Phil Scott said:
I mean that, on the face of it, it is Platonistic.
But I'd like to ask what you mean by "it is Platonistic". I thought it might be something like:
Platonistic-1: Assertions using that language, and in particular the assertion that "there exists a line", etc., imply (by the meaning of the words + classical? logic) that Platonism is true.
However, given that you're saying that the assertion does not commit Hilbert to Platonism, I'm not sure. Are you saying that it does not commit him because he said otherwise elsewhere, or because Platonistic 1 is not what you have in mind by "Platonistic"?
At any rate, I don't see anything Platonistic-1 about the language on the face of it, so if you mean Platonistic-1, I would like to ask why you find it so.
If you did not mean Platonistic-1 (and now it seems more likely that you did not), I'm not sure how to construe "it is Platonistic" in your sentence above. I can try some options, such as:
Platonistic-2: Assertions using that language, and in particular the assertion that "there exists a line", etc., provide strong evidence that the person is a Platonist.
But in that case (i.e., if you mean that), I disagree that it's Platonistic-2, and I would like to ask why you think it is so on the face of it. If you did not mean that, I could try:
Platonistic-3: Assertions using that language, and in particular the assertion that "there exists a line", etc., provide at least some evidence that the person is a Platonist.
Given that the language is pretty standard, and appears clearly compatible with Platonism, it seems extremely improbable that a Platonist would refrain from using that language. On the other hand, some non-Platonists think that the language is or in some sense looks "Platonistic", and refrain from using it for that reason
.
Hence (by Bayes' Theorem), if the only new piece of information that we add to our background evidence (as just described) is that a person uses the language in question, one should update upwards the probability that the person in question is a Platonist. So, given this background, it does seem to provide at least
some evidence that the person is a Platonist. However, the language is definitely not Platonistic-3
on the face of it. Rather, in order to realize it's Platonistic-3, one needs information about the existence of non-Platonists who prefer to avoid the language, no Platonists (very probably, or at least a minuscule percentage) who prefer to avoid it, etc.
In short, if by "Platonistic" you mean "Platonistic-3", then I would agree it's Platonistic, but disagree that it is so on the face of it. Moreover, if that's what you mean, then I think the use of the expression "Platonistic" to describe the language is misleading.
That said, I think it's very improbable that you meant that, in particular because of your assessment that it's Platonistic "on the face of it", so I'm still in doubt as to what you mean.
Granted, usually we don't need to define the expressions we use in a stipulative manner. Rather, we do so ostensively. So, if you prefer to define "Platonistic" ostensively, I'm okay with that.
Phil Scott said:
By contrast, Euclid's language is absolutely not Platonistic. I'm confused as to why you ignored the Euclid quote, given that my whole point here is to contrast the two ways of saying what purports to be the same thing.
I did not address Euclid's quote because I did not see how you were trying to get to that. I suppose one could say that Euclid's language is
incompatible with Platonism, since in a Platonic realm, one can't draw a line if one takes that literally. Is that what you meant?
Phil Scott said:
As for whether you can desugar the Hilbert quote to something else, and since you used an example from moral philosophy: I'm pretty sympathetic to non-cognitivist metaethics, while freely admitting that our moral language is, on the face of it, cognitivist and realist. I am perfectly happy to think that the surface realism and cognitivism of "factory farming is wrong" does not in any way commit me to a realist or cognitivist metaethics, and that it might just be a useful way to say "Boo factory farming." But I will readily admit that the language is both realist and cognitivist.
Without further input, I would have understood the claim that the language is cognitivist as implying that it is not a useful way of saying "Boo factory farming". Clearly, either you mean by "cognitivist" something I'm not familiar with (in which case, I'd like to ask for a definition), or you are saying that the language appears on the face of it cognitivist (in the sense I would understand "cognitivist"), but it might turn out not to be so. That, I can understand. But then, translated to the Platonism case, it would be something like Platonistic-1, and by saying that the language is Platonistic, you're saying that on the face of it, it is Platonistic-1, though it might turn out not to be so for some other reason. If that's what you're saying, as I mentioned, I don't see anything Platonistic-1 about the language on the face of it (but if I misunderstood what you mean by "Platonistic", please let me know what you mean and I will address that).
