tea.mathoverflow.net - Discussion Feed (What is the “Physically Consistent” proper subset of arithmetic?)

Scott Morrison comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13794)

Scott Morrison — Thu, 24 Mar 2011 14:04:10 -0700

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

Mark Meckes comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13792)

Mark Meckes — Thu, 24 Mar 2011 08:48:33 -0700

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.

Simon Wadsley comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13791)

Simon Wadsley — Thu, 24 Mar 2011 07:57:18 -0700

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

HJRW comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13731)

HJRW — Sun, 20 Mar 2011 18:10:16 -0700

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.

markvs comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13729)

markvs — Sun, 20 Mar 2011 13:34:12 -0700

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.

an_mo_user comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13728)

an_mo_user — Sun, 20 Mar 2011 12:16:41 -0700

Andrew Stacey, the 'research level' might be hard to judged. For example, if I remember correctly the 'first collaboration' question had an add-on question along the lines 'why was joint research uncommon' or 'why did it become common' or something like this. And, I could well imagine somebody writing a book or thesis on this question; most likely something like this even exists already. Actually, one could say this is way too broad as a question. For example, I know of recent manuscripts (two I believe, at least one published in a scholarly journal) analysing just the history of the collaboration of two mathematicians on *one joint paper* (Albert-Hasse). So, for certain A,B,C the mere question 'why did A and B write paper C' is a research question.

Perhaps, it is sufficient and more viable to insists on higher standards for motivation and presentation than for proper math questions. To put it bluntly, if such a question is too short/vague, a well-meaning observer can always say 'no, look at it in this way this is interesting', and actually be right (as documented for this question). Then, the research-level criterion becomes rather void.

Andrew Stacey comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13725)

Andrew Stacey — Sun, 20 Mar 2011 09:44:28 -0700

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.

Andy Putman comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13715)

Andy Putman — Sat, 19 Mar 2011 18:57:31 -0700

@Andrew Stacey : If I recall correctly, you were the only person arguing that history of math was inappropriate for MO. That sounds like as much of a consensus as one could get among mathematicians (but I might be remembering the thread inaccurately).

Andrew Stacey comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13707)

Andrew Stacey — Sat, 19 Mar 2011 11:08:56 -0700

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:

What is the questioner's motivation for asking this question? In particular, how will it help with their research?
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.)

Pete L. Clark comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13705)

Pete L. Clark — Sat, 19 Mar 2011 10:39:42 -0700

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.

Zev Chonoles comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13704)

Zev Chonoles — Sat, 19 Mar 2011 09:47:29 -0700

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.

an_mo_user comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13702)

an_mo_user — Sat, 19 Mar 2011 06:31:26 -0700

I cannot vote and most likely would not have voted to close, but still I feel that this question combines two features in an unforunate way.
Namely, the questions is philosophical and I cannot help but feel, and I might be wrong here in which case I appologize, that the questioner is an expert in none of the subfields this question involves (philosphy, maths, physics).

To make it short: personnaly I appreciate math.-philos. questions from experts (in a broad sense) in at least one of mathematics or philosophy. And, I also appreciate mathematical questions from non-experts (even if vague and not well-written). However, the combination of non-expert and philosophy starts to be a bit too much, as such things tend to be very vague and open-ended, and the outcome of such things that I saw on other places on the web (though I have no good concrete example, but might well be sci.math mailing list) is such that I'd prefer not to see it on this site, at least not too much.

François G. Dorais comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13701)

François G. Dorais — Sat, 19 Mar 2011 04:54:43 -0700

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.

dan petersen comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13700)

dan petersen — Sat, 19 Mar 2011 04:39:12 -0700

@Zev. I agree that very few questions on philosophy of math have a definitive answer. But many questions get posted to MO for which no absolutely correct answer is possible: on the front page we have
http://mathoverflow.net/questions/58554/preferably-rare-audio-video-recordings-of-famous-mathematicians
http://mathoverflow.net/questions/7627/set-theory-and-alternative-foundations
http://mathoverflow.net/questions/58870/what-should-be-taught-in-a-1st-course-on-smooth-manifolds
http://mathoverflow.net/questions/37610/demonstrating-that-rigour-is-important
and arguably several more, depending on how strictly you interpret a lack of an objectively right answer.

DL comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13699)

DL — Sat, 19 Mar 2011 01:07:04 -0700

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 Chonoles comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13698)

Zev Chonoles — Sat, 19 Mar 2011 01:01:36 -0700

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 Chonoles comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13697)

Zev Chonoles — Sat, 19 Mar 2011 00:39:01 -0700

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 petersen comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13696)

dan petersen — Sat, 19 Mar 2011 00:38:43 -0700

I would be very happy to see more questions on philosophy of mathematics on MO and I have voted to reopen although the question was poorly written.

This discussion has some resemblance to the debate a couple of weeks ago on whether history of mathematics is appropriate for MO questions, where I think the consensus was that it is.

DL comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13694)

DL — Fri, 18 Mar 2011 23:13:32 -0700

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.

François G. Dorais comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13692)

François G. Dorais — Fri, 18 Mar 2011 22:22:40 -0700

Zev,

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.

Scott Morrison comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13690)

Scott Morrison — Fri, 18 Mar 2011 21:52:50 -0700

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 Chonoles comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13689)

Zev Chonoles — Fri, 18 Mar 2011 21:45:54 -0700

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.

François G. Dorais comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13686)

François G. Dorais — Fri, 18 Mar 2011 21:21:08 -0700

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 ?" 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...

markvs comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13685)

markvs — Fri, 18 Mar 2011 20:53:15 -0700

I voted to close and I think the question can't be saved. For example, what is "apple"? How do you define it? And what does quantum theory have to do with Peano arithmetic or apples? And what is "quantum theory"? And the word "thisness" crowns all that. I think that the question is just a senseless collection of speculations which cannot be rigorously formulated.

François G. Dorais comments on "What is the “Physically Consistent” proper subset of arithmetic?" (13684)

François G. Dorais — Fri, 18 Mar 2011 20:16:29 -0700

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.