/sci/ - Science & Math

Math is just a joke invented by atheists so they can pretend to be smart when they really don't know what they are talking about

Nice rare pepes friends, I'll post mine.

>>7363212
Read Cauchy anon.

>>7363212

Serious question:

Can we construct the derivative (of a single-variable real function) in a purely abstract way, without reference to infinitesimals and limits?

>>7363272
approximate function with polynomial

define derivative as linear operator on finite dim vector space of degree n polynomials

b00m gauB BTFO
cauchy BTFO
newton BTFO
Wildberger reigns as king

>>7363298
>approximate function with polynomial
How do you prove you can do this to arbitrary precision without using the tools of analysis?

>>7363298
Limits are approximations though. You are just taking a rigorous concept and making it less rigorous.

>>7363310
i didn't say anything about arbitrary precision nigga.

i can use numerical methods to create poly-FUCKIN-nomials that come as close as i want to satisfy whatever properties i want out of a fuckin function m8, then i fuckin PROVE its close enough for my own liking so my bitch ass approximation dont get outta line. i do all this shit indepedent of any faggot ass limits or arbitrary precision shit. i stay silent about all dat shit. if theres some gay ass function that i cant even polynomialize it then fuck that function i don't even care about dat function

then i choose a bomb ass basis for polynomials of degree <=n, and make a big ass FINITE matrix that is all linear and shit and THAT is my derivative

ITT its 1687.

If the earth attracts the moon with gravity, why doesn't the moon fall into the earth? Check mate, Newton.

>>7363332
One away lad. Newtonian physics is the fucking truth, if you disagree you are inferior to me.

>>7363272
Taking the derivative of a polynomial is easiest.
The others I can't really get into because approximations.
Let <span class="math"> f(x) = x^n [/spoiler] be a monomial (you can either do this with any polynomial or you can show that the derivative of monomials).
<span class="math"> f(x) = x^n [/spoiler]
Replace x with x+y
<span class="math"> f(x+y) = (x+y)^n [/spoiler]
By binomial expansion (I can't \choose or \binom)
<span class="math"> f(x+y) = \sum_{k=0}^n \frac{n!}{k!(n-k)!} x^k y^{n-k} [/spoiler]
Replace the new x with x-y
<span class="math"> f(x-y+y) = f(x) = \sum_{k=0}^n \frac{n!}{k!(n-k)!} y^{n-k} (x-y)^k [/spoiler]
which is the taylor series around point at point <span class="math"> y [/spoiler].
The <span class="math"> k[/spoiler]th derivative is defined to be the <span class="math"> k [/spoiler] coefficient of the <span class="math"> x-y [/spoiler] polynomial
<span class="math"> f^{(n)}(y) := \frac{n!}{k!(n-k)!} y^{n-k} [/spoiler]
for the first derivative
<span class="math"> f'(x) = \frac{n!}{1!(n-1)!} y^{n-1} = \frac{n!}{(n-1)!} y^{n-1} = n y^{n-1} [/spoiler]

>>7363310
depends on how you define a derivative.
if you define the nth derivative as the nth coefficient of the taylor series, then you're not in trouble.

>>7363359
Lol defining the derivative to be what it is from calculus after doing a bunch of bullshit algebraic manipulations

This has to be wildberger

>>7363359
> show that the derivative of monomials
is linear

>>7363212
>saying two numbers are equal even though the definition explicitly says they differ by non-zero number and therefore are not equal
What is this referring to?

>>7363367
Actually it was Lagrange.
Wildberger stole it.

>>7363381
bitch people knew the power rule for polys before newton even

>>7363387
> power rule
??

They knew to define derivative from the binomial theorem?

>>7363250
>not posting rare mochis

>>7363402
Oh my gawd he is SO HAWT

>>7363390
correcta-mundo

watch dis shit: lets define a magical symbol "dx" and i'm gonna declare dx isn't zero but has the bitchin' property that dx^2=0.

now some polynomial like f(x)=x^n if we add a little dx: f(x+dx)=(x+dx)^n by the binomial fuckin theorem is x^n + (n choose n-1)x^(n-1)dx + (a bunch of shit that turns into zero because dx^2=0) which equals x^n+nx^(n-1)dx.

oh shit nigga but now f(x+dx)-f(x) = nx^(n-1)dx and now out derivative [f(x+dx)-f(x)]/dx = nx^(n-1) DAYUM HOW DID DAT HAPPUN THATS DA RITE ANSWER

magic

now we all like

>>7363415
except the question was how to do it without infintessimals.
> dx isn't zero but has the bitchin' property that dx^2=0.
There's no number that that's true for.


