Tim Gowers describes the two cultures of mathematics – combinatorics, e.g., Erdos/theory-building, e.g., Grothendieck’s algebraic geometry.
Paseau’s argument (in Studies in History and Philosophy of Science review) against my claim that we should want a philosophy of mathematics resembling philosophy of science is that the foundations of the latter are described in two mutually inconsistent languages, whereas mathematics has one, namely, set theory. Could we argue that two heuristically incompatible mathematical cultures can play that role? (It’s unclear his argument works anyway. If QFT and GTR are reconciled, would philosophy of physics need to be done differently?)
The Harvard math curriculum leans heavily towards the systematic, theory-building style; analytic number theory as usually practiced falls in the problem-solving camp. This is probably why, despite its illustrious history (Euclid, Euler, Riemann, Selberg, … ) and present-day vitality, analytic number theory has rarely been taught here. … Now we shall see that there is more to analytic number theory than a bag of unrelated ad-hoc tricks, but it is true that partisans of contravariant functors, adèlic tangent sheaves, and étale cohomology will not find them in the present course. Still, even ardent structuralists can benefit from this course. … An ambitious theory-builder should regard the absence thus far of a Grand Unified Theory of analytic number theory not as an insult but as a challenge. Both machinery- and problem-motivated mathematicians should note that some of the more exciting recent work in number theory depends critically on symbiosis between the two styles of mathematics.
Heuristic incompatibility as in my heuristic falsifier.
I find Tim Gowers’ “two cultures” distinction to be relevant here. PDE does not have a general theory, but it does have a general set of principles and methods (e.g. continuity arguments, energy arguments, variational principles, etc.).
Sergiu Klainerman’s “PDE as a unified subject” discusses this topic fairly exhaustively.
Any given system of PDE tends to have a combination of ingredients interacting with each other, such as dispersion, dissipation, ellipticity, nonlinearity, transport, surface tension, incompressibility, etc. Each one of these phenomena has a very different character. Often the main goal in analysing such a PDE is to see which of the phenomena “dominates”, as this tends to determine the qualitative behaviour (e.g. blowup versus regularity, stability versus instability, integrability versus chaos, etc.) But the sheer number of ways one could combine all these different phenomena together seems to preclude any simple theory to describe it all. This is in contrast with the more rigid structures one sees in the more algebraic sides of mathematics, where there is so much symmetry in place that the range of possible behaviour is much more limited. (The theory of completely integrable systems is perhaps the one place where something analogous occurs in PDE, but the completely integrable systems are a very, very small subset of the set of all possible PDE.)
It does seem that categorification and similar theoretical frameworks are currently better for manipulating the type of exact mathematical properties (identities, exact symmetries, etc.) that show up in structured objects than the fuzzier type of properties (correlation, approximation, independence, etc.) that show up in pseudorandom objects, but this may well be just a reflection of the state of the art than of some fundamental restriction. For instance, in my work on the Gowers uniformity norms, there are hints of some sort of “noisy additive cohomology” beginning to emerge – for instance, one may have some sort of function which is “approximately linear” in the sense that its second “derivative” is “mostly negligible”, and one wants to show that it in fact differs from a genuinely linear function by some “small” error; this strongly feels like a cohomological question, but we do not yet have the abstract theoretical machinery to place it in the classical cohomology framework (except perhaps in the ergodic theory limit of these problems, where there does seem to be a reasonable interpretation of these informal concepts). Similarly, when considering inverse theorems in additive combinatorics, a lot of what we do has the feel of “noisy group theory”, and we can already develop noisy analogues of some primitive group theory concepts (e.g. quotient groups, group extensions, the homomorphism theorems, etc.), but we are nowhere near the level of sophistication (and categorification, etc.) with noisy algebra that exact algebra enjoys right now. But perhaps that will change in the future.
When defining the concept of a mathematical space or structure (e.g. a group, a vector space, a Hilbert space, etc.), one needs to list a certain number of axioms or conditions that one wants the space to satisfy. Broadly speaking, one can divide these conditions into three classes:
. closed conditions. These are conditions that generally involve an or a sign or the universal quantifier, and thus codify such things as algebraic structure, non-negativity, non-strict monotonicity, semi-definiteness, etc. As the name suggests, such conditions tend to be closed with respect to limits and morphisms.
. open conditions. These are conditions that generally involve a or a sign or the existential quantifier, and thus codify such things as non-degeneracy, finiteness, injectivity, surjectivity, invertibility, positivity, strict monotonicity, definiteness, genericity, etc. These conditions tend to be stable with respect to perturbations.
