/sci/ - Science & Math

That's why mathematicians first show a bunch of shit from stronger theorems and branches, and see what properties are actually needed to prove them, and generalize to spaces were you have less restrictions. But there's no branch of mathematics that contradicts anything you can check empirically. Go and measure right angle triangles like an idiot and see if pythagoras was incorrect. Or better, go and compute areas under a curve numerically and see if you find a curve that follows the requirements of the fundamental theorem of calculus and see if it's incorrect. Mathemstics is inherently grounded in what we can observe and everything must work for these cases so that your system is consistent.

>>9537573
>this hypothesis kind of emulates part of reality
yep, guaranteed pure mathematical truth

>>9537580
No, the rigorous truth is in the system, but I'm saying there's nothing in our physical world up to know that contradicts mathematics and that mathematicians take counterexamples as more fundamental than proofs. Our most modern physical theories are exposed through mathematics, so there's no doubt it more of "kinda emulates reality".

>>9537585
*much more than

>>9537585
>rigorous

>>9537593
XD, eric

>>9537561
>>9537580
you couldn't even prove you existed if you were defined to

More like
>observe and analyze the real world
>identify as-simple-as-possible invariants
>formulate a general abstract version of them
>postulate those as axioms
>explore what your theory predicts using those and the rules of classical or intuitionistic logic
>test whatever prediction is testable
Really deactivates those neurons, doesn't it?

>>9537601
>if I can't disprove x it has to be fact

>>9537561

You can prove anything only if your axioms contradict eachother

>>9537561
>you can now prove or disprove anything
Yeah, but if your axioms are just random bullshit you threw together then you probably won't get much use out of your system.
In fact if you read up on the axioms that actually get used for the majority of modern mathematics you'll see they're pretty insanely conservative and assume as little as is reasonably possible while still allowing for useful system to be built from them.

>>9537619
More like
>If x has fitted all of our observations despite attempts to refute it, it has to be a good approximation of reality

>>9537636
wrong
>>9537641
prove that any axiom is more true than any other axiom

>>9537646
>find a pattern that (probably) approximates a tiny part of reality
I FOUND A NEW MATHEMATICAL LAW GUISE

>>9537652
More like
>find a pattern that generalizes empirical data without contradicting it
I FOUND A NEW INTERPOLATION OF FACTUAL INFORMATION GUISE

>>9537652
Mathematical laws are independent of emirical observation. The argument is that since you seem to not trust pure rationalism, well that leaves empiricism and the point is neither these two schools of thought contradict mathematics.
While this doesn't exactly is the ultimate proof of the consitency of mathematics, further skeptisim can't bring up alternatives or proof of math inconsistency. For all I care you could also say
>hurr durr logic isn't necesarilly true
And you may be correct, but disecting everything till you arrive to solipsism is a zero sum game. So unless you bring me a counterexample in math or a contradiction with empirical patterns, I'm going to keep learning and trying to support this wonderful field.

>>9537656
Ah yes, who could forget Newton's three interpolations of factual information, Avogadro's interpolation of factual information, or Tarski's interpolation of factual information?
No mathematician has ever said that, you pink brainlet wojak. Probably because that isn't even the correct usage of interpolation. We're discussing generalizations, not data sets.

>>9537648
>prove that any axiom is more true than any other axiom
I clearly wrote "useful," not "more true."
Come up with an axiomatic system more useful than ZFC and then everyone will use that one instead.

>>9537666
You're going to keep learning what, exactly? Do you intend to memorize bits of the speculative construct some people use to describe reality in hopes of finding knowledge or truth?
protip: you won't

>>9537678
Useful implies true, brainlet, unless you want to invent some mathematics with no bearing on reality.

>>9537674
Newton formulated his mechanics and gravity theories out of empirical data of the orbits of planets. The basic laws of ideal gases (Mariotte, Avogadro, etc.) were derived from empirical observations and parsimonious assumptions that hadn't been refuted. All the logic tools are based on patterns studied by philosophers since ancient Greece.

>>9537679
No, I intend to learn different topics because it's pretty fucking cool and interesting and it a way I can do work doing something I love.
>in hopes or finding knowledge or truth
Not in your pedantic way, but knowledge into a lot of abstract problems within that preconsived system of truth, and I'm not too troubled about justifying why I want to ground myself into these kind of problems/truth systems.

>>9537688
nice goal shifting

>>9537681
>Useful implies true
No it doesn't.
>no bearing on reality
"Not true" doesn't mean "no bearing on reality."
Most thought experiments aren't true (e.g. there's no actual Chinese Room and Einstein never really observed phenomena around him while moving at the speed of light) and the whole point of them is to make a point about reality.

>>9537695
Nice strawman.