Also who was it that figured this out before Newton? I'm curious.

>>7363425
>There's no number that that's true for.
Bro, I dunno what kind of math you're doing, but I do commutative algebra all day 'er day, and we have to bend over backwards to prevent situations like this.
Also:
>Knowing nothing about differential forms or exterior algebra
>commenting anyway

>>7363430
>Knowing nothing about differential forms or exterior algebra
>commenting anyway
You were suppose to prove me wrong.
I can't preface every post with "I know that I know nothing, but...."

>>7363425
i was showing you how they used to do it back in the day.

lots of partial results were known before newton mang, fermat, cavalierri, james gregory, barrows. newton didnt pull all this shit outta nowhere

it is in spirit the same shit as what was posted up in da thread. dx is just a symbol for the intuition "small enough to matter but not enough to matter when squared"

this thread reeks of the berger

>>7363516
no one shall expel us from the paradise that wildberger has created for us

>>7363516
>Can we construct the derivative (of a single-variable real function) in a purely abstract way, without reference to infinitesimals and limits?
I wonder why.

>>7363548
What does this have to do with this wildberger guy? I just wanted to know if there was a standard way of doing it, that's all. No need to get all butthurt and throw your /sci/-maymays at me.

>>7363212
>saying two numbers are equal even though the definition explicitly says they differ by non-zero number and therefore are not equal
> >adding infinitely many infinitely small quantities to obtain finite quantity
Am I missing something? Doesn't that explain itself?

>making zero division valid by simply telling the divisor isn't zero but it's indistinguishable from zero
If it were defined as anything less it could be A: an entirely different number line, or B: The opposite spectrum of the number line

>>7363568
> I just wanted to know if there was a standard way of doing it
there isn't.
That's why Wildberger tries to do it.

>>7363586
Don't worry, we've already derailed the thread.

>>7363568
>>7363590
Yes there is. The standard way of doing it is with limits.

>>7363326
>i didn't say anything about arbitrary precision nigga.

>i can use numerical methods to create poly-FUCKIN-nomials that come as close as i want to satisfy whatever properties i want out of a fuckin function m8.

Kek but that's what arbitrary precision means, anon.

>>7363597
> Can we construct the derivative (of a single-variable real function) in a purely abstract way, without reference to infinitesimals and limits?
Exactly. There isn't one without limits (or infinitesimals).
Try to keep up.

>>7363212
Proof by induction works for " two numbers are equal even though the definition explicitly says they differ by an IMMEASURABLY small number"

>>7363601
>I just wanted to know if there was a standard way of doing it
>there isn't
Wrong: there is. With limits. try to keep up

>>7363597
Read the whole thread before posting. We were talking about standard ways other than limits and infinitesimals, in response to >>7363272

>>7363608
for fuck's sake, read the thread.

As I said earlier,
>Can we construct the derivative (of a single-variable real function) in a purely abstract way, without reference to infinitesimals and limits?

>>7363608
did you read the post he responded to?
it was a standard way WITHOUT limits.

>>7363615
You asked two questions, even if you're too retarded to know it:

1.) Can we construct the derivative (of a single-variable real function) in a purely abstract way, without reference to infinitesimals and limits?

2.) I just wanted to know if there was a standard way of doing it

I invite you to analyze these two sentences and turn in for 5 extra credit points a paragraph or two explaining why they are in fact distinct questions.

>>7363626
> autism speaks

>>7363373
I gauss it refers to 1 = 0.999...

>>7363626
>I just wanted to know if there was a standard way of doing it
meaning,
>I just wanted to know if there was a standard way of constructing the derivative (of a single-variable real function) in a purely abstract way, without reference to infinitesimals and limits.

I thought that would be obvious, but I guess I forgot how much autism there is in this board.

>>7363359
>if you define the nth derivative as the nth coefficient of the taylor series, then you're not in trouble.
How do you define the taylor series, in such a way that you can actually calculate it?

>>7363669
>of an arbitrary function
idk. I haven't gotten that far yet
> of a monomial
by expanding the polynomial via the method shown there.
> of a polynomial
by showing that addition and scaling holds

>>7363516
assbergers or wildbergers?

