/sci/ - Science & Math

[Anonymous No.6935626]

What is some must-read literature for every mathematician?

[Anonymous No.6935647]

>>6935626

How to Prove it, and The Princeton Companions to Mathematics are two off top of my head. I have a few more specialized recommendations for the 'intelligent layman' such as Visions of Infinity, The Golden Ticket, and anything by Smullyan. Journey Through Genius has some nice basic proofs along with the history of the mathematicians involved in them. Godel's Proof is another recommendation.

How is GEB? I specialize in the field of TCS/Logic and never bothered to read it.

[Anonymous No.6935675]

>>6935626
I have GEB but had no time to read it so far, is it good?

[Anonymous No.6935726]

I'm glad I read:
Zagier - Newman's Short Proof of the Prime Number Theorem
If you know basic complex analysis but never saw a complete proof of the PNT, read these 3 pages.

Other people usually mention these:
Thurston - On proof and progress in mathematics
Lockhart - A Mathematician's Lament
Hadamard - The Psychology of Invention in the Mathematical Field
Kant - Critique of Pure Reason

[Anonymous No.6935750]

>>6935626
>asks about math books, posts a CS book
ishygddt

[Anonymous No.6935755]

>>6935626
>How to Prove it
I was just working through this not 5 minutes ago.

So good.

[Anonymous No.6935761]

>all that fedoracore
Must read is one basic analysis text, like big Rudin, and one basic algebra text, like Jacobson I&II or Lang. Rest is optional.

[Anonymous No.6935773]

>>6935626
Flatland

[Anonymous No.6935865]

>>6935750
>calling a pop sci book for high schoolers a "CS book"

Don't get me wrong, CS is high school level and can be learned by any 14 year old, but GEB is still a few levels lower.

[Anonymous No.6935883]

>>6935865
Never change /sci/

[Anonymous No.6935890]

>>6935755
I just started reading through How to Prove it as well. I think it is one of the best Intro to Proofs books I've ever came across.

[Anonymous No.6936069]

The VNR Concise Encyclopedia of Mathematics.

[Anonymous No.6936105]

>>6935626

[Anonymous No.6936106]

Pic related.

[Anonymous No.6936143]

Rudin
Lang
never been in any prof's office and not seen at least those two.

[Anonymous No.6936148]

>>6935675
I haven't read the whole book yet, but I absolutely love it

[Anonymous No.6936149]

>>6935761
shiggy
>>6936143
diggy

[Anonymous No.6936152]

>>6936143
Which Lang? Dude was prolific.

[Anonymous No.6936153]

David Foster Wallaces' "Everything and More"

[Anonymous No.6936154]

>>6936152
Not him, but Algebra I guess.

[Anonymous No.6936179]

>>6935675
One of my all time favorites. Even if you're too smug to admit it's interesting all his thought experiments and little stories are extremely entertaining and well crafted.

[Anonymous No.6936403]

Spivak or Apostol are also really good if you want to see a lot of shit that your book probably ignored and every math professor at my school has both in there office. If you are reading any books that aren't specifically text books before your finished with your post doc you probably aren't a mathematician.

[Anonymous No.6936408]

Also Stanley's combinatorics text books are a real must have, there is nothing like them in the subject.

[Anonymous No.6938490]

bump

[Anonymous No.6938501]

>>6935626
The Bible.
Barnett's Proof.

[Anonymous No.6939724]

bump I'm really interested in this thread

[Anonymous No.6939736]

>>6935626
>implying mathematicians read pop-math/sci books

You don't know any actual mathematicians do you?

[Anonymous No.6939739]

>>6938501
>2014
>magic

/x/ plz go

[Anonymous No.6939753]

>>6935865
Fuck you dude.

[Anonymous No.6939756]

>>6939739
>bible
>book that gave birth to western culture
>magic

pleb pls

[Anonymous No.6939759]

>>6935865
The same can be said about a math degree. How is calc, abs algebra,complex analysis, or diff-eq any more difficult?

