/sci/ - Science & Math

[Anonymous No.9409664]

What is /sci/ studying over winter break? I'm in a coffee shop reading The Book of Proof.

[Anonymous No.9409674]

>>9409664
Measure theory, about halfway through this guy's lecture series: https://www.youtube.com/watch?v=qaCOTKh8o4w&list=PLDf7S31yZaYyKJaq9ETuIh1TK_uGHaHHt

based professor UMA

[Anonymous No.9409679]

>>9409674
Are you calling that professor uma delícia? What are you, gay?

[Anonymous No.9409683]

>>9409679
well his accent is pretty nice

[Anonymous No.9409701]

>>9409683
I don't know why, but I cringe everytime I see a native portuguese speaker speaking English.
B-But if you like it, that's ok.

[Anonymous No.9409704]

>>9409664
How do you like it? Was just thinking of buying a copy.

[Anonymous No.9409727]

>>9409664
why did they print the euler's proof twice?

[Anonymous No.9409744]

>>9409727
to state that each sentence should end in a period regardless of whether it ends in a mathematical symbol or expression. this chapter seems to be focused on the syntax and formatting of writing proofs

[Anonymous No.9409751]

>>9409664
Papers

[Anonymous No.9409753]

>>9409664
writing a bunch of god dam fucking job apps and god dam fucking papers

[Anonymous No.9409770]

>>9409704
It's decent to read. It feels like a professor talking to you, not like a text book. Here's another page so you can see what I mean

[Anonymous No.9409779]

>>9409664
More like using the book as cover for skeezin on coffee shop cuties

[Anonymous No.9409788]

>>9409770
>garbage misinterpretation of Euclid's proof by using "contradiction" when it's not needed

[Anonymous No.9409795]

>>9409664
Italian /lit/. I just got my master in math, so I'm taking a pause.

I ended up on /sci/ out of nostalgia: I was thinking of those plebs who believe that Dirac's \delta is a function R->R that sends 0 to infinity and every other value to 0. Losers don't know no distribution theory
\[ \sadface \]

[Anonymous No.9409801]

>>9409664
The Principles of Quantum Mechanics and Philosophiæ Naturalis Principia Mathematica. Kind of easy.

[Anonymous No.9409804]

>>9409664
>studying during my break
Maybe if I was a brianlet. I studied everything I had to during the semester, now is my time off.

[Anonymous No.9409820]

>>9409788
>given any finite set of prime numbers, you can build a new one, so we can conclude (by induction) that there are infinite prime numbers

>if there was a finite set containing all prime numbers, you would be able to build a prime number that lies outside that set, giving a contradiction

the difference is minimal, really

[Anonymous No.9409822]

>>9409788
how would you prove it then?

[Anonymous No.9409823]

>>9409664
>time off
If studying is not something you do during your "time off," you're not going to make it.

[Anonymous No.9409832]

>>9409664
>Coffee shop and 'reading'
>Coffee shop and taking a picture of what you are 'reading'
OP you're not a girl right? Your not just refreshing this thread and waiting for attention right?

Just to remind you OP this isn't your Twitter account. You are most likely trying to get more attention from people but do you mind finding it somewhere else, like reddit instead of here please.

[Anonymous No.9409857]

>>9409832
I got 11 (you)'s. Try again.
You missed the point, anyway. What have you studied?

[Anonymous No.9409861]

Primarily this

[Anonymous No.9409862]

>>9409664
Just read the section in that pic. Are there other standard guidelines for writing clear mathematics? The only I can think of is paying attention to the reading direction and write "choose x such that P(x)" instead of "P(x) for some x" for instance.

[Anonymous No.9409867]

>>9409862
Chapter 1 covers this

[Anonymous No.9409901]

>>9409770
Looks nice.And it’s cheap. Thanks anon.

[Anonymous No.9409913]

>>9409664
The Book of Pussy?
You can't just whip it out like nothing mang

[Anonymous No.9409942]

Got this as a recommendation from /lit/

[Anonymous No.9410201]

>>9409862
Paul Halmos wrote a pretty good essay on mathematical writting. And its pretty easy to find it on google

[Anonymous No.9410219]

>>9409664

>book of proof
>teaches you grammar

I always, wondered. what the fuck all these Intro, to basic proofs -courses, etc, were?

lmao at americoons learning gramar in universitys

[Anonymous No.9410224]

>>9409861
Oh look, it’s [math]\mathbb{C}[/math] for engineers.

