What is Realistic Mathematics?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.)

Answer 5

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.

  

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.

Answer 6

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.

Answer 7

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.

Answer 8

Update: Having read below a second time, I think you should change your usage from 'realistic mathematics' to 'empirical mathematics'. I think this is much more apt, plus it has the advantage that everyone can give it their own interpretation, which I suspect is about as much as you can hope for.

You mention Von Neumann. Actually he wrote:

"…mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetically motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from ‘reality’, it is beset with very grave dangers. It becomes more and more purely aestheticising, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so for from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganised mass of details and complexities. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration."

I not sure I agree with all of this (as much as I am a judge), especially the tone. But I thought it was relevant.