Phil Scott said:
If you're interested, there is no context to the Hilbert quote. These are foundational axioms for geometry, as Euclid's are. He explicitly does not want to be building on top of something like R3. There is philosophical commentary that he was trying to axiomatise Kantian a priori notions of space.
When one gives axioms, there is still a context. If - say - someone explains Peano's Postulates, the context generally is the set of natural numbers, already intuitively grasped at least to some extent by the people the explanation is meant to. The same applies to Hilbert.
Phil Scott said:
Mathematics and computer science require lots of mathematicians and lots of computer scientists: manpower. My preference for intuitionism means that I want more folk on the ground doing intuitionistic maths. However, I didn't arrive at my preferences by way of a philosophical argument, and I'd be suspicious of those preferences if I had, so I'm not going to attempt anything of the sort here. I think I'd get the job done quicker by gunboat.
Fair enough (I don't understand the "by gunboat" expression, and I would also suggest the argument would not need to be philosophical, but I think those are details in context; I get you aren't making that argument).
Phil Scott said:
Intuitionistists accept ¬¬P->P and (P v ¬P) when reasoning in any finite domain, and they'll often accept that reasoning in simple finite domains. Infinite domains of infinite domains only seriously appear in the 1800s, and immediately kick-off controversy as to whether reasoning about such objects goes awry. And it does, as shown by the set theory paradoxes. Intuitionism is one solution to those paradoxes. ZF set theory is another. I will happily argue that intuitonism is a more intuitive take on infinite domains than the hack-job that is the ZF axioms.
What I mean is that they don't accept ¬¬P->P or (P v ¬P) in all domains, which is very counterintuitive. The set theory paradoxes show that
some reasoning goes awry, not that all reasoning about infinite domains goes awry. ZF theory is an answer to the paradoxes in the limited context of set theory - not in all infinite domains -, and it has some advantages and disadvantages. But I think I will take you up on that offer (i.e., to argue that intuitionism is a more intuitive take on infinite domains that the ZF axioms are). At any rate, I'm saying that it's counterintuitive to reject (P v ¬P), etc., in general mathematic contexts, such as analysis (in many, or rather most of its variants).
Phil Scott said:
Classical foundations of analysis require AC. Intuitionistic foundations do not. We don't know which the physicists are using, because they largely don't care about mathematical foundations. It is a point up for debate, and mentioned in the video I linked, as to whether physicists implicitly assume classical or intuitionistic mathematics. I currently have little evidence either way.
They do not care about the foundations, but physics students, like math students, (at least implicitly) use AC when they learn math, and - more to the point -, they also use classical logic. That seems to come intuitively to them. It's not that they require it for physics, but rather, that it would seem weird to reject it (i.e., classical logic; I prefer to leave AC aside since it's another matter, though I also think it's intuitive).
Phil Scott said:
Here's an anecdote: I have a friend who got his physics degree from Cambridge, and one evening, I was berating physicists for not minding their epsilons and deltas, and thus ignoring potential pathological edge cases that are counterexamples to their reasoning. He responded quite fairly: we don't care about being too rigorous about epsilons and deltas, because we assume that, once you get to a small enough scale, the epsilons are literally 0. This isn't intuitionistic: it's basically finitistic.
I don't think it's finitistic (based on that quote; of course, you might have a lot more evidence about your friend), because physicists do not seem to assume that the number of galaxies, stars, etc., is finite. If anything, it's an assumption (perhaps justified, though I wouldn't bet on it) that space is discrete, there is a minimum actual distance, etc.
Phil Scott said:
As for the intuitiveness of excluded-middle, it's absolutely absent if you phrase things like Euclid does.
But it's intuitive when you don't restrict the language in that manner. What is counterintuitive is to reject it, as far as I can tell.