[Anonymous No.9410228]

>>9409664
>in a coffee shop reading The Book of Proof

>>9409832
thissssssssssssss

[Anonymous No.9410231]

>>9409861

>picture of a desert on the cover

What did they mean by that? I mean, it's no secret that complex analysis is fucking pointless and a terrible waste of time, but now they're just making fun of their readers.

[Anonymous No.9410239]

Instead of going through an intro to proofs book can I skip the crap and go directly to abstract algebra and learn how to do proofs that way? Otherwise I’m wasting a lot of time learning “how to do proofs” when all I want to do is learn abstract algebra

[Anonymous No.9410245]

>>9410239

Yes. Intro to proofs is for brainlets who don't understand basic logic.

[Anonymous No.9410248]

>>9410245
Okay thanks, I’m skipping intro to proofs junk then & diving right into algebra.

[Anonymous No.9410289]

>>9410239
>Instead of going through an intro to proofs book can I skip the crap and go directly to abstract algebra and learn how to do proofs that way? Otherwise I’m wasting a lot of time learning “how to do proofs” when all I want to do is learn abstract algebra

[Acbn !7CUUUUUUUU No.9410454]

Finished this up

Been screwing with an Atmel AVR ATmega328p in C now, learning PWM and stuff

[Anonymous No.9410827]

>>9409861
Visual Complex Analysis is better IMHO

[Anonymous No.9410905]

>>9410239
>>9410245
you do have to be rigurous in not assuming things when doing proofs (a very common mistake); at least make sure you thoroughly study some proofs to get the gist of it

that, or you can skim through the second (maybe third) part of Book of Proof before starting: https://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf

[Anonymous No.9410917]

I need to learn MATLAB in three days

[Anonymous No.9411152]

>>9410228
At least she's using version control

[Anonymous No.9411213]

>>9410239
>can I skip the crap and go directly to abstract algebra and learn how to do proofs that way?
yes

[Anonymous No.9411215]

>>9410917
Is it doable? I've been meaning to learn MATLAB for a long time now but if it's not very complicated I'll finally do it.

[Anonymous No.9411217]

>>9410917
For what? You can learn the basics fairly fast.

[Anonymous No.9411240]

>>9409664
I'm finishing Genslers intro to logic and then I was planning on reading Book of Proof or How To Prove It to get a more mathematical perspective on proofs. I haven't thought too far beyond this though I just wanna get better at proofs before I take on anything else.

[Imnot Arobot No.9411246]

>>9409744
a period after a mathematical symbol or expression looks retarded

[Anonymous No.9411249]

>>9409664
knowledge or get the fuck out, i'm tired of these non-sense posts rooted in the insecurities of high functioning people - man the fuck up, or get the fuck out

[Anonymous No.9411343]

>>9409664
https://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf

[Anonymous No.9411374]

>>9409664
you act like I study for things

[Anonymous No.9411466]

>>9410239
You are going to fail. Proof books are not only there to show you the idea of how to do proofs, but to actually understand the full implications of a theorem.

If you don't understand any of: logical quantifiers, how to show equivalence of sets (a really big one for beginners), how to formulate a negation of a statement, what the difference between contrapositive and contradiction is, the difference between a constructive and existence proof, what uniqueness implies and how to show it in general, then you are going to go nowhere.

[Anonymous No.9411487]

>>9410239
>Instead of going through an intro to proofs book can I skip the crap and go directly to abstract algebra and learn how to do proofs that way?
Proof books are essentially a meme, you pick up what you need while working through actual math books anyway (and some have this material in an appendix)

I've never even met a mathematician that's actually used any of those proof books

[Anonymous No.9411495]

>winter break
I am taking a winter class (english composition) and it's so shit. an essay a day, I feel actual intelligence (STEM) being sucked out of my mind and being replaced with humanities fluff.

[Anonymous No.9411568]

>>9409801
>Kind of easy
Then challenge yourself with the Discourse on Revelations

[Anonymous No.9411854]

>>9411466
I dunno, my algebra book taught me like half of those.