. hybrid conditions. These are conditions that involve too many quantifiers and relations of both types to be either open or closed. Conditions that codify topological, smooth, or metric structure (e.g. continuity, compactness, completeness, connectedness, regularity) tend to be of this type (this is the notorious “epsilon-delta” business), as are conditions that involve subobjects (e.g. the property of a group being simple, or a representation being irreducible). These conditions tend to have fewer closure and stability properties than the first two (e.g. they may only be closed or stable in sufficiently strong topologies). (But there are sometimes some deep and powerful rigidity theorems that give more closure and stability here than one might naively expect.)
Ideally, one wants to have one’s concept of a mathematical structure be both closed under limits, and also stable with respect to perturbations, but it is rare that one can do both at once. Instead, one often has to have two classes for a single concept: a larger class of “weak” spaces that only have the closed conditions (and so are closed under limits) but could possibly be degenerate or singular in a number of ways, and a smaller class of “strong” spaces inside that have the open and hybrid conditions also. A typical example: the class of Hilbert spaces is contained inside the larger class of pre-Hilbert spaces. Another example: the class of smooth functions is contained inside the larger class of distributions.
As a general rule, algebra tends to favour closed and hybrid conditions, whereas analysis tends to favour open and hybrid conditions. Thus, in the more algebraic part of mathematics, one usually includes degenerate elements in a class (e.g. the empty set is a set; a line is a curve; a square or line segment is a rectangle; the zero morphism is a morphism; etc.), while in the more analytic parts of mathematics, one often excludes them (Hilbert spaces are strictly positive-definite; topologies are usually Hausdorff (or at least ); traces are usually faithful; etc.)
So what of the impact of nonstandard analysis? Tao writes
One of the features of nonstandard analysis, as opposed to its standard counterpart, is that it efficiently conceals almost all of the epsilons and deltas that are so prevalent in standard analysis, as discussed in my blog post linked below. As a consequence, analysis acquires a much more “algebraic” flavour when viewed through the nonstandard lens.
Consider for instance the concept of continuity for a function on the unit interval. The standard definition of continuity uses a bunch of epsilons and deltas:
is continuous iff for every and in there exists such that for all in with one has .
The nonstandard definition, which is logically equivalent, is as follows:
is continuous iff for every in and in with , one has .
Here is the ultralimit extension of the original function to the ultrapowers and , and denotes an infinitesimal, i.e. a nonstandard real whose magnitude is smaller than any standard . (See the blog post Ultrafilters, nonstandard analysis, and epsilon management for more discussion.) It is a good exercise to test one’s understanding of nonstandard analysis to verify that these two definitions are indeed equivalent.
There is an analogous nonstandard characterisation of uniform continuity:
is uniformly continuous iff for every in and in with , one has .
One can now quickly give a nonstandard proof of the classical fact that continuous functions on a compact interval such as are automatically uniformly continuous. Indeed, if in are such that , then and have the same standard part , which lies in . If is continuous, then and , hence . (It is instructive to see how the nonstandard proof and the standard proof are ultimately just reformulations of each other, and in particular how both rely ultimately on the Bolzano-Weierstrass theorem. In the nonstandard world, Bolzano-Weierstrass is needed to demonstrate existence of the standard part.)
One can also use nonstandard analysis to phrase continuity (on compact domains, at least) in a very succinct algebraic fashion:
is continuous if and only if commutes with the standard part function .
Note how the number of quantifiers required to define continuity has decreased all the way to zero. (Of course, they have all been hidden in the definition of the standard part function.) It is this elimination of quantifiers that allows the theory to be algebrised; as a zeroth approximation, one can view algebra as the mathematics of quantifier-free statements.
If nonstandard analysis eliminates quantifiers, does this change the status of any conditions expressed using it (as you explained here) or are the quantifiers just hiding?
To some extent, yes; as discussed above, a hybrid condition such as continuity can become a closed condition once one gains the ability to quotient out the infinitesimals. (In part this is because having the notion of an infinitesimal is enough to change the “topology” that determines which conditions are closed and which are open.)
Nonstandard analysis shuffles around all the quantifiers, and so blurs distinctions such as that between existential and universal statements, or open and closed statements; almost everything becomes closed algebra as a consequence. For instance, the property of a set being closed in nonstandard analysis becomes the algebraic statement that (i.e. in whenever in ), or equivalently that ; conversely, the property of being open is the assertion that (i.e. whenever ), or equivalently that .
With the nonstandard perspective, the distinction is no longer between algebra and analysis, or between closed conditions and open conditions; it is between closed algebra without infinitesimals and closed algebra with infinitesimals.