godelian wrote: ↑Thu Jun 23, 2022 3:39 am
Immanuel Can wrote: ↑Thu Jun 23, 2022 2:36 am
That's again only half the story. The idea of a totally inapplicable mathematics is futile. Mathematics is vindicated not merely by its formal elegance, but also by its efficacy and accuracy in representing empirical situations.
Yes, but this efficacy is a concern only for scientists, engineers, and anybody else who desires to apply it to real-world situations. It is not a concern for mathematics.
But that's the point: maths isn't a thing that "just exists" somewhere out in an imaginary Platonic realm. It was always a product of empirical situations in the first place. It's a tool we use in order to get a handle on the empirical, and in any matter in which it remains only self-relevant, it's not actually relevant at all.
Science and engineering seek to be meaningful and even seek to be useful. Mathematics does not.
Mathematics does not "seek." It's not a person. It has no teleology of its own.
But what's interesting, and most remarkable, is that we live in a universe that can be rationally described through the tool, mathematics. That's really astonishing, if we suppose the universe is nothing but the product of a cosmic accident. Why shoulld we be in a universe that was amenable to rational analyses at all? And why should we human beings be capable of utilizing that tool, mathematics, to unpack the universe as we do?
...mathematics itself...strives to be meaningless and useless
You're anthropomorphizing again. Maths do not "strive" for anything. Nor do they aspire, intend, aim, labour, try, or long. If mathematics has any ultimate purpose, any teleology, it is not anything that the mathematics themselves tell us anything about. It has to be some sort of purpose external to the equations. And this is what makes maths seem "meaningless": that the meaning of the equations and symbols is not at all available from within that system of equations and symbols itself.
It comes from beyond them, if it comes at all.
Another anthropomorphism, surely. Maths do not "purpose" anything, anymore than they "strive."
After proper axiomatization, this source of inspiration became completely irrelevant.
Irrelevant to what
You mean "irrelevant to whether or not the maths can perform operations with their closed symbol system"? That might be true. But that's just a further indication that the teleology or purpose of mathematics does not come from within mathematics itself.
The correspondence theory of truth is therefore inapplicable to mathematics.
I think you misunderstand me. I'm not plugging for a "correspondence theory." I'm pointing out that the lack of a purpose or meaning for maths is a product of the fact that mathematics has no teleology discernable within the system itself.
The same could be said of any symbol system. They are tools employed by conscious entities, not self-justified, Platonic realities. Their purpose has to be discerned from outside the symbol system itself.
Maths has no internal claims about its own
purpose, goal or outcome. That does not mean maths has none.
Operator precedence is an issue that only exists in the infix notation, which is indeed ambiguous. It does not exist in the postfix or prefix notations, because these alternative notations are not ambiguous.
Infix notation along with the Eulerian notation for function application are costly conventions, because they are so ambiguous. These things tremendously complicate the construction of compiler front ends. If you switch to prefix notation, such as in Lisp, the compiler has a much simpler core. If you switch to postfix, such as in assembler, there is not even a need for a real compiler front end.
Sorry...you've lost me.
Perhaps you're into refined technical concepts here with which I lack familiarity.