[Anonymous No.9411886]

>>9409788
So his proof is incorrect?

[Anonymous No.9411974]

>>9411466

You don't need to justify your lack of brains here. Not everyone needs to be spoonfed that shit.

[Anonymous No.9412002]

Textbook for my linear algebra course. Some excercises are pretty hard- Motl's fixation on string theory seeped into the book.

[Anonymous No.9412041]

>>9411886
>So his proof is incorrect?
The proof is correct, but the claim that it's Euclid's argument is incorrect.

[Anonymous No.9412068]

>>9409664
Working on this beauty, just finished all the exercises in chapter 1, moving on to chapter 2 now.

[Anonymous No.9412069]

>>9412068
>that disgusting handwriting

[Anonymous No.9412073]

>>9412069
It's notes for myself, not homework to turn in

[Anonymous No.9412089]

>>9409664
These notes
http://www.math.harvard.edu/~lurie/252x.html

[Anonymous No.9412096]

>>9412069

>m-muh handwriting

Face it, cuck, no one gives a shit about handwriting at this point when you actually type out anything that actually matters. Your faggy handwriting is worthless and you wasted time perfecting that shit.

[Anonymous No.9412098]

>>9412096
>faggy
Why the homophobia?

[Anonymous No.9412099]

>>9412098

Used it as an adjective. Hope it didn't offend you.

xoxo

[Anonymous No.9412103]

>>9412096
writing is better for taking notes
There are studies

[Anonymous No.9412109]

>>9412103

Hey, friend, that's pretty sweet, although I fail to understand how it relates to the quality of your handwriting mattering at all. Did the studies also claim that your notes are even better if you have neat and pretty handwriting?

[Anonymous No.9412136]

>>9410239
Yes skip it. If you really need it then you can always go back to a proof book.

[Anonymous No.9412141]

>>9412109
Neat handwriting is neat. Stop making a fuss.

[Anonymous No.9412147]

>>9412096
>cuck
kek. you got em good. fucking handwritingcucks. who needs that shit.

[Anonymous No.9412910]

>>9409861
kek, try Lang

[Anonymous No.9412919]

>>9410219
>universitys

>>9412002
>lumo is still using his rutgers email because he can't get a position worth shit in czechia

[Anonymous No.9412926]

>>9410231

[Anonymous No.9412928]

>>9409901
$20 cad dollat softcover so it's cheap

[Anonymous No.9412958]

>>9412910
>Lang
Lang is a meme.

[Anonymous No.9413015]

>>9410231
>it's no secret that complex analysis is fucking pointless and a terrible waste of time
>tfw only learned this after one whole semester-long course
fucking branch cuts man

[Anonymous No.9413025]

>>9413015

It's okay, we all make mistakes. At least you got some arithmetic routine if nothing else.

[Anonymous No.9413040]

>>9413015
>fucking branch cuts man

[Anonymous No.9413049]

>>9409788
Proof by 'contradiction' is flim flam math anyway
http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/
>Proof by contradiction, or reductio ad absurdum

[Anonymous No.9413051]

I've only read literature this winter. I don't give a fuck about science or math during my breaks.

[Anonymous No.9413196]

>>9413051
>I've only read literature this winter. I don't give a fuck about science or math during my breaks.

[Anonymous No.9413217]

>>9409779
It's just you

[Anonymous No.9413500]

>>9409664
Machine learning and convolutional neural networks in particular for my final project (explaining DNNs).

[Anonymous No.9413509]

Grinding out my research paper

[Anonymous No.9413652]

>>9413196
>literature is for brainlets

[Anonymous No.9413681]

>>9413652
literature is for brainlets

[Anonymous No.9413713]

>>9409664
> I'm in a coffee shop
Fuck off you retarded hipster shit

[Anonymous No.9413735]

>>9413713
lol

[Anonymous No.9413744]

>>9411487

Lol, do you ask every mathematician you meet if he/she took a course dedicated to proofs in their undergrad?

[Anonymous No.9413796]

>>9413744

Not directly, no. I just make a snarky remark about the brainlets who have to take a course like that. Every single time they join in and we have a good laugh.

