/sci/ - Science & Math

[Anonymous No.6906266]

If the "essence" of science is to better understand how the world works, and use rigorous techniques to explain natural phenomenons, and if the "essence" of, say, history, is to learn the truth about the past and apply it to current events, then how would you describe the "essence" of mathematics?

What does it mean to "do mathematics"?
What does it mean to come up with a mathematical proof?
What exactly do mathematical proofs "prove"?
What goal or purpose do mathematicians strive for in their work?

I'm asking this because I have never really known mathematics outside of "learn these concepts, don't question them, apply them on the test" that happens in my math classes. Lately, I've been reading essays and articles by mathematicians trying to explain that this isn't really the point of math, but I don't seem to understand what they are talking about.
Could you guys help me understand what mathematics really "is"?

Pic unrelated.

[Anonymous No.6906272]

The point of mathematics is to prove those concepts are valid in the first place

[Anonymous No.6906275]

The essence of science is understanding phenomena that can be used to develop practical technologies.

[Anonymous No.6906281]

The essence of mathematics is curiosity and boredom. Same goes with all the sciences.

[Anonymous No.6906282]

>>6906275
Engineer pls go

[Anonymous No.6906305]

Mathematics is just art in the medium of rigor. The same way music is art in the medium of sound, or literature is art in the medium of written language

[Anonymous No.6906308]

>>6906305
In other words, we do math because it's beautiful, and because it's relevant in the context of itself

[Anonymous No.6906312]

>>6906305
>art in the medium of rigor
This sounds interesting, what does "rigor" mean in this context, and what qualities of math make it an art that science doesn't have?

[Anonymous No.6906317]

I feel that I fucked up big time when it came to my math education.

I always have memorized how to do problems to get good grades in class, but I haven't paid attention to how everything works.

If I want to spend time to learn about mathematics the correct way, where would be the best place to start?
It seems that sites like khan academy and patrickjmt are more geared towards helping people do well on their math tests than actually "learn" math (there's nothing wrong with that, but it's not what I want to focus on).

Are there any categories of books in the /sci/ sticky to help out with this specific task?
What books/sites/videos/materials would you recommend to me to help me gain a better understanding of how math works, and how it is interconnected?

[Anonymous No.6906324]

>>6906317
Gelfand is nice for the early stuff imo.
http://4chan-science.wikia.com/wiki/Math_Textbook_Recommendations

[Anonymous No.6906326]

>>6906317
learn to do proofs, for one. "How to prove it" is a great book for this.

but also experiment. What really prepared me in high-school was my sense of curiosity. I'd play around with functions, learning what the different variables meant and they'd shape the function. Shit, you can probably learn a lot just playing around with matlab or wolfram alpha.

also, a lot of purists may shriek at this, but learn what kind of real world phenomena you can model with different branches of math. definitely helps with understanding.

[Anonymous No.6906331]

Mathematics is essentially the rigorous study of quantities. Different branches cover different properties of these quantities (algebra : relationships ; calculus : change/accumulation)

[Anonymous No.6906341]

>>6906324
Thanks for the tip. Algebra and coordinate systems do seem really early, but I want to get a holistic understanding of everything, so I'll certainly plan on including those books.

>>6906326
I'll take a look at that book, thanks.

>experiment
I haven't really experimented with these concepts before, and it is a really great idea.

When you experiment with different concepts, do you follow a set of guidelines in order to figure something out?
If you experiment with, say, a function, what do you try to look for in the function to get a better intuitive understanding of it?
What kinds of experimentation can you do to understand certain aspects of the function make it easier to "grasp" with your mind?

[Anonymous No.6906350]

>>6906341
Are you looking more for calculus stuff?
Try Rudin or I prefer Pugh or any other analysis book.
Spivak or Apostol if it gets too rough

[Anonymous No.6906358]

>>6906350
I'm really looking for everything from the basics to more advanced topics, because I want to build up my math knowledge again.

>Rudin
That means Principles of Mathematical Analysis, right?

>if it gets too rough
I'm just curious, but if spivak's book is easier to digest than rudin, does that mean spivak is less rigorous, or does it mean that spivak explains the concepts better than rudin?

[Anonymous No.6906367]

>>6906341
>When you experiment with different concepts, do you follow a set of guidelines in order to figure something out?
I'm not sure how much background you have, just start by asking "what if" or "what happens if...". Say, for instance, we have a sine function and we know absolutely nothing about sine functions. Just by asking "what happens if we stick a coefficient in front of the function as in f(x) = 3 sin x. Just by dicking around, we've discovered amplitude, as in how extreme f(x) goes for every value of x and that this defines its range. This is pretty obvious, but it's amazing what doing things like this will do for your understanding, as opposed to just rote learning steps for problem solving.

