> Does it even matter if Math is invented or discovered?
It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
And if you believe art is about creativity, you can find beauty and art in in mathematics and other sciences.
Many great mathematicians were also art lovers, they appreciated music, they appreciated paintings and other forms of art.
I don't think there's a dichotomy between art and sciences.
The significance of the creation (if it has any at all) is not necessarily the concern of the creator.
Some people do Mathematics for its sheer beauty.
You could say anything about anything.
> no course of action could be determined by a rule, because any course of action can be made out to accord with the rule
If I had my own way of (mis?)interpreting him, I think he was alluding to what we now call "strict evaluation" in Programming Language Theory. A diligent rule-follower.
The significance of the creation (if it has any at all) is not necessarily the concern of the creator.
Some people do structural engineering for its sheer beauty.
The significance of the creation (if it has any at all) is not necessarily the concern of the creator.
Some people do [Scientific discoveries|Mathematics|Ethics|Political science] for its sheer beauty.
It backfired on you.
It contains an unjustified assumption that “logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics" and [other?] art belong in different categories.
It also takes an incomplete view on "critical thinking". Drawing distinctions is complemented by abstracting similarities.
The creation of knowledge (in all its forms) is itself a form of artistic self-expression. It is essential to humans, and therefore essential to society.
As a programmer my medium of self-expression is software. I am an artist as much as I am a logician and a scientist.
What I do is create. It also happens to be useful to others, which is why it pays fucking well too.
> It matters for people who find it valuable, the same way art, music or literature matters.
Key phrase "the same way".
There's nothing wrong with being wrong, yet you strike me as a person who doesn't like it.
A major movement (or several, depending on how you slice it) in modern philosophy takes aesthetics as first philosophy - so yes, arguably they matter exactly the same way art does.
It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?
It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.
At the very least, we know it isn’t parsimonious, though that wasn’t clear until after it was developed (and the philosophical inclination to preserve locality was reasonable).
The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.
Your entire argument attempting dismiss philosophy is philosophy.
I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.
Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...
I was thinking more of tangible implications. Maybe there are some.