[Anonymous No.9413930]

>>9413744
>Lol, do you ask every mathematician you meet if he/she took a course dedicated to proofs in their undergrad?
No, but I've never seen a book like that anywhere in several maths department buildings, never heard a mathematician recommend one during a lecture, never seen it listed on a course website as a supplement, or even just seen anyone reading one.

Mathematicians read math books, does it really surprise you that they don't waste their time on something like that?

[Anonymous No.9413943]

>>9413744
>Lol, do you ask every mathematician you meet if he/she took a course dedicated to proofs in their undergrad?
Also none of the universities I've studied at even offered a course dedicated to proofs

[Anonymous No.9414095]

>>9413744
>a course dedicated to proofs
Is this actually a thing somewhere? What's the point?

[Anonymous No.9414211]

>>9414095

You know how they've been lowering the standards everywhere (+ diversity quotas)?

That's the reason. They have to go out of their way to actually separately teach that shit so all the sub-100iq future geniuses won't drop out at calc 1.

[Anonymous No.9414431]

>given any finite set of prime numbers, you can build a new one, so we can conclude (by induction) that there are infinite prime numbers
>if there was a finite set containing all prime numbers, you would be able to build a prime number that lies outside that set, giving a contradiction

you only get a new number which is relatively prime to all your primes, not a new prime

[Anonymous No.9414435]

>>9414431
>you only get a new number which is relatively prime to all your primes, not a new prime
If the new number is relatively prime to all your primes, then you have a new prime.

[Anonymous No.9414449]

>>9414435
proof?

[Anonymous No.9414453]

>>9414449
>proof?
Any prime dividing the new number is your new prime.

[Anonymous No.9414622]

>>9409788
>anti-contradiction-ist

Why does /sci/ attract all the loons?

[Anonymous No.9414629]

>>9414622
It's not being "anti-contradiction", it's just pointing out that Euclid didn't use "contradiction".

[Anonymous No.9414675]

>>9413681
>literature is for brainlets

[Anonymous No.9414724]

>>9412919
No. It was written when he was still at rutgers. But his reluctance towards academia is sad to me, his style of teaching suits me very well and, he's only an aßhole online- he might be the most friendly and down-to-earth person in this autism-ridden field.

[Anonymous No.9414734]

>>9411466
>how to show equivalence of sets
show they're subsets of each other
>how to formulate a negation
it's obvious
>difference between contrapositive and contradiction
obvious
>what uniqueness implies and how to show it in general
well let's assume there's two. oh shit they're equal

[Anonymous No.9414735]

>>9410224
[math]\mathbb{C}++[/math]?

[Anonymous No.9414754]

>>9414734
Yikes. The absolute state of undergrads on here.

[Anonymous No.9414762]

>>9409664
>I'm in a coffee shop reading The Book of Proof.
what are you fucking gay lol

[Anonymous No.9414773]

>>9414734
>>how to show equivalence of sets
>show they're subsets of each other

>carrying about actual equality

[Anonymous No.9415577]

>>9409770
Literally any self-study book dude. Have you not read any math books not provided by your class? They're all like this and their purpose is to make it feel like you have a professor explaining everything.

[Anonymous No.9415583]

>>9410827
>>9410224

I don't have a choice, my prof likes this book. I have already taken most proof based courses and am about to graduate.

Probably will use a better book in the future.

[Anonymous No.9416429]

Friedberg's linear algebra because fuck wasting 6 months of my life on a piss-easy class.

[Anonymous No.9416619]

>>9411466
>If you don't understand any of: logical quantifiers, how to show equivalence of sets (a really big one for beginners), how to formulate a negation of a statement, what the difference between contrapositive and contradiction is, the difference between a constructive and existence proof, what uniqueness implies and how to show it in general, then you are going to go nowhere.
lmao, who let the freshman in here?

[Anonymous No.9416708]

>>9410917
The syntax should come fairly smoothly, especially if you already know a language. The hard part is what you're actually using it for.

[Anonymous No.9416973]

>>9409664

[Anonymous No.9416987]

>>9416973
I love reading montrosities like that gravitation book. Feels almost like reading one of those fancy ornate Bibles.
It's a shame I only read math nowadays and nobody writes anything longer than 700-800 pages.

