“if what our brains can do is computable then there is no way we can do mathematics that avoids both inconsistency and incompleteness.”
This sentence illustrates the formalist essentialism that I’m criticizing. If we consider mathematics as a social activity, as Brouwer did, then the notion of completeness doesn’t come up in the first place, and it’s useless to worry about such a thing. This perspective, in part, influenced Gödel to make his discoveries in the first place.
Much of the point of Hilbert’s program (and the wider goal of formalism/logicism) was to prove mathematics in entirety consistent by providing a formal logic which could be considered mathematics itself. Without that, there’s no meaningful sense in which mathematics is actually founded on a formal logic. After all, that would mean that everything outside of your chosen logic wouldn’t be part of mathematics, which is obviously wrong. After incompleteness was established, this situation was shown to be terminal. I think calling the whole project untenable after the publication of Gödel’s incompleteness theorems is a fairly reasonable read of history.
If what the universe does is computable then there is no way the whole community of mathematicians can do mathematics that avoids both inconsistency and completeness.
Now, of course you’re at liberty not to worry about completeness. Nothing wrong with that. But in that case I don’t see that you can fairly say that formalism is untenable on account of incompleteness. If it’s OK not to get answers to all mathematical questions then it’s OK for formalist mathematics not to deliver answers to all mathematical questions. You might contemplate a strong version of formalism one of whose tenets is “all mathematical questions must be soluble by these means”, but I claim formalism shouldn’t be committed to that.
I take it your last paragraph is suggesting that in fact formalism should be committed to that. I disagree, or more precisely I think formalism-without-that is prima facie a reasonable position. I don’t think I understand what you say about not being “part of mathematics”, because (1) something can still be “part of mathematics” even if the axioms you’re working with leave it open whether it’s true (one can still prove theorems like “if the axiom of choice holds, then X” even if working in a system that doesn’t decide AC) and (2) a formalist can still choose to work with different formal systems on different occasions, and regard both as part of mathematics.
“If whatthe universedoes is computable then there is no waythe whole community of mathematicianscan do mathematics that avoids both inconsistency and completeness. ”
I don’t think you understand what I’m getting at. It’s not that completeness shouldn’t be worried about, it’s that it doesn’t make sense if you aren’t already assuming that mathematics is a formal logic. If you worry about formal logic then you worry about completeness. If you don’t assume that mathematics is a formal logic, then worrying about mathematics does not lead one to consider completeness of mathematics in the first place. I’m saying that it does not make sense to talk about mathematics being complete or incomplete in the first place, since mathematics isn’t a formal logic. Yes, it’s impossible for the community of mathematicians to create a formal logic (of sufficient expressivness) which avoids inconsistency and incompleteness, but since mathematics isn’t a formal logic, that doesn’t matter.
“a formalist can still choose to work with different formal systems on different occasions, and regard both as part of mathematics.”
Implicit in that statement is the assumption that mathematics is not, at its core, a formal logic. Yes, it contains formal logics, but it isn’t one. I think you’re using a very weak (and very modern) definition of “foundation of mathematics”, being something capable of doing a significant chunk, but not all, of, mathematics. I think I’ve been clear in what I mean by “foundation of mathematics”, being something that should be capable of facilitating ALL of mathematics. My point is that such a thing doesn’t exist. If you disagree, feel free to argue against what I’m actually saying.
I do not take issue with the idea that one can do a significant chunk (perhaps most of in practice) mathematics using a formal logic. That logic would not then be mathematics, though. That’s all I asserted.
I’m getting the feeling that you didn’t read my post because you’re ascribing beliefs to me that I do not hold. I will quote myself;
“[...] many people seem to think that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic[...]”
That is the statement you’re taking issue with, yes? Do you think that mathematics is, in fact, a formal logic? If not, then you agree with me. Do you think that mathematics has to be founded on a formal logic? If not, then you agree with me. What are you actually disagreeing with? Are you going to support the assertion that mathematics is a formal logic? Are you going to support the assertion that mathematics has to be founded on a formal logic?
“You might contemplate astrongversion of formalism one of whose tenets is “all mathematical questions must be soluble by these means”
I don’t know what version of formalism you think I’m referring to, but my explicit reference to Hilbert should have clued you into the fact that I’m talking about Hilbertian formalism. I’d personally prefer it if you didn’t waste time arguing with a straw-man.
You feel like I’m strawmanning you. I feel like you’re strawmanning me. I propose that we make the obvious assumption that neither of us is deliberately constructing strawmen (I promise I’m not, though of course you don’t have to believe me) and see if we can come to a better understanding.
What follows is rather long-winded; I apologize for not having had time to make it shorter. I hope I’ve at least been able to make it clear.
1 What is “formalism”?
“Formalism” can mean a bunch of things. Let me list a few.