There's an entire school of thought on this

http://mathworld.wolfram.com/ExperimentalMathematics.html
http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en

[Anonymous No.6906371]

>>6906367
And by "discovered" you make the inference from multiple trials that multiplying the sine of x affects its range

[Anonymous No.6906385]

>>6906367
>>6906371
I think I better understand what you mean now, thank you for the explanation. I will certainly keep this in mind.

[Anonymous No.6906406]

>>6906312
"Rigor" means anything that can be formally proved. Math is art in a way that science isn't because mathematicians are able to be as creative and imaginative as possible, as long as it status within the bounds of the medium. Also math and art are self-referential in a way that science isn't. What makes for good math is original ideas that are unlike things that came before, but still interrelated and connected to past work so that the new theorems add to the larger context of math as a whole. It's structured like art

[Anonymous No.6906442]

>>6906266
>What does it mean to "do mathematics"?
>What does it mean to come up with a mathematical proof?
>What exactly do mathematical proofs "prove"?

Read a book on Mathematical Logic and Metamathematics

>What goal or purpose do mathematicians strive for in their work?

To prove tasteful theorems

[Anonymous No.6906450]

>>6906266
When I try to solve a programming problem. I never consult a math book immediately. I start going at it like a frustrated 5 year old.

Generally, I can brute force a solution with what math experience I have, but other times, I need to consult a larger framework of understanding to lead me in the right direction.

So.. For example. For a solution to a problem I think I need a recursive structure that embeds itself in several vector spaces. I lack the knowledge of how to deal with this, how others have delt with this, or whether or not some greater concept in math can help me break the problem down into tangible components I can understand.

I search recursive vector spaces and find papers about Recursive Model theory,

I read about it. Then I go about brute-force zero restraint problem solving.

If through this brute-force problem solving I notice a pattern, or know that my approach can be re-applied in other contexts. I notify someone in my department and try to get them interested and involved.

The proper learning process for mathematics isn't about learning abstraction to solve problems.
Its about solving problems that force the need for abstractions on you.

Math is tricks people have conjured up to solve their problems. Some people go as far as look for similarities between each trick and describing them. This helps you understand and categorize the problems and helps you identify what bag of tricks to use to solve it.

Mathematicians are the anal retentive curators of our bag of tricks. They organize, analyze, sort create or destroy our tricks. They find newer, more interesting ways to communicate how it is they've organized them, why they organized them that way, and what other tricks and solutions go in each bag.

To consistently understand how to handle all of these tricks, they've built an solid internal understanding of how to categorically think of the world in terms of tricks.

[Anonymous No.6906456]

>>6906331
I think the term, "quantities," is far too restrictive; mathematicians may study structure (like in group/ring/field theory, universal algebra, model theory, category theory...), general spaces (topology, cohomology, homotopy) or metric spaces (geometry, vector calculus, functional analysis), quantities as you had said, which can be discrete (combinatorics, classical number theory) or continuous (analysis, analytic number theory).

Mathematics is the product of restricting your domain of discourse to philosophical objects and restricting your methods to deductive reasoning.

[Anonymous No.6906460]

>>6906450
I love this. We are tricksters. I need to learn an ironic snicker to accompany my smirk.

[Anonymous No.6906461]

>>6906450
I like the way you explained that, but could coming up with an eloquent proof showing something very interesting be really considered a simple "trick"?

[Anonymous No.6906469]

>>6906450
In elementary school, through most of high-school, you're learning several tricks to help you solve problems as well as putting together elementary bags.

First you learn the add two numbers trick 1 + 1 = 2, Then you learn the add three numbers trick 1 + 1 + 2 = 4. Then you learn how to categorize these tricks in the "add as many numbers as you want to each other bag.

Likewise when you go shopping, you do not look at every item in the store trying to determine if the item you're currently focusing on is the chocolate milk you're looking for.
That would take forever.
You look for a better trick. You develop a trick where you look for the refrigerated area. You then apply your trick of examining every single Item in the refrigerated areas to find your chocolate milk. While doing so, you start realizing patterns in your counting.
1) Like Items are grouped together.
2) Most sections of groups are about a meter long.
This is awesome news. You immediately develop a trick to help you scan for Choco milk a lot faster. You look at an Item. If its not there look half a meter to the left, If that is not it, look at the item half a meter to the left. So on and so forth until you find milk. At that point you make the assumption that you have stumbled upon the milk group then start looking at other milk in the area.
This general solution to finding things in the store is what mathematicians like to call "abstraction". Instead of getting very specific, we look for big-picture patterns. We describe the patterns such that other people can use them and explore them to find choco milk.
Then another mathematician discovers that this can also be used to find pringles.
A third discovers that it also works with finding books in a library.
A fourth mathematician sees the pattern and bags these tricks and declares them to fall under the category of "finding things by looking at other things nearby and seeing if they're related to the thing I'm looking for"

[Anonymous No.6906514]

>>6906469
A proof:
Its a careful defense of your trick categorization.

