/sci/ - Science & Math

understanding is for brainlets, real intelligence is knowing

What did he mean by this

>>9738281
3blue1brown

lol calculus is easy shit and inuitive as all hell

>>9738301
t. doesn't understand algebra

>>9738302
Linear algebra is also intuitive. I don't understand the "wahh calculus hard" and "wahh linear algebra gonna kill myself" memes.

>>9738308
t. amerimutt who learns the pledge of allegiance in math class

>>9738310
Lol. Half a planet off. Try again.

>>9738281
Wtf? As someone who graduated years ago and has gone back and reread and practiced all my old analysis textbooks...yeah you can understand calculus, all the way down to set theory and ZFC and all that shit. It's not even hard.

alot of small things maek big thing

>>9738339
>set theory is calculus

>>9738339
>>9738364
everything wrong with modern math. fuck set theory

>>9738364
>>9738589
what's wrong with set theory? is it wrong? not useful?

>>9738611
>is it wrong?
Quite

>>9738611
it's only use is destroying intuition and promoting mathematical autism (rigor)

As h approaches 0, my understanding approaches its limit.

>>9738659
kys

>>9738705
not an argument

>>9738702
dam i feel bad for u son, keep at it though

>>9738708
There are literally hundreds of books of proofs (the ultimate argument) that you don't accept because your blubbering shit brain is incapable of understanding the simplest concepts in math.
>muh infinity don't real
>muh there is only natural numbers
kys

>why does a wronskian work? the world may never know

>>9738749
>>Why does the determinant work? The world may never know.

>>9738756
>four tau minus det squared? why not?

>>9738785
What?

>>9738793
the shortcut for finding what the roots of a linear system do or whatever. You plot tau vs det, and then place your "point" on it. Positive tau and det above the curve = unstable spiral, etc

My andean genes make me understand these simplicities easily.

>>9738659
>thinks this is bad

>>9738659
Intuition is for faggots and homosexuals (that is, physicists and engineers), rigor is for men, that is mathematicians

I'll teach you.

Let h be a small number (small compared to the smallest length scale of the function you are considering).

A derivative is just (f(x+h)-f(x))/h. f(x+h) is very close to f(x), so I'll typically call f(x+h) = df(h) + f(x) (here, df is a function of h, or df(h)). You'll notice that f'(x) is df(h)/h.

Let's prove the product rule. d/dx(f*g)(x) = (f(x+h)*g(x+h)-f(x)*g(x))/h = ((f+df)*(g+dg) - gf)/h = g*df/h + f*dg/h = gf' + fg', dropping the small df*dg term. Can do a similar thing with the chain rule. All the differentiation formulas are just algebra like this.

The fundamental theorem of calculus is also just algebra. Let's establish that if you integrate a derivative, then you get the original function. I'll define int(f) = sum_{n} f(n*h) * h. This defines how we integrate from 0 to x = N*h (the sum over n stops at N). What's int(df/h)? Well it's just sum_{n} df(n*h)/h * h = sum_{n} df(n*h) = sum_{n} f(n*h) - f((n-1)*h) = f(N*h) - f((N-1)*h) + f((N-1)*h) - f((N-2)*h) + f((N-2)*h) - f((N-3)*h) + ... - f(0) = f(N*h) - f(0) (it's a telescoping sum) = f(x) - f(0).

Okay, so that's it. Notice how it was all just algebra.

You guys are awful at explaining shit.
Calculus is just the rate that curves shift at.

>>9738914
Why do I need to kill myself with deltas and epsilons when, to find the instantaneous rate of change for a given point, I can evaluate a second point .01 units away from it and take the slope of the two points? It literally gives you the derivative without performing gay mental gymnastics.

>>9738925

that is exactly what epsilon and delta is

>>9738300
This. If you cant understand what its doing after watching their vids then you are legitimately a brainlet.

>>9738883
Intuition is for men. Rigor is for autists.