[Anonymous No.6939761]

>>6939759
You do learn all that shit in high school.

[Anonymous No.6939765]

>>6939739

*tips meme*

[Anonymous No.6939777]

>>6935626
Euclid's Elements
The Bible
The complete proof of the Classification of Finite Simple Groups
The Elements of Style

[Anonymous No.6939795]

>>6939761
Wait are you talking about CS1 in highschool? Because realistically its barely a freshman course. I thought the guy was talking about complicated things like compiler design, advanced operating systems, digital logic, or advanced graph theory.

[Anonymous No.6939801]

>>6935626
Why doesn't school teach math literature?

[Anonymous No.6939832]

>>6939795
>complicated things
>compiler design, advanced operating systems, digital logic, or advanced graph theory

pick one

[Anonymous No.6939855]

>>6939756
>bible
>book that set back western culture for a millennium

[Anonymous No.6939863]

>>6939855
>muh renaissance

pleb pls, you don't know what you're talking about

[Anonymous No.6939885]

>>6935675
I read about half of it. It's terribly dumbed down and uses this structure that alternates between chapters containing interesting content and "dialogue" between a tortoise and Achilles (or something) that plays out as a dumb story. The idea is that the dialogue/story is supposed to provide some analogies for the concepts before you're actually introduced to them.

Personally I found it very tedious and so dumbed down that eventually I just got fed up with it and never finished it.

This book is kind of similar and much more interesting. I highly recommend it.

[Anonymous No.6939891]

>>6939863
>Implying humanity would not be immensely advanced if the attentions of billions of intelligent minds were not wasted on fairy tales

[Anonymous No.6939895]

A Mathematician's Apology

[Anonymous No.6939898]

>>6939777
Euclid's Elements aren't a good text on modern geometry since it was discovered that the there are statements Euclid uses that aren't provable with his Axioms. Any modern geometry text from the 1900s would be better because logic and mathematics weren't formalized till the late 1800s/early 1900s.

The rest of your post is just trolling.
>Bible
No rigorous foundations, lots of inconsistencies.

>The complete proof of the Classification of Finite Simple Groups
Way too long and scattered across multiple papers. Unless you have a specific text you're suggesting then you can't seriously expect someone to take you seriously.

>The Elements of Style
One shouldn't write mathematics in the same way they write English. This leads too far too much handwaving and as a result the mathematics suffers. Even dealing with common/simple things like lots of nested quantifiers will quickly lead to problems in comprehension. There are specific texts out there for writing mathematics.

[Anonymous No.6939900]

Eulid: Element
Ptolemy: Algamest
Rene Descartes: Geometry, Discourse of Method

[Anonymous No.6939901]

>>6939900
Euclid*

[Anonymous No.6939911]

>>6939863
>>6939756
>implying the bible isn't full of magical fairytale bullshit

>implying Christianity didn't set back western culture by centuries

>implying modern Christians actually follow the bible or their supposed dogma


lrn2WorldHistory101

Christianity set the world way back. And the Christianity practiced today is extremely different than the original barbaric versions. If modern Christians actually followed their bible, they would be just a bad as Islamic terrorists.

The majority of the shit in the bible is completely illegal and objectively immoral by western standards.

[Anonymous No.6939913]

I think every mathematics student should get an introduction to formal logic since it's how we build axiomatic systems and properly formalize proofs. For this I recommend students read

The Logic Book supplemented with Simpsons lecture notes. The book is like first year undergrad level and the notes are much more formal.

http://www.amazon.com/The-Logic-Book-Merrie-Bergmann/dp/0078038413
http://www.personal.psu.edu/t20/notes/logic.pdf

I also recommend Halmos Naive Set Theory book. It's very introductory level (afterall, it's not an Axiomatic Set Theory book) but it gives the reader a good solid introduction to set theory that they can use for other mathematics.