F0: Hilbert’s original programme of finding a single perfectly formalized system, simple enough that no mathematician could reasonably object to it, powerful enough to determine the answers to all mathematical questions.
It is (I think) uncontroversial that F0 turned out to be impossible. No one who is actually thinking about these things is a F0-formalist now.
F1: The idea that we should pick some single formal system (maybe ZFC, perhaps augmented by some large cardinal axioms or something) and say that mathematics is the study of this system and its consequences. (This implies, e.g., accepting that some mathematical questions simply have no answers. It doesn’t mean that we can’t talk about those questions at all, though; we can still say things like “X is true if Y is” where X and Y are both undecidable within the chosen system.)
Obviously F1 is hopeless if it’s taken to mean that mathematics always was precisely the study of the properties of ZFC or whatever, since there was mathematics before there was ZFC. But if it’s taken as a proposal for how we should currently understand the practice of mathematics, it’s defensible, and I think quite a lot of mathematicians think in roughly those terms.
(That’s compatible with saying, e.g., that later on we might decide to switch to a different formal foundation. And the best way to think about how that decision is made might well be in terms of mathematics-as-social-activity. But advocates of F1 might prefer to say that mathematics is a formal activity, but that philosophy of mathematics is a social one, and that what happens is that sometimes we switch for partly-social reasons from doing one sort of mathematics to another sort of mathematics.)
F2: The idea that mathematics is the study of formal systems, of which (e.g.) ZFC is just one. Different mathematicians might work with different and mutually incompatible formal systems, and that’s fine. Most mathematicians work at a higher level than the underlying formal system, but what makes the stuff they do mathematics is the fact that it can be implemented on top of one of these formal systems. The ancient Greeks didn’t have a decent underlying formal system, but what they did could still be layered on top of (say) ZFC and is therefore mathematics.
Versions of F1 that countenance the possibility of changing formal system are clearly shading into F2; what distinguishes F2 is that F2-ists are comfortable with the idea that multiple different systems of this type can be around concurrently and equally legitimate. (Think of it as shifting a quantifier. F1 says “there exists a formal system P such that doing mathematics = working in P” and F2 says “doing mathematics = having there exist a formal system P such that you’re working in P”. Kinda.)
I don’t claim that F0, F1, and F2 exhaust the range of things one could call “formalism”, but they seem reasonably representative. And I do claim that all of them can reasonably be called “formalism”. They both assume that what distinguishes mathematics from other activities is that it is grounded in formal-system calculations. They do, however, both admit that the choice of formal systems, and the expectation that their calculations are worth doing, may in turn rest on something else.
I think you may disagree with calling them “formalism”, since when I gestured earlier towards something like F1 or F2 you said: “Implicit in that statement is the assumption that mathematics is not, at its core, a formal logic” (etc.). Well, obviously what matters isn’t exactly what definition we should give to the word “formalism” but what sorts of positions mathematicians actually (explicitly or implicitly) hold, and how coherent and fruitful those positions are.
What you originally said was this:
Many of the discussed philosophical problems, as far as I can tell, stem from the assumption of formalism. That is, many people seem to think that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic, especially a classical one.
I don’t think “many people seem to think” F0, even though that was what Hilbert originally had in mind. And I don’t think F0 is required in order for philosophical questions about (e.g.) what numbers really are to arise. So that’s why I didn’t take you to be talking about F0, but about something more like F1 or F2. In the context of what “many people seem to think”, it seems to me that F0 is itself a strawman; not because no one ever embraced F0 (Hilbert did, and he was no fool) but because so far as I know no one explicitly does now, and I don’t think anyone does implicitly (in the sense that they think things that only make sense if one assumes F0) either.
It’s true that F1 and F2 allow for the possibility that one thing mathematicians do may be to change what formal system they study, and that they will do that on the basis of something not obviously reducible to formal-system calculations. But I can’t agree that that means that they aren’t truly formalist positions on account of not making “ALL of mathematics” be about formal systems; I think an F1-ist or F2-ist can perfectly well say that while choosing a formal system is something a mathematician may sometimes do, making that choice isn’t mathematics but something else closely related to mathematics.
2 Is “formalism” untenable because of the incompleteness theorems?
F0 is, for sure. F1 and F2, not so much. Someone may embrace F1 or F2 but be quite untroubled by the fact that mathematics (for an F1-ist) or the particular variety of mathematics they happen to be doing (for an F2-ist) is unable to resolve some of the questions it can raise.
Such a person’s position would be notably different from yours as I understand it—you say “it does not make sense to talk about mathematics being complete or incomplete in the first place”; an F1-ist would say it absolutely does make sense to talk about that, and as it happens mathematics turns out to be incomplete; an F2-ist would say that at any rate we can talk about whether a particular system we’re working in is complete or incomplete, and when doing mathematics we’re always working within some system and it’s always incomplete. (But they might e.g. say that we are always at liberty to use a different system, and that if we run across an instance of incompleteness that troubles us we can go looking for a system that resolves it; their position might end up resembling yours in practice.)
