Believing the Conjectures - MathOverflow

Believing the Conjectures

2012-10-25T15:45:52Z

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the development of set theory and the techniques of natural science. As such, applications of these rules should be found in all branches of mathematics. The most natural context for these to manifest themselves is through conjectures that are obtained by applying one of these rules of thumb in some context. I would like to hear about such conjectures (open or closed, big or small, true or false) in your area.

Maddy's five rules of thumb are:

Maximize: This is the opposite of Occam's Razor. The idea is that the universe should be as large as possible, anything that is likely to occur should actually occur.
Inexhaustibility: This is the idea that the universe is too rich to be generated by a handful of basic building blocks: there should be transcendental objects.
Whimsical identity: An object is unlikely to be the unique object satisfying a property that does not directly pertain to the object in question.
Uniformity: The richness of the universe should not localize in one particular part, similar richness should be found in all suitably large parts.
Reflection: If there is one object with a given property then there must also be a small (or otherwise simple) object with that property.

The above brief descriptions are mine. These were formulated by Maddy in the set-theoretic context, I attempted to phrase them in a way that would make sense in a lot of other contexts. Interpret them loosely: object, property, universe can be anything you want.

Note that these rules of thumb are not always good ideas and their negatives are sometimes plausible too. Although I am mainly looking for conjectures formulated in the positive sense, I think negative conjectures are also acceptable if the main reason to disbelieve the conjecture is one of the five rules above. For example, I think the Poincaré Conjecture can be understood as a negative example of the whimsical identity rule.

Standard Big List rules apply... One example per answer please! Try to include some brief context for the benefit of people outside your area.

Answer by Gerhard Paseman for Believing the Conjectures

2012-10-25T16:32:02Z

Rather than give an example with characterization, I will give an example and invite characterization.

Harry Altman gives a nice description of $c(n)$, what I call the one-complexity of an integer $n$, at this accepted answer of MathOverflow question 75698.

Using $\lg$ to mean $\log$ to the base $3$, it is clear that for $n>1$, one has $5 \lg(n) > c(n) \ge 3 \lg(n)$. I conjecture that $5$ can be replaced by $4$. Is this a negative instance of maximize?

Gerhard "Enquiring Minds Want To Know" Paseman, 2012.10.25

Answer by Terry Tao for Believing the Conjectures

2012-10-25T16:49:45Z

In number theory, I would say that the counterpart of the "Maximise" principle is the "Local to global principle": if there is no local obstruction to solvability of some number-theoretic problem (e.g. solving a Diophantine equation), then there is no global obstruction either. In the case of Diophantine equations, this becomes the Hasse principle. In the case of patterns in the primes, this leads to the prime tuples conjecture and its generalisations. And so forth. (But bear in mind that this principle sometimes fails, due to non-obvious algebraic structure beyond the obvious "local" ones.)

EDIT: The Riemann zeta function (and other L-functions) also exhibit the "maximise" principle, a phenomenon known as zeta function universality. (But it may well be that whimsical identity fails; as pointed out in comments below, Selberg conjectured that standard axioms such as Euler product, analytic continuation, functional equation, and the Ramanujan conjecture may, when combined, become just strong enough to describe the class of all known L-functions without introducing any really exotic ones (and in particular, avoiding the artificial examples of "fake" L-functions which do bad things such as violate RH).)

Answer by Terry Tao for Believing the Conjectures

2012-10-25T17:03:43Z

In algebraic geometry, I would say that the counterpart of the "reflection" principle is the Lefschetz principle, as discussed in this previous MathOverflow question: if something is solvable in a "big" field, then it is also solvable in a "small" field.

As for the maximise principle in algebraic geometry, perhaps Ravi Vakil's "Murphy's law in algebraic geometry" qualifies.

