Thread 16747719

I feel like Lagrange was onto something when he built differential calculus from the ground up using nothing but algebra. Why did everyone decide to depart from his short-lived methods and go on with limits? It feels as though modern calculus/analysis is a Frankenstein monster where every problem is solved by throwing the kitchen sink at it.

>>16747719 (OP)
Limits just got their proverbial boots on quicker

>go to Wikipedia article on limits
>see they were adopted far after Lagrange's time
>has a motivation section
>it's all mathematical formalism in casual language
>says nothing about why mathematicians have a hardon for this formalism
What the fuck is their problem? Why can I not find ANYTHING about why mathematicians adopted this retardation? Before any of the cultists pop up, I have a PhD in physics and a BS in math. I know how to use them. I spent my time in the weeds with real analysis and complex analysis. Yet I was never given a historical reason for why this epsilon delta crap took foothold in the field of calculus when calculus was well understood and functioned fine for centuries.

>>16747755
Math was hijacked by people who sucked at or had no interest in physics I believe.

>>16747719 (OP)
What the fuck are you on about, retard? Notions from calculus can be easily formulated purely algebraically.
Also, limits are algebraic tools.
How about you go and learn some actual math, instead of wasting your time wading around in high-school level garbage?

>>16747755
Limits were conceived in the 300s BC.

>>16747794
>Notions from calculus can be easily formulated purely algebraically.
No it's not that easy. The entire subject of calculus revolves around the nonsensical idea of finding two consecutive points. How about you go and learn some math yourself. Limits are not algebraic tools, you might use algebra to solve limit problems but you're confusing the tool for the concept and vice versa.

>>16747818
>The entire subject of calculus revolves around the nonsensical idea of finding two consecutive points
No it doesn't.
>Limits are not algebraic tools
Yes they are https://math.stackexchange.com/questions/60590/category-theoretic-limit-related-to-topological-limit

>>16747755
One disadvantage that can be levied against the power series development of calculus is that the Taylor polynomials are not guaranteed to describe the actual function at a given point. One can always find pathological examples where the Taylor remainder shrinks to zero as you add more and more terms, hence the sum converges to a function, but it may not be the correct function. However, we can ask ourselves if these pathological functions are legitimate to begin with. I mean, do they even arise anywhere in the outside world or are they just a gimmick? Dunno if you get my point. Would really love to have your input anon.

In other words, can you explain physics with just analytic functions?

>>16747836
>can you explain physics with just analytic functions?
Consider the following.
>Taylors theorem devised in late 1600s/early 1700s
>Formalization of limits via epsilon/delta arises late 1800s/early 1900s
So you told me: what do you make of physics from approximately 1700-1900? Does it hold any explanatory power?

>>16747861
I stand corrected. Will math curriculum ever recover? What kind of paradigm shift is required here

>>16747834
Anon that's like saying because you can stick a hammer up your ass that a hammer is formally a dildo. Technically correct, practically retarded and socially autistic (i.e. completely irrelevant)

>>16747861
Thoughts on this guy?
https://www.youtube.com/watch?v=l7LvgvunVCM

>>16747871
kek

>>16747872
If you had asked me 15 years ago, I would have said he's a schizo. Today, I think he's based and every day I grow more sympathetic toward the finitist arguments.

>>16747794
completely unrelated but what are some good resources for learning calculus, anybody know? my background in math is basically only highschool algebra, euclidian geometry and some trig. i haven’t touched math since highschool two years ago but id like to be able to participate in these threads cause this stuff seems pretty interesting.

>>16747891
Calculus is boring, and relatively useless besides giving some motivation for certain concepts down the line. You're better off learning some linear algebra first.

>>16747891
I would learn calculus using the notion of infinitesimals. Perhaps begin here: https://intellectualmathematics.com/calculus/
Or 3b1b’s Essence of Calculus series and then solve some physics problems.
When you find that the infinitesimal approach is ‘handwavy’ or you are skeptical, read Lagrange and you can convince yourself (with nothing but high school algebra) that calculus with infinitesimals yield the correct solution. The other anon with Bsc in Math and Phd in physics might have a different opinion and I would honestly listen to him instead lol.