3 What do we actually disagree about?
Probably about whether philosophical assumptions made by “many people”, to the effect that mathematics is fundamentally about formalized (or at least formalizable) logic, are untenable on account of the incompleteness theorems. I think it isn’t, because positions like F1 and F2 involve such assumptions and aren’t made untenable by the incompleteness theorems, and I think those rather than F0 are the positions held by “many people”.
Perhaps about whether “many people” make assumptions that are more or less equivalent to F0. I think they don’t. Perhaps you think they do.
Perhaps about whether F1 and F2 really say that mathematics is fundamentally about formalized logic. I think they do. Perhaps you think they don’t.
You asked some questions that (I think) assume that I am endorsing “formalism” in some sense, even if not yours. That’s not what I’m doing—nothing I’ve said above is intended to claim that “formalism” in any sense is right. (I find F2 tempting, at least, but I’m not sure I would actually endorse it.) It is possible that it will turn out that I agree with you about, say, whether mathematics “has to be founded on a formal logic”, while still disagreeing about whether those poor misguided souls who think it does are taking a position that is untenable because of the incompleteness theorems.
I don’t know yet whether we disagree about the extent to which mathematics is a social activity. Perhaps we do.
“I think it isn’t, because positions like F1 and F2 involve such assumptions and aren’t made untenable by the incompleteness theorems ”
I think it’s ironic that you’re arguing with me over the meaning of a word, considering the content of my essay. I stated at the begining of my essay what I meant by “formalism”. If you don’t think that word should be used that way, that’s fine, but I’m not interested in arguing about the meaning of a word. By pretending that I’m arguing against any and all forms of what may be called formalism, you are replacing what I actually said with something else. That’s not a substantive disagreement with any position I actually endorsed.
“F1: The idea that we should pick some single formal system [...] ”
In my original quote I said “has” for a very specific reason. “Should” is a matter of opinion. I don’t think it’s unreasonable to choose a safe window from which to study the universe of mathematics, but one shouldn’t speak as if that window is the universe itself.
“I don’t think “many people seem to think [...]”
When someone states something of the form “mathematics turns out to be incomplete” they are ascribing properties of a formal logic to mathematics. When someone states that mathematics is an activity involving, on occasion, a decision “to switch to a different formal foundation”, they are ascribing properties of an activity which do not hold for formal logics. This is the central contradiction I’m fixating on. When I say “many people seem to think” I don’t mean that many people explicitly endorse, but rather that many people implicitly think of mathematics as a formal system. Saying “mathematics is incomplete” is a form of synecdoche, saying “mathematics” but meaning only a part of it. Failure to realize that this is being done leads people to say silly things.
″ F2: The idea that mathematics is the study of formal systems ”
“making that choice isn’tmathematics ”
A field of study can’t be incomplete in the way a formal logic can. Saying “mathematics is incomplete” is incompatible with the view that mathematics is a field of study, and yet I’ve seen many people endorse such a view. If you say that mathematics is the study of formal systems, I’d say that’s wrong, but that’s not relevant to any of my earlier points.
I think this might actually be the main point of disagreement. Making that choice involves mathematical reasoning and intuition which is certainly part of mathematics, not least because it’s part of what mathematicians, in particular, actually do. Excluding such things from being mathematics is arbitrary and artificial. If you’re going to make such a designation, then it seems the ultimate goal is to make mathematics mean “the study of formal systems”, but I have no interest in talking about such a thing. This is, again, arguing over a definition.
Incidentally, I stated that the position which was untenable after Gödel’s Incompleteness Theorems is the assumption, implicit in the statement “mathematics is incomplete”, that mathematics is a formal logic. However, that doesn’t appear as either your F0, F1, or F2.
I’ve found this discussion to largely be a waste of time. I won’t be responding beyond this point.
I am sorry that you haven’t found the discussion useful. For my part, I am also disappointed by how it’s turned out, and especially by how ready you seem to be to assume bad faith on my part.
Since obviously you don’t want to continue this, I won’t respond further except to correct a few things that seem to me to be simply errors. One: I am not (deliberately, at least) “pretending” anything, and in particular I am not “pretending that [you’re] arguing against any and all forms of what may be called formalism”. I thought I went out of my way to avoid making any claim of that sort. What I am claiming is that the things you said about “many people” apply only to “weaker” versions of formalism, while at least some of the objections you make apply only to “stronger” versions. The point of listing some particular versions was to try to clarify those distinctions. Two: Once again, although I am discussing and to some extent defending some kinds of formalism, I am not endorsing them, which much of what you’ve written seems to assume I am. Three: the position you actually said was untenable because of the incompleteness theorems was “that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic, especially a classical one” (emphasis mine), and it still seems to me that all of F0, F1, F2 say pretty much that.