>>9537681
>>9537696
And incidentally this works back to the original point pretty well that axioms are about usefulness, not truth. Truth is a concern for what you're doing within the system, but the axioms themselves aren't anything you prove true by definition. You pick them based on utility. You can use any axioms you want, but most axioms you could come up with would probably not lead to anything particularly useful or interesting.

>>9537696
prove that any axiom is more useful or descriptive of reality than any other
>>9537701
good try

You don't start with axioms.
You start with the first principals of Logic.
Here:
1. Fist Principals
2. Formulation of premises based on observations (induction)
3. Cross check via propositional calculus (fallacy-check, consistency-check, congruency-check)

Axioms are used when debating premises.

>>9537720
>man who can't even spell principles tries to differentiate logical principles from axioms
you can't make this shit up
In mathematics, first principles are referred to as axioms or postulates

>>9537730
I'm referring epistemology, which precedes mathematics.
And I'm typing on a Acatel Onetouch. I can barely see.
Your criticism is invalid.

>>9537740
Ah, a phoneposter. I should have known you faggots would arrive eventually. And no, an axiom, assumption, or first principle all refer to the same thing.

>>9537705
>prove that any axiom is more useful
You know that computer you're using to communicate through right now?
Set theory was at the foundation of their invention.
Russell and Gödel were also friends with that guy von Neumann whose name you might have heard in the context of von Neumann architecture.
So name an axiomatic system that's produced more wealth than the one which yielded computers. Because money is the physical embodiment of value and utility is a prequisite for value (nothing valuable can be without utility, even something not directly useful like an autographed piece of cardboard with a photo of a baseball player on it is in fact useful in less direct ways if valuable because you can sell it to the party who values it and get money from them to use for any other purpose you need or want).And certainly the invention of computers has created a massive amount of wealth.

>>9537782
>the most useful system is one that makes the most shekels
>this is somehow analogous to proof

>>9537790
That's the literal definition of money, it's an agreed upon proxy for the Platonic Form of value.

>>9537782
Holy shit dude

>>9537596
Phenomenological proof of existence is inherent to sentient existence. We've known this shit since the 17th century. One of the most famous quotations in the fucking world.

>>9537800
That's only the definition of unbacked currency from the perspective of only some people.

>>9537919
Gold backed currency is also an agreed upon proxy for the Platonic Form of value. Anything you use as money / currency is, even cigarettes in a prison or pogs in a 1990s middle school playground. And precious metals are no exception. You can exchange gold for another person's time and/or labor, or for physical goods that have nothing to do with gold in itself because it's a proxy for value.
>>9537917
Descartes' argument isn't universally accepted as logically sound.

>>9537573
>Go and measure right angle triangles like an idiot and see if pythagoras was incorrect.
Are you saying the theory of relativity is wrong? Pythagoras only holds in euclidean spaces.

>>9538036
as far as we know, so far

>>9538036
>Pythagoras only holds in euclidean spaces.
Just because something is true in a specific set of circumstances doesn't mean the contexts where it isn't applicable prove it wrong.

>>9538036
If you are stationary and have a paper and pen, pythagoras will hold. Also, the empirical verification is just a sanity check for people who could find mathematical proofs "scetchy", but obviously is not a fucking proof. Also you are dumb.

Yes, you will disprove some fact in your theory, but it doesn't make them false in other theories

>>9537678
ZFC+large cardinals

>>9537601
>>observe and analyze the real world
>>identify as-simple-as-possible invariants
axiom of power set, axiom of infinity, axiom of choice... yeah sure bud

>>9538302
>axiom of power set
>implying you can't imagine a collection of parts of a set

>axiom of infinity
>implying you can't imagine building an infinite object out of performing the same operation on something over and over

>axiom of choice
>implying you can't choose an element in a set

If any axiom of standard mathematics really came from thin air, be certain we'd doubt much more about its coherence.

>>9538353
>>>implying you can't imagine a collection of parts of a set
>implying you can when the set is infinite
>let alone the power set of the power set ... (ad infinitum) ... of the starting infinite set

>>implying you can't imagine building an infinite object out of performing the same operation on something over and over
>implying you aren't merely imagining that you can imagine an infinite set

>>implying you can't choose an element in a set
>implying you can do that for infinite sets or god forbid uncountable ones

>>9538378
>>>implying you can't choose an element in a set
>>implying you can do that for infinite sets or god forbid uncountable ones
Why can't you, exactly?

>>9537561
You can't. Gödel's incompleteness theorem.

>>9538353
you do realize that that bit of brains at the bottom still has to hold all that weight above, right? so stupid

>>9537573
>>9537573
>you can

>check empirically.

>>9538419
Because no number can exceed [math]10^{200}[/math] or something.