> certainly you have to be able to do mathematical reasoning about natural numbers, but that doesn't necessarily imply determining their ontological status

That's true, of course you can use something without understanding it, but can you use something, and then deny its existence?

I want to correct what I said earlier about the natural numbers and infinity. Of course you could add infinity to the natural numbers, if you wanted to. This can be done in different ways. It's just that natural numbers without infinity are a simpler starting point.

if i use monopoly dollars to buy the b&o railroad, does that entail that i believe that monopoly dollars exist? surely not in the same platonic sense that you and i believe that 53 exists. monopoly dollars are, i think uncontroversially, merely a socially constructed consensus hallucination, like national borders, google, or adultery; as far as i can tell, formalists claim that 53 is too. for that matter, subjective idealists like berkeley claim that so are physical objects like electrons

i haven't seen a satisfying formalist explanation of why mathematicians of different cultures come to identical conclusions about 53 but vastly different conclusions about national borders and adultery. (borges' tlön, uqbar, and orbis tertius famously makes a similar point satirically about subjective idealism.) but i am forced to admit that the formalists themselves do exist and do make this assertion about 53, and while many working mathematicians are unwilling to wholeheartedly embrace formalism (and even fewer constructivism, though, as you point out, for example in type theory constructivists are ubiquitous), fairly few of them are willing to wholeheartedly reject it and declare certainty about platonic realism, as you are

what i was saying about the natural numbers and infinity is that the "common sense" of many children (though perhaps not your own childhood self) insists to them not that a maximum natural number is logically consistent, which i agree with you that it is, but that it is logically necessary, which of course the peano axioms reject. this, to me, is good evidence that common sense is not a solid enough foundation for reaching mathematical truth

by the way, i want to express gratitude for your willingness to discuss epistemology and ontology without descending into the kinds of vicious personal attacks i generally expect on this website, or more generally when discussing epistemology

i see that hn included trailing punctuation in this link that i posted above, causing it to fail: https://plus.maths.org/content/philosophy-applied-mathematic...

A most interesting book in this context is [1]. It has introductory discussions of logicism, intuitionism, and formalism by Carnap, Heyting, and von Neumann, and interestingly, they were not aware yet of Gödel's results at that point which were published in the same year. After that, Heyting's "Dispute" is a nice exposition of the different view points, and I find myself agreeing with much of the intuitionist point of view, namely that mathematical reasoning cannot be reduced to formalism (but I don't agree that intuitionistic reasoning is the only allowed form of mathematical reasoning). I mean, Gödel's incompleteness result is as hard a proof of this as you will ever get! And again, in a later piece by Paul Bernays, "On platonism in mathematics", Bernays states: 'Is it possible to draw an exact boundary between what is evident, and what is only plausible? I believe that one must answer [this question] negatively'. In other words: Common sense is a necessary part of mathematical reasoning. Fundamentally, there is no way around this.

[1] Philosophy of Mathematics, Selected Readings. https://doi.org/10.1017/CBO9781139171519

thank you very much!