maths is fake

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maths is fake

Postby earthonator » Mon Jan 30, 2017 1:10 am

i couldn't find a math thread and this probably doesn't fit in the school thread so here's my question. why do integrals work?

like if im just trying to find the integral of a line with 2 points A and B, I split it up into infinitely small pieces. so how am gonna get from point A on my line to point B? I wouldn't get anywhere the pieces are infinitely small! we are now finding the volume and surface area of 3-D objects in my calc class but this one fundamental concept doesn't make sense.

also why are integrals exact answers? obviously in the real world they're all estimations as nothing is perfectly smooth. but how are integrals not estimations. when I do an integral on my calculator I know it's an estimation. it's some program doing a reimann sum. but why is it when I do it on paper manually I get an EXACT answer? this is related to my first question.

I'm scouring the web right now, reading reddit explanations and YouTube videos, maybe I'm just rubbish at googling, but it still doesn't make sense. can anyone explain this or link me to something that would help? im asking my calc teacher tommrow hopefully she'll know and not give me an answer like "wait for sixth form"...
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Re: math is fake

Postby fun_yunchables » Mon Jan 30, 2017 4:14 am

earthonator wrote:i couldn't find a math thread and this probably doesn't fit in the school thread so here's my question. why do integrals work?

like if im just trying to find the integral of a line with 2 points A and B, I split it up into infinitely small pieces. so how am gonna get from point A on my line to point B? I wouldn't get anywhere the pieces are infinitely small! we are now finding the volume and surface area of 3-D objects in my calc class but this one fundamental concept doesn't make sense.

also why are integrals exact answers? obviously in the real world they're all estimations as nothing is perfectly smooth. but how are integrals not estimations. when I do an integral on my calculator I know it's an estimation. it's some program doing a reimann sum. but why is it when I do it on paper manually I get an EXACT answer? this is related to my first question.

I'm scouring the web right now, reading reddit explanations and YouTube videos, maybe I'm just rubbish at googling, but it still doesn't make sense. can anyone explain this or link me to something that would help? im asking my calc teacher tommrow hopefully she'll know and not give me an answer like "wait for college"...


i work in a branch of applied maths so i guess i can try to answer this however i can although it seems like maybe a weird topic for a fashion forum. idk

warning long winded explanation below
Spoiler:
this is along the lines of a more advanced topic (measure theory, typically taught in upper div uni or even postgrad courses in maths), but I don't wanna bore you with a mathematical definition.

Intuitively, if you split the domain [A,B] into 10 equal parts, you'd get a length (B-A)/10 for each part. For 100, (B-A)/100. And so on. If you inspect how this length shrinks as you take more (finite) parts, it approaches zero (but is never zero!). You also have to sum a growing number of parts as well. These two things "balance" each other in some sense -- as you make smaller parts, you must also sum more parts. In the limit (and I need to stress this, you need to understand the concept of limits well to go anywhere conceptually with calculus!), you are summing infinitely small pieces, but infinitely many of them. Like this, they still "balance" each other out.

as for your second answer, it also ties back to the notion of a limit. physically, we can't sit around and compute sums to infinity, nor can we compute with precision infinitely small numbers. We can very well approximate the sum with a big number (like billions, maybe trillions of terms), and similarly with the small numbers. For "well-behaved" functions (things that almost all physical things in the world are represented by), we get an even stronger property that as we take this approximation with more and more terms in the sum, we inch closer and closer to the "true" value (in the limit to infinity we recover the exact value!).

We can find the exact value when we do it on paper, since a function is a representation of all values at all points. If you consider an example:
f(x) = x^2
No matter whatever point you pick, x = 0.13534624, 0.345, 1000000, etc, you and I both know how to calculate x^2. There are an infinite number of points to pick! But for a computer, it can't keep track of an infinite number of points (as it would require infinite memory). It would instead have to approximate this function with a finite number (could be a large number of them, but still finite). This is the central concept behind numerical integration. For simple cases (like the one given in fact), computers actually can compute analytically the integral like we can (this is how wolfram alpha can do some integrals!)
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Re: math is fake

Postby maj » Mon Jan 30, 2017 6:48 am

this is the content i crave on caretags
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Re: maths is fake

Postby parastexis » Mon Jan 30, 2017 10:00 am

To add onto f_y's reply: WRT the first question, look up Zeno's paradox. For question 2, keep in mind that Riemann integration is a definition, not a proof, and the idea of smaller and smaller rectangles is an illustration of why it makes intuitive sense to define integration this way.
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Re: maths is fake

Postby DeafIdiotGod » Mon Jan 30, 2017 11:07 am

Anyone watch the YT channel 3blue1brown? Excellent maths content with good animation to make it super easy to understand while being a step above NumberPhile in terms of depth.

This one's a banger

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Re: math is fake

Postby earthonator » Mon Jan 30, 2017 2:28 pm

fun_yunchables wrote:We can find the exact value when we do it on paper, since a function is a representation of all values at all points.

