Believing the Conjectures

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.

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One should make a distinction between the fields of mathematics that study "structured" or "rigid" objects (e.g. objects with large group actions or which extremise some functional), and those that study "arbitrary" objects; generally, the principles above only apply to the latter fields of mathematics, and their negations apply to the former fields. (This is part of what I like to call the "dichotomy between structure and randomness".) –  Terry Tao Oct 25 '12 at 15:58
–  Terry Tao Oct 25 '12 at 16:45
Very nice question. How is the conjecture that 2^{ALEPH_0}=ALEF_2 fits into Maddy's philosophy? (See mathoverflow.net/questions/23829/… ) –  Gil Kalai Oct 25 '12 at 20:56
"A great truth is a truth whose opposite is also a great truth." --- Niels Bohr –  Gerry Myerson Oct 25 '12 at 21:35
Dear Gil, that's another good one... Intuitively, the more general ide of ampleness suggests that AC is false: there is such a rich variety of sets that it's impossible that, up to isomorphism, they are all sets of ordinals. However, AC can also be viewed as a maximality principle, one where there is a maximal amount of bijections between sets, so many that the cardinal numbers collapse to a linear order (which is equivalent to AC). I personally think that the ¬AC argument is stronger but it's not strong enough to definitely win... –  François G. Dorais Oct 27 '12 at 19:36
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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.

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Nice examples in both answers. –  Abdelmalek Abdesselam Oct 25 '12 at 18:12

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.

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I like this one, Gil. I had never thought of $P \neq NP$ as an application of maximize, but it makes sense... –  François G. Dorais Oct 26 '12 at 0:48

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).)

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Dear Terry, I'm curious as to why you don't think the Selberg class does the job of axiomatizing the correct family of L-functions (for RH etc.). Regards, Matthew –  Emerton Oct 25 '12 at 19:02
Huh, it appears my information here is out of date (I knew about RH counterexamples to previous axiomatisations of L-functions, but didn't realise that Selberg's formulation manages to avoid all known counterexamples.) I'll update the text accordingly. (I'm still skeptical that these sorts of axioms aren't just abstracting what we already know how to do with L-functions, rather than being the way forward to discover new proof methods that might make progress on problems such as RH, but I would of course be very happy to be proven incorrect on this point.) –  Terry Tao Oct 25 '12 at 21:19
Selberg's conjectures are very interesting! Thank you, Matt and Terry, for bringing those up. –  François G. Dorais Oct 26 '12 at 1:01

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.

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Regarding the reals over the rationals, mightn't the fact that any real (or countable collection of reals) is contained in countable real-closed subfield of $\mathbb{R}$ be regarded as fulfilling the reflection principle? That is, $\mathbb{R}$ is reflecting down to its countable elementary substructures. –  Joel David Hamkins Oct 25 '12 at 22:41
I understand your point of view, Tim, but I think there is a counterpoint here. For the sake of argument, I contend that these are both successes of reflection. Note that the wording is small (simple) and not smaller (simpler). A basis for $\mathbb{R}$ over $\mathbb{Q}$ cannot be any smaller nor simpler than any other, so this is a case where reflection says nothing interesting. [...] –  François G. Dorais Oct 26 '12 at 0:26
The fact that there ought to be a separable example is too trivial to count as a success of Reflection, since if there is any example at all, you can take a separable subspace of it and then you've got a separable example. –  gowers Oct 26 '12 at 8:30
The general point I'm making here is that in this context it's the mathematics that tells us to what extent Maximize is an appropriate principle, rather than the principle that is guiding our mathematical expectations. –  gowers Oct 26 '12 at 8:42
I agree that this instance of reflection is too trivial to be called a leap, but it is nevertheless an example of reflection. (Reflection is rooted in the Löwenheim-Skolem Theorem, the Compactness Theorem, and relatives. Direct applications of these are too trivial to ever form conjectures, but the reflection idea is what drives me to try these theorems in relevant contexts.) –  François G. Dorais Oct 26 '12 at 12:02
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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.

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Very interesting, Tim! I did expect examples in physics but this is beyond what I had imagined. This is also tied to renormalization - en.wikipedia.org/wiki/Renormalization_group –  François G. Dorais Oct 26 '12 at 0:16

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:

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

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

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Interesting. My (limited and naïve) experience led me to believe that holomorphic objects are simpler than continuous ones. I am eager to know more... –  François G. Dorais Oct 26 '12 at 0:51
It depends what is meant by simple". One can view a holomorphic function just as the sum of a (convergent) power series, so it is quite easy (=simple) to give plenty of examples of such functions and manipulate them. But these functions satisfy many properties that plain continuous functions do not: the identity principle, the Cauchy- Riemann equations, they can be recovered through the integral formulas etc., so they are more elaborate (=non-simple). This refers to functions, but functions make up maps, give coordinates on manifolds, so they are a proxy for other holomorphic objects. –  Margaret Friedland Oct 26 '12 at 16:05
Given that Larusson is a colleague of mine, I can add some modern spin to this. The viewpoint he and others take in this is somewhat homotopy-theoretical, in that the space of holomorphic maps between certain types of complex manifolds is homotopic to the space of continuous maps, and even a deformation retract. The conditions on the manifolds are roughly like 'having enough maps to C' and something dual to that. –  David Roberts Oct 27 '12 at 0:03
Indeed Larusson has but a simplicial model structure on the category of complex manifolds, and the conditions I allude to are expressed as fibrancy and cofibrancy, but I'd have to double check how it relates to the Oka principle(s). There are still a lot of interesting questions to answer. –  David Roberts Oct 27 '12 at 0:18
Of course I just scratched the surface of Oka theory, trying to answer a philosophical question. I am not an expert on it (although my background is in complex analysis); for reading about the developments you mention I can recommend Larusson article for Notices of the AMS: ams.org/notices/201001/rtx100100050p.pdf –  Margaret Friedland Oct 28 '12 at 1:52

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).

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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.

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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

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I just added the link (and removed the excuse for it not being there), and some dollars and backslashes –  quid Oct 25 '12 at 16:48
Thanks for the assist. Gerhard "Should Replace Smartphone With Computer" Paseman, 2012.10.25 –  Gerhard Paseman Oct 25 '12 at 17:35