Actually, I will say one other thing, though I’m not terribly optimistic that it will help. The main point I’ve been trying to make, though perhaps I haven’t been as explicit about it as I should, is that I think you are ascribing to “many people” a position more extreme, and sillier, than they would actually endorse, and that the bits of that position that lead to bad consequences are exactly the bits they wouldn’t actually endorse. E.g., the idea that mathematicians do nothing other than formal manipulation (of course they don’t, and everyone knows that, and no I don’t think the things people say about formal systems imply otherwise). Or the idea that if someone says “mathematics is incomplete” this means that they don’t know the difference between a field of study and a formal logic, rather than that they are saying that we should think of the practice of that field as in principle reducible to operations in a formal logical system, which is incomplete. Etc.
>If you don’t assume that mathematics is a formal logic, then worrying about mathematics does not lead one to consider completeness of mathematics in the first place.
To make sure I understand this right: This is because there are definitely computationally-intractable problems (e.g. 3^^^^^3-digit multiplication), so mathematics-as-a-social-activity is obviously incomplete?
No. I’m not advocating for some sort of finitism, nor was Brouwer. In fact, I didn’t actually mention computability, that’s just something gjm brought up. It’s irrelevant to my point. Mathematics is a social activity in the same way politics is a social activity. As in, it’s an activity which is social, or at least predicated on some sort of society. Saying that mathematics is incomplete is as meaningful as saying that politics is incomplete in the same way a formal logic might be. It just doesn’t make sense.
Note that the intuitions which justify the usage of a particular axiom is not part of an axiom system, but those intuitions would still be part of mathematics. That’s largely the intuitionist critique of “old” formalism. It was also used as a critique of logical positivism by Gödel.
I get that old formalism isn’t viable, but I don’t see how that obviates the completeness question. “Is it possible that (e.g.) Goldbach’s Conjecture has no counterexamples but cannot be proven using any intuitively satisfying set of axioms?” seems like an interesting* question, and seems to be about the completeness of mathematics-the-social-activity. I can’t cash this out in the politics metaphor because there’s no real political equivalent to theorem proving.
*Interesting if you don’t consider it resolved by Godel, anyway.
Mathematics is a social activity in the same way politics is a social activity. As in, it’s an activity which is social, or at least predicated on some sort of society.
Are you sayig that nothing a hermit would ever do can be called mathematics? That doesn’t seem right.
I’m pretty sure Anthony isn’t claiming that mathematics is social in the sense that every mathematical activity involves multiple people working actively together. But that hermit would be doing mathematics in the context of the mathematical work other people have done. Suppose the hermit works on, say, the Riemann hypothesis: they’ll be building on a ton of work done by earlier mathematicians; the fact that they find RH important is probably strongly influenced by earlier mathematicians’ choices of research topics; the fact that they find other things RH relates to important, too. Suppose they think of what they’re doing in the context of, say, ZFC set theory; there are lots of possible set-theoretic foundations (note: Anthony would probably prefer to avoid this notion of “foundations”) one could use, and the particular choice of ZFC is surely strongly influenced by the foundational choices other mathematicians have made.
(Anthony might perhaps also want to say, though here I don’t think I could agree, that if e.g. the hermit writes proofs then what those look like will be largely determined by what sorts of arguments mathematicians find convincing: that the point of a proof is precisely to convey ideas and their correctness to other mathematicians, which is a social activity even if those other mathematicians happen not to be there at the time. I don’t agree with this because I think our hermit might well pursue proofs simply for the sake of ensuring the correctness of their conclusions, and a sufficiently smart hermit might come up with something like the notion of proof all on their own with only that motivation.)
That’s a pretty low bar. Is wiping your ass a social activity too? Because, presumably, your mom taught you how to do it, and the fact you’re doing it with paper is strongly influenced by earlier ass wiper’s choices.
But never mind that. Suppose the hermit never learned any math, not even addition. Will you say that his math would still be social, because he already knew the words “zero”, “one”, “two”, which hint at the set of naturals? Then suppose that the hermit has not seen a human since the day he was born, was raised by wolves, developed his own language from zero, and then described some theory in that (indeed, this hermit might be the greatest genius who ever lived). Surely that’s not social. But is it not math?
Personally, I’d be perfectly happy to say that our hypothetical hermit is doing mathematics despite the complete absence of social connections; but I wasn’t endorsing the claim that mathematics is a social activity, merely explicating it. (And of course it’s possible that my explication fails to match what Anthony would have said.) I am not confident enough of my understanding of Anthony’s position to guess at his answer to your hypothetical question.
(But, for what it’s worth, if for some reason I were required to defend the mathematics-is-social claim against this argument, I think I would say that it suffices that mathematics as actually practiced is social; making political speeches is fairly uncontroversially a social activity even though one can imagine a supergenius hermit contemplating the possibility of a society that features political speeches and making some for fun.)