[Anonymous No.9416999]

>>9416987
>I only read math nowadays and nobody writes anything longer than 700-800 pages.

Lurie does

[Anonymous No.9417013]

>>9410917
how did it work out?

[Anonymous No.9417023]

>>9411215
the idea behind matlab is that you focus on the shit you wanna do, while not having to care too much about programming in matlab itself

[Anonymous No.9418050]

>>9414431
using the method of poduct+1, you get a relative prime
e.g. claim 2,3,5,7,11,13 are the only primes
2*3*5*7*11*13 + 1 = 30031
30031 = 59*509

[Anonymous No.9418079]

>>9416999
So does thomas/stewart/larson

[Anonymous No.9418137]

>be me
>23 years old college dropout multiple times
>never worked in school but managed
>no prospect or skills
>decide to start learning math
>start from the beggining
>prof leonard videos on prealgebra plus textbook a few hours every day
>im not great at it but making good progress and enjoying it

No bully

[Anonymous No.9418223]

>>9418137
keep at it mate, you could get your math major before 30 so go for it.
Don't give up!

[Anonymous No.9418238]

>>9418223
Thanks man ! Do you think a math major is worth it ? I was thinking of trying CS but dont know what kind of major could get me a job

[Anonymous No.9418252]

>>9418238
Mh, math is pretty diverse. You could get into crypto, finance, perhaps something more akin to engineering, of course you'll have to do a lot of legwork and make a vast array of contacts in whichever field you're interested, if you want to make money I'd suggest you to stay out of academia because unless you end up teaching in a good school, you wont earn much I think.

CS will give you a job in your first year but you have to enjoy it, are you familiar with the stuff discussed at /g/?

[Anonymous No.9418264]

>>9418252
Well the main reason I got into math again is that I have an entrance exam for a CS school in a few month so Im learning C / Unix alongside Math. I think im not very good at it, maybe because I lack the mathematical / logical part for problem solving and coming up with algorithms to solve programs. Im hoping I can get better at it rather than look up the answers to problems everytime

[Anonymous No.9418270]

>>9418264
How much time do you have left? Because after you finish with leonard vids, I suggest you to check this: https://www.youtube.com/watch?v=L3LMbpZIKhQ
It's got a book under the same name, I don't recall exactly the contents but if you know well your HS math you're good to go.

[Anonymous No.9418272]

>>9418270
its*

[Anonymous No.9418278]

>>9418270
Thanks ! By finishing Leonard do you mean all the way to calculus 3 ? My exam is in 6 month and I have nothing to do but to study until then. I wanted to finish prealgebra - intermediate algebra - statistics and maybe get to calculus 1 before then

[Anonymous No.9418280]

>>9418278
No no, up until calculus 1 and the course I gave you I think you'll be good to go.
Always remember to use a book to find a more formal (not so much) explanation and more exercises, math is not a watching sport!!

Good luck.

[Anonymous No.9418293]

>>9418280
Yes and thank you ! My method is always to watch the video and then do whole notion in the textbook plus all exercices, I'll do my best

[Anonymous No.9418300]

>>9418293
I hope you do, anon.

[Anonymous No.9418610]

>starting perturbation theory next term
>don't even know dirac notation yet

[Anonymous No.9419180]

>>9418137
Where did you get that book Anon?

[Anonymous No.9419703]

>break
>studying

The fuck, my boy.

[Anonymous No.9419740]

>>9409664
This. /sci is not your fucking twitter, cunt. Go compensate for your 2nd-tier brain somewhere else.

[Anonymous No.9419749]

How am I doing sci
Im watching the nfl playoffs too though so I cant really concentrate

[Anonymous No.9420031]

>>9410219
Students who end their mathematical statements in periods are often thinking, reading, and writing them as complete sentences which leads to more organized thinking. Students who aren't thinking of their mathematics in complete sentences often tangle themselves up in proofs and have trouble organizing their proofs logically and coherently. Some universities actually offer entire courses focused entirely on mathematical writing and there exist popular textbooks and lecture notes on the subject.

Personally I don't like the way a period looks at the end of a line of mathematics but I do think it has many benefits.