As far as pop-math books I recommend The Art of the Infinite since it has a lot of neat geometry history and clever proofs (as well as a good coverage of constructive geometry along with it's philosophical implications back in 300 BC) and Concepts of Modern Mathematics by Ian Stewart since it's kind of a nice sampler on lots of different branches of mathematics (at a very introductory level).

>>6935755
>>6935647
>>How to Prove it

I like this book but recently I ran into The Book of Proof. It's available for free online and is really cheap to buy. I own both but I think as an intro book I prefer The Book of Proof. What do you guys think?

http://www.people.vcu.edu/~rhammack/BookOfProof/

[Anonymous No.6939917]

>>6939913

I think you recommended good into to logic books. I'd like to add another text for a primer in logic:

http://mathsci2.appstate.edu/~jlh/primer/hirst.pdf

Hirst is one of Simpson's former students.

[Anonymous No.6939922]

>>6939913
My impression is the Book of Proof is better suited for the beginner student in math than Intro to Proofs. The Book of Proof seems to truly assume no background knowledge, while Intro to Proofs seems to be a bit higher brow.

[Anonymous No.6939948]

>>6939911
>Christianity is the sole responsible for the Dark Ages
>lrn2WorldHistory101

You are beyond pathetic.

[Anonymous No.6939981]

>>6939911
>Report submitted! This window will close in 3 seconds...

[Anonymous No.6939990]

>>6939898
Hating Euclid's Elements is the surest sign of a terrible mathematician.

>This leads too far too much handwaving and as a result the mathematics suffers

Incomprehensible prose around the mathematics is just as bad.

[Anonymous No.6939991]

>>6939911

>THE HOLE LEFT BY THE CHRISTIAN DARK AGES

>AND DEFINITELY NOT BECAUSE A BUNCH OF NOMADS SHANKED THE ROMAN EMPIRE RIGHT IN THE GIBBLETS

>GLORIOUS ROME WOULD NEVER HAVE FALLEN IF THEY HADN'T CONVERTED TO CHRISTIANITY

>THEIR EMPEROR CULT WOULD HAVE BEEN MUCH BETTER FOR SCIENCE

>EMPEROR CULT - SO SCIENTIFIC

[Anonymous No.6940004]

>>6939911
>If modern Christians actually followed their bible, they would be just a bad as Islamic terrorists

No.

[Anonymous No.6940084]

>>6939911
>Christianity set the world way back
I've been an atheist for many years, and I am sick of simple-minded bullshit statements like this. How the fuck would you know what the current world would be like if Christianity had not dominated the dark ages? Ever heard of chaos theory? You gonna rewind time, change the past, and then fast forward time to see the results? Plus, to ignore all other factors in history and solely concentrate on religion merely to express your anger is retarded.
Also, let's say purely for argument's sake you are right? So what? Ever heard of cause and effect? Christianity is the effect, and its appearance and domination had causes. Therefore Christianity's rise was, in a sense, unavoidable/inevitable [ignoring QM randomness for argument's sake]. And its causes had causes, and these causes also had causes, and so on all the way back to the big bang. "Duh the The Big Bang set the world way back duh derp herp".
I bet if you were a 10th century European living centuries before the invention of microscopes, telescopes, electricity, chemistry, etc. then you would have believed that shit too. Does knowing more than someone who lived 1000 years ago make you feel superior? Let's see how well you can discover stellar nucleosynthesis or DNA with only 10th century technology.
tl:dr grow up faggot

[Anonymous No.6940148]

>>6939922
I'm not sure what you mean. Do you care to elaborate?

>>6939917
Thank you, I'm skimming through this and it does seem like a very good text. Though I disagree with way it introduces proof systems and some of the vague wording in that section. On the plus side it has a lot of introductory examples that students can actually play with and build intuition with (something that Simpsons lacks).