Answer by Margaret Friedland for Believing the Conjectures

2012-10-25T19:01:38Z

An example for the negation of the reflection rule in complex analysis/geometry is the Oka principle, often informally expressed as ``whatever can be done continuously (on Stein manifolds), can be done holomorphically" The starting point is the following theorem from 1939 by K. Oka: The second Cousin problem on a domain of holomorphy can be solved by holomorphic functions if it can be solved by continuous functions.

This negates the rule (or so I think), because holomorphic objects are more complicated than continuous ones.

More applications of Oka principle can be found in the survey by Forstneric and Larusson, located here:

http://nyjm.albany.edu/j/2011/17a-2v.pdf

and in G. Elencwajg's answer to the MO question

http://mathoverflow.net/questions/3079/most-helpful-heuristic

The answers to that question can be probably mined for more examples/counterexamples.

Answer by Gil Kalai for Believing the Conjectures

2012-10-25T21:11:37Z

An obvious area to check is computational complexity. For example, "Maximize" can refer to the conjecture that computation complexity classes can be separated. Occasionally there is an unexpected collapse but generally speaking the animals in the complexity zoo are genuinly distinct.

Answer by gowers for Believing the Conjectures

2012-10-25T22:16:45Z

I initially wrote this as a comment, but it got too long and it sort of contains an example, so here goes. Reflection seems false in a number of contexts, since there are many properties that can't be satisfied in any canonical way. For example, there isn't a small or simple basis for the reals over the rationals.

But maybe more in the spirit of the question than constructions that require the axiom of choice are a number of strange Banach-space counterexamples that are built using tools such as a sufficiently fast-growing sequence, a concave function that tends to infinity more slowly than any power, an injection from finite sets of rationals to the positive integers, etc., where the properties you need can be achieved reasonably simply, but not canonically, and the combination of the various elements is best viewed not as a single example but as a technique for building examples, where the precise details of the implementation clearly don't matter.

I'm making a slightly stronger point than may immediately be apparent, which is that for some of these strange Banach-space properties (a famous example being the property of not containing $c_0$ or any $\ell_p$ space, which was first shown to be possible by Tsirelson), not only is there considerable flexibility in how you build counterexamples, but it appears that this flexibility is in some sense "necessary". One way of making that assertion semi-precise is to say that there don't seem to be additional (sensible) properties you can insist on that cause the flexibility to go away.

I'm not saying that Reflection is definitely false for this kind of property, but it does seem to be, and I see no reason to suppose that it would be true.

Answer by Timothy Chow for Believing the Conjectures

2012-10-25T22:40:56Z

If I understand correctly what "Uniformity" means, then I think that the universality principle in statistical mechanics, percolation theory, and related areas is an example.

Answer by Benjamin Steinberg for Believing the Conjectures

2012-10-26T03:08:54Z

I think a good example of maximize is Gromov's principle that there is no non-trivial result about all finitely generated groups. Monster groups like Burnside groups show that things you cannot trivially prove from the group axioms are not true for finitely generated groups.

I think the theory of finite groups shows the opposite of the maximize principle and perhaps all these principles (except perhaps the theory of p-groups).

Answer by anonymous for Believing the Conjectures

2012-10-26T05:28:35Z

An example of Whimsical identity is the global Langlands correspondence: elliptic curves should not be the only objects from arithmetic geometry with L-functions related to harmonic analysis on algebraic groups. In fact many conjectures in arithmetic geometry are of this kind: Deligne-Beilinson conjectures on special values of L-functions, description of the image of Galois on etale cohomology groups in terms of the Mumford-Tate group, etc.

Answer by Timothy Chow for Believing the Conjectures

2012-10-26T19:05:13Z

Given that you used the word "transcendental" in describing inexhaustibility, Schanuel's conjecture seems to be an obvious instance. In effect, Schanuel's conjecture implies that numbers such as $e+\pi$ that have "no reason" to be algebraic are indeed not algebraic. See also the conjectures of Kontsevich and Zagier about periods, which have a similar flavor.