>analysis
>not an irrelevant garbage field that was borne out of the cold-war-stalemate of oneupmanship that is `mathematical rigor' started by Hilbert in the 1900s
pick 1
>>7944206
my point is that mathematics can advance appreciably more quickly if we didn't have, as a field, this childlike obsession with rigor
math was created to explore, not to dwell on trivialities, fringe scenarios, and arcane consequences
>>7944210
Is this rigorous?
>>7944216
define your operators property and follow the definitions.
end of discussion: no need to make mathematicians waste a century because of weak definitions
>>7944228
sounds like you've drank the Hilbert cool-aid
enjoy wasting your time while others make appreciable discoveries
>>7944232
you're going to make discoveries in what? in calculus? you can't seriously believe this.
mathematicians need to go through a rigorous phase in their development, in order to develop real intuition. the bullshit handwavy shit you get out of highschool with is NOT intuition. great article by an eminence:
https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/
>>7944216
If f in infinitely differentiable and that series converges, then sure.
>>7944238
I think we're looking at this from different perspectives, though I appreciate you're actually responding. Admittedly my tone was inflammatory since im on 4chan and desperately want to get attention or even upset someone... though this is something I have an interest in
Imagine I am able to come up with a fantastic result with amazing applications but can only show it is convergent in probability (and not almost sure convergence)
Any modern self-respecting school will accept the result, but look down and dismiss it as unimportant or irrelevant since the author didn't waste the rest of his career proving almost sure convergence
Meanwhile the rest of planet earth can enjoy the result for what it is (in spite of its weaker convergence)
>>7944247
I should note that if analysis, or other `foundational' fields, are something that particularly interests a person then they should 100% go for it and be encouraged to do so. But treating legitimate results as babby-tier because they fail some test over the reals but not the rationals is harmful & wasteful
>>7944247
well then you don't really care about mathematics, you care about its applications. and that's fine.
>>7944252
if it only holds on the rationals then it fails at almost every point. that means you don't expect it to ever work in a real setting. you know this, right?
>>7944259
i'd say that im mostly butt-hurt at the dismissive culture in mathematics of `non-rigourous' results
>>7944266
give an example? the examples you gave all sound very rigorous, just not strong enough.
>>7944264
> implying the universe is not quantized
>>7944267
if i'm going to be honest, this is more of an imaginary scenario i have in my head while stuck in traffic driving
...
essentially i wanted to vent and hear why i'm wrong
>>7944271
well I gave you my take on it, tao's article is a good read. rigor shapes your intuition. obviously you want to navigate and explore ideas through intuition, but you can't do that without a rigorous preparation or you'll just go blind and get lost.
we like to shit on other disciplines for lacking rigor, sure. and since last year undergrads / starting grads are in the peak of the "rigorous phase" while high school / freshmen are in the prerigorous stage, you'll see banter and mocking on that vein. but that's just human nature
>>7944247
>Meanwhile the rest of planet earth can enjoy the result for what it is
>the rest of planet earth
>can enjoy
>the result
>>7944229
Dude, this is like super basic shit, integration by parts, exponents. What more is there to define?
>>7944216
Yes if you use a proof by induction. it seems like you are tending towards a topic called the method of stationary phase when evaluating fourier integrals. what you have posted is basically the argument for why non-stationary phase is not worth considering, because one can integrate by parts repeatedly as many times as the smoothness of f allows.
>>7944206
>bitches about analysis
>has at least the emotional maturity of a ten year old
Pick exactly one.
Jesus, its like all you haters were molested by analysts as children or something
>>7944390
>implying none were
>>7944381
Lmao no. You can't use induction to prove infinitely many operations are valid. You have to use limits.
Also using integral to denote antiderivative is just fucking stupid. FTC does not hold for whatever [math]f^{(\infty)}[/math] is.
>>7944396
> assume f is n time differentiable
> proceed with inductive proof
> holds for all n in N
what now
>>7944396
oh wait i see your point now. you'd only be able to show that this holds for a function of n times differentiable (n arbitrarily large but finite)
im sorry m8
>>7944403
>let [math]\left\{q_n\right\}[/math] be a sequence of rationals
>[math]\sum_{n=1}^{N}q_n \in \mathbb{Q}, \quad \forall N \in \mathbb{N}[/math]
>but holy shit [math]\sum_{n \in \mathbb{N}} \frac{1}{n^2} = \frac{\pi^2}{6} \notin \mathbb{Q}[/math]
>>7944435
Nah just giving you a classic example of why your thought process is flawed.
