Hilary Putnam is dead. I read through the Wikipedia page and found that he has changed his stance on the ontology of mathematics: "How real are mathematical things?"—like integral numbers, real numbers, functions, sets, deduction rules, proofs, whatever? For some reason, this makes me nervous: I thought, first of all, that this question is more or less settled—I might not know the answer, but some great philosophers—like Putnam—certainly did and do. But even if they do not, I thought that the various possible answers are so clearly described that you would not switch from one to the other. Rather, you would hold to one conviction and argue why the other ideas were wrong, or irrelevant, or not sufficient, or deficient in some other way.
And in the first place, I thought that the accepted answer is clearly that mathematical objects are in some sense "real," at least in the (or some) sense that mathematical object are "out there" even if no one is in this world or the whole universe who actually does mathematics: See e.g. Linnebo, Øystein, "Platonism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.).
But, in the second place, I know that I somehow feel confident that this is not true: That mathematics is a purely human construction, or at least a construction of minds doing mathematics. In other words, there is no mathematics without mathematicians.
This is, altogether, unsatisfactory. I am probably wrong, or vague enough so that my ideas count as wrong—but why? And what is a better answer?
I could set out to find the answer—the answers?—in the books. But I fear that I would quickly lose my own ideas, being overwhelmed by the precise arguments of the professionals. And I am not yet there. I want to know why, and how, I am wrong, not what is right.
So—I'd like to think a little bit on my own, probably with a huge bunch of implicit assumptions, at least some of which might become explicit in this process: And only then see whether this matches with someone else's ideas—and then, ideally, find refutations for these! I do not seem to be too humble, probably.
I start out with realism, about what I think Popper said: The actual material things in this world are real. Many of them (at least all the atoms) were here before the first humans and will be here after all of mankind has gone, for whatever reason. I must be careful here: That things were here does not mean that they were identifiable as things. That it is possible to classify certain of the "things in the world" as being similar, and hence be able to call them atoms, stars, trees, might already be mathematics, at its core. So I should probably rather say: The whole world, the universe was, is, and will be there independently of humans and other beings with a reflective, thinking mind. But actually, this has not solved anything: This universe changes. So even talking about a continuing existence or "behaving" of something that—by being continually there—can be seen as (or thought of?) as being something singular might be doing mathematics. So, what is this "realism" actually?
I did not think I would stumble so early. I thought that I could deal with the physical world in a paragraph or two, and then proceed to mathematics. Great. Should I now consult all the books and papers on realism to find out about how one, and we, and I should think about these "real," "observer-independent" world? Yes, I should. But I don't know where to start, and how to make sure I am not again drowned by other ideas and arguments and refutations.
Clueless.
But there is something (that's my given which I would not want to argue about). And there happens something: Some energy gets transformed—yes, we might call it a lightning stroke or a nerve impulse, but even if we would not categorize it (risk: this is already math), it was there. So I have to talk about it without categorization. I try the concept of a "description": This is some more or less precise notation that can be mapped to that world. And a description of something is "good" or "good enough" if someone else who reads the description will agree that it is a description of what it claims to describe.
Now, I have planted myself deep in a big hole full of shit: I wanted to find out what "real" means as in "something is real if it exists independently of a mindful observer"—and I end up answering this by introducing "more or less" precise "notations" that are "read" by "someone" who "agrees" to a "claim." Absolutely ridiculous, isn't it?
First, yes.
Second, I have not better idea (read the books, someone says—yes, I will, um, soon).
Third, I think I can defend at least a bit of this, at least with my target in mind, namely to show, or at least argue, that mathematics is man-made. See, descriptions, in the sense above, can be taken by suitably capable whatevers and compared to the world. And if many such whatevers do that and will e.g. converge in their predictions of what happens next, this is a good description.
Kuhn, and others, would tell me that "compare" and "converge" and "predict" are totally theory-dependent concepts. For my train of thought, however, I want to believe that they are not "totally arbitrary" in the sense that what in many discourses is "converging" will be "not converging" in ("similarly") many other discourses, but that in "most" discourses the predictions will converge. Maybe, to agree on this, it is also necessary (for the "whatevers" that do the reading and comparing and agreeing) to have the descriptions of what converging means, which again are compared and do converge, and maybe a whole hierarchy of descriptions of comparing and converging is necessary. All this would most probably be doing mathematics—but this is only an accident in my description of "realism"—it has, right now, nothing to do with my target problem of, more or less, how mathematical things come into the world. Rather, I leave it totally to the whatevers to do that as they like, and not think about what they do. But I suppose that they will agree, as we humans do over time.
The important aspect behind this notion of realism is that it is fuzzy: It is enough for me if some descriptions of some events are somewhat good. No more can be hoped for before starting with mathematics: No more is found by us and anyone else when looking into the world. And for at least thousands of years, our ideas about that reality were, for concrete aspects, completely wrong; but many were somewhat ok, and this alone did not keep us from doing math. So, it must be possible to base mathematics, at least as a human activity, on such a shaky and largely wrong view of the world. The question is whether all of mathematics must be based so. I think—or rather: feel—that yes, from these muddy waters, mathematics can be "argued up." But I have to argue this a little more.
I have to think about this.