See this SEP section.
“if what our brains can do is computable then there is no way we can do mathematics that avoids both inconsistency and incompleteness.”
This sentence illustrates the formalist essentialism that I’m criticizing. If we consider mathematics as a social activity, as Brouwer did, then the notion of completeness doesn’t come up in the first place, and it’s useless to worry about such a thing. This perspective, in part, influenced Gödel to make his discoveries in the first place.
Much of the point of Hilbert’s program (and the wider goal of formalism/logicism) was to prove mathematics in entirety consistent by providing a formal logic which could be considered mathematics itself. Without that, there’s no meaningful sense in which mathematics is actually founded on a formal logic. After all, that would mean that everything outside of your chosen logic wouldn’t be part of mathematics, which is obviously wrong. After incompleteness was established, this situation was shown to be terminal. I think calling the whole project untenable after the publication of Gödel’s incompleteness theorems is a fairly reasonable read of history.
If what the universe does is computable then there is no way the whole community of mathematicians can do mathematics that avoids both inconsistency and completeness.
Now, of course you’re at liberty not to worry about completeness. Nothing wrong with that. But in that case I don’t see that you can fairly say that formalism is untenable on account of incompleteness. If it’s OK not to get answers to all mathematical questions then it’s OK for formalist mathematics not to deliver answers to all mathematical questions. You might contemplate a strong version of formalism one of whose tenets is “all mathematical questions must be soluble by these means”, but I claim formalism shouldn’t be committed to that.
I take it your last paragraph is suggesting that in fact formalism should be committed to that. I disagree, or more precisely I think formalism-without-that is prima facie a reasonable position. I don’t think I understand what you say about not being “part of mathematics”, because (1) something can still be “part of mathematics” even if the axioms you’re working with leave it open whether it’s true (one can still prove theorems like “if the axiom of choice holds, then X” even if working in a system that doesn’t decide AC) and (2) a formalist can still choose to work with different formal systems on different occasions, and regard both as part of mathematics.
“If what the universe does is computable then there is no way the whole community of mathematicianscan do mathematics that avoids both inconsistency and completeness. ”
I don’t think you understand what I’m getting at. It’s not that completeness shouldn’t be worried about, it’s that it doesn’t make sense if you aren’t already assuming that mathematics is a formal logic. If you worry about formal logic then you worry about completeness. If you don’t assume that mathematics is a formal logic, then worrying about mathematics does not lead one to consider completeness of mathematics in the first place. I’m saying that it does not make sense to talk about mathematics being complete or incomplete in the first place, since mathematics isn’t a formal logic. Yes, it’s impossible for the community of mathematicians to create a formal logic (of sufficient expressivness) which avoids inconsistency and incompleteness, but since mathematics isn’t a formal logic, that doesn’t matter.
“a formalist can still choose to work with different formal systems on different occasions, and regard both as part of mathematics.”
Implicit in that statement is the assumption that mathematics is not, at its core, a formal logic. Yes, it contains formal logics, but it isn’t one. I think you’re using a very weak (and very modern) definition of “foundation of mathematics”, being something capable of doing a significant chunk, but not all, of, mathematics. I think I’ve been clear in what I mean by “foundation of mathematics”, being something that should be capable of facilitating ALL of mathematics. My point is that such a thing doesn’t exist. If you disagree, feel free to argue against what I’m actually saying.
I do not take issue with the idea that one can do a significant chunk (perhaps most of in practice) mathematics using a formal logic. That logic would not then be mathematics, though. That’s all I asserted.
I’m getting the feeling that you didn’t read my post because you’re ascribing beliefs to me that I do not hold. I will quote myself;
“[...] many people seem to think that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic[...]”
That is the statement you’re taking issue with, yes? Do you think that mathematics is, in fact, a formal logic? If not, then you agree with me. Do you think that mathematics has to be founded on a formal logic? If not, then you agree with me. What are you actually disagreeing with? Are you going to support the assertion that mathematics is a formal logic? Are you going to support the assertion that mathematics has to be founded on a formal logic?
“You might contemplate a strong version of formalism one of whose tenets is “all mathematical questions must be soluble by these means”
I don’t know what version of formalism you think I’m referring to, but my explicit reference to Hilbert should have clued you into the fact that I’m talking about Hilbertian formalism. I’d personally prefer it if you didn’t waste time arguing with a straw-man.
You feel like I’m strawmanning you. I feel like you’re strawmanning me. I propose that we make the obvious assumption that neither of us is deliberately constructing strawmen (I promise I’m not, though of course you don’t have to believe me) and see if we can come to a better understanding.
What follows is rather long-winded; I apologize for not having had time to make it shorter. I hope I’ve at least been able to make it clear.
