Here's a very important question: Is math science? Please elaborate in the replies if you want!
@treppenverstand I said yes because it's somehow part of science but I more see it as a TOOL for science.
Just like a programming language is a tool for software development.
@treppenverstand I don't believe mathematics to be reflective of reality, and I believe it to be largely invented, hence why it is one of the few sources of objective fact
I believe that while mathematics is immensely useful and largely accurate to reality, I believe it is also effectively fabricated and is oftentimes the study of hypotheticals (see: Banach-Tarski, Gabriel's Horn, etc)
and I believe that while reality keeps cropping up things that are immensely similar to mathematics, I don't believe that to go on forever
@treppenverstand in short I believe a lot of mathematical questions to serve the same purpose as a lot of philosophical questions -- to study that which is unknowable or largely hypothetical
@treppenverstand no, because it's not empirical (if there are empirical parts of math, those count as both math _and_ science imo. but idk if there are)
@if A funny example that could fall into that weird space would be the first "proof" of the four color theorem for which they exhaustively checked finitely many finite problems the problem could be reduced to on a big computer. The empirical aspect was that back then, correct execution of the program was not as certain as nowadays so in the end, they couldn't actually be sure there weren't some rogue bit flips along the way. The whole history of that problem is super interesting if you find time to look into it!
@lm I guess the question could be rephrased more clearly as "Is the field of mathematics a scientific field?". I think that question certainly makes sense.
@treppenverstand i think that math is primarily a language for very specifically describing certain kinds of phenomena; mathematics is a bit more difficult to describe and certainly some of the phenomena we invent math words to talk about are scientific in nature. but i think there is more to mathematics than that.
for what it's worth, while i'm being contrarian, i think the notion that "math isn't physics because physics applies to physical things and math applies to ideas" is fishy: look around all you want, you'll never find "entropy" or "acceleration" as "real things"; physics is as much about physical stuff as it is about creating language and concepts to describe phenomena we observe, as is math.
@lm @treppenverstand Also, the problem with describing math as (just/merely) a language is only acknowledging the definitions it introduces, but not the deep connections between the notions which were defined.
The definitions are only a small technical aspect of what math ought to be about.
- Indeed say mechanics introduces the notion of a frame, momentum, and force, but not just for the sake of having a term: these are a means to define the first and third laws of motion, from which non-trivial conclusions may be drawn.
- And e.g. geometry is no different in this regard: it does introduce the notion of a point, a line, a plane, and the axioms; not just for the sake of it, but to show how to find center of a circle etc.
- Same for arithmetic: numbers are introduced not just to have them, but to describe the algorithms and how the geometric problems may be solved without geometric constructions themselves.
- Modern branches/fields follow this pattern as well: even category theory not only introduces (an awful lot of) definitions but also proves some lemmas with these, which are then utilized to show breakage of invariants, e.g. in algebraic topology.
@treppenverstand What's the difference between science and technology anyway, and why should it even matter?
@amiloradovsky No idea, but math and science seem quite different in terms of method and results to me.
@treppenverstand theorems are a specifications, their proofs are a constructions, theories are mental models, and even fundamental research is expected to bring practical benefits, otherwise it's being rightfully dismissed
@treppenverstand I tend to equate math with technology, and natural sciences are just that + hand waving.
@amiloradovsky What were the practical benefits of Fermat's last theorem? Or can we rightfully dismiss its proof as unimportant?
@treppenverstand IDK, but they're expected. I wouldn't dismiss it, or works on similar poorly motivated problems, although I may have reservations about whether it's a good use of one's own time to spend it with these. Exploring the area, not just solving an outstanding problem in isolation, that's the approach I would prefer.
Solutions for hard problems in general are expected to have a lot of practical applications, just because those are the boundaries we were able to identify, and therefore are somehow limited by those.
@wolf480pl @treppenverstand It's the case where the very hardness of a problem is the subject of interest (BTW, we're simply assuming there is no effective classical factorization algorithm). And it's not as much of number theory itself as of computational complexity that's involved.
The main contribution of NT is not the problems which were solved, but the tools developed within it (e.g. modular arithmetic, later ring and module theory: ideals etc.).
What I mean about the reservations is the priorities: there are stupid hard unmotivated problems (e.g. homotopy groups of spheres, general case), the solutions for which will produce a lot of noise, even if they're practically incomprehensible (too large, too poorly organized; much of the proofs is just trash and could as well be generated by a computer), but praising that will be a survivor's bias, because it won't account for all the unrecognized failed attempts where people have spent a lot of time without getting any results.
The issue with this kind of strategy is that it doesn't really let the researchers collaborate: either it's The Solution(TM), or it doesn't matter, thus isn't worth publishing~ And that's why developing the nearby area, instead of fixating on the hard(est) problems, is the way to go.
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