“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.
“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.