This thread is about question 58880.

I'm perplexed why this was closed. The implicit "hopeful" part of Voevodsky's talk is that 2+2=4 is an empirical truth. It seems quite natural to ask how far does this empirical truth reach -- this is what this question is about... except for the quantum bend...

It is very hard to even formulate quantum theory without presupposing Peano arithmetic (and a lot more). Nevertheless on can ask whether quantum theory itself implies much of Peano arithmetic. As far as I know, this is completely unknown and it deserves some attention.

Albeit the question is not formulated in the way an expert in logic would, but, in any case, I think this is a valid and important question to which experts could provide some important input.

Mark,

Well, it's pretty obvious that apples are irrelevant to the question itself, so I'll disregard those critiques as irrelevant.

What does Peano arithmetic have to do with quantum field theory? That is pretty much the second question as stated by the op. The question "what does Peano arithmetic have to do with <fill in the blank>?" is pretty much the generic question of reverse mathematics... What's wrong with that?

I'm not entirely sure what "thisness" meant to the op, but the concept of "definite entity" has been very well studied by foundations experts and I can't think of any other meaning for "thisness."

Since I'm far from an expert, I can't say what defines quantum field theory (in a sense usable by logicians) but I'm confident that there is such a thing and that experts can define it rigorously if pressed to do so...

I don't really have a strong opinion on the question itself, but just wanted to point out something about people's attitudes towards it. It seems mathematicians have widely varying views on the importance, pertinence, and meaning of questions on foundational issues, especially in regards their "real world" interpretations. I hope that everyone can be respectful of these widely varying views, and tread lightly. I worry that we're prone to getting a bit emotional about all of this stuff!

Zev and Mark,

Speaking as someone with little knowledge of the mathematical side of this question, but a pretty reasonable education on the philosophical side, I think the attitude expressed here and in the comments of those who voted to close are really oddly dismissive of entire branches of philosophy.

The idea that we can't talk about apples in a serious way is oddly common in mathematics, but is really far from accepted as philosophical truth among those who actually think about these issues rather than simply dismissing them. I recommend Rota's essay "The Pernicious Influence of Mathematics on Philosophy" for a mathematician's take on this, but really, tons of very smart people write about "thisness" and observations of the real world in a pretty serious way, and dismays me to see that a large part of this forum believes the issue to be closed.

As for the question itself, it's certainly not very well-written, but I do think it is a real question, and likely of interest to many research mathematicians. I've voted to reopen.

DL, I'm not aware of any "accepted philosophical truth".

François and DL, my complaints about the first part of the question include the unexamined assumption that empirical consistency exists (i.e. "It nevertheless remains true that when you have 2 apples and 2 apples, you have 4 apples"), the unexamined assumption that empirical consistency (of experiments) has anything to do with mathematical consistency (of formal mathematical systems, chosen by humans, in a fallible way, to model those experiments), and the conceptual error about empirical consistency that Qiaochu points out (i.e. "... the result of an experiment, which cannot be inconsistent" doesn't make any sense). I also made the point that if the OP has doubts about the consistency of first-order arithmetic, even though (I assume) the people who developed it were doing so in order to capture some aspect of their empirical experiences, then I don't see why should the OP believe that the logical system he is proposing is consistent (much less "'necessarily' consistent") when it is based on exactly the same process.

The second part of the question is presented as a specific case of the first. However, asking that arithmetic model reality and asking that arithmetic model a (human-created) physical theory are different. Now, we instead get the problems of what the formal mathematical definition of "quantum field theory" is, and whether that itself is consistent (which seems like something to worry about before getting started on arithmetic consistently modeling it). Without those two things, I don't see how the second part of the question is reasonable. Finally, irrespective of how many very smart people do write seriously using the words "primitive thisness" or how open the issues regarding it are (neither of which I have any idea about), I remain to be convinced that it is in any way a mathematical concept.

Could you explain why you think that these issues have been addressed? or, why you think that they don't need to be?

Zev,

I claim only that the "assumption that empirical consistency exists" is not unexamined at all--there's tons of literature about it. So one shouldn't dismiss claims about it on the grounds that a question invoking this notion doesn't recapitulate the entire literature. Especially if the objection is that we can't define "apple," or cannot trust experimental observation, both issues which are really entirely unsettled. (I guess you've already observed there are *no* accepted philosophical truths; I'd say that there are actually plenty of extremely widely accepted truths, but we tend not to notice them anymore. In any case, this is a digression. By the way, people really do talk about "primitive thisness," but there are many less silly-sounding synonyms.)

