(Part 3) Reddit mentions: The best number theory books

I haven't read this book cover to cover, but I've flipped through it a number of times at my uni library. Seemed like a pretty good introductory text. https://www.amazon.com/Invitation-Mathematics-Fermat-Wiles-Yves-Hellegouarch/dp/0123392519

In the math world we used the 2nd edition of books all the time even when the current edition was like 8th edition solely because for math books nothing changes except sometimes the order of the sections or some of the numbers on the homework. A few of my classes we knew would only have 5 people and the teacher had 3 extra copies of the book. I had my classes mapped out till graduation in an excel spreadsheet so I would order my books 2 months before the semester started, so in the case of those tiny classes I just ordered my book.

Then again math books are absurdly expensive. Case and point 150 page book that is 6 by 9, the book store wanted close to $200 for it. I joke that the smaller a book is the more it costs.

This book by Rosen seems to be the one that held it's value the best. https://www.amazon.com/Elementary-Featured-Kenneth-Published-Hardcover-dp-B00HQ0Y4Q8/dp/B00HQ0Y4Q8/ref=mt_hardcover?_encoding=UTF8&me=&qid=1571925354

The books by Lions and Magenes are next to impossible to find but highly cited.

I think that a number of Grothendieck's thousand-page manuscripts are impossible to get. I don't know what is the state of SGA.

Some famous mathematicians had notebooks that have been reprinted. For example, there's Ramanujan. Also, Gauss is said to have discovered all mathematics that was to happen 100 years after him. Joseph Fourier's Théorie Analytique de la Chaleur is also available. (I've briefly looked at some of Fourier's work and it's remarkably easy to read.)

The opposition to String Theory boils down to it being "not even wrong" on the basis that it cannot be falsified by experiment. Pretty well summed up in the book of the same title.

Scientists have been successful in ruling out some forms of a String theory (there are almost boundless forms), but the most "successful" forms don't really yield any unique predictions that can be tested (in the real world).

edit: I also found the book I linked to be a very approachable way to understand the mathematics of the Standard Model (irrespective of String Theory).

Yup, I get that. I'm not advocating for the imperial system in general, but a base-12 instead of base-10 system for metric. It's the reason the guy you were replying to is advocating for feet/inches, I presume.

Edit: for this and more, check out Here's Looking at Euclid, a look at how our numbering systems came to be and how they interact with the human mind! Good read.

So it doesn't violate the quick-reject sniff tests. Now what?

I'll let someone smarter than me make the arguments. If you're really interested, go check this out: http://www.amazon.com/Not-Even-Wrong-Failure-Physical/dp/0465092764

I want to get this: Introduction to the Theory of Numbers by G.H. Hardy. Anyone been through it?

This is a very good book: http://www.amazon.com/Introduction-Analytic-Number-Undergraduate-Mathematics/dp/1441928057/ref=sr_1_1?ie=UTF8&qid=1371966075&sr=8-1&keywords=introduction+to+analytic+number+theory

If you mean lattices as in geometry of numbers, Cassels' book is pretty good:

http://www.amazon.com/Introduction-Geometry-Numbers-mathematischen-Wissenschaften/dp/B0000BH2Z9/ref=sr_1_4s=books&ie=UTF8&qid=1377400524&sr=1-4

If you mean lattices as in ordered sets, I don't know, but someone else posted a suggestion.

Not exactly. For an incredibly long time, string theory has dominated the field of physics over a small minority of objections that it cannot be tested - that it wasn't even a theory, it was "not even wrong" as Peter Woit has written; Lee Smolin wrote a similar book around the same time. Smolin and Woit were mocked by hordes of theorists who just knew the evidence for string theory was going to show up any day now. But every time it didn't show up at the LHC, all these same theorists had to do was tweak their work a bit and move the goal post to a new energy level - this gimmick has been repeated, ad nauseam, for years. Only recently have some people finally started to come around to the possibility that string theory might not be the solution to figuring out the last pieces of the Standard Model.

So the analogy goes something like this:

Woit and Smolin:Scholze and Stix :: string theorists:Mochizuki and his inner circle.

I like Solved and Unsolved Problems in Number Theory by Daniel Shanks. It takes a unique approach, showing how particular problems led to the development of number theory.

For a more "standard" approach I like An Introduction to Number Theory by Harold Stark, which was the textbook used in the course I took as a sophomore.

めちゃくちゃ面白いでこの本

Unknown Quantity - John Derbyshire

Fearless Symmetry - Avner Ash

i assume you were looking at this book? it's not correct, but it's probably a step in the right direction

Woit's blog is a good source on all of that:

https://www.math.columbia.edu/~woit/wordpress/

Woit and Lee Smolin's books on the subject:

https://www.amazon.com/Not-Even-Wrong-Failure-Physical-ebook/dp/B00JLMMEQQ/

https://www.amazon.com/Trouble-