physics are elegantly encoded in formal structures provided by
mathematics (a school of mathematics known as constructivism posits this is
in a sense a tautology because
we construct such formal structures in terms of what we know).
However, at least as far as we know, most formal structures thought up
by mathematicians, or even theoretical physicists, have nothing whatsoever
to do with the real world, and assuming they do because they are formal
could mean a false lead.
For instance, in the speculation section of your paper you
mention six-dimensional superconforman N=(2,0), which is indeed being
studied for its formal properties. But I never saw any hint that such
a theory has anything to do with the real world.
Likewise, the objects being touted as being relevant to the
number theory connections you mention in section 6 are nothing like
realistic quantum field theories, for pretty fundamental reasons.
Perhaps this way of thinking is too superficial and reductive, but on the
other hand perhaps there are good reasons why such theories are unrealistic,
and focusing on their “elegant properties” connected to mathematics will make
us miss them: In the 19th century the leading minds of mathematics studied
potential inviscid flow without asking themselves why this tended to be
such a poor model for “real” fluids, especially low viscosity ones where
such methods should apply best, because potential flow was so elegant and so
connected to vector calculus. It turns out that real fluids are much more
elegant and complicated and fascinating, for reasons that you would never
understand focusing on potential flow.
Mayer A. Landau, Qwertz, Art,
Thanks, typos fixed, some references added. Not sure whether it’s worth trying to refer to other points of view and draw distinctions, that’s a long story. For the MUH, I’d say the difference is that it also posits an identity of math=physics, but that all mathematical structures are equally good. My interest is in the question of why certain very specific mathematical structures appear in this context.
lun,
Yes, there are lots of mathematical structures and one real world. But the evidence is that there are certain more fundamental structures, and the puzzle is that these show significant relations to fundamental physics. The example of the N=(0,2) 6d theory was given mainly to illustrate the fact that it is becoming widely accepted that we need new ways to define qfts (beyond picking fields and a Lagrangian). Finding new such techniques might give added insight into the fundamental physical theory.
I mostly agree with tomate, and your excuses will really not fly. A good essay makes advanced subjects feel accessible, this one doesn’t. And further the thesis seems to be: The mysteries in physics are linked to some even more mysterious mysteries in mathemathics. The smart mathematicians, without any physical insight; knowing any facts about nature, or doing experiments, thought or otherwise, will be able to solve it all for us. You attack string theory for much less.
Radioactive,
I wish I could do better at explaining some of these concepts in a few pages, but am not so sure it’s possible. There is no royal road….
Most of what I’ve actually written here is a long list of examples of things we DO understand, linking deep ideas about fundamental mathematics and deep ideas about physics. Yes, I expect more of this to come, illuminating mysteries both of mathematics and physics. You’re very defensive and skeptical at the idea that help from mathematics will solve problems in physics, but I can assure you that the reaction of many of my mathematician colleagues about my views is to be equally defensive and skeptical about the idea that anything coming from this kind of physics will help solve problems in mathematics. Personally I believe information flow in either direction is about equally likely.
As for the idea that all physicists need is “physical insight”, I don’t think this has historically worked very well in the absence of any experimental results to provide data for physical insight to operate on. If nothing new turns up at the LHC, the problem of no data is going to be a very significant one going forward, and I don’t think physicists should refuse to consider that what mathematicians know might be of help. And, hey, even if that doesn’t work out, they might find themselves solving the Riemann hypothesis problem and getting a million dollars…
Peter,
Regarding the remarks of tomate and radioactive, anyone who seriously follows fundamental physics generally, rather than just a particular approach, will understand the main import of most of your essay, and can easily look up the details. Nor is it inappropriate for the context; an essay about exotic smoothness structures by Torsten Asselmeyer-Maluga in the 2012 contest was probably much steeper from the perspective of the typical non-expert, but was nevertheless one of the winners, and now appears as one of the chapters of the resulting Springer book.
A quibble I have is the generality implied by “mathematics.” No one will argue against the centrality of number theory, geometry, and representation theory, but mathematics is vastly broader than that, and I don’t think this approach has any unique claim to represent the essence of mathematics, or to pair it with physics in a way that is necessarily superior to other approaches. However, some degree of hyperbole in advertising shouldn’t offend anyone.
Typos: group of \mathbf{F}_p, move commas and periods inside quotes (illogical, I know!)
Peter,
IMHO section 8 should have an additional point of the following kind:
\item (or \bullet) If an unusual space-time structure, such as quantum foam, occurs around the Planck length or some other very small length scale, completely new mathematical structures could appear in QFT.
I think that you’re overcomplicating things.
The reason there are so many links between current math/physics research is the probable existence of a “universal geometry” which is the true geometry of space-time and the source of all the geometries encountered by mathematicians. I expect Grothendieck’s favorite toys (topos, motives) to play a significant role in its eventual discovery.
You say: ‘I wish I could do better at explaining some of these concepts in a few pages, but am not so sure it’s possible. There is no royal road….’
So is it not possible for a not specialist to undertendimg anything about the question why it seems unreasonable that mathematics, which is a creation of the human mind, should be effective in understanding the world?