[Anonymous No.6940160]

>>6939990
>Hating Euclid's Elements is the surest sign of a terrible mathematician.
Euclid's great achievement is that he introduced the idea of axiomatization. Most of the content in Euclid's Elements had been known for hundreds or thousands of years, he just gave it structure. In the late 1800s and early 1900s people further formalized logic, mathematics, and axiomatic systems. They separated the notion of the model from the axiomatic system and showed that given a certain axiomatic system it was often possible to construct several different models satisfying it. At this point many flaws were discovered in Euclid's Elements (that had largely been the standard text for about 2000 years) and a lot of work went into fixing them.

I'm not saying I hate Euclid's Elements, just that I wouldn't recommend it as anything more than a historical reference. There are better more precise ways of doing things nowadays.

>>This leads too far too much handwaving and as a result the mathematics suffers
>Incomprehensible prose around the mathematics is just as bad.

Exactly. You're forced to either
1) Handwave away important details because they make the writing a mess.
2) Add a bunch of incomprehensible prose to the writing.
3) Just write the math as precise mathematical statements, the way they were meant to be written (i.e. in a formal language like some predicate logic and not in a natural language like English).

[Anonymous No.6940375]

>>6939913

Halmos's is terrible, I remember trying to read it and I couldn't get past the 10th page without a headache, his writing is incomprehensible in this book.
And then I got Thomas Jech's Introduction to set theory and set theory felt easy even though it's more advanced, it deals with axiomatic set theory and has a lot more content.

>>6940160
>a formal language like some predicate logic
No one writes math like that.

[Anonymous No.6940379]

>>6940004
> quoting Jesus instead of that crazy guy saul

[Anonymous No.6940433]

>>6939885

This. GEB is tiring as fuck, it could be 1/3 of the length without losing anything.

The dialogue parts are supposed to be witty and funny, but they are lame and cringeworthy (with like 2 exceptions, but he just tried to pump out way too many of them and didn't have enough funniness in him to pull it off).

The whole premise of the book, of Bach, Escher and Godel's workd having literally something in common and not just by a way of some metaphore is utterly ridiculous.

Hofstadter later rewrote the whole thing leaving the relevant parts in and called it "I am a strange loop". He said that people missed the whole point of GEB, so he had to write it again. No shit, I wonder why.

One might learn things from GEB I guess, but it will be too much effort for what it's worth.

[Anonymous No.6940452]

>>6935675
It's okay. Get his newest book I Am A Strange Loop

[Anonymous No.6940534]

>>6940148

I haven't read too much of the "Book of Proof", I merely skimmed various sections of it. It probably wouldn't be wise for me to jump to such conclusions so quickly. I'll reserve judgement until I can read more of both books. Thus far, the Into to Proof book is excellent; it is the quintessential book on learning proofs. I'll read through the Book of Proofs to see if it tops that. I spoke a bit too soon!

[Anonymous No.6940539]

>>6940148
>Though I disagree with way it introduces proof systems and some of the vague wording in that section.

What do you disagree with specifically? What do you find vague about it? I read from that text maybe 5+ years ago, so my memory of it may be somewhat vague.

From what I remember the text introduces a system with a set of axioms. It also introduces a set of lemmas. From those axioms and lemmas you can prove the standard theorems in logic. As the text goes further and further your axioms become more equipped / powerful and you can prove deeper and deeper results.

Also, I met Simpson. Nice guy.

[Anonymous No.6940541]

>>6940534
Intro* for some reason either autocorrects keeps messing it up or I can't "r" key.

[Anonymous No.6940566]

>>6940433
I am reading it for enjoyment, not to learn things

[Anonymous No.6941523]

>>6940539
Honestly I have several problems with it. It describes the system with a set of axioms. However, an axiomatic system is typically described as a set of sentences in a predicate logic. So, someone introduced to proof systems in this way would feel there is some circular logic going on (this resolves itself as long as the student's first introduction to proof systems isn't as axiomatic systems, similarly a student should note that the sets and functions that Simpson uses aren't really sets and functions). It mainly only introduces modus ponens and introduces other elements of the classical proof system as valid sentences. This is a problem because there are other types of proof systems out there that differ from the classical system such as the constructive logics. My preferred way of introducing a proof system is as rules of inference. This is also the way The Logic Book does it.

