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Timeline for Believing the Conjectures

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Oct 26, 2012 at 13:22 comment added François G. Dorais I think you're right in your last comment. My criterion for negative maximize is not sufficient. There ought to be some reason to negate maximization in the formulation of the conjecture. Some more thought is needed here too...
Oct 26, 2012 at 13:06 comment added François G. Dorais Your general point is intriguing. I find miself disagreeing to a certain extent with both alternatives. I'll have to reflect more on that. Thank you, Tim, for this exchange. One of the indirect reasons for asking this was a seed of doubt in Maddy's rules. I'm glad you took the time to explain your own doubt.
Oct 26, 2012 at 13:02 comment added gowers Going back to your hypothetical Tsirelson story, I still don't see what Maximize adds to it. Why not just say that if you've tried hard to prove a conjecture, it becomes more reasonable to doubt it, regardless of what that conjecture looks like? (This doesn't apply to all conjectures: for example, our failure to prove the twin prime conjecture doesn't suggest that it might be false, since there are good heuristic reasons to expect both that it is true and that it is hard to prove.)
Oct 26, 2012 at 12:59 comment added gowers This discussion, which I find very interesting by the way, leaves me with the feeling that I don't understand very well what the rules of thumb are really saying and what their purpose is. I agree about Banach space theory: in some ways it is a very structureless subject, because any old bunch of functionals can be used to define a norm (as long as you've got enough of them that you don't have just a seminorm), but from time to time it springs surprises -- the almost negligible constraints nevertheless interestingly restrict what you can do.
Oct 26, 2012 at 12:26 comment added François G. Dorais (PS: It's not clear exactly how Banach space theory fits in Terry's dichotomy from the comments to the question. It seems to have elements of both sides.)
Oct 26, 2012 at 12:21 comment added François G. Dorais There was a remark about Maddy's rules that I deleted when posting the question. These are all ampleness rules, which makes sense since these rules were meant to justify large cardinal axioms, but they don't make sense in all contexts. These are not universal rules. As Terry pointed out, there seems to be an interesting dichotomy between fields that permit these ampleness rules and fields that permit their negative forms. (Aside: I wish I had a good set of positive terms for all the negative rules.)
Oct 26, 2012 at 12:13 comment added François G. Dorais [...] It's not clear (and it might never be) that maximize was the main reason to doubt the original conjecture, but that would make it a negative maximize conjecture (according to my definition above).
Oct 26, 2012 at 12:12 comment added François G. Dorais I think Tsirelson's space may be an example of doubting a negative maximize statement (sorry for the double negative). The conjecture (please correct my interpretation of history if it's wrong) was that "every Banach space contains $c_0$ or $\ell_p$." Then, probably after trying to prove this for a long time, Tsirelson decided to doubt the conjecture thinking that it's not as unlikely as he (and others) might have originally thought for there to be a Banach space containing no $c_0$ or $\ell_p$. That's a (modest) application of maximize. [...]
Oct 26, 2012 at 12:02 comment added François G. Dorais 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.)
Oct 26, 2012 at 8:42 comment added gowers 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.
Oct 26, 2012 at 8:41 comment added gowers Is Tsirelson's space an example of the success of Maximize? Again, I have my doubts. Before his example, it was reasonable to try to prove that every space did contain $c_0$ or $\ell_p$ -- all known spaces did, sometimes for quite non-trivial reasons. I would contend that it was only after (i) a failure, despite considerable efforts, to prove positive theorems and (ii) Tsirelson's example that it became reasonable to believe quite strongly that if you can't easily prove something about a general Banach space, then it is probably false. And there have been counterexamples to that principle ...
Oct 26, 2012 at 8:33 comment added gowers As for Maximize, I find it unhelpful in this context, since it is not clear what is "likely to occur". For example, it is still open whether there exists an infinite-dimensional Banach space such that every operator defined on it is a multiple of the identity plus a nuclear operator. I feel as though the answer could go either way, and no principle like Maximize is going to alter that perception. On the other hand, the existence of examples with comparable properties, such as Argyros and Haydon's construction where "nuclear" is replaced by "compact", does have an impact.
Oct 26, 2012 at 8:30 comment added gowers 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.
Oct 26, 2012 at 0:45 comment added François G. Dorais (Note that the above was "for the sake of argument" and not a genuine disagreement with Tim. Retrospectively, I think my interpretation of reflection is closer to Maddy's, though Tim's is perfectly reasonable.)
Oct 26, 2012 at 0:31 comment added François G. Dorais [...] Tsirelson's example is more interesting. I view it as a success of both maximization and reflection. First, the belief that there is a Banach space that does not contain $c_0$ or any $\ell_p$ is an example of maximize. The fact that there ought to be a separable example is reached through reflection since that is as small as such as space could be. To go from there to actually constructing such as space is a major leap, but these two initial steps seem to fit Tsirelson's own account of that leap: tau.ac.il/~tsirel/Research/myspace/remins.html
Oct 26, 2012 at 0:26 comment added François G. Dorais 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. [...]
Oct 25, 2012 at 22:41 comment added Joel David Hamkins 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.
Oct 25, 2012 at 22:16 history answered gowers CC BY-SA 3.0