1 What is “formalism”?
“Formalism” can mean a bunch of things. Let me list a few.
F0: Hilbert’s original programme of finding a single perfectly formalized system, simple enough that no mathematician could reasonably object to it, powerful enough to determine the answers to all mathematical questions.
It is (I think) uncontroversial that F0 turned out to be impossible. No one who is actually thinking about these things is a F0-formalist now.
F1: The idea that we should pick some single formal system (maybe ZFC, perhaps augmented by some large cardinal axioms or something) and say that mathematics is the study of this system and its consequences. (This implies, e.g., accepting that some mathematical questions simply have no answers. It doesn’t mean that we can’t talk about those questions at all, though; we can still say things like “X is true if Y is” where X and Y are both undecidable within the chosen system.)
Obviously F1 is hopeless if it’s taken to mean that mathematics always was precisely the study of the properties of ZFC or whatever, since there was mathematics before there was ZFC. But if it’s taken as a proposal for how we should currently understand the practice of mathematics, it’s defensible, and I think quite a lot of mathematicians think in roughly those terms.
(That’s compatible with saying, e.g., that later on we might decide to switch to a different formal foundation. And the best way to think about how that decision is made might well be in terms of mathematics-as-social-activity. But advocates of F1 might prefer to say that mathematics is a formal activity, but that philosophy of mathematics is a social one, and that what happens is that sometimes we switch for partly-social reasons from doing one sort of mathematics to another sort of mathematics.)
F2: The idea that mathematics is the study of formal systems, of which (e.g.) ZFC is just one. Different mathematicians might work with different and mutually incompatible formal systems, and that’s fine. Most mathematicians work at a higher level than the underlying formal system, but what makes the stuff they do mathematics is the fact that it can be implemented on top of one of these formal systems. The ancient Greeks didn’t have a decent underlying formal system, but what they did could still be layered on top of (say) ZFC and is therefore mathematics.
Versions of F1 that countenance the possibility of changing formal system are clearly shading into F2; what distinguishes F2 is that F2-ists are comfortable with the idea that multiple different systems of this type can be around concurrently and equally legitimate. (Think of it as shifting a quantifier. F1 says “there exists a formal system P such that doing mathematics = working in P” and F2 says “doing mathematics = having there exist a formal system P such that you’re working in P”. Kinda.)
I don’t claim that F0, F1, and F2 exhaust the range of things one could call “formalism”, but they seem reasonably representative. And I do claim that all of them can reasonably be called “formalism”. They both assume that what distinguishes mathematics from other activities is that it is grounded in formal-system calculations. They do, however, both admit that the choice of formal systems, and the expectation that their calculations are worth doing, may in turn rest on something else.
I think you may disagree with calling them “formalism”, since when I gestured earlier towards something like F1 or F2 you said: “Implicit in that statement is the assumption that mathematics is not, at its core, a formal logic” (etc.). Well, obviously what matters isn’t exactly what definition we should give to the word “formalism” but what sorts of positions mathematicians actually (explicitly or implicitly) hold, and how coherent and fruitful those positions are.
What you originally said was this:
I don’t think “many people seem to think” F0, even though that was what Hilbert originally had in mind. And I don’t think F0 is required in order for philosophical questions about (e.g.) what numbers really are to arise. So that’s why I didn’t take you to be talking about F0, but about something more like F1 or F2. In the context of what “many people seem to think”, it seems to me that F0 is itself a strawman; not because no one ever embraced F0 (Hilbert did, and he was no fool) but because so far as I know no one explicitly does now, and I don’t think anyone does implicitly (in the sense that they think things that only make sense if one assumes F0) either.
It’s true that F1 and F2 allow for the possibility that one thing mathematicians do may be to change what formal system they study, and that they will do that on the basis of something not obviously reducible to formal-system calculations. But I can’t agree that that means that they aren’t truly formalist positions on account of not making “ALL of mathematics” be about formal systems; I think an F1-ist or F2-ist can perfectly well say that while choosing a formal system is something a mathematician may sometimes do, making that choice isn’t mathematics but something else closely related to mathematics.
2 Is “formalism” untenable because of the incompleteness theorems?
F0 is, for sure. F1 and F2, not so much. Someone may embrace F1 or F2 but be quite untroubled by the fact that mathematics (for an F1-ist) or the particular variety of mathematics they happen to be doing (for an F2-ist) is unable to resolve some of the questions it can raise.
Such a person’s position would be notably different from yours as I understand it—you say “it does not make sense to talk about mathematics being complete or incomplete in the first place”; an F1-ist would say it absolutely does make sense to talk about that, and as it happens mathematics turns out to be incomplete; an F2-ist would say that at any rate we can talk about whether a particular system we’re working in is complete or incomplete, and when doing mathematics we’re always working within some system and it’s always incomplete. (But they might e.g. say that we are always at liberty to use a different system, and that if we run across an instance of incompleteness that troubles us we can go looking for a system that resolves it; their position might end up resembling yours in practice.)