http://en.wikipedia.org/wiki/List_of_rules_of_inference

>As the text goes further and further your axioms become more equipped / powerful and you can prove deeper and deeper results.
Consider this. If you remove axioms then your system is weaker and you can prove less results, correct? But then there exist sentences that you can't disprove. So, what if you take one of these "false" sentences and use them alongside others to construct an axiomatic system. Then the system (your set of sentences) will be consistent because your proof system is no longer strong enough to prove the contradiction. This is what you do in constructive mathematics, you weaken your proof system by removing double negation, law of the excluded middle, axiom of choice, etc... (all things that can be used to imply the law of the excluded middle) and then you assume sentences like "every function is continuous" and you go on to do useful mathematics with it.

Here is a talk explaining it much better than I could.
https://video.ias.edu/members/1213/0318-AndrejBauer

(cont.)

[Anonymous No.6941555]

kindof pop science but i enjoyed choas by james gliek a lot.

[Anonymous No.6941557]

>>6941523
(cont.)
Anyway, if your focus is just on strengthening your system then you miss out on the powerful side effects of weakening it.

All that aside I still think it's a very good text. Many people probably won't have the same issues with it that I do.

>>6940375
I've never heard anyone saying they've had trouble reading Halmos. Almost everyone usually complements his writing.

Either way, if you can get you're actually interested in Set Theory then an axiomatic set theory book is much better. I'm not familiar with the book you mentioned (I used Patrick Suppes Axiomatic Set Theory its a Dover Book) but I'll check it out. Suppes book is nice but often feels antiquated.

As far as math, I much prefer it when texts write out quantifiers as quantifiers instead of writing them out in English.

>>6940534
I pulled them off my shelf and compared them some more. I think I see what you mean. How To Prove It has a more formal "structured approach" to proofs which is identical to how one writes a formal argument in formal logic. The Book of Proof seems to focus more on building an intuition to how to read and write proofs. Though, they cover much of the same content I can see how Book of Proof is aimed as more of an intro to proof based math while How To Prove It is aimed as more of a guide on how to formally write proofs.

>>6940433
I thought I Am A Strange Loop was a "sequel" covering more modern topics. It's all the same content without the dialogue?

[Anonymous No.6943420]

>>6940433
>The whole premise of the book, of Bach, Escher and Godel's workd having literally something in common and not just by a way of some metaphore is utterly ridiculous.

[Anonymous No.6943456]

>>6935626
I'm currently half-way through GEB, and although it's a great book, it's a bit cumbersome, for me at least. I study CS/maths so I'm already familiar with most of the stuff - albeit not familiar with Gödel's incompleteness proof.

So yeah, gained some new insights, but it was an easy read so far tbh. I can imagine a layman getting his mind blown though.

>>6940433
>The whole premise of the book, of Bach, Escher and Godel's workd having literally something in common and not just by a way of some metaphore is utterly ridiculous.

Now that's just nonsense.

[Anonymous No.6943457]

>>6943420
>>6943456

Seriously, anyone who thinks that a Bach's seemingly-looped recording and Godels incompletness theorem literally have something in common has a problem with precise thinking.

What's next? A Mobius strip? Maybe a bicicle rim?

[Anonymous No.6943480]

>>6943457
>Seriously, anyone who thinks that a Bach's seemingly-looped recording and Godels incompletness theorem literally have something in common has a problem with precise thinking.

Of course it's not 1:1 the same, but that's not the point. They all include concepts like self-reference, undecidability, paradoxes, multiple meanings, recursion etc. It's quite neat actually. Don't be so close-minded.

[Anonymous No.6943500]

>>6943480

They are sort of related metaphorically, if you stretch your imagination. It has some poetic value, I guess, but it does not enrich your mathematical, strict understanding.

