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This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers will understand this post and the nature of the question.

I will first try to explain what I would call Realistic Mathematics. Let us say that mathematics is about the formalization, organization and expression of thought. At the same time one could have the feeling that thought is usually trying to capture some aspect of physical reality; and let that be anything from a feeling, an impression of something, to experimental data of an experiment, or the observed geometric properties of lines and points in two-dimensional space. Of course thought itself (as described above) can be about anything and hence anything axiomatizable could then be seen as mathematics (that is David Hilbert's point of view). At the other hand if thought is primarily about physical reality then the focus of mathematics should be more restrictive. (I remember that Arnold argued in favor of this view; also von Neumann.) Let us call this restricted part of mathematics for the moment Realistic Mathematics. I am not saying that such a restriction of focus would be good or necessary; and I do not want to start a discussion about this. I just want to find out and discuss, whether mathematicians could agree on what we could call Realistic Mathematics. Let us suppose for a moment that we made some sense of the concept of Realistic Mathematics, and observed that it is a science that is about some part or aspect of physical reality. My naive question is now:

Question: What is Realistic Mathematics about from a mathematical (or model theoretic) perspective?

or

Question: Is there any mathematical structure which serves as the part of mathematics which is about physical reality?

Just to give some examples: I consider anything related to finite-dimensional geometry (manifolds, simplicial complexes, convex sets etc.), number theory, operators or algebras of operators on separable Hilbert spaces, differential equations, discrete geometry, combinatorics (under countability assumptions) etc. as being part of observed physical reality or potentially useful for the study of physical reality. At the other side, existence of large cardinals, non-measurable subsets of the reals, etc. are not (immediately) useful for such a study. In particular, my view is that the Axiom of Choice does not add anything to the understanding of physical reality. It produces highly counter-intuitive statements (not observed in nature) about subsets of finite-dimensional euclidean space and has its merits (in Realistic Mathematics) only through short proofs and the knowledge that many statements are provable in ZFC if and only they are provable with more realistic assumptions.)

What about $L(R)$? (See Wikipedia for definitions.) That would be a concrete model and maybe a realistic mathematician just studies properties of $L(R)$? Maybe a realistic mathematician studies what can be proved using $ZF + DC$? Is there any other canonical candidate which arises? My question here is mainly about opinions or some sort of vision which explains why this or that model or object of study arises naturally.

Question: Is there any mathematical application to the study of physical reality which is not captured by the study of the model $L(R)$?

More specifically: What about concrete statements which are undecidable in $ZF$? Does the Continuum Hypothesis belong to the statements we want to be true in Realistic Mathematics? What about the Open Coloring Axiom? Here, I am also asking for opinions or some consistent perspective on the realistic part of mathematics which captures my imprecise way of describing it.

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Noting that there is already a vote to close, I would vote for this to stay open. –  Steve Huntsman Aug 7 '10 at 16:21
    
(I have deleted an off-topic comment that used to be here.) –  S. Carnahan Aug 7 '10 at 23:04
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Probably at least all Pi01 sentences should be considered realistic. So provability in some formal system doesn't seem to mean a lot (consider "ZF is consistent"). Note that certain large cardinal axioms also decide some Pi01 sentences, but AC and CH do not (see esp. the work of Harvey Friedman). –  Sebastian Reichelt Aug 7 '10 at 23:33
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7 Answers

up vote 30 down vote accepted

When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theorists that analysts (and others doing what you call realistic mathematics) would adopt ZF + DC + "all sets of reals are Lebesgue measurable" as their preferred foundational framework. There would be no more worries about "pathological" phenomena (like the Banach-Tarski paradox), no more tedious verification that some function is measurable in order to apply Fubini's theorem, and no more of various other headaches. But that expectation wasn't realized at all; analysts still work in ZFC. Why? I don't know, but I can imagine three reasons.

First, the axiom of choice is clearly true for the (nowadays) intended meaning of "set". Solovay's model consists of certain "definable" sets. Although there's considerable flexibility in this sort of definability (e.g., any countable sequence of ordinal numbers can be used as a parameter in such a definition), it's still not quite so natural as the general notion of "arbitrary set." So by adopting the new framework, people would be committing themselves to a limited notion of set, and that might well produce some discomfort.

Second, it's important that Solovay's theory, though it doesn't include the full axiom of choice, does include the axiom of dependent choice (DC). Much of (non-pathological) analysis relies on DC or at least on the (weaker) axiom of countable choice. (For example, countable additivity of Lebesgue measure is not provable in ZF alone.) So to work in Solovay's theory, one would have to keep in mind the distinction between "good" uses of choice (countable choice or DC) and "bad" uses (of the sort involved in the construction of Vitali sets or the Banach-Tarski paradox). The distinction is quite clear to set-theorists but analysts might not want to get near such subtleties.

