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Comment by edanm | Hacker News Reader

> When you say "given ZFC", you're assuming a lot.

Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing.

> In what sense do [numbers that can't be finitely specified] exist?

In the sense that we can describe rules that lead to them, and describe how to work with them.

I understand that you're trying to tie the notion of "existence" to constructability, and that's fine. That's one way to play the game. Another is to use ZFC and be fine with "weird, unintuitive to laypeople" outcomes. Both are interesting and valid things to do IMO. I'm just not sure why one is obviously "better" or "more real" or something. At the end, it's all just coming up with rules and figuring out what comes out of them.

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My point is that going from a lay understanding of mathematics to "just accept ZFC" means jumping past a variety of debatable philosophical points, and accepting a standard collection of answers to them. Mathematicians gloss over that.

Yeah, I think that's fair.

On the other hand, I think it's really cool to teach laypeople about things like "sizes of infinities", etc. They are deep math concepts that can be taught with relatively simple analogies that most people understand, and they're interesting things to know. I know that I personally loved learning about them as a kid, before I had almost any knowledge of math - it's one of the reasons that while I initially didn't connect with other areas of math, I found set theory delightful as a kid.

I just feel like if you need to first walk people through a bunch of philosophical back and forth on constructionism, you'll never get to the fun stuff.

We each find different things delightful. What I like, you may not. And vice versa.

But it is easy to present deep ideas from constructivism, without mentioning the word constructivism. Or even acknowledging that the philosophy exists.

For example the second half of https://math.stackexchange.com/questions/5074503/can-pa-prov... is an important constructivist thing. It shows why everything that a constructivist could ever be interested in mathematically, can be embedded in the natural numbers. With all of the constructions needing nothing more than the Peano Axioms. (Proving the results may need stronger axioms though...)

From my point of view, https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach does something similar. That book got a lot of people interested in basic concepts around recursion, computation, and what it means to think. Absolutely everything in it works constructively. And yet that philosophy is not mentioned. Not even once.

The only point where a constructivist need discuss all of the philosophical back and forth on constructivism, is in explaining why a constructivist need not accept various claims coming out of classical mathematics. And even that discussion would not be so painful if people who have learned classical mathematics were more aware of the philosophical assumptions that they are making.

> We each find different things delightful. What I like, you may not. And vice versa.

To be honest, I don't feel like I know enough about the constructivist philosophy. What would be a good place to start if I want to learn more about it?

I haven't yet read your PA proving Goodstein sequences article, though I have skimmed it and it is, indeed, super interesting.

And for the record, Godel, Escher, Bach was probably the single most important influence on me even starting to get interested in computation, etc.

> Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing.

ZFC (and its underlying classical logic) is precisely the problem here though

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