As a physicist, analysis was the most useful thing I learned in undergrad. I'm honestly curious in what context it's "irrelevant."
>>7944210
I kind of agree with this, but you need pretty enormous talent to get away with it
Ramanujan is the perfect example of a mathematician who was totally non-rigorous and someone (might've been the man himself) even said something like if he saw one or two examples and he had an intuition or justification as to why something should be true he'd not even bother with a proof and just take it as true. But obviously he was like an Euler-level intellect and very rarely will any other mathematician get away with that shit
>>7944247
ive seen a lot of bullshit in my short time, and i am thankful for the stigma placed on those that neglect the burden of proof
>>7944206
Taking an analysis class where the final exam is basically hoping you get lucky in the lottery of the 7 proofs that come up on the exam are 7 proof that you know very well out of the 55 that were demonstrated during the semester is fucking retarded.
Doesnt take any understanding at all and the memetards in my class with photographic memories did well.
Analysis is a good thing to study but taking an exam on it is fucking retarded.
>>7944216
[math] \dfrac{1}{1-z} = \sum_{n=0}^\infty z^n [/math]
so
[math] \frac {d} {dx} \left( e^x \dfrac{1}{1+\frac {d} {dx} } f(x) \right) [/math]
[math] = e^x \dfrac{1}{1+\frac {d} {dx} } f(x) + e^x \frac {d} {dx} \dfrac{1}{1+\frac {d} {dx} } f(x) [/math]
[math] = e^x \dfrac{1}{1+\frac {d} {dx} } \left( 1+\frac {d} {dx} \right) f(x) [/math]
[math] e^x f(x) [/math]
so yes.
>>7944652
that exam sounds idiotic
they should have set a new theorem that you have to prove otherwise it's just an exercise in memory (although it is a pretty dismal statement if you can't remember 55 basic analysis proofs).
>>7944652
You took an exam that was just a recitation ? Wtf
Although, you should have been able to do well since many analysis proofs look a lot alike
ITT: OP mad that he got BTFO when he found out he couldn't major in triple integrals
>>7944266
Mathematicians speak in non-rigorous terms all the time. However, you expect rigor at the end of the day, because this is what math is. Being unrigorous or loose with definitions led to issues in set theory and analysis, in particular with infinite series.
>>7944206
I'm rereading rudin right now. That book is great.
The fuck makes you think "analysis" = "rigor". If anything, analysis is not rigorous at all compared to the other fields of math, like topology. In analysis, you make all sorts of topological assumptions and don't bother to explain yourself because "the context is understood."
Are you like 18 OP?
>>7944206
Let me clue you in faggot.
Analysis is basically the study of what happens when we take numbers going to infinity. Series, limits, derivatives, integrals, all that. It's all about the infinite and the infinitesimal. That's analysis. Doesn't have jack shit to do with "rigor," which is a concept of logic and equally inherent in all mathematics.
Enjoying babby's first real analysis class?
>>7944247
>only show it is convergent in probability (and not almost sure convergence)
What the fuck do you think "convergence" means? Read the book in your own OP for fuck's sake OP, you sound retarded.
>>7944390
OP thinks "analysis" = "math with rigor"
He doesn't understand that the "analysis" part refers to certain concepts and not to the actual difficulty of those concepts.
>>7945050
https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_probability
https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence
Not that OP doesn't sound like a retard, but honestly so do you.
>>7944210
No one really thinks about the rigor at the highest levels. We explore basically the same way we always have, but also know how to make the arguments rigorous after the fact.
>>7944266
That isn't true.
Mathematicians are very interested the methods physicists employ in fields like quantum field theory and string theory despite those methods being non-rigorous, even by physicist standards. They don't dismiss these results - they embrace them! They won't accept results demonstrated using these methods as theorems though, but that is exactly why they are interesting. After all, a lot of these techniques can be used to make experimental predictions that have been confirmed to an incredible degree of accuracy, so there must be some sort of well defined, rigorous computation underneath it all.