@fun_yunchables the balancing thing makes a bit more sense but I asked my calculus teacher and she said even the integral we derive on paper is still an estimate. so that would mean it's NOT an exact value. is she wrong?

also in physics we do more "application in the real world" problems so when we take the integral of something, these pieces aren't infinitely small correct? they're actually as small as an electron or quark right; they're finitely small. because nothing in the real world can be smaller than that. right?
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Re: maths is fake

Postby fun_yunchables » Mon Jan 30, 2017 4:06 pm

The act of computing an integral analytically is exact (which is why we use an equal sign, and not an approximation sign). What comes into question as an estimate is intimately related to your second point. How well does a function approximate the real world? That, I think, is where the "estimation" comes into play. A function is, at best, an "estimate" of the real world. The table surface you write on isn't actually completely smooth, you can always zoom in really close to see it is jagged. So in that sense, a plane function is not the best estimate of a table surface. But from a practicality standpoint, the error we find between a plane function and a table surface is so ridiculously small and thus we approximate it as a plane. That is essentially where the approximation comes in -- we do not have a perfect representation, but we have something close, and it is "well-behaved." However, the act of taking an integral given a function is exact -- it is a mathematical construct. IF we had a function that perfectly modeled the real world, then the integration would give us a perfect value of whatever we wanted to find.

this kind of question kinda gets into the philosophy of mathematics to a degree -- its possible to see different intepretations & still be right about it. kind of like how in statistics & probability, there are two philosophies: bayesian & frequentist, and neither is wrong.

that being said, maths is beautiful but maths also sucks. i have a huge love-hate relationship w/ it.
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Re: math is fake

Postby DeafIdiotGod » Mon Jan 30, 2017 4:49 pm

earthonator wrote:also in physics we do more "application in the real world" problems so when we take the integral of something, these pieces aren't infinitely small correct? they're actually as small as an electron or quark right; they're finitely small. because nothing in the real world can be smaller than that. right?

This brings this quote to mind:
A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness.

It basically sums up the role of maths in Physics, it's used as a map of the real world that fits well enough to be useful but isn't so precise as to be a massive hassle to use, that's why physical theories are referred to as "models".

It's still pretty crazy that maths can be used to predict the universe so effectively though, as a number of very clever people have also noticed.
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Re: maths is fake

Postby JewTurk » Mon Jan 30, 2017 6:08 pm

A good xkcd comic about math/study.

Image

I think my favorite math quote on usefulness is something to the effect of no model is perfect, but some are useful.

I fucking love math, I found myself auditing so many math classes freshman year I just declared an additional major and started getting credit for them. Math is bonkers fun at my university, so many people join the math program we have, stick with it through their masters, and end up working towards their PhD at other universities. Tons of undergrad research opportunity etc.

Curious, for those who study math/enjoy it. What class made you fully realize how much you enjoyed it? For me Calc 3 was an awesome payoff for getting through Calc 1 and 2 and the visualization/models you could create with Calc 3 were just absolutely amazing. I find myself enjoying differential equations and seeing what you can model with those now more than anything. Probably because I get to see the applications in my circuit analysis from various projects I'm working with.
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Re: maths is fake

Postby khayandhi » Tue Jan 31, 2017 1:43 pm

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Re: maths is fake

Postby blanket » Wed Feb 01, 2017 8:41 am

JewTurk wrote:Curious, for those who study math/enjoy it. What class made you fully realize how much you enjoyed it?

i got into maths because i read a lot of these when i was little
Image Image
they had funny jokes and the parents were not very keen on them so they were perfect
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Re: maths is fake

Postby rjbman » Wed Feb 01, 2017 10:52 am

Great math game: Frog Fractions - trust me, keep playing.
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Re: maths is fake

Postby fun_yunchables » Mon Jun 05, 2017 10:15 pm

Image

quality math memes
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Re: maths is fake

Postby fun_yunchables » Fri Nov 22, 2019 12:09 pm

reviving this just to say that working in PDE and functional analysis/fredholm theory is laugh worthy because it relies on two main arguments: integration by parts and sweeping all our problems under the “except on set of measure zero” rug

also started an industrial research job landing people on the moon and i miss doing real math. nobody cares for proofs here
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Re: maths is fake

Postby turkeyschmitt » Sun Dec 01, 2019 6:57 pm

fun_yunchables wrote:also started an industrial research job landing people on the moon and i miss doing real math. nobody cares for proofs here


been working in software for 3 years since studying cs+math, and i yearn for even the idea of a proof. the work does sometimes does end up being surprisingly satisfying, but its not the same. then again, i am no way smart/dedicated enough to stay in the magical land of academia, so can't really complain. ideally i would work through some books in my free time - he says as cauchy-schwarz master class stares at him from his desk, covered in a light layer of dust and guilt...
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Re: maths is fake

Postby fun_yunchables » Mon Dec 02, 2019 12:46 pm

turkeyschmitt wrote:
fun_yunchables wrote:also started an industrial research job landing people on the moon and i miss doing real math. nobody cares for proofs here


been working in software for 3 years since studying cs+math, and i yearn for even the idea of a proof. the work does sometimes does end up being surprisingly satisfying, but its not the same. then again, i am no way smart/dedicated enough to stay in the magical land of academia, so can't really complain. ideally i would work through some books in my free time - he says as cauchy-schwarz master class stares at him from his desk, covered in a light layer of dust and guilt...



lemme tell u smthn, u dont gotta be smart to hack it in academia. just tenacious. in fact, being too smart and principled may work against u (see this famous case)

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