>>9738925
Imagine if you had a function that was constant throughout its domain, but between x values of 0 and 0.01 its behavior was very erratic. Using your method, you would completely miss any nuance of the function.
So by making your x difference infinitely small, you can find the slope at any point, regardless of the function's behavior.
Thats what the deltas and epsilons attempt to do

>>9739099
>Imagine if you had a function that was constant throughout its domain, but between x values of 0 and 0.01 its behavior was very erratic.
Just take a point that is .0001 away from the point you want to find the derivative at.

>>9739107
Imagine if you had a function that was constant throughout its domain, but between x values of 0 and 0.000001 its behavior was very erratic.

>>9739022
Rigor is the result of having the correct intuition since intuition is often wrong.

>>9738356
Or many smaller thing make no thing

Calculus is a means by which to resolve space positions over time. Given any initial point, we can observe the relationship that it will have with any other given point. Defining these points around objects allows for one to describe their motion relative to the other objects perceived.

Ezpz.

>>9739127
toast to that lovely roast anon

>>9739148
why would you even bother trying to encapsulate the essence of calculus, retard? This is done by every book, teacher, YouTubed and math page concerning the topic (Wikipedia included). Supplying a definition when it is so easily and literally copied and pasted shows nothing of your own intelligence nor does it help those who do not understand, as they have certainly read many similar phrases regarding the topickf calculus.

>>9738281
>if you sum rates of change over a range you get the end value minus the start value
WOW SO HARD

>>9738281
I mean you do understand it intrinsically, your brain does it every time it calculates the trajectory of a thrown object or attempts to find the best balance between a set of flavors in a dish. Understand all you want but if you don't know what's going on then you are useless all the same.

>>9739162
Because that's what intrinsically understanding it is, that's what the OP posted about. What else are you going to post about it?

Oh, you're actually just a retard who doesn't understand what the word "intrinsic" means -- aren't you?

I get it best when it's explained while looking at a coordinate system. You have two points on a graph. P1(x,f (x)) and P2(x+h,f(x+h)). If you draw a line through those two points you get a secant(touches two points) which is sort of an approximation of the slope of the graph at the point P1. You can calculate slopes of lines as change in y(f(x)) over change in x. This gives us f(x+h) - f(x) over (x+h) - x. The x'es in the denominator cancel out so you get f(x+h)-f(x) over h.
This is literally just the slope of our constructed line. What calculus does is say what happens if we let P2 approach P1, the secant become a tangent, let h become smaller and smaller. It's all sort of the same way to say that you want to move from an approximation (secant/two points/h is big) to instantaneous rate of change(tangent/"one point"/limit of h->0).
If you can follow that far the rest is just plugging the functions in and doing some juggling (this is called derivatives by first principle). Rest just adds onto that knowledge.

It's actually really fun because if you've been just doing derivatives by memory this can help you a lot.
You can try finding some common derivatives of functions in minutes. f (x)=x^2 is easy and fun to practice and you can do for example f (x)=ax+b <---- you should be able to guess what the derivative becomes in this one, since it is a linear function and the derivative is all about finding slopes of tangents (hint: slope of a line is constant).

Some students have a hard time following it, but I think everybody has the mental capacity to understand this if you just apply yourself a tad

>>9739178
Your post did not display an instrinisic understanding. It displayed only familiarity with the common definitions and descriptions of the word. Even someone who hasn’t done a lick of math could post such a description.

You could post resources like 3B1B, which intentionally aim to cultivate said level of understanding. Or recommend books like Spivak (among others such as Apostol and Courant), that leave the reader with no other choice but to rigorously and intuitively understand the subject.

>>9739198
Linguistically you can't convey intrinsic understanding because it's intrinsic.

You're literally too autistic to understand language, you should stop.

>>9739198
I realize that I'm going to need to beat this point over your head before you get it.

>
You could post resources like 3B1B, which intentionally aim to cultivate said level of understanding. Or recommend books like Spivak (among others such as Apostol and Courant), that leave the reader with no other choice but to rigorously and intuitively understand the subject.