Third, in ZF + DC + "all sets of reals are Lebesgue measurable," one lacks some theorems that analysts like, for example Tychonoff's theorem (even for compact Hausdorff spaces, where it's weaker than full choice). I suspect (though I haven't actually studied this) that the particular uses of Tychonoff's theorem needed in "realistic mathematics" may well be provable in ZF + DC + "all sets of reals are Lebesgue measurable" (or even in just ZF + DC). But again, analysts may feel uncomfortable with the need to distinguish the "available" cases of Tychonoff's theorem from the more general cases.

The bottom line here seems to be that there's a reasonable way to do realistic mathematics without the axiom of choice, but adopting it would require some work, and people have generally not been willing to do that work.

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Is it really "ZFC+there is an inaccesible cardinal" in Solovays theorem? I always thought the large cardinal axiom that has to be used is a measurable cardinal... Also: Functional analysis would be pretty ugly without the Hahn-Banach-theorem (I think that it is equivalent to the AC, but I'm not sure) and with ugly Functional analysis, the theory of PDEs would become (even more) ugly. This could be another reason why analysts don't prefer ZF+DC+"P(IR)=Lebesgue sets" over ZFC. –  Johannes Hahn Aug 7 '10 at 20:32
    
Hahn-Banach is strictly weaker than AC. –  Gerald Edgar Aug 7 '10 at 20:41
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Yes, Solovay's theorem really needs only (the consistency of) an inaccessible cardinal. As for Hahn-Banach, it is, as Gerald Edgar says, strictly weaker than AC, but it's nevertheless false in the Solovay model. I think what I said about Tychonoff's theorem may apply here as well: The particular uses of Hahn-Banach that are needed in "realistic mathematics" might be available even though the general theorem isn't. And people might not want to keep track of which uses are available and which aren't. –  Andreas Blass Aug 7 '10 at 21:33
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"One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work." :P –  Mariano Suárez-Alvarez Aug 8 '10 at 14:37
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Is there any other canonical candidate which arises? My question here is mainly about opinions or some sort of vision which explains why this or that model or object of study arises naturally.

I'm surprised that nobody mentioned the following two "canonical" candidates for "realistic mathematics":

1) ZF+AD (the Axiom of Determinacy). This proves DC (but disproves the full AC) and that all sets of reals are Lebesgue measurable; an advantage over the Solovay model (see Andreas Blass' answer above) and over $L(\Bbb R)$ is that AD is quite easy to state and can be seen to some extent as a philosophical principle. ZF+AD also has other plausible consequences, including the Continuum Hypothesis in its original formulation (every uncountable subset of the reals has cardinality continuum) and on the other hand that the continuum is not well-orderable. Descriptive set theory has a lot of "nice" results proved under AD or under a weaker axiom PD (Projective Determinacy) which does not contradict the full AC. With recent work of Woodin, the "politically correct" form ZFC+PD of the original theory ZF+AD seems to be gaining considerable prominence in set theory. AD implies the consistency of ZF (so ZF+AD is substantially stronger than ZF), but on the other hand ZF+AD is consistent relative to ZF + a large cardinal axiom (in fact, $L(\Bbb R)$ satisfies AD modulo this axiom).

2) Positive set theories. An intuition for "realistic mathematics", which also reminds of $L(\Bbb R)$, could be that all sets should have something to do with the continuum, insofar as they are not definable in pure logical terms. A more specific intuition, which also reminds of AD, could be that all sets should have something to do with a topology (or a uniform structure).

In a positive set theory, "sets" are thought of as closed subsets of some space, and non-closed subsets are the "proper classes". Discrete subsets can be identified with a model of ZFC. The complement of a set is a class (possibly proper; this is where "positive" comes from), and the closure (with respect to the topology) of every class is a set. So classes are not "bigger" than sets, somewhat unexpectedly. The Russel class $\{x\mid x\notin x\}$ is a proper class whose complement is a set, $\{x\mid x\in x\}$. I find this topological explanation of proper classes and Russel's paradox quite convincing.

For an introduction to positive set theories see survey notes by Holmes, who in particular explains how to obtain a model of such a theory by collapsing a type theory. There is a variety of positive set theories; at least one of them is equiconsistent with a second-order ZFC (Kelley-Morse-Tarski theory with choice).