>>16747719 (OP)
>>16747755
Fourier theory.

>>16747960
>plotwist: Euler was already aware of Fourier series long before Fourier and gave several examples
https://www.sciencedirect.com/science/article/pii/S0022314X14003497

>>16747960
Which is a theory of linear algebra fundamentally and not calculus…

>>16747960
Fourier's work (late 1700s/early 1800s) far predates the formalized epsilon delta treatment of limits (late 1800s/early 1900s). Unless you're trying to say fourier's work wasn't practical or applicable until the formalization of limits a century later? I would find that hard to believe.

>>16747871
>hammer is formally a dildo
>Technically correct, practically retarded and socially autistic
Okay, but why bring up engineering?

>>16747755
simply to avoid infinitesimals, and the method became the standard, hence why the one with infinitesimals that was well formed what termed nonstandard

>>16748060
>>16748063
The emergence of Fourier theory inspired much investigation into convergence problems, and intuitive arguments involving infinitesimals and unjustified power series expansions no longer cut it.

>>16747719 (OP)
His methods were unrigorous and left behind when Cauchy and Weierstrass formalized calculus successfully. You cannot build mathematical theories on shaky foundations, especially not when a more rigorous alternative is presented. Infinitesimals have finally been made rigorous in the 1960s with nonstandard analysis and they are fun but don't have much use currently.

>>16747755
>I have a PhD in physics and a BS in math.
Doubtful considering you say
>Yet I was never given a historical reason for why this epsilon delta crap took foothold in the field of calculus
You cannot finish a bachelor in maths without having heard of the loose reasoning style around limits in the early days leading to misconceptions, such as that limits of continuous functions are continuous, which were then starting to be resolved following the introduction of the epsilon delta definition.
>when calculus was well understood and functioned fine for centuries.
It was not well understood, an extremely basic notion such as the limit of a sequence of continuous functions wasn't even understood. If it functioned fine is a subjective matter, I would say it cannot function fine if elementary foundational questions are not understood, but I suspect you will be of the opinion that matters such as these are not relevant for most physical questions. On the other hand, I don't value math exclusively for its usefulness to physics and consider it its own field, but that is again subjective.

>>16748180
>loose reasoning style
no need to continue reading. so much for that "rigor" mathematicians pretend to have lmfao

>>16748180
So people like Jacobi, Euler, Lagrange, Fourier, Gauss, Bernoulli, Bernoulli, et al. were all just incapable of working with continuous functions or infinity and were plagued with faulty math? And Euler but no mathematical theories because he was working with broken math devoid of rigor? Spare me.
>Doubtful
Bartle & Sherbet say nothing about this. Nor did any of my texts on complex analysis and real analysis. You're free to quote where in a standard intro book such statements are made. It should be easy, since you remember it so well you summarized it as loose reasoning.

Your argument amounts to exactly what I critiqued from the wikipedia page on limits about alleged motivations. What specific problem are you claiming math was ill equipped to tackle such that a fix was needed? Because as it stands it sounds to me like you're talking out of your ass, and have a sophomore understanding of the topic. Like you heard a hand wavy explanation from your professor and took it as gospel. You yourself called it loose reasoning and dropped vague generalisms. You're going to have to be specific.

