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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.
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...
Well put, Mark. However, I feel there is something to be said about the first part of the question:
For the question to even make sense, rather large philosophical assumptions need to be made that I (and I imagine others too) take issue with, and in my opinion a real MO question should not require this. Qiaochu made this point in the comments: a single experiment is not consistent or inconsistent, but rather repeated experiments are consistent or inconsistent with each other; and the first problem I have with this question is that I don't see why repeated experiments couldn't be inconsistent with each other. For example: if tomorrow, we empirically observed that two apples and two apples made five apples (and I'm purposely not using numerals here, to emphasize that these are "real world two" and "real world five"), what exactly could we complain had "gone wrong"? Nothing. We might not like it if our current analogy between mathematical symbols and reality failed, but then, we never got a guarantee that it wouldn't. Secondly, even if one does take the view that the universe must be "consistent", there is still no reason to expect that the process of observing reality (already questionable provenance!), mentally processing, distilling, and abstracting it as humans are wont to do, and writing down symbols that we feel represent this mental abstraction, must preserve this "consistency". That is, even if you believe that two apples and two apples will always make four apples (or any statement about empirical facts, since "apples" are irrelevant), how is that belief supposed to grant you knowledge about the consistency or inconsistency of the purely formal system of glyphs you have written down? Indeed, I imagine people invented first-order arithmetic because they felt it represented or modeled their experiences with reality in some way - if first-order arithmetic really is inconsistent, how can we ever know that we're not making the same mistake they did?
(toned down after Scott's comment) So, in regards to the first part of the question, there are certainly some issues in the philosophy of mathematics that are raised, but I feel that MO is not the right place to discuss whether one philosophy of mathematics or another is "actually true" (as I think this question would require), precisely because it will engender more emotional responses.
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!
I think there is some confusion here. The role of empiricism in mathematics is a purely philosophical question, but this is not what the op asked about (though it is implicit in the formulation that empirical validities should correspond to mathematical validities). As far as I can tell, the op asked what part of Peano arithmetic is validated by quantum field theory (or other physical theories). This is a mathematical question, albeit "quantum field theory" (or other physical theories) needs to be rigorously defined in order to properly address it.
As far as I know, the answer to this question is presently unknown. The main problem is that all formulations of quantum field theory (and physical theories in general) that I have seen presuppose the validity of Peano arithmetic in some way or another. However, this does not necessarily invalidate the question. Indeed, one of the main issues in reverse mathematics is to correctly formulate the question. For example, the reverse mathematics of general topology is not very well understood simply because it is very hard to formulate what is a topological space without undue presuppositions. I suspect the same problem underlies physical theories such as quantum field theory, but to the best of my knowledge this has not been studied at all and this question illustrates one of the many gaps in our current understanding.
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?
Dan, I would absolutely agree that the history of mathematics is an acceptable subject for MO. But I'm curious as to what kinds of philosophy of mathematics questions you think can be definitively resolved, and why you think debating them on MO is going to generate more light than heat (keep in mind that we specifically advise people that MO is not for discussions). I enjoy thinking and arguing about the philosophy of mathematics plenty, but that doesn't also prevent me from being of the opinion that there isn't a right answer.
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.
I agree with an_mo_user here: math philosophy questions should be acceptable on MO provided they meet certain standards of consideration and expertise on the part of the questioner (and it is true that the majority of research mathematicians do not have philosophical expertise, even the ones who do not have outright disdain for mathematical philosophizing). This question strikes me not as a philosophical question but as a(n even to me...) philosophically naive question. Anyone who thinks that "two apples plus two apples equals four apples" is an unquestionably true physical fact is missing out on lifetimes of philosophical subtleties (Mill, Kant, Frege...). It's not a good question for an expert-level site like MO.
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:
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.
Regarding the history of mathematics, the thread under discussion is this one.
Looking over it, I see at least six people explicitly arguing that history of mathematics is appropriate on MO, while only one arguing that it's off topic. Various others were critical of the question under discussion, but of course that's not the same thing at all. I'll leave it to others to decide whether or not that constitutes a 'consensus' in favour of the history of mathematics.
I agree with 95% of what Andrew says on meta, including on the issues discussed here. I don't generally bother to post saying so, because it usually wouldn't contribute much to the discussion for me to write "I agree with Andrew" over and over.
And I agree with 83% of what Andrew says on meta.