In ZF-like theories, sets (e.g. those in Goedel's constructive universe $L$) unfold in a fashion that reminiscent of direct limits (=colimits), beginning with an initial object, $\emptyset$. In positive set theories, there is the universal set, and the unfolding of sets is more in the spirit of inverse limits (=limits). Here is Hinnion's description (from the chapter "Alternative Set Theories" in this book) of a model (I think it is this model that is also known as the "$\omega$-hyperuniverse") of a positive set theory.

image

In conclusion let me cite from this review by R. Holmes of a paper by R. Hinnion:

In this paper, the author introduces a general method of converting an arbitrary first-order structure into a uniform space, and studies the resulting notion of Cauchy completion of a first-order structure. This is a generalization of constructions used to build models of positive set theories.

The paper itself (as well as its erratum) seems not easy to find in libraries, but the construction should be similar to, if not same as, as in a previous (or rather subsequent?) paper by Hinnion.

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Thanks a lot for this answer. The Axiom is Determinacy is really what I was looking for. Some axiom which makes sense on a philosophically level and at the same time implies the real stuff, while avoiding the non-realistic pathologies. Really striking however, that the consistency of ZF can be proved within ZF+AD. –  Andreas Thom Feb 21 '11 at 14:34
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May I give a humorous answer? Some may find that it expresses a realistic view on mathematics. It's from V.I. Arnold, and I found it in MATHEMATICS: FRONTIERS AND PERSPECTIVES, p. 403:

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA.)

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This is more a philosophical question, and therefore hasn't a definite answer.

But if you want to make plausible that AC isn't realistic mathematics, you might reason something like the following:

There is a set of mathematical sentences, that has a direct (so, not indirect yet) relation with physics. Call this set B (of Basic). Personally, I think they are in the following four areas:

  • Computation
  • Probability
  • Geometry
  • Topology

Note, I count arithmetic as part of computation, since numbers are not a physical entity, but computation is. But, likely many people will disagree.

Also note, that the sentences in B, might be far simpler than the mathematical sentences suggested in your question or in one of the answers.

Now, we have more complex mathematical sentences, that are still "realistic", if they can be converted or "instantiated" to sentences of B. Call this extended set be E. These more complex sentences capture a higher principle, which can be powerful in the science of physics. Still, there is no direct link with physics, the mathematic sentence first needs to be instantiated, to make a direct link with physics. Example, any sentence with a real number, is not be part of B, because we can not observe real numbers in physics, but they can be part of E.

Consider that there is a sentence s1 ∈ E and s2 ∈ E. Furthermore, that s2 follows from s1, but with a rather difficult proof. Suppose there is a proposed axiom a, such that s1 + a leads more directly to s2 (a simpler proof). However, a ∉ E. So, axiom a is independent from sentences in B.

From above concept it follows that axioms can exist that are "useful", because they make proofs shorter, but have nevertheless no "meaning". I do believe that AC is such axiom.

About CH, I think it is not useful and not having a meaning.

But again, this is more an opinion.

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At the other side, existence of large cardinals, non-measurable subsets of the reals, etc. are not (immediately) useful for such a study.

I don't know about non-measurable subsets, but large cardinals are definitely useful for ordinary applied mathematics of the kind I do (theoretical computer science). The reason is twofold.

First, I design programming languages, and modern programming languages are type theories -- so assertions about whether all programs in a given language are total or not can boil down to assertions about the existence of large cardinals. Meta-mathematically, this is a trivial point, but often people don't immediately see that large cardinals amount to facts about the termination of computer programs.

Secondly, we need things like functor categories to organize the semantics of programming languages, which call for (weak) large cardinal axioms. Here's a simple example.

In a language like Java, the execution of a program can create objects which live in the computer's memory. However, even though the concrete memory is addressed by integers, Java only lets you test whether two objects are equal or not, and doesn't let you compare objects for their relative position in the address space. So, when you give a semantics to such a language, we want to ensure that the meaning of a program is invariant under permutations of the address space (ie, you can move objects around, without changing the observable behavior of the program).

One approach to this problem (invented by Stark and Pitts) is to give semantics in terms of functor categories. To model this permutation business, we start with the category $I$ of finite sets and injective maps. The idea is that each object of the category is a set of locations, and an injection gives you a map renaming some locations and allocating some more. Then, you model the types of your programming language as presheaves on $I$, and model the programs as natural transformations between the functors $[I^\mathrm{op}, \mathrm{Set}]$. This way, the semantics can't even speak about ill-behaved elements which don't respect the invariant, since every denotation is by construction natural.

You can ask whether this is necessary, and of course it isn't: we could just use a simple transition relation to model the semantics. The trouble is that this misses the point! The purpose of the semantics is to make it easy to reason about our programs, and to that end we want to work in a way where the properties we want to use are "come for free" rather than being laboriously pushed through every proof that might involve them.