>>16748192
If you don't want to reply to my comment, then don't.
>>16748198
>So people like Jacobi, Euler, Lagrange, Fourier, Gauss, Bernoulli, Bernoulli, et al. were all just incapable of working with continuous functions or infinity and were plagued with faulty math? And Euler but no mathematical theories because he was working with broken math devoid of rigor? Spare me.
That's not what I said so I will disregard this. Reread what I was quoting regarding Lagrange's differential calculus.
> You're free to quote where in a standard intro book such statements are made.
Why would I need to? A bachelor in math comes with lectures and interactions with mathematicians and in that setting I consider it unlikely that one would spend a whole bachelor without being told or coming across some historical context and intuition on epsilon delta limits.
>What specific problem are you claiming math was ill equipped to tackle such that a fix was needed?
I never said math was ill equipped to tackle calculus without epsilon delta definition, in my post I mention the introduction of nonstandard analysis which is a different but also correct way of doing calculus. Historically speaking however, epsilon delta was the first successful attempt to make limits and calculus rigorous and it led to a very productive century of analysis. Much of those century's results turned out to be essential for physics, such as functional analysis for quantum mechanics or Fourier series for PDEs.
>You yourself called it loose reasoning and dropped vague generalisms. You're going to have to be specific.
What? I specifically gave the example of a limit of a sequence of continuous functions.
It's fine if you dislike the epsilon delta formalism, it is a matter of taste and perhaps you were indeed never made aware of the context behind it, I consider that unlikely but unlikely does not mean impossible.

>>16748221
your post reeks of reddit. nobody will take you seriously except other predditors

>>16748255
Your post reeks of I have nothing meaningful to reply so I resort to insults to cope.

>>16748255
>your post reeks of reddit. nobody will take you seriously except other predditors

>>16747872
Hero, he saved /sci/ from horsefuckers.

>>16748221
> Why would I need to? A bachelor in math comes with lectures…
Cool, so because you got a side-comment in a lecture once, everyone else must have too. My pure math BSc didn’t have some “here’s the historical brick wall this fixed” moment, just “here’s the tool, now do contrived problems that make it look essential.” That’s post-hoc sales pitch, not history.
>example of a limit of a sequence of continuous functions
And you just leave that hanging? Here’s the actual chain since you can’t:
Cauchy (1821) says “limit of continuous funcs is continuous” -> people swap limits/integrals wrong for decades -> Weierstrass (1850s–70s) drops counterexamples (pointwise ≠ uniform convergence, nowhere-diff func) -> ε–δ formalism nails it down. Done.
>led to a productive century of analysis… essential for physics… Hilbert spaces… QM
Yeah, and Navier–Stokes is “essential” for airplane design too, yet Boeing isn’t filling hangars with PDE proofs. You’re confusing “the blueprint exists” with “we actually use it.”

Bring names, dates, and the broken theorems, or keep cosplaying as a Wikipedia page with ADHD.

>>16748686
>My pure math BSc didn’t have some “here’s the historical brick wall this fixed” moment, just “here’s the tool, now do contrived problems that make it look essential.”
So you went to a shitty school and did the bare minimum to get the degree. No wonder you're still struggling with this.

>>16748689
If that's all you have to say, then I accept your concession. I went to a top 50 uni, so lick my ass.

>>16748691
Uh huh. But you seem to have never touched any algebra or geometry, and have admitted that your supposedly pure math classes forsook any of the culture of pure math.

>>16747719 (OP)

I believe Lagrange's methods only really worked for analytic functions: stuff like polynomials, e^x, cos(x), sin(x). This being said, if you like to think of calculus from algebra, I recommend reading a book on nonstandard analysis. They typically use algebraic structures to formalize a notion of infinitesimal that doesn't rely on the common notion of limits, and then use that formalization to do "old timey" calculus

>>16748691
If yout top 50 uni didn't contain any explanation on why a rigorous notion of a limit was needed I pity you.

Consider how many of those old school mathematicians despite their brilliance thought that a function continuous everywhere would certainly be differentiable for example. Rigor allows us to consider these pathological examples with certainty.

>>16748131
>nonstandard analysis
that's not exactly the reason for the name. it's called that because it uses an extended version of N and the extra numbers are termed "nonstandard numbers"

>>16748755
You don't understand. Obviously the weirdstrass function was taught to him. He's saying instead of it being presented as a counterexample to conventional wisdom at the time, it was presented merely as an oddity to caution one against assuming that a function continuous everywhere can be differentiable nowhere. I had the same experience. Open the book you used in your class, index search the weirdstrass function, flip to the page, read the surrounding context and you'll have the same experience. You did read your textbooks, didn't you?

