User:Boris Tsirelson/sandbox2

Tao

4. Summary

Let me summarise with a brief list of pros and cons of switching to a nonstandard framework. First, the pros:

Many “first-order” parameters such as {\epsilon} or {N} disappear from view, as do various “negligible” errors. More importantly, “second-order” parameters, such as the function {F} appearing in Theorem 2, also disappear from view. (In principle, third-order and higher parameters would also disappear, though I do not yet know of an actual finitary argument in my fields of study which would have used such parameters (with the exception of Ramsey theory, where such parameters must come into play in order to generate such enormous quantities as Graham’s number).) As such, a lot of tedious “epsilon management” disappears.

Iterative (and often parameter-heavy) arguments can often be replaced by minimisation (or more generally, extremisation) arguments, taking advantage of such properties as the well-ordering principle, the least upper bound axiom, or compactness.

The transfer principle lets one use “for free” any (first-order) statement about standard mathematics in the non-standard setting (provided that all objects involved are internal; see below).

Mature and powerful theories from infinitary mathematics (e.g. linear algebra, real analysis, representation theory, topology, functional analysis, measure theory, Lie theory, ergodic theory, model theory, etc.) can be used rigorously in a nonstandard setting (as long as one is aware of the usual infinitary pitfalls, of course; see below).

One can formally define terms that correspond to what would otherwise only be heuristic (or heavily parameterised and quantified) concepts such as “small”, “large”, “low rank”, “independent”, “uniformly distributed”, etc.

The conversion from a standard result to its nonstandard counterpart, or vice versa, is fairly quick (but see below), and generally only needs to be done only once or twice per paper.

Next, the cons:

Often requires the axiom of choice, as well as a certain amount of set theory. (There are however weakened versions of nonstandard analysis that can avoid choice that are still suitable for many applications.)

One needs the machinery of ultralimits and ultraproducts to set up the conversion from standard to nonstandard structures.

The conversion usually proceeds by a proof by contradiction, which (in conjunction with the use of ultralimits) may not be particularly intuitive.

One cannot efficiently discern what quantitative bounds emerge from a nonstandard argument (other than by painstakingly converting it back to a standard one, or by applying the tools of proof mining). (On the other hand, in particularly convoluted standard arguments, the quantitative bounds are already so poor – e.g. of iterated tower-exponential type – that losing these bounds is no great loss.)

One has to take some care to distinguish between standard and nonstandard objects (and also between internal and external sets and functions, which are concepts somewhat analogous to measurable and non-measurable sets and functions in measurable theory). More generally, all the usual pitfalls of infinitary analysis (e.g. interchanging limits, the need to ensure measurability or continuity) emerge in this setting, in contrast to the finitary setting where they are usually completely trivial.

It can be difficult at first to conceptually visualise what nonstandard objects look like (although this becomes easier once one maps nonstandard analysis concepts to heuristic concepts such as “small” and “large” as mentioned earlier, thus for instance one can think of an unbounded nonstandard natural number as being like an incredibly large standard natural number).

It is inefficient for both nonstandard and standard arguments to coexist within a paper; this makes things a little awkward if one for instance has to cite a result from a standard mathematics paper in a nonstandard mathematics one.

There are philosophical objections to using mathematical structures that only exist abstractly, rather than corresponding to the “real world”. (Note though that similar objections were also raised in the past with regard to the use of, say, complex numbers, non-Euclidean geometries, or even negative numbers.)

Formally, there is no increase in logical power gained by using nonstandard analysis (at least if one accepts the axiom of choice); anything which can be proven by nonstandard methods can also be proven by standard ones. In practice, though, the length and clarity of the nonstandard proof may be substantially better than the standard one.

In view of the pros and cons, I would not say that nonstandard analysis is suitable in all situations, nor is it unsuitable in all situations, but one needs to carefully evaluate the costs and benefits in a given setting; also, in some cases having both a finitary and infinitary proof side by side for the same result may be more valuable than just having one of the two proofs. My rule of thumb is that if a finitary argument is already spitting out iterated tower-exponential type bounds or worse in an argument, this is a sign that the argument “wants” to be infinitary, and it may be simpler to move over to an infinitary setting (such as the nonstandard setting).