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I agree. If some mathematical notion or concept is useful for some practical purpose like the design of programming languages, then I would would call it realistic. I think, my statement about large cardinals was to broad. Maybe I should have asked instead what properties of large cardinals are the useful ones, when it comes to any sort of practical applications or any relation with the real world. The question is: Is it possible to agree on a mathematics which could be called realistic. More concretely: What properties of large cardinals do you consider useful. –  Andreas Thom Aug 8 '10 at 11:56
    
Great answer. You want higher infinities for much the same reason you want countable infinity: every real computer is a finite machine but that is missing the point, and indeed the halting problem has more useful things to say about the limits of computers than the pumping lemma. The termination of any given computable function of practical interest relevance can be proven by induction with small ordinals. The ones that individually require large ordinals are part of the package, byproducts of a more convenient category; they're like non-measurable sets in analysis in that sense. –  Per Vognsen Aug 8 '10 at 14:28
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I think you need to be clearer about how directly useful you need your mathematical concepts to be to the study of the real world. For example, suppose you find PDEs useful for modelling physical phenomena, certain concrete Banach spaces useful for studying PDEs, and the abstract theory of Banach spaces useful for understanding the concrete ones, then you'll have to allow the Hahn-Banach theorem, since that is undeniably useful in the abstract theory of Banach spaces. The Hahn-Banach theorem is slightly weaker than the axiom of choice, but the usual proof of it uses the axiom of choice. Does that make the axiom of choice realistic after all?

In the other direction, one might say that even very large positive integers are not realistic. For example, the number 123871205412470874297947938271423698765734564756028492656 has no direct role to play in physics.

For this kind of reason, I think it may be very hard to come up with a precise characterization of realistic mathematics, but I'd be interested to see some attempts.

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+1 for the fact that you typed out that number. =)! –  Harry Gindi Aug 8 '10 at 4:19
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For most purposes, only separable Banach spaces are relevant. Cannot Hahn-Banach can be proved with Countable Choice of Dependent Choice in this case? Of course, non-separable Banach spaces such as $\ell^{\infty} {\mathbb N}$ are relevant. For example, I see that for the existence of Banach limits or means on amenable groups (both are elements in the dual of $\ell^{\infty} {\mathbb N}$), Krein-Milman (which uses Tychonoff) is practical and seems right. However, the non-constructive nature of the mean and the unintuitive consequences give me the feeling that these objects are not realistic. –  Andreas Thom Aug 8 '10 at 5:12
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Maybe I would even go a bit further and say that the consequences of the Hahn-Banach theorem are not part of the percieved reality. Nobody has ever encountered an invariant mean on ${\mathbb Z}$! On the other side, nobody ever found it necessary to consider non-continuous linear operators from one Hilbert space (defined everywhere) to another. Doesn't it seem more realistic to strive for continuity of all such operators and not for the existence of the mean. Of course, it is difficult to come with a definition. –  Andreas Thom Aug 8 '10 at 5:27
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I agree with you that some parts of mathematics are more "realistic" than others. What I was questioning was your apparent use of the "is helpful for studying" relation to capture this intuition. My example may not be perfect, but I think that this relation will let in mathematics that is not realistic. Also, if you say that only separable Banach spaces are relevant, why don't you also say that only rational numbers with numerators and denominators less than 10^100 are relevant? Or do you? –  gowers Aug 8 '10 at 13:26
    
@gowers: I think Ed Nelson is suggesting the plausibility of something not entirely different than what you are saying. docs.google.com/… –  Steve Huntsman Aug 8 '10 at 16:40
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I may be wrong, missing the point and in way over my head, but given that category theory is useful in physics (is it?) and that category theory deals with classes, then I don't think that set theories can say everything that we can say about physical reality.

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Category theory doesn't (necessarily) deal with classes. You can express most of it (certainly the part that is useful in geometry/physics) in the language of ZF with possibly C. –  Andy Putman Aug 7 '10 at 17:56
    
You can also express most of it in type theory and thus in constructive mathematics, IIRC. (No, I don't know the details.) One uses classes just to avoid thinking about what one actually uses. –  darij grinberg Aug 7 '10 at 19:04
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This answer prompted me to ask the following question: mathoverflow.net/questions/34861/… –  Steve Huntsman Aug 7 '10 at 19:53
    
You can also generalize L(R) to include proper classes and the question evaporates.So this isn't really a challenge to the idea,just how it's stated above. –  Andrew L Aug 7 '10 at 21:20
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