>>16748686
How do you suggest someone ought to learn calculus? You have an important vantage point assuming you’re the phys/math anon. Mind posting an infograph or two on tackling physics and math ‘correctly’

>>16748707
Is this true? Maybe for Taylor series, but Taylor polynomials with finitely many terms can deal with non-analytic functions.

>>16748872
https://archive.org/details/foundations-of-differential-calculus-euler/mode/2up
Euler, the goat, before the widespread corruption of math as aptly described by anon
>Math was hijacked by people who sucked at or had no interest in physics I believe.

>>16748909
What order should his books be read?

>>16747891
Spivak's Calculus is what I used for my stats undergrad. Don't think you can go wrong with any textbook though, it's all the same math

>>16748686
>Cool, so because you got a side-comment in a lecture once, everyone else must have too.My pure math BSc didn’t have some “here’s the historical brick wall this fixed” moment
It was much more than a side comment and yes I do consider it unlikely that someone could go through a maths bachelor without coming across historical context and motivation of the definition. Again, unlikely does not mean impossible.
>And you just leave that hanging? Here’s the actual chain since you can’t:
So now it's not about a specific example anymore but a chain of events?
>Yeah, and Navier–Stokes is “essential” for airplane design too, yet Boeing isn’t filling hangars with PDE proofs. You’re confusing “the blueprint exists” with “we actually use it.”
An odd statement to make for a physics PhD. Do you not agree that e.g. functional analysis is essential for QM and that QM is a worthy field of study? Never mind the fact that I was not talking about practical use in engineering at all.
>Bring names, dates, and the broken theorems, or keep cosplaying as a Wikipedia page with ADHD.
Why? You made your point that you didn't get any historical context when you learned epsilon delta, I suggested that was unlikely in a maths degree. Maybe you are or aren't lying, at this point I don't care either way because you bring up tangential points to keep arguing. Seems like you have strong feelings about the abstract nature of parts of modern physics and math, cool.

>>16749061
bro was right to call you out for being peak reddit. you are responding to chatgpt lmao

>>16748255
leave the kekistani alone tranny

>>16749198
>kekistan
I almost forgot about that cringe autism, /pol/ getting together with r/t_d during 2016 was a fucking travesty

>>16748897
You can't use Taylor polynomials to construct partitions of unity.

>>16749313
cool, name one physical problem that muh partitions of unity solve.

are the partitions of unity in the room with us right now?

>>16749313
>what are complete orthonormal bases
get the fuck out of my thread

in physics if your hypothesis says that gravity is repulsive instead of attractive your hypothesis is discarded for contradicting observation

in mathematics saying that 0.9999... = 1 is called a quirk of mathematics even tho it violates identity. And people get away with it mostly because of modern definition of real numbers and limits

>>16749452
>even tho it violates identity
How? It just says that base 10 expansions are unique up to the geometric series

>>16748824
>You did read your textbooks, didn't you?

My analysis textbook also waxed lyrical about how the genius Cauchy saved analysis by devising a rigorous defitnition for a limit.

Also OP has a hardon against the rigorous definition of a limit yet offers no alternatives. Maybe he is a prof at my uni that teaches a first year course using ultrafilters to define limits.

>>16749365
Any physical problem using Stokes' theorem.
>>16749424
Nigger, what are you even trying to say here?

>>16749577
>a prof at my uni that teaches a first year course using ultrafilters to define limits.
kek based.

>>16747880
>If you had asked me 15 years ago, I would have said he's a schizo.
that was also my judgment back then. what do you like about finitist arguments?

