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.
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.