3 What do we actually disagree about?
Probably about whether philosophical assumptions made by “many people”, to the effect that mathematics is fundamentally about formalized (or at least formalizable) logic, are untenable on account of the incompleteness theorems. I think it isn’t, because positions like F1 and F2 involve such assumptions and aren’t made untenable by the incompleteness theorems, and I think those rather than F0 are the positions held by “many people”.
Perhaps about whether “many people” make assumptions that are more or less equivalent to F0. I think they don’t. Perhaps you think they do.
Perhaps about whether F1 and F2 really say that mathematics is fundamentally about formalized logic. I think they do. Perhaps you think they don’t.
You asked some questions that (I think) assume that I am endorsing “formalism” in some sense, even if not yours. That’s not what I’m doing—nothing I’ve said above is intended to claim that “formalism” in any sense is right. (I find F2 tempting, at least, but I’m not sure I would actually endorse it.) It is possible that it will turn out that I agree with you about, say, whether mathematics “has to be founded on a formal logic”, while still disagreeing about whether those poor misguided souls who think it does are taking a position that is untenable because of the incompleteness theorems.
I don’t know yet whether we disagree about the extent to which mathematics is a social activity. Perhaps we do.
“1 What is “formalism”? ”
“I think it isn’t, because positions like F1 and F2 involve such assumptions and aren’t made untenable by the incompleteness theorems ”
I think it’s ironic that you’re arguing with me over the meaning of a word, considering the content of my essay. I stated at the begining of my essay what I meant by “formalism”. If you don’t think that word should be used that way, that’s fine, but I’m not interested in arguing about the meaning of a word. By pretending that I’m arguing against any and all forms of what may be called formalism, you are replacing what I actually said with something else. That’s not a substantive disagreement with any position I actually endorsed.
“F1: The idea that we should pick some single formal system [...] ”
In my original quote I said “has” for a very specific reason. “Should” is a matter of opinion. I don’t think it’s unreasonable to choose a safe window from which to study the universe of mathematics, but one shouldn’t speak as if that window is the universe itself.
“I don’t think “many people seem to think [...]”
When someone states something of the form “mathematics turns out to be incomplete” they are ascribing properties of a formal logic to mathematics. When someone states that mathematics is an activity involving, on occasion, a decision “to switch to a different formal foundation”, they are ascribing properties of an activity which do not hold for formal logics. This is the central contradiction I’m fixating on. When I say “many people seem to think” I don’t mean that many people explicitly endorse, but rather that many people implicitly think of mathematics as a formal system. Saying “mathematics is incomplete” is a form of synecdoche, saying “mathematics” but meaning only a part of it. Failure to realize that this is being done leads people to say silly things.
″ F2: The idea that mathematics is the study of formal systems ”
“making that choice isn’t mathematics ”
A field of study can’t be incomplete in the way a formal logic can. Saying “mathematics is incomplete” is incompatible with the view that mathematics is a field of study, and yet I’ve seen many people endorse such a view. If you say that mathematics is the study of formal systems, I’d say that’s wrong, but that’s not relevant to any of my earlier points.
I think this might actually be the main point of disagreement. Making that choice involves mathematical reasoning and intuition which is certainly part of mathematics, not least because it’s part of what mathematicians, in particular, actually do. Excluding such things from being mathematics is arbitrary and artificial. If you’re going to make such a designation, then it seems the ultimate goal is to make mathematics mean “the study of formal systems”, but I have no interest in talking about such a thing. This is, again, arguing over a definition.
Incidentally, I stated that the position which was untenable after Gödel’s Incompleteness Theorems is the assumption, implicit in the statement “mathematics is incomplete”, that mathematics is a formal logic. However, that doesn’t appear as either your F0, F1, or F2.
I’ve found this discussion to largely be a waste of time. I won’t be responding beyond this point.
I am sorry that you haven’t found the discussion useful. For my part, I am also disappointed by how it’s turned out, and especially by how ready you seem to be to assume bad faith on my part.
Since obviously you don’t want to continue this, I won’t respond further except to correct a few things that seem to me to be simply errors. One: I am not (deliberately, at least) “pretending” anything, and in particular I am not “pretending that [you’re] arguing against any and all forms of what may be called formalism”. I thought I went out of my way to avoid making any claim of that sort. What I am claiming is that the things you said about “many people” apply only to “weaker” versions of formalism, while at least some of the objections you make apply only to “stronger” versions. The point of listing some particular versions was to try to clarify those distinctions. Two: Once again, although I am discussing and to some extent defending some kinds of formalism, I am not endorsing them, which much of what you’ve written seems to assume I am. Three: the position you actually said was untenable because of the incompleteness theorems was “that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic, especially a classical one” (emphasis mine), and it still seems to me that all of F0, F1, F2 say pretty much that.
