can /sci/ explain to a humanities student why fermat's last

I can't pretend to know a lot about the subject, but as far as my understanding goes, the theorem itself doesn't have many important applications (compared with what a proof of P=NP or Riemann Conjecture could bring, for an example).

It got famous for being a problem which is very easy to pharse, but exceptionally hard to solve. Many mathematicians have dedicated their lives into working on it, and the proof itself uses very deep, complicated results from different branches of math.

The theorem itself isn't much of a big deal in terms of practicality. What's important is that something so seemingly simple took hundreds of years to prove, with new techniques that are applicable to algebraic geometry and other fields.

FLT in itself isn't important. The important things are the techniques developed while searching for the proof, mainly in number theory and algebraic geometry.

>>7941273
it's like a mathematical arms race, developing new math to solve the problem ends up being useful in other fields like

>>7941319


said

It's like those Plato's problems. Dividing an arbitrary angle into three equal parts with just a straightedge and a compass is not mathematically that important, but it took over 2000 years to prove it can't be done.

I am not an expert on this subject, but I can try to explain a bit. Algebraic number theory is a branch of mathematics that was essentially started by the search for the proof. It basically deals with enlargements of the integers, we add other "numbers" to our normal integers and study the "ring" formed by these. Some of the properties of integers are preserved in this process, while others are not. This gives a huge amount of algebraic systems not known before, sort of freeing us from our previous 'box', where there were just the integers. This is somewhat similar thing to what complex numbers did to analysis, and usually when we add something to the integers, we add complex numbers (in algebraic number theory, of course).

Other thing hugely influenced by FLT is algebraic geometry. Originally it began as the study of solution sets of multivariate polynomials over the real numbers. Such solution sets include planes, lines, circles, parabola and so on. The interesting thing is that the techniques can be used in context,where there is (naively) no geometry, for example over integers modulo a prime number. This gives a bridge between number theory and geometry, which allows to transfer problems and theorems from the other world to the other, sometimes making hard problems much easier. This applies to much more than just the FLT.

>>7941346
That's a pretty good explanation, thanks anon.

For anyone interested, take a look at the geometric interpretation of the ring of polynomials over the integers and its various number field extensions. A rudimentary (and false) proof was essentially based on adjoining the n-th roots of unity to the integers so that the z^n term could be factorised accordingly. Sadly, unique factorization fails for n=23 and more cases.

If anyone is willing to expand and correct, please go ahead.

>>7941316
>what a proof of P=NP or Riemann Conjecture could bring, for an example).
what a Riemann Conjecture proof could bring ?

>>7941273

Hello humanities student. I would like to add to this in a way that may appeal to your unscientific sensibilities of philosophy, beauty, etc.

Other posters have already mentioned how Wiles' search to prove FLT led to developing other interesting areas of math. Now, it is true that mathematicians want among other things to develop techniques and objects that are useful in multiple ways, prove theorems, solve problems, formulate newer, more interesting problems, actually work with other people in the sciences, and so on. But there is a simpler aspect to this that most people can appreciate. Wiles himself has talked about it, multiple times. I will also talk a little bit about how 'mathematics culture' is different from the culture in other sciences (there are people on this board who would argue that math is not strictly speaking a science, which is another point that goes into my thought process). I'm going to get into the more /impractical/ side of why we do math, a little bit.

Some of what mathematicians do is thought to be very /impractical/ by scientific standards. About a year ago, researchers found a new type of pentagon that tiled the plane. I made a thread on the topic, and although some people were happy to hear about the result, most dismissed it as 'having no meaningful applications', 'stamp collecting', 'unscientific; and so on. Basically, in many instances when mathematicians do math for its own sake, this bewilders some, and actively angers a few. "Why are you trying to solve that problem? It's pointless!" etc.

cont.