>>16747891
ok, so imma level with you and say that it will take quite a time investment until you're ready to participate in these kinds of threads. i have a master's in mathematics and i don't know what OP is on about because i have no clue what lagrange did since that's not the way it's been taught and i'm not even one of these guys who are completely oblivious to the history of mathematics.

so if you want to study mathematics on your own for philosophical reasons, i think that can be worth while but you're looking more at 2-3 years of self-study at minimum before you are at a level where you can actually reason about this stuff and it's also not quite clear what exactly you have to study in order to get there.

bird's eye perspective: modern mathematics is build on flaky set-theoretic foundations that have arisen during the foundational crisis in the beginning of the last century. the essence however are still the ideas themselves and so the general attitude among mathematicians is to do mathematics which is phrased within a set-theoretic framework that itself, however, is then largely ignored. on these foundations real analysis and linear algebra are first taught in "bourbaki style" (rigorous, based on zfc set theory), which then serve as basis for most other mathematics.

so grab any recommended book on real analysis and linear algebra for starters, but also read princeton companion to mathematics and watch the yt channel "intellectual mathematics" in order to avoid brainwashing. i'm german, so i can recommend amann / escher for real analysis and bosch for linear algebra, they are absolutely excellent, but i would also take on a second book for real analysis because amann / escher is way too formal about the topic and the comparison will be instructive. (so far, there's sadly no english translation for bosch's linear algebra.)

>>16747891
oh btw since i recommended intellectual mathematics, try out this one for learning calculus with a historical perspective:
https://intellectualmathematics.com/calculus/
that one is absolutely stellar.

>>16749661
Buy an ad, Viktor.

>>16749684
>buy an ad
new to /sci/ and i've seen this twice today. is that some sort of meme whenever someone recommends a paper or something? some sort of /sci/-internal mechanism to deal with shills?

>>16749811
It's said throughout the site whenever someone tries to shill their shit.

>>16748327
What the hell is this?

>>16749811
Lurk moar

>>16749659
>i have a master's in mathematics and i don't know what OP is on about because i have no clue what lagrange did since that's not the way it's been taught and i'm not even one of these guys who are completely oblivious to the history of mathematics.
why not have a look? the book is free and still very readable 200 years later. Lagrange uses Taylor series to define derivatives since they appear as coefficients in a power series. historically Brook Taylor took the interpolation polynomial, divided by [math]\delta t[/math] and looked at the limiting case. Lagrange on the other hand avoided limits altogether and derived the Taylor series by working only with finite quantities, thus avoiding any infinities. he never speaks of ratios of infinitely small numbers either since for him derivatives are purely derived functions and nothing else, hence his suggestive notation for derivatives [math]f^{'}[/math], [math]f^{''}[/math], [math]f^{'''}[/math], etc. You can read a summary here https://cfraser.artsci.utoronto.ca/lagrangetheorie.pdf

it's actually a very enlightening and logical approach

>>16750149
damn what an elegant way to derive taylor's theorem

>>16749880
The 4chan football tournament

>>16750149
i'm not really into analysis tho and am bad at reading. that sad, i've skimmed through your linked pdf, it's indeed interesting and i agree that lagrange's conception of derivatives is indeed very logical.

the obvious critique from a modern stand point is then that this conception only applies to locally analytic functions, the modern concept of derivatives is hence more general. what lagrange's approach is hence missing is not only application to pathological set-theoretic concoctions, but also semi-pathological functions such as exp(-1/x) on the positive real ray extended to the rest of the real line by zero.

of course, it generalizes more naturally to the notion of derivatives as continuously varying multilinear maps in higher dimensions (but again only for analytical functions).

in any case, the observation that differentiability at a certain point is equivalent to the continuity of an approximating linear (or multilinear) map at that point opens the notion of differentiability up to topological inspection. if you want to ignore that you deprive yourself of the application of very fundamental insights about continuity in your understanding of differentiability.

>>16750333
>insights
who gives a fuck? i'll take descriptions of reality over prescriptive insights.

>>16750333
ok, wait, he of course uses remainders, so it works more generally. the latter point about the applicability of topology still stands tho.

>>16750335
>descriptions of reality
not quite sure, what you're on about, but clearly the notion of topological limits are one of the true gems of mathematical descriptions of reality?

>>16750341
is this not something that can be achieved with algebraic topology?

>>16750341
To be fair Lagrange was very old when he published this and Leçons. And the applicability of his methods are quite wide (eg. calculus of variations). If he lived longer or if someone had taken over his work, we’d be looking at a different math, maybe a preferable one.