>>7944662
why was this omitted from Rudin?
>>7945045
>not rigorous at all compared to the other fields of math, like topology.
You mean "hold on let me draw a convincing picture and ask you to accept it as a proof on faith"-ology ? That's like the worst example you could have given.
>>7945358
Found the shitposter who never took a topology course
>>7944662
Holy shit, can you really do that? I mean if this is a troll how can this work so well. But this almost looks like abusing the reality, the logic, like a gamer exploiting a bug in a video game. Where can I learn how to handle d/dx in such a way?
>>7944662
AHAHAHAHAHAHAHAHAHAHA
>>7945362
You know, he's right. That's what everyone does after they take the intro topology course.
engineer detecdted
>>7945373
Sure
Just define an inverse operator [math] \frac{1}{{1 + \frac{d}{{dx}}}} = {\left( {1 + {{\mathbf{D}}_x}} \right)^{ - 1}}[/math]
>>7944206
The Italian School of Algebraic Geometry is what happens when mathematical rigor is not upheld.
https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school
>>7945700
well that only works rigorously if you can prove that the operator norm of D_x is smaller than 1.
hint: it's unbounded
>>7946290
First lesson my advisor taught me was 'never trust a physicist'. Second was 'never trust an italian'.
>>7946290
That's a general problem with Italians desu, they are nearly universally incompetent.
>>7947908
>tfw italian
kill me
>>7945373
>I mean if this is a troll how can this work so well.
You can write down
[math] (I-A) \, (I+A+A^2+A^3+A^3+...) [/math]
[math] = (I+A+A^2+A^3+A^3+...) - A \cdot (I+A+A^2+A^3+A^3+...) [/math]
[math] = (I+A+A^2+A^3+A^3+...) - A - A^2 - A^3 - A^3 - A^3 - A^4 - ... [/math]
[math] = 1 [/math]
which you know works for I=1 and A=z with |z|<1.
You can ask yourself what other models fulfill these manipulations. That is you can ask which mathematical objects I, A, +, and infinite sums behave like that.
The picture in >>7944216 made it clear that the operator [math] A = \dfrac {d} {dx} [/math] allowed for [math] (1+A) g = h [/math] to be solves as [math] g = (1+A+A^2+A^3+A^3+...) h [/math] as [math] (I-A) (1+A+A^2+A^3+A^3+...) h = h [/math] works out for those objects.
Whether that would work is not something I know off-hand, because as the guy above points out, the derivative living in as simple a space as C, and existence of infinite sums depend on a norm.
You can read up on Banach space theory to get to know the vocabulary.
But the take away message here is that if you know a relationship from some model and you deal with a new mathematical framework that shares some axioms (e.g. numbers, as differential operators, can permute and can be squared in some sense), then wondering how certain theorems carry over to the new theory is worth it. (Many many mathematical theorems are extensions of number systems, for example. Just think of any group or ring and how that's an abstraction from certain properties of the integers.)
Finally, writing [math] \dfrac { 1 } { 1- \frac { d} {dx } } [/math] is a priory a game, but you can actually also get a whole bunch of theorems by setting [math] \left( \frac { d} {dx } \right)^{-1} = \int dx [/math].
If you have a diff equation like the one for thermal conductivity
[math] \dfrac { d} {dt } f(t,x) = - \Delta f(t,x) [/math]
then it's solved by [math] f(t,x) = e^{-t \Delta} f(0,x) [/math] and you may make sense of the e.
>>7944627
>claiming the post rigorous stage needs no rigor
read the tao article will you?
People only study analysis today because Important White Men have told generations of students to study analysis because they were forced to learn it by their teachers.
After 1900, nothing good or important has come from Analysis. We could have stopped with Lebesgue and nobody would have been the wiser. Every student today would be better off learning subjects like Algebra or Combinatorics where they can actually develop an aesthetic sense.
>>7944662
This might be the best thing I have ever read
>>7944206
Analysis was made rigorous by weierstrauss, not hilbert
And analysts is the subject of infinity so it needs to be rigorous because intuition often fails.
Infact all pure fields of math need to be as rigorous as each other, it's just sometimes it can be easier to reach that level in other fields that aren't about infinity.