>>7941597


The motivation, of course, like the desire to climb Everest, is extremely simple. "Because it's there." Wiles just plain wanted to solve this problem, because it had fascinated him since he was a kid. I was once in a room with other students, and a professor told an anecdote of some long mathematical proof which wasn't particularly useful, but which he personally 'liked'. A student asked why, and the professor replied "Because it's beautiful. It's like... poetry." This is the impractical side of math. There is an /aesthetic/ component to math, not just pretty geometric drawings, but the development of a train of thought, the exposition of a proof, etc.

The way that math also tends to abstract away from the physical world, plus the above Muh Aesthetics sketch, enrages some /sci/ posters, to the point that they would like to hold math apart from the other sciences.

In my personal view, mathematics proper, although part of the sciences, is still a speculative 'bridge-field' somewhere between the hard/physical sciences on the one hand, and the unscientific human endeavors on the other. Philosophy proper is analogous to mathematics in the sense that it is a "bridge-field" between practical fields of unscientific endeavor (law, education, politics etc), various non-fictions, and "speculative" thought, certain fictions, thought experiments, etc. They are areas of thought and human endeavor where people intentionally sperg out for its own sake, frequently (and frequently not) coming up with things that may be applied in practical life. This frustrates certain people who would like their activity to always be more practical, though as we've seen (the moon landing), impractical endeavors can bear happy unintended consequences, unintended fruit elsewhere.

>>7941541

There are several conjectures or theorems of the form: "If the Riemann Hypothesis is true, then this other thing over here is also true." Or, similarly, "this one goofy thing here is true if AND ONLY IF the Riemann hypothesis is true." In other words, it has been established that certain things rise (or fall) together, depending on their truth or falsity. The trouble is that no one knows whether they're actually true or not.

One example of "something else" equivalent to RH, is certain statements involving "Farey Sequences". So if someone somewhere could prove that thing (or vice verse), the one would get at the other. This investigation of seemingly unrelated things, by the way, is common in math, and was exactly Wiles' personal working process with FLT.

https://en.wikipedia.org/wiki/Farey_sequence#Riemann_hypothesis

fermat pretty much wrote down it is provable, so people have wanted to know since if it was. what fermat wrote was akin to "nvm proved it".

there is a good documentary made about it by the bbc.

I'm surprised they gave him the abel prize so long after, he proved FLT 8+ years ago..

It's interesting that FLT and the poincare conjecture that perelman solved both had similar stories:they both worked in secrecy for a lot of years( i think around 7)

>>7941273
what if
n=3
x = any negative number
y = |x|
z = 0

>>7941829
for that matter if n = any odd integer

the statement in the picture would be false

>>7941829
>>7941831
amirite?

>>7941608
>>7941597
I enjoyed these posts. Thanks anon.

>>7941316
P = NP for N = 0, 1

Q.E.D.

>>7941791
>It's interesting that FLT and the poincare conjecture that perelman solved both had similar stories:they both worked in secrecy for a lot of years( i think around 7)

But for different reasons, I think. Wiles was afraid that he would be mocked for actually trying to prove FLT. Perelman is a shut-in autist.

>>7941871
I know right. Computer science nerds are fucking retarded.

>>7941871
>for N = 0

>>7941316
It didn't help that Fermat himself wrote that he knew the solution, but he couldn't fit the proof in the margin of the book he was scribbling in. Considering how many pages it took to prove, I wonder if Fermat was a master ruseman.

>>7941829
>>7941831
>>7941835
In the actual conjecture its stated that x, y and Z are positive integers so no.

>>7941273

If you ever wanna now more about teh mathz, check out, "Mathematics for the non-Mathematician."

>>7941273
X=1;Y=1;Z=1;N=197361877

>>7942014
We have some idea what Fermat would have likely tried and honestly believed to be correct.

>>7941273
It's just dickwaving.
>"look at me! I solved a 300 year old problem that had no significance to even other pure maths fields!"

>>7942443
how is middle school going?

>>7941791

In mathematics, there is a long history of two or three people doing roughly the same thing at the same time. IMO, a large part of the reason why that is, is that the thus-far developed historical mathematics and mathematical notions of a certain period become capable of The Next Big Thing, whatever it is.