>>9410248
Unless you already know off the top of your head how to prove statements of the following forms (as well as how to negate them and prove the negation) then you would really be better off not skipping it.

Prove [math](P\Rightarrow Q)\Longleftrightarrow(R\Rightarrow S)[/math]
where [math]P,Q,R,S[/math] are statements.
In intro courses you're very likely to see this broken down into bite size chunks but that won't always be the case and if you can't prove a question of that form it can be a showstopper for you on a test:
>Recall theorem 1:
[math]P\Rightarrow Q[/math]
>and theorem 2:
[math]R\Rightarrow S[/math].
>Prove that theorem 1 and theorem 2 are equivalent.

Given a set [math]S\subseteq\mathbb{R}[/math] and a function [math]f\colon S\to\mathbb{R}[/math], prove
[math](\forall x\in S)(\forall \varepsilon\in\mathbb{R^+)}(\exists \delta\in\mathbb{R^+})(\forall y)\in S\left[
\left\lvert x-y \right\rvert<\delta
\Longrightarrow
\left\lvert f(x)-f(y)\right\rvert<\varepsilon
\right]
[/math]
where
[math]R^+\colon=\{x| x>0\land x\in\mathbb{R}\}[/math].
This one is actually the definition of a continuous function and lots of students struggle with it (and its variations) so much that they only pass analysis by mimicking the patterns in class. You'd likely see it phrased:
>Prove that [math]f[/math] is continuous on [math]S[/math].

[Anonymous No.9420033]

>>9420031
Small typo fix:
[math](\forall x\in S)(\forall \varepsilon\in\mathbb{R^+)}(\exists \delta\in\mathbb{R^+})(\forall y\in S)\left[ \left\lvert x-y \right\rvert<\delta \Longrightarrow \left\lvert f(x)-f(y)\right\rvert<\varepsilon \right][/math]

By the way, negating a sentence like this should be trivial for you if you actually understand some basic formal logic.

[Anonymous No.9420045]

I'm currently reading some books on martial arts and physical training, as well as Kant's Critic of Pure Reason.

[Anonymous No.9420335]

>>9409664
Rip lil [math] p\in P [/math]

[Anonymous No.9420349]

>>9409664
You are such a pretentious soy boy.

[Anonymous No.9420533]

Wall of text incoming!

>>9409770
It's one of the best intro to proofs books out there. I only have two gripes with it (and 99% of all other books do these too).

1) It never gives a clear and quick run down of all the different proof rules. It's especially annoying because the book almost sets itself up to do so since it contains a (single page) section talking briefly about logical inference (p.61, section 2.11).

So, in place of that please refer to pic related. The rules are laid out in a grid with a row for each 'logical connective' (eg. "[math]\land[/math]") and two columns labeled "introduction" and "elimination". Each rule is then given a short name depending on where it falls in the grid (eg. "conjunction introduction" is "[math]\land i[/math]"), and subscripts are introduced for cells containing more than one rule (eg. "[math]\land e1[/math]" and "[math]\land e_2[/math]").

Each rule contains a space-separated list of assumptions on top, and a conclusion on the bottom (separated by a horizontal bar). It can be thought of as a recipe, eg:
[eqn]
\frac{\text{things I have}}{\text{thing I can make}}
[/eqn]
In short,
>The introduction rules tell you what you require in order to prove a sentence with that connective. Sometimes called "production rules" because they produce something.
and
>The elimination rules tell you what you can produce given a sentence with that connective. Sometimes called "deduction rules" because they deduce something.

(cont.)

[Anonymous No.9420537]

>>9420533

Some notes:
>[math]\bot[/math] just means contradiction and you should interpret it as logic giving you the finger.
>Negation elimination/deduction [math][\lnot e][/math] says: "If you have [math](P\land\lnot P)[/math] then you've got a contradiction".
>Contradiction elimination/deduction [math][\bot e][/math] says: "Given a contradiction you can deduce anything".
>The tall rectangles are subproofs where you suppose the sentence at the top and conclude the sentence a the bottom
>Conditional introduction/deduction [math][\rightarrow i][/math] says: "Suppose [math]\varphi[/math] and conclude [math]\psi[/math] to deduce [math]\varphi\rightarrow\psi[/math]".
>Negation introduction [math][\lnot i][/math] is "(negative) proof by contradiction".