These are EXTRINSIC. Cultivation of knowledge and understanding is not an intrinsic understanding, it is a LEARNED (i.e. not from within) understanding, which means that it comes from a source which inherently implies outside (extrinsic) knowledge.

Intuitively understanding something merely means that you are aligning what you already intrinsically understand with knowledge that allows you to express this to others by explaining the objective constants.

Intrinsic understanding has nothing to do with that, and you are going off because you literally have no (ironically) intrinsic understanding of the phrase itself. OP is posing literally 'I wish I had a framework by which to integrate knowledge', and you are responding with "DURR JUST INTEGRATE THE KNOWLEDGE" and "WOW DONT GIVE HIM A FRAMEWORK"

Cherry on top?

>Your post did not display an instrinisic understanding.
> It displayed only familiarity with the common definitions and descriptions of the word.

Yeah that's what that is you fuckshit.

intrinsicness is a spook
empiricism is all we have

>>9738281
>tfw i thought this too
>but then I took analysis
come into the light, anon.

>>9739263
You’re only proving my point further.

>>9739278
You should probably drop the deragatory tone otherwise you will only look more foolish later. Anyhow, I know these are extrinsic resource (it really should go without saying, but alas, here we are), but they help cultivate an intrinsic understanding. If you are unfamiliar with 3B1B, I suggest you watch his videos as cultivating an intrinsic understanding is practically his mission statement.

Also, your cherry on top is, still, sadly not a demonstration of intrinsic understanding. If you’d like, find a definition, or set of definitions, that you agree with for the word and we can work from there.

Other than that, your post was a lot poorly written jibberish and responding to individual lines would be far too pendantic and tedious, esp since it was mostly a pointless rant (just formally define intrinsic instead of going nowhere fast about what isn’t intrinsic). It’d be much easier to simply agree upon a definition (or set of) and work from there.

>>9739358
Dude you're fucking awful at English. Is this your third language or some shit? Are you literally Chinese?

YOU LITERALLY CANNOT LEARN AN 'INTRINSIC UNDERSTANDING'

YOU

CANNOT

LEARN

HOW

TO

UNDERSTAND

SOMETHING

INTUITIVELY

BY

NOT

INTUITIVELY

UNDERSTANDING

IT

YOU DENSE MOTHERFUCKER. YOU LITERALLY CAN DO NO OTHER THING BUT GIVE YOUR 'hey this is what my take is' and HOPE THAT THEY GET A TAKE THAT HELPS THEM.

THAT'S FUCKING IT.

>>9739278
And to be clear, your post is implying that, in your opinion, an intrinsic understanding cannot h
be aided in development from tangible resources? That an instrinic understanding cannot be formed from a learned understanding?

A warning if this is seriously your stance: even intuition is learned, my friend.

>>9739373
reddit spacing.

>>9739383
>>9739374
>posting about reddit in the first place

Eat a dick you uneducated faggot.

https://www.jstor.org/stable/3326788

>>9739373
First of all, you’re deviating from the primary argument into this new tangent.

Secondly, if you only have the intuition you were born with, I feel sorry for you. But in all likelihood you don’t, and this essentially breaks your arguement. Intuition can be learned and developed, and often is - just like anything else. What’s the example about the fireman in a room that was about to collapse?

To be clear, your claim is that you cannot learn understand something intuitively without already understanding something intuitively. Thus intuition can not be learned. I healthily and calmly disagree.

>>9739387
Hey at least I can learn to understand things intuitively

stay intellectually poor faggot

>>9739387
reddit spacing

>>9739390
>trolls on /sci/

get a life

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if you bring up reddit, you're the cancer. the only thing you intrinsically understand is sucking cocks, clearly.

>>9739396
I’m not the one posting about reddit, but it’s become clear you’ve forfeited the arguement. Good luck, anon

>>9739406
>arguement

There's no argument, you don't understand English and either as a godawful troll or your lack of understanding you continually fail to understand the linguistic implications at hand. You never meaningfully posited anything other than your own inability to understand things. That's not my job and it's of no importance to things at hand.

decision time: gonna write you off as a garbo troll, 1/10 for making me respond, you can get the other 9 when you aren't just baiting responses by being a stupid asshole.