As for formally defining QFT, that's a pretty thorny issue. But I would argue that neither this issue, nor the deep epistemic issues you raise, need to be addressed for this to be an acceptable question. There is some consensus that certain type of finitary reasoning (see e.g. Zeilberger's stuff) is reflected in nature, and a reasonable answer could address this. On the QFT (or even more crazy physics) side, there's good reason *not* to insist that we pin down such things rigorously; for example, physicists were able to compute certain Gromov-Witten invariants and discovered lots of cool things in symplectic geometry well before mathematicians, not to mention Witten's more recent work with Chern-Simons stuff and Khovanov homology, which is pretty non-rigorous but clearly awesome. Insisting on rigor, I guess, often gets in the way of discovery. And I guess it's reasonable to ask to what extent those non-rigorous results rely on rigorous work, and the extent to which they rely on e.g. the full strength of Peano Arithmetic.

In any case, there are some candidate rigorous approaches to QFT, see e.g. Borcherds's recent preprint here: http://math.berkeley.edu/~reb/papers/renorm/renorm.pdf

Zev, I've been trying to read the question from your point of view. I think I've found one key point where our readings diverge. From my point of view, the discussion on apples is simply motivation for the question and not part of the question itself. It never occurred to me to challenge the op's motives. I now see how "physically consistent" in the first question can be read as referring to these motives. I interpreted it differently after reading the second question where the term "physical theory" is introduced for the first time. To me, both questions were about analyzing the logical strength of physical theories.

François, yes, that sounds like a key difference. I was reading the first part of the question as, "What pieces of arithmetic do we know must be mathematically consistent because they model reality (which is empirically consistent)?", which I take issue with for the reasons I've described. I would propose asking the OP to clarify, but he doesn't appear to have ever followed up on his previous two MO questions, so I'm not optimistic about it.

I guess I'm resigned to the question remaining open, despite what I feel are numerous reasons "physically consistent subset of arithmetic" is meaningless. I have no doubt that someone (usually JDH?) will divine a meaning in the question, and post an answer that is amazingly informative and interesting, while the question's dubious merits are ignored. EDIT: this question is a perfect example of this phenomenon.

Minor comment: the "consensus" was most definitely not that history-of-mathematics is acceptable on MO. My recollection was that no consensus was reached. I'm much more inclined to allow philosophy-of-mathematics, but as usual I want to know:

1. What is the questioner's motivation for asking this question? In particular, how will it help with their research?
2. What are the criteria for assessing an answer to this question? That is, how can an answerer know that they are actually answering the question; and how can an interested observer assess whether or not an answer is valid?

Neither of these is visible in this question.

(And appealing to the fact that other dire questions are still open is never a good idea! It just makes me go back to the front page and vote to close all those ones again.)

Andy, your recollection is different to mine. The argument was conducted in a thread about a specific question, so many may not have seen it as such, and the voices heard split along "party lines" but with few from the "purist party" so I would guess that many who often agree with me didn't actually spot that thread. There was certainly no resolution, and none in favour of H-of-M actually put forward an argument as to why a "general interest" question in H-of-M should be considered equivalent to a research level question on, say, algebraic geometry.

Let me offer a compromise on this. If a question like this one, or a history-of-mathematics one, is research level then I would accept it on MO. That is, a person whose field is history-of-mathematics, or philosophy-of-mathematics, would accept it as a valid research question. I have not seen any evidence that either this question, or the "first collaboration" question, is of that ilk.

In defense of the history question, I can say that the question was completely reasonable, one of the answers contained a link to a research paper on that subject (along the lines from an_mo_user's comment). The question certainly was of interest to research mathematicians, and even trying to make it more precise (specifying what "joint research" meant 300 years ago, etc.) was interesting and non-trivial. On the other hand, the question we are discussing here is a meaningless collection of "scientific looking" words. It is as close to serious philosophy as to agriculture. Nevertheless, if five people want to see this question open, it is fine with me. I think MO can contain (limited amount of) garbage.

For the record, since I finally got around to registering for meta, I agree with Andrew Stacey.

And I agree with 83% of what Andrew says on meta.