Monday, September 10, 2012

Limits of math

My FQXi essay is now being publicly discussed and rated, as contest submissions are closed. I am not sure I addressed these objections fully enough. The argument is:
Using mathematics to model physical reality cannot be a limitation because mathematics allows arbitrary structures, operations, and theories to be defined.

No matter what new phenomena we discover, it will always be possible to describe our observations mathematically. It doesn’t matter how crazy nature is because mathematics is just a language for expressing our intuitions, and we can always add words to the language.
This is an incorrect view of math.

Math has been axiomatized as Zermelo–Fraenkel set theory (ZFC). The content of math consists of the logical theorems of ZFC. It is not just some language or some arbitrarily expandible set of intuitions.

I wonder whether many physicists understand this point. I doubt it.

I thought that everyone understood that math has limits, and the point of my essay was to argue that there might be no ultimate theory of physics within those limits. But if someone does not even accept that math has limits, then the rest of the argument is hopeless.

If you agree that math has limits and that they may not include possible physical theories, then it ought to be obvious that nature may not have a faithful mathematical representation. Saying that there is such a representation is an assumption that may be unwarranted.

So I guess I should have explained the ZFC issue better to the physicists who will be judging my essay.

An example of a limit of math is the unsolvability of quintic polynomials by radicals. That is what keeps us from having something like the quadratic formula for more complicated equations. This fact does not necessarily stop us from solving the equations, but certain kinds of formulas just won't work.

My essay "public rating" is not very high, but my "community rating" must be much higher. If you look at the list of essays sorted by community rating, then you will see that mine is near the top.

Update: Bob Jones argues below that ZFC is not enough because we need mathematical constructs like the Grothendieck universe that are outside ZFC. That is an interesting example, as noted here:
Colin Mclarty says that the big modern cohomological number theory theorems, including Fermat’s Last Theorem, were all proved using Grothendieck’s tools, making use of an axiom stronger than Zermelo–Fraenkel set theory (ZFC). He says that there is a belief that these theorems can be proved in ZFC, but no one has done it.
I would really be surprised if cohomological number theory needs something more than ZFC. And it would be even more amazing if theoretical physics found some mathematization of the universe that could be formalized in an extension of ZFC, but not ZFC.


  1. "The content of math consists of the logical theorems of ZFC."

    In a similar way, a mathematician living in ancient Greece could have defined math to consist of those statements provable from Euclid's axioms. Back then, such a definition might have made sense, but today we know that there's much more to math than Euclidean geometry. For example, we now have non-Euclidean geometry, which formalizes geometric intuitions that the Greeks never considered. Non-Euclidean geometry is also useful for describing the physical world.

    "It is not just some language or some arbitrarily expandible set of intuitions."

    You can define mathematics however you like, but this statement does not reflect how mathematics is developed and applied to physical problems. Mathematicians and physicists are rethinking the foundations of their subjects all the time. In the twentieth century, mathematicians began to develop new approaches to mathematics that did not assume the law of excluded middle, one of the basic assumptions of classical mathematics. Some physicists have experimented with new logical foundations for quantum mechanics based on topos theory. While these are not ideas that I personally find interesting or compelling, they show that it is at least conceivable that physics might require radically new mathematical ideas.

    "I thought that everyone understood that math has limits"

    Well, certainly math does have limits. For example, we never study things that are vague, ill-defined, or contradictory. But that can't stop us from constructing a faithful mathematical representation of physics because physical observables are always precisely defined, and it makes no sense to say that our observations of nature are contradictory. It's always possible to observe something confusing (like quantum particles), but we can always construct a consistent framework in which those observations can be understood (like the theory of Hilbert spaces and linear operators).

    "An example of a limit of math is the unsolvability of quintic polynomials by radicals."

    This is not a "limit" of math but a reflection of the fact that not everything can be true. There's nothing extraordinary about the fact that you can't have a "quadratic formula" for the roots of a quintic polynomial. There are also systems of linear equations that have no solutions. The unsolvability of the general quintic is a necessary consequence of the consistency of mathematics, and if physics is based on some mathematical framework in which this result is true, then you can expect nature to respect the unsolvability of the quintic.

  2. You make some good points, but I think that even the ancient Greeks would have understood that there is more to math than Eudlidean geometry, even if they did not anticipate curved geometry.

    Yes, mathematicians have tried denying the law of excluded middle, and taken some other unusual approaches. Some of these approaches have some merit. But there is a broad consensus about what math is all about. I really don't think that math is going to be reinvented to explain theoretical physics.

  3. "I really don't think that math is going to be reinvented to explain theoretical physics."

    I'm not so sure. Higher categories show up in several parts of mathematics (like homotopy theory and derived algebraic geometry) and mathematical physics (topological and conformal field theory and string theory), and such categories are already outside of the foundation that you're using to define mathematics. You may argue that we can supplement ZFC with additional axioms for proper classes, but I don't think the problem can be solved so easily. Developing a natural foundation for category theory is a difficult problem, and I think it's fair to say that it has not been solved in a satisfactory way.

  4. No, I don't believe that znything beyond ZFC is needed for any of those things.

  5. From a practical point of view, if you want to study large categories, then yes, you do need more than just ZFC. It is possible to incorporate classes into ZFC through some ad hoc device like defining a class to be a collection of sets of sets defined by some formula in the language of set theory, but this is a very unnatural way of thinking about classes. It's much better to use Grothendieck universes or work in an axiomatic framework that distinguishes between large and small collections.

    In any case, if you're trying to argue that nature has no faithful representation in ZFC, then you should agree with me when I tell you that ZFC is limited...

  6. You're implying (among other things) that ZFC is a choice, necessary for "good" math and that "bad math" is an oxymoron -- whereas "bad physics is still physics". Pardon my laymanship, I jumped in sans reading comments. Wanted to do so, then get caught up and check back.

  7. Or: Math must be revealed / discovered, while all of physics is by way of contrast constructed. Bad math turns out "not to be math" and can lead to "bad" physics? (I'm working on it). Primarily, though, I just thought I'd weigh in on Bob's comment "we never study things that are vague, ill-defined, or contradictory" -- actually, my dad got a Ph.D in Math and I believe he taught me otherwise. He was speaking to us one day about mathematicians being at odds over a definition of "neighborhood" (set theory). And (my words, not his): some problems seem intractable and/or ill-posed, but as-yet-interim "solutions" are still ... "mathy"?

  8. Apologies for one last "bad" comment. I haven't resolved anything, but due to the seriousness of all the commentary I thought I'd share a bad pun which came to mind re the topic. A bit of a stretch ... well, perhaps a giant stretch. Preface: "a-wild" signifies "natural" and "woice" Russian accent for "voice", voice implying choice as in "he voiced his (chosen) opinion" "choice" implies "artificial" i.e. not revealed or discovered f/ nature. Ahem (like I said, stretch).

    " ZFC ... it's not a-wild, it's a Woice." :-0