The most famous example of this is Newton and Leibniz each developing calculus independently, but some other examples that come to mind are Ferro, Cardano, Ferrari, Tartaglia etc all variously competing and (slightly) co-operating with each other in order to solve cubics and quartics. Once the vague, misty notion of a complex number was admitted (though they barely knew what those were), they could proceed without the benefit of what we would call modern algebra, let alone calculus (which is not even necessary to their derivations). However, modern algebraic techniques would have to be developed later, and this is what Galois and Abel did in order to treat of higher cases. Also, mathematical logic was booming in the late 19th/early 20th century.

My point being that if your account of the time-scale of both Wiles' and Perelman's working processes is accurate, then this similarity would be a familiar theme for two contemporary mathematicians (who are themselves, contemporaries). They were working on different problems, but each required about a decade of work. It is not unreasonable to hold both working processes common, as generally representing (part of) the present upper limit of human mathematical ability. Mochizuki-kun and Tao belong in this convo, probably.

>>7942443
>1+1=1

>>7942443

BILL GATES BTFO

>>7941791
>I'm surprised they gave him the abel prize so long after, he proved FLT 8+ years ago..
There are old dudes like Serre who have arguably been more influential and could kick the bucket at any minute at this point that they've been trying to give the award to.

File: 1425345110594.jpg (43KB, 400x258px) Image search: [Google]

43KB, 400x258px

>>7942443
highkek

Just came here to shit post.

>>7941273
It's a literal meme.
Fermat wrote that he found a marvelous proof for the problem, but unfortunately the margin of the book was too small to contain it.

>>7941320
What if we use FLT to obtain FTL?

>>7941608
>impractical endeavors can bear happy unintended consequences, unintended fruit elsewhere

Very nice way to wrap up some well thought out posts, kudos anon

>>7941608
>though as we've seen (the moon landing), impractical endeavors can bear happy unintended consequences, unintended fruit elsewhere.

Would you like to expand on this? Which solutions to pointless problems allowed the moon landing?

>>7942552
>I can't see the significance
>Therefore it's insignificant!

One implication is in combinatorics. If we call n the number of objects, z the number of bins, x the number of non-red bins, and y the number of non-blue bins, then fermat's last theorem is equivalent to stating that the number of ways of sorting the objects into bins without using any red or blue bins can never be equal to the number of ways of sorting them that use both red and blue bins, when there are more than 2 objects.

>>7944930

You've got it backwards, relative to what I was suggesting. I might have better said "the space program" generally, but the idea of putting a man on the moon, specifically, was of itself just "cool", and a highly /impractical/ general-principles dickwaving thing. Like the Everest thing, only higher up. And I'm very happy that we did it. Just like I'm happy that the Russians landed on Venus, and the Europoors landed on a comet.

The conventional argument against space exploration is "we have enough problems down here." This is why Gil-Scott-Heron wrote his resentful song "Whitey's on the Moon". And yet, as a single link suggests, a bunch of highly practical tech resulted from the problems that needed to be solved by the space program, generally.

http://www.techradar.com/us/news/world-of-tech/10-tech-breakthroughs-to-thank-the-space-race-for-617847

Now, some nabob somewhere is going to (and, it seems, could correctly on one case) call the accuracy of one or more of the above into question, and shit on the mere mention of a popsci listicle. But a moment's further searching pulls up some better links of what I want: NASA's claimed "spinoff" technologies into the commercial sector over the years (searchable dbase), and even a wiki link which debunks certain of the misattributed technologies.

https://en.wikipedia.org/wiki/NASA_spin-off_technologies#Mistakenly_attributed_NASA_spinoffs

http://spinoff.nasa.gov/

In my treatment, the impractical feat was the moon landing itself. The practical fruit that the space program more generally has borne out, are the "spinoff" technologies that NASA cites, of which there are hundreds.

The larger point is that similarly, it sometimes happens that a piece of impractical mathematical knowledge turns out to be useful elsewhere. But it may take time and historical development for this to become clear.