Actually, I will say one other thing, though I’m not terribly optimistic that it will help. The main point I’ve been trying to make, though perhaps I haven’t been as explicit about it as I should, is that I think you are ascribing to “many people” a position more extreme, and sillier, than they would actually endorse, and that the bits of that position that lead to bad consequences are exactly the bits they wouldn’t actually endorse. E.g., the idea that mathematicians do nothing other than formal manipulation (of course they don’t, and everyone knows that, and no I don’t think the things people say about formal systems imply otherwise). Or the idea that if someone says “mathematics is incomplete” this means that they don’t know the difference between a field of study and a formal logic, rather than that they are saying that we should think of the practice of that field as in principle reducible to operations in a formal logical system, which is incomplete. Etc.
>If you don’t assume that mathematics is a formal logic, then worrying about mathematics does not lead one to consider completeness of mathematics in the first place.
To make sure I understand this right: This is because there are definitely computationally-intractable problems (e.g. 3^^^^^3-digit multiplication), so mathematics-as-a-social-activity is obviously incomplete?
No. I’m not advocating for some sort of finitism, nor was Brouwer. In fact, I didn’t actually mention computability, that’s just something gjm brought up. It’s irrelevant to my point. Mathematics is a social activity in the same way politics is a social activity. As in, it’s an activity which is social, or at least predicated on some sort of society. Saying that mathematics is incomplete is as meaningful as saying that politics is incomplete in the same way a formal logic might be. It just doesn’t make sense.
Note that the intuitions which justify the usage of a particular axiom is not part of an axiom system, but those intuitions would still be part of mathematics. That’s largely the intuitionist critique of “old” formalism. It was also used as a critique of logical positivism by Gödel.
I get that old formalism isn’t viable, but I don’t see how that obviates the completeness question. “Is it possible that (e.g.) Goldbach’s Conjecture has no counterexamples but cannot be proven using any intuitively satisfying set of axioms?” seems like an interesting* question, and seems to be about the completeness of mathematics-the-social-activity. I can’t cash this out in the politics metaphor because there’s no real political equivalent to theorem proving.
*Interesting if you don’t consider it resolved by Godel, anyway.
Are you sayig that nothing a hermit would ever do can be called mathematics? That doesn’t seem right.
I’m pretty sure Anthony isn’t claiming that mathematics is social in the sense that every mathematical activity involves multiple people working actively together. But that hermit would be doing mathematics in the context of the mathematical work other people have done. Suppose the hermit works on, say, the Riemann hypothesis: they’ll be building on a ton of work done by earlier mathematicians; the fact that they find RH important is probably strongly influenced by earlier mathematicians’ choices of research topics; the fact that they find other things RH relates to important, too. Suppose they think of what they’re doing in the context of, say, ZFC set theory; there are lots of possible set-theoretic foundations (note: Anthony would probably prefer to avoid this notion of “foundations”) one could use, and the particular choice of ZFC is surely strongly influenced by the foundational choices other mathematicians have made.
(Anthony might perhaps also want to say, though here I don’t think I could agree, that if e.g. the hermit writes proofs then what those look like will be largely determined by what sorts of arguments mathematicians find convincing: that the point of a proof is precisely to convey ideas and their correctness to other mathematicians, which is a social activity even if those other mathematicians happen not to be there at the time. I don’t agree with this because I think our hermit might well pursue proofs simply for the sake of ensuring the correctness of their conclusions, and a sufficiently smart hermit might come up with something like the notion of proof all on their own with only that motivation.)
That’s a pretty low bar. Is wiping your ass a social activity too? Because, presumably, your mom taught you how to do it, and the fact you’re doing it with paper is strongly influenced by earlier ass wiper’s choices.
But never mind that. Suppose the hermit never learned any math, not even addition. Will you say that his math would still be social, because he already knew the words “zero”, “one”, “two”, which hint at the set of naturals? Then suppose that the hermit has not seen a human since the day he was born, was raised by wolves, developed his own language from zero, and then described some theory in that (indeed, this hermit might be the greatest genius who ever lived). Surely that’s not social. But is it not math?
Personally, I’d be perfectly happy to say that our hypothetical hermit is doing mathematics despite the complete absence of social connections; but I wasn’t endorsing the claim that mathematics is a social activity, merely explicating it. (And of course it’s possible that my explication fails to match what Anthony would have said.) I am not confident enough of my understanding of Anthony’s position to guess at his answer to your hypothetical question.
(But, for what it’s worth, if for some reason I were required to defend the mathematics-is-social claim against this argument, I think I would say that it suffices that mathematics as actually practiced is social; making political speeches is fairly uncontroversially a social activity even though one can imagine a supergenius hermit contemplating the possibility of a society that features political speeches and making some for fun.)