Faustino:
As a non-specialist in these areas I would say that analogies can only take you so far, sometimes off in the wrong direction – e.g. the Higgs Field/British Prime Minister moving through a room one.
Hi Peter,
From the contest guidelines:
“Foremost, the intellectual content of the essay must push forward understanding of the topic in a fresh way or with new perspective. While the essay may or may not constitute original research, if the core ideas are largely contained in published works, those works should be the author’s. At the same time, the entry should differ substantially from any previously published piece by the author. ”
What is new in your essay?
Isn’t it meant to be a review?
I’ve never been surprised at the effectiveness of mathematics in describing and predicting the ‘real world’, either. I take it that they both are a description, or reflection, or condensation, of the same thing. I imagine there is some kind of of ‘Ur’-reality that is not accessible to us. That’s not very useful I suppose from a technical aspect, but as a layman that’s how I think of it.
Ben,
I agree that what I’m writing about is just one particular aspect of mathematics, just as I’m also writing about just one particular aspect of physics (which I awkwardly call “fundamental”, with no corresponding word for mathematics). In the case of both physics and mathematics, the great majority of the subject and the questions people are addressing aren’t about “fundamentals”, and that’s a good thing. It’s only because people think about specific “non-fundamental” problems and discover new phenomena in math and physics that we find out about unexpected ideas about fundamentals. Just sitting around thinking about fundamentals themselves is typically an activity that goes nowhere.
Faustino,
I think non-specialists can certainly can understand the general picture I’m trying to advocate. Understanding the specifics of the aspects of quantum field theory and number theory I’m writing about does just require a major investment in time, you’re not going to understand these things by reading an 8 page popular essay.
Maurice Carid,
One comment would be that no ideas are ever completely new, with just about every sensible thing anyone comes up with having appeared in some form earlier.
I think some of the ideas about the relationship of QFT and number theory are new (they’re new to me, I didn’t understand them a few years ago, and don’t know anywhere else they’re written down). There are very few details in what I wrote, I hope to write something with specifics later this year.
I suppose the “grand unified theory of math and physics” claim isn’t new, since I get claims of that sort in my e-mail or mailbox every few days. But maybe a non-crackpot such claim would be new…
There may be no royal road to geometry, but someone who has been down it can usually give a brief travel guide. You have undoubtedly read John Baez’s and Terrence Tao’s writings on very advanced topics that are neither trivial nor alienating.
A flow of ideas from mathemathics to physics and vice versa is welcome and, dare I say, necessary. But historically significant breakthroughs have come from experimental results, or in the case of GR thought experimental, from thinkers with incredible insight into the physics and the mathemathics involved. The historical precedent makes me very skeptical that progress in physics will come from what is basically pure math, but of course doesn’t preclude it. However there is no argument that there is a ‘lack of experimental results’ when there are exciting developments from Plank and others coming quite regularly. The pessimism generated by the LHC’s failure to drown us in new particles is also quite misguided. Null results have had an important place in physics. It is the lack of new ideas in light of it that is most disappointing.
As for the unreasonable effectivness of mathemathics: in math, where ideas are generalised as much as possible and no more, it is not surprising to me that connections can be found between different broad categories. Nor that complicated objects (like gauge theories) may have different aspects illuminated more or less by looking at them in different lights, using different, possibly very complicated, mathematical tools.
Radioactive,
Tao and Baez are great expositors, but I think you’ll find that their expositions are great because they write at the necessary length. I had a 9 page limit, not 90, so was not doing exposition. If you want exposition, try the book I’m writing, which seriously tries to do that (although surely not as well as Tao or Baez). It’s at 450 pages and more to come…
Sorry, but claims that all is fine with fundamental theoretical physics research because of “exciting developments from Plank and others coming quite regularly.” is just hype. There’s a reason for little to no progress beyond the standard model over the last 40 years, I don’t think physicists are doing themselves any favors by being in denial about it.
Peter,
your essay looks like a winner to me, definitely in the spirit of past winners such as: It From Bit or Bit From It 2013.
Personally, I would have liked to have seen more on the extraordinary interplay in the 80s, starting with Instantons from QFT leading to Donaldson theory, Donaldson invariants, simplified using Seiberg–Witten gauge theory, but I loved what you put in all the same.
Peter:
> But maybe a non-crackpot such (Grand Unified) claim would be new…
Yeah, but this hasn’t this claim already been made by Frenkel in the
quote of your paper? He calls it “grand unified theory
of mathematics” but he includes quantum field theory so
I guess he means the same?
> I think some of the ideas about the relationship of QFT and number theory are new
You mean the three sentences after Frenkel’s quote dealing with
adele rings and the sentence
“In a hazy analogy…” further down the page?
I hold a PhD in physics but never heard about “adele rings”. I just asked
a fellow mathematician and he said “yeah that’s something from number
theory, but I’d had to look it up”.
I fear this doesn’t cut the following requirement from the contest guidelines:
“Accessible to a diverse, well-educated but non-specialist audience…”
What is new in your essay seems to be accessible only to specialists
in number theory.
