/sci/ - Science & Math

>>9106321
don't have an answer for you mate, just bumping out of interest. been wondering about the same thing lately.

>>9106321
Isn't that what second order logic is?

I mean, I'm sure a pure maths proof could be understood without human input. However more applied maths like combinatorics would be a little more difficult(??)

>>9106356
>second order logic
What I'm saying is, all of these math concepts require guides, teachers, and human spoken language instructions to comprehend. Nobody opens up a textbook on second order logic or any mathematical concept, and it becomes apparent that it is correct and what the meaning of the proofs are without human language explanations... Is this okay though?

What I'm asking, is:

Do we require a mathematical language that we could share with sufficiently advanced aliens who have vision, and they would be able to read it and comprehend it without much effort and realize that it is math and that we are correct, and quickly and just intuitively comprehend it? Is that something we need for mathematics to be well founded? Is it useful?

>>9106369
>Textbook on second order logic
>Aliens
I'm going to bed

>>9106381
I'm not talking about "aliens" per se, I'm talking about the generalization of intelligence and mathematical reasoning into an ideal form, a form which requires no cultural predispositions, or incidental knowledge, or knowledge of the results of incidental events.

For instance, there's no good reason a "+" sign should represent addition. And there's no good reason why elementary arithmetic should be organized into "+" and "-" first and foremost, then "*" and "/". There's nothing more or less elementary about this. Breaking a stick into what seems to be half is a more "primitive" action than breaking off a part of a stick equal to the length of another stick, in some senses.

This is not to say that there is anything inherently higher-echelon about multiplication and division in comparison to addition and subtraction, but that this is the method our culture uses to teach these concepts. And then you go up to higher and higher level abstractions

What I'm saying is, are these higher level abstractions actually higher? Or is there a way to conceptualize mathematics, with regards to the language used to put the points across, which would firmly mount complex topics like calculus into the minds of a sentient being initially, and from calculus you then abstract off to discrete addition and subtraction?

I feel like these are reasonable questions but I feel that they can easily also be misconstrued as crackpot rambling.

Anyway thanks for the replies anons. Hopefully someone sees what I'm talking about

>>9106321
Why would you think people have fewer predispositions towards shapes than words?

>Do we require a mathematical language that we could share with sufficiently advanced aliens who have vision, and they would be able to read it and comprehend it without much effort and realize that it is math and that we are correct, and quickly and just intuitively comprehend it?
No. Pic kind of related though.

>>9106419
With arabic numerals though there's nothing "about" them that gives their meaning. You can't figure out what 3 means just by looking at it, unless you're also given the rest of the digits and their relative ordering.

>>9106420
There's also nothing about any picture that necessarily gives it a mathematical meaning. Three dots doesn't have to mean three, it could mean anything. All that matters is that the meaning is obvious to the reader.

>>9106419

1 3
3 2

>>9106321
t-minus 3 dots before this discussion spirals into the numerical equivalent lots-of-dots-represented-by-succinct-symbols representing the many other topics this can devolve into

Mathematics is a game of symbol manipulation. At its lowest level, the (few) axioms are combined to form other statements. These statements are 'true' in the sense that they are tautologies.

Alien mathematics can therefore be completely different, depending on which axioms they choose. However, all we know is that they share the same universe with us. Some sets of axioms can be shared because they describe physical processes. If aliens encounter discrete objects, it is possible they have concepts like natural numbers.

Anyway, communication through pure mathematics is very likely futile. Only the combination of physical reality with mathematics can lead to meaningful exchange.

OP, must most "proofs" are not rigorous as you point out.

Math can be formalized to a syntactic system: a set of symbols and symbol manipulation rules, so that a computer can do without any insight of its semantics.

Read up about Hilberts program, Godels theorems, Computability and Automata Theory, and the computational and mathematical approaches to Logic.

Read about proof assistants, like Coq.