>>7363676
noone cares about your fuqin nomial das fuqin 3rd grade shit

>>7363598

nah bro. where i choose to remain silent is a subtlety that speak VOLUMES. i remain silent about arbitrary precision, altho i see were i am being unclear

when my ass is sayin "as close as i want" i am talkin' subjectively. i am sayin if i find a poly thats good enough for me, then i stop there and its deriv will be a linear ass operator, finite as shit.

as in, if i make a some polys C and S that are bounded by 1 and satisfy C(0)=1 and S(0)=1 and C^2+S^2 ~ 1 and a coupla otha approximate functional relations dat characterize trig functions

now when i constuct C and S i can measure precisely how nicely they emulate the properties i want, i.e. their fuckin error. now we both know i am approxing sin and cos but if i am psychologicaly asspained by R and limits and infinitesimals then i can simply choose to ignore that notion.

NOW it just so FUCKITY-DO-DA HAPPENS that the 'best' C and S poly's follow a simple ass pattern and that pattern can be continued easy a fuck to make a bigger poly and the error from emulating them functional relations satisfied by 'circle functions' gets smaller. this is a nice cherry on top. it also turns out dat any function worth a SHIT can be analyzed in this bomb ass pattern type way

if a nigga decides to extrapolate the shit outta this decreasing error pattern shit and call it a limit then whatevs. however if said nigga receives psychological butt-blastation from talkin' about limits, they can choose to make no further assumptions or extensions. they can now do non-limit calculus as follows

1.) be happy that any function that matters is computable
2.) compute it approximately
3.) do fuckin calculus on it, which is now just linear operators on finite fuckin vector spaces, no analytic horseshit to worry 'bout.

voila, no more epsilons and shit. now dis bitchin' approach has da upside and/or downside dat every object a nigga works wit has to be constructed explicitly, none of that axiom of choice shit.

>>7363701
also, in this 'numerical' calculus, whenever a nigga proves some shit about a function, for example its error with respect to some ideal property he wants a function to have, that nigga is actually providing a fuckin procedure for how its done. you give and you take

>>7363686
it's like all of calc 2 and the start of calc 1

>>7363707
you should take yo kronecker ass back to the 1800s and prove deez nuts nigga

>>7363708
polynomials is literally pre calc yo ass

>>7363212

For the purposes of calculation at 1/1000th of an inch, useful for rocket scientists maybe.

>>7363714
but calculus on polynomials is calc 1 and all those polynomial series including the taylor series is calc 2

>>7363709
bitch kronecker was small time. my mind is nimble enough to see dat behind the interplay between da concrete and da abstract and da formal and da intuitive lies the sublime platonic Object invariant with respect to all and fashioned str8 from the mind of God or Natura Naturata

>>7363721
>but calculus on polynomials is calc 1
I'm sorry for your loss. I went to a crummy junior college, and even we were not so bad as you.
>and all those polynomial series including the taylor series is calc 2
That isn't relevant, we're talking about calculating the Taylor series of non-precalc functions.

>>7363373
Infinitesimals in definitions of limit, derivative, integral etc

>>7363729
> I went to a crummy junior college, and even we were not so bad as you.
it was in high school.
> we're talking about calculating the Taylor series of non-precalc functions.
All Taylor series are polynomials...

>>7363212
Because it just werks.

I still dislike it though.

>>7363212
Because it just werks.

I still dislike it though.

>>7363978
There are no infinitesimals in any of those definitions.

>>7363982
An infinite power series is not a polynomial idiot.
polynomials have a finite order

>>7363326
>>7363415
>>7363701
>>7363722

top jej

>>7363684
I'm not even a programmer and this picture pains me deeply

>>7363701
regular calculus uses linear operators you fucking pleb

>>7363212
OP does not understand how a limit is defined.

>>7363359
>f'(x)=n*y^(n-1)

>>7363326
Kek'd hard

>>7363212

You lack understanding of the "infinite"... "finite" properties of numbers are NOT the same as "infinite" properties of numbers. This is a HUGE mental leap to make, but once done, calculus is understandable.

>>7364084
you bitch nigga not on a FINITE DIMENSIONAL SPACE you imprecise language using motherfucker

>>7364082
It's a /g/ meme

>>7363359
>depends on how you define a derivative.if you define the nth derivative as the nth coefficient of the taylor series, then you're not in trouble.

Circular reasoning is circular. Nth derivative is the nth coefficient of the Taylor series. But Taylor series is defined by setting the Nth coefficient equal to the Nth derivative.

Also Taylor series do not necessarily converge. If it diverges it is completely useless.