>self-reference, undecidability, paradoxes, multiple meanings, recursion

Which one of those can you find in the Bach's seemingly-endlessly-rising-but-really-looped piece?

Which can you find in Escher's impossible figures, like the stairs? Maybe paradoxicality, in a lame way.

That is my problem with GEB. The more poetically leaning will love it, those who are more autisic and nitpicking like me will get angry at those kinds of comparisons.

[Anonymous No.6943512]

>>6943500

>those who are more autisic and nitpicking like me will get angry at those kinds of comparisons

well thats why synthesis is a higher order of thought than whatever rung autism puts you at

[Anonymous No.6943546]

>>6943500
>Which one of those can you find in the Bach's seemingly-endlessly-rising-but-really-looped piece?

Recusion. rise(){ octave++; rise();}

For every mathematical concept introduced in the book, one similar musical figure by Bach and a drawing by Escher get introduced as well. So you can find everything you want. They don't correspond exactly, but just the fact that there is some similarity was beyond my expectation.

I just enjoy reading it. It never claimed to give you any "strict mathematical understanding", that's what textbooks are for.

[noko No.6944224]

>>6940084
>How the fuck would you know what the current world would be like if Christianity had not dominated the dark ages?
That's a myth propagated endlessly by people like you. Feel free to disagree with Christianity as much as you like, but this myth has to die.
Please, please take the time to read this (if only the last paragraph):
http://www.strangenotions.com/gods-philosophers/

[Anonymous No.6944230]

>>6943546
No one says it's not strict mathematical understanding because it's obvious. It's not math at all. It's not science. It's shit. We don't need le fractals, le fibonacci or le Tesla. Fuck off.

[Anonymous No.6944252]

>>6940084
>dark ages

Dark ages didn't exist. it's just a bullshit meme that was popularized by fedora-wearing fаggоts.

Consult pic related.

[Anonymous No.6944331]

>>6939753
stop crying because he's right

[Anonymous No.6944377]

>>6939756
Lel the works of Plato and Aristotle gave birth to western civilization. Not that psychopathic, delusional, superstitious shit people call the Bible. Besides the Bible didn't really come about until like the third century AD.

[Anonymous No.6944382]

>>6944377
The works of Plato and Aristotle are psycopathic, delusional superstitious shit. This whole thread is a mess. Nothing scientific in either side, it's fedora vs pseudoscience

[Anonymous No.6944389]

>>6944252
>popularized by fedora-wearing fаggоts.

I didn't realized fedoras existed in the 1300s (you retard).

[Anonymous No.6944392]

>>6944252
The point of the Dark Ages is that the Roman Catholic Church burned and destroyed and suppressed the prior pagan Northern European Culture, books, etc. The Darkness refers to all of that was lost as the Romans took cultural hegemony over the majority of Europe.

[Anonymous No.6944416]

>>6935626
Foundation by Asimov

[Anonymous No.6944418]

>>6944382
So are you telling me that modern science would have emerged in the way and manner it did if it wasn't for Plato and Aristotle? You're obviously not very familiar with men like Descartes, Leibniz, Newton, etc. All the great thinkers of the Enlightenment studied and were influenced by Plato and Aristotle. Sure a lot of the claims of Plato, Aristotle, and other ancient Greek philosophers may seem primitive and incorrect to modern man, but if it weren't for these guys asking the questions they did, we would live in a totally different world today. Or perhaps you assume human history didn't start until the 1600s?

[Anonymous No.6944425]

>>6944382
>it's fedora vs pseudoscience

you mean fedora and pseudoscience

[Anonymous No.6946251]

>>6944331
>>6935865

wow. You must be really clever. Understanding everything so early. Put another notch in your bedpost tallying all the anons you've impressed with your intellectual prowess

[Anonymous No.6946562]

>>6935626
That depends on what you mean. First off, do you mean textbooks, books about the general philosophy (small p) of mathematics or papers? What do you mean by a mathematician? So some things are crucial for one field, but not so important for others (e.g. EGA).