Lets say I try looking for my car in a parking lot under the assumption that I can simply "find things by looking at other things nearby and seeing if they're kind of like the thing I'm looking for".
All cars I look at to find mine belong in the category of cars. Crap. This trick is no longer a valid trick because it is contradicted by a parking lot search!!
Mathematician 4 is not pulling his weight in the mathematics world and lives out his life in shame and embarrassment.

That is, until he comes up with a better way to describe how we categorize our "how to find things tricks". Our mathematicians revisits finding cars in a parking lot.After much research and consideration he refers to the work and research of a brilliant young mathematician in Japan whos managed to come up with a trick to find cars. "Look for nearby shopping cart receivers, remember how close you're parked to them. Remember what parking lot row you parked in"
Excellent. You immediately start seeing the bigger picture here. You come up with an abstraction to describe the universal trick to finding things.

"Look for a reference point. If you find the reference point, the object you're looking for is nearby, look at every single object nearby until you find your object, if your object is not nearby, look for other desirable reference points".

This solves the problem of finding things."Cars have isles and shopping-cart receivers as refferences, milk has other milk and refrigeration as refferences".
Great.
but to save you the embarassment of being proven wrong again, you need to both be sure about what you've come up with, and to also start figuring ways to defend your ideas.
If you can escape going into enough detail, you rely on the logic of other categorizers/mathematicians to save you the work.(Under the assumption that their logic is based on solid reasoning)

[Anonymous No.6906538]

>>6906514
Mathematics is sometimes about looking at other tricks, weighing and calculating how much you trust the other trick to help you come up with your trick.
It helps with your decision greatly if you can put absolute trust in tricks defended by other tricks that are defended by similar tricks/structures all the way down to the most elementary universal tricks like "finding something means that the object we're looking at is the object we're looking for".

Now. Lets talk about math a bit more. Math is not numbers. Math is tricks and the world.We strip the world of everything we find is useless so that we're not distracted when we're trying to come up with tricks. You learn only the most basic tricks like measuring physical dimensions, solving for x, describing properties of cyclical structures using sin,cos,tanh are all very small tricks. Teachers are trying to set you up with an arsenal of small, seemingly unrelated tricks.
In college, you start learning about the masterful categorizations and abstractions of tricks, that can lead to useful sub-categorizations, as well as trick generation. You get so good at trick solving, you eventually find your tricks automating robots, recognizing speech, flying space ships, managing economies and banks.

Utlimately mathematics is not about learning the every trick of a trade to make someone money.

Its about CREATING a trick of a trade for other to live by.

We do this by studying the intrinsic nature and properties of a trick in context of other tricks, and trick categorizations, then making inspired guesses and attempts to create meaningful categorizations that generate more tricks than an average trick user can come up with on his own.

[Anonymous No.6906542]

>>6906461
Yes. One presupposed by the understanding of other simpler tricks.

[Anonymous No.6906936]

>>6906275
Bacon pls go

[Anonymous No.6906948]

>>6906442
And also, to create tasteful proofs

[Anonymous No.6906959]

>>6906266
>What does it mean to "do mathematics"?
To solve logical questions that employ signs and numbers with assigned values and functions in a logical way.
>What does it mean to come up with a mathematical proof?
To show, in a logical and mathematical way, that something is true or false. It's basically like solving an equation, but writing the details. If the proof is for a broad spectrum of of numbers, which is usually the same, the proof must show that this applies to all numbers or some numbers and state why.
>What exactly do mathematical proofs "prove"?
They prove a mathematical statement that weren't already proven.
>What goal or purpose do mathematicians strive for in their work?
Mathematicians (Not Engineers or some other bullshit that employs mathematics) usually merely do it for the love of it and the joy you get afterwards. It also is done to advance human knowledge and make use of it in other physical or 'real' fields that employ mathematics heavily (Physics, Engineering).

[Anonymous No.6906963]

>>6906317
If you want to go from the beginning and up, Spivak's Calculus is top notch, as well as Mathematics: Form and Function.