So there you have it, all of your rules in a single page in succinct notation along with some major theorems.

(cont.)

[Anonymous No.9420540]

>>9420537

2) My second gripe is that its explanation of constructive math is woefully lacking and part of the reason so many undergrads fail to understand what it is or how it works.

Here is an attempt at a basic explanation focusing on negation and how it relates to contradiction and double negation. So, take a close look at [math][\lnot i][/math], [math][\lnot e][/math], [math][\bot e][/math], and [math][\lnot\lnot e][/math].

Consider the following:
First, [math][\lnot e][/math] is the only way to obtain a contradiction.
Second, [math][\bot e][/math] tells us that we never actually want to 'directly' introduce a contradiction or it will break our system. See:
https://en.wikipedia.org/wiki/Principle_of_explosion#Use
Third, the only time we want to 'indirectly' introduce a contradiction is as the subproof conclusion for [math][\lnot i][/math]. In other words, a negation is like a red flag telling us that we 'indirectly' introduced a contradiction (and the proof is indirect).
Fourth, if you suppose [math]\lnot P[/math] and conclude a contradiction then [math][\lnot i][/math] will deduce [math]\lnot\lnot P[/math]. In order to deduce [math]P[/math] (as in "(positive) proof by contradiction") you need to apply double negation elimination [math][\lnot\lnot e][/math] afterwards.
Fifth, applying [math][\lnot\lnot e][/math] has a trade-off where one loses the ability to distinguish between direct and indirect proofs but obtains statements that 'ought' to be true.

This trade-off is the difference between classical logic (normal math) and intuitonistic logic (the logic used for constructive math). To make things more precise, there are two paths we could take.
Normal math:
>Apply double negation elimination (this yields classical logic), focus on what 'ought' to be true, and try not to worry too much about whether the proof was direct or indirect.

(cont.)

[Anonymous No.9420543]

>>9420540

Constructive math:
>Throw out double negation elimination so that it is no longer a rule in our system (this yields intuitionistic logic, since LEM is equivalent to double negation elimination). Now we're free to really embrace the proof structure by noting that the rule for [math]\lnot P[/math] doesn't just say there was a contradiction after supposing [math]P[/math], it says that there does not exist a proof of [math]P[/math]. In other words negation is nothing like the other logical connectives.
>[math]P\land Q[/math]: "There exists a proof of P and there exists a proof of Q."
>[math]P\lor Q[/math]: "There exists a proof of P or there exists a proof of Q."
>[math]P\rightarrow Q[/math]: "If there exists a proof of P then there exists a proof of Q."
>[math]P[/math]: "There exists a proof of P"
>[math]\lnot P[/math]: "There exists a proof that (there does not exist a proof of P)"
>[math]\lnot\lnot P[/math]: "There exists a proof that (there does not exist a proof that (there does not exist a proof of P))"
In short, constructive math is interested in whether or not a statement is provable by a direct proof, not whether or not it 'ought' to be true.

tl;dr: Double negation elimination is weird compared to the rest of classical logic and it introduces is convoluted and indirect reasoning, basically this:
https://www.youtube.com/watch?v=4ZU56vBluow
Intuitionistic logic throws it out and focuses on applying straightforward direct reasoning. Also, there's two very different types of "Proof by contradiction" and one of them is okay in constructive math.

[Anonymous No.9420649]

>>9419180
I bought it on amazon. Its the textbook for prof leonard prealgebra videos

[Anonymous No.9420655]

>>9409664
Nothing because it's important to have a vacation.

[Anonymous No.9420693]

>>9420543
Interesting. How is the LEM equivalent to the double negation elimination?

[« No.9420752]

None at all. Doing as many drugs as possible to reward myself for getting straight A's before the new semester starts tomorrow and I have to quit cold turkey.

[None !NiggerFgtE No.9420764]

>>9409664
C++ Templates, Hashing, sorting algorithms, processor architecture and Calc II + LinAlg because I was too lazy to lrn2matrix.
All in all, a horrible time to be alive.

[Anonymous No.9421192]

>>9409664
Poser / PUA