>>9739396
reddit spacing

>>9739117
Kek

>>9739396
the spam filter should've caught this one

>>9739413
Ok, again, you’ve avoided addressing anything I’ve said. Nighty night you permabrainlet

>>9738308
>algebra is linear algebra
Retard

>>9738611
>>9738589
>>9738364
I didn't say set theory was calculus. I meant that as a foundation for mathematics, from which you can develop an understanding of calculus, it's really not that difficult.

>made As in calc 1,2,3
>have no idea what i'm actually calculating at any given point or its implications, just solving problems using strategies i've rote-memorized

>>9740372
>american education

>>9740490
>>9740490
t. Euro brainlet who can't deal with his inadequacies so he assumes an education system he's never dealt with must be gimped

I didn't take easy mode calculus, the shit they teach in high schools. I got As in college calculus classes with 33% to 50% dropout rates.

I've learned to build Taylor series, test for divergence, integrate by parts, construct triangles to use trig sub, build a parametric equation and sketch surfaces, trace a normal vector at a given point, find the volume of an intersection of two surfaces, find functions using FTC, find curl/divergence with Green's theorem, etc etc

But I'd be lying if I said I totally understood everything I was doing at every point and every implication of every value I got back. I get the right answers, I just couldn't explain wtf most of it meant.

>>9739117
I've never even had a function just appear in front of me from any proof I've every published that using a point .01 wouldn't work for

>>9740572
https://www.youtube.com/watch?v=M2fHQ9eULzk

>>9740594
>who is the vice president of america
>bin laden
mega kek

>>9740596
>Didn't know something before he knew it
>mega kek
Off yourself brainlet

>>9740603
>amerilard defending his "education"

>>9740572
>>9740372
The same thing happened to me. When I went took a real analysis course, I was surprised at how much I actually understood just from solving problems, though.

>>9740603
mutt butblasted lmao

>Just completely fucked up my fourier analysis exam
I'm sorry /sci/, I couldn't make it. I'll off myself now.

>>9738281
Assuming you "intrinsically understand" functions:

Limits = given an interval of outputs ( y - e , y + e ) , there exists an interval of inputs ( x - h , x + h ) whose outputs are within the given interval.

Derivatives = slopes "at a point" defined using limits.

Integrals = inverse derivatives.

Series = infinite sums defined using limits.

Best explanation by far >>9739196

>>9740572
It sounds like you are proud of taking the exact same courses that euros with the exception of having to prove shit
t. euro and if you couldnt prove shit you would fail the course

>>9740671
>t. brainlet europeasant who conflates the goals of mathematics aimed at engineers with those aimed at students of pure mathematics
FTFY.

PS. Proving the power rule by induction doesn´t make you any less of a brainlet.

>>9738310
t. retard

>>9739127
/thread

>>9740894
>triggered amerilard

>>9738281
no one does. But people can learn it.

>>9740925
>t. triggered euroshit whose life is controlled from cradle to grave by the almighty State
I can´t wait for your leaders to instigate a third world war, during which you will be drafted to die, government-issued weapon (which you are under no circumstances allowed to own in peacetime) in hand, on the front lines.

What does it feel like to be a useful idiot?

>>9740880
>american engineers can't prove themselves out of a wet paper box
Why am I not suprised by this, every great """"american"""" scientist or engineer has been european

>>9740933
why don't you just move abroad if you want to learn something in school or walk on the streets without getting shot and having to tip the shooter?

>>9738884
Thanks.

>>9740638
Thanks, I think one of the difficulties of learning calculus online/on boards like this is the multiple different notations for the derivative. It's not until you get to differential equations you usually get exposed to more than one kind of notation.

>>9740957
We're learning partial derivatives in classes but I never fully understood what dx meant, I use it as a variable when doing substitution, can you explain what it is really?

>>9740975
it is an infinitesimally small increase in the variable x

>>9741055
and why is it there?