Hi Peter,
I enjoyed the essay and look forward to its final form. One thing I would like to see better addressed is just what a “grand unifying theory of mathematics and physics” would even look like. Not in its specifics, but what such a beast even IS. For example, Tegmark might say that his “Multiverse IV” is just such a theory. I imagine you would not find it so, but I don’t think your article is clear enough about what the words even mean to say why not (or, perhaps, why). It focuses mostly on specific mathematical and physical objects or theories.
Another question that comes to mind comes from the fact that mathematics is far larger than those things that turn out to be important in physics. SO(3) is an important Lie group in physics, for example; if SO(73) is, I’m not aware of it, but nobody would be too shocked if SO(73) turned out to have some at least moderately interesting mathematical properties (I’m thinking along the lines of 1729 / Ramanujan), or even to be really important in some far-future mathematical theory. Physics thus seems irreducibly more concrete, and any “grand unified theory” of physics and math seems (to me, right now) like it will have to fall quite a bit short of explaining a lot of interesting features of mathematics just by virtue of its being at most coequal with physics.
I also fear you may come close to dancing with a category error; but it’s a speculative essay about a very interesting and fun thing to think about, one that might be fruitful, and it doesn’t claim to be rigorous at all; so I think this is pretty forgivable.
Thanks for posting.
One of the questions that I’ve always been fascinated with is : Is there ‘more’ in mathematics than in physics, in the following sense — Is there a mapping between everything provable in mathematics and some facet of the ‘real’ world of physical laws. Whether there is or is not such a mapping seems to me to be a key feature of a unified theory.
I don’t know whether it would be more mysterious if there weren’t such mapping or if there were, but either way it seems fundamental to me.
Interesting Essay.
FWIW, I am not a physicist or a mathematician and found many of the pronouncements descriptive and believable, but over my head. I don’t really know what the target audience is for this, but how many outside the area really even know what a Lie Group is?
Two points of interest (to me anyway) arose from this essay. The first is the assertion that experimental evidence has been elusive for the past few decades, so we really need either mathematical path finding or perhaps, some GR-type “deep thinking” to get off the plateau we are stuck on. The essay seems to argue for the former.
Which brings me to my second point. I risk getting into the String Theory issue here, but isn’t that precisely an example of mathematical path finding NOT working? Hasn’t String Theory been driven/seduced by the beauty and elegance of the math? Even if one is more supportive of the String Theory issue, one can see the risk of exploring an area of mathematics in the hopes of furthering our understanding of physics, but which has nothing to do with how our world works or functions at all. String Theory has been wandering in a box canyon for 30+ years, due in some part to periodic revelations of mathematical elegance.
So, to me, the essay has pushed me to believe the next step forward will be from some kind of “deep thought” insight, like which Einstein achieved to give us GR. Maybe mathematics will get lucky, but there are obviously risks involved.
Finally, it would seem to be that mathematics is completely abstract but limited (by our own intellects) whereas the physical world is somewhat less abstract and likely somewhat less limited. So neither field encompasses the other. We seem to be near some kind of boundary that we can’t quite surmount. My gut tells me the deep thinker will best the mathematical explorer.
I also couldn’t figure out the HTML stuff…..
Jim Beyer,
The “string theory is elegant math” claim is endlessly repeated by everyone, and it seems to be hopeless to get most people to understand the problem with it. “string theory” encompasses a wide range of different things, some of which are elegant math and some of which very much aren’t. The efforts to use string theory to unify physics are now based on postulating an infinity of ugly possibilities, the antithesis of elegant math.
There’s lots of elegant math that has no relationship to fundamental physics, I’m not arguing at all that one just needs to look for elegant math. One is looking for elegant math that has explanatory power about physics. The essay points out past examples of this in the past and argues there will be more in the future. String theory just isn’t especially relevant.
There is a typo in the first sentence of section 8.
Overall a pleasant summary of very beautiful topic, the study of symmetries.
In the introduction it would be helpful to clarify the problem that is meant in the statement “grand unified theory of physics and mathematics”.
Do you mean strictly looking for a theory that has both QM and GR as the limiting cases?
Or is this a more ambitious program of looking for a single theory from which the results of any experiment can be derived at least in principle?
The former being a reasonable expectation of mathematics and physics, while the latter might be fundamentally forbidden within the context of mathematical logic (i.e. could the latter goal be shown to be equivalent to the halting problem? If you had a theory that predicted everything then could it predict when any algorithm halts?).
So you think actual experiments like Planck/Bicep/LHC2 are just hype but some fuzzy hope that number theory is going to help with fundamental physics is a solid path? I don’t deny that there are some problems with fundamental theoretical physics research, part of the reason I left the field and the reason I read this blog, but not knowing enough math and paying too much attention to experimental results are not problems I ever observed. Quite the opposite in fact.
The essay is supposed to be about the surprising unity of math and physics but reads more like an essay on the surprising unity of math and math.
Dear Peter,
You might be interested in checking out this contribution to the essay contest: http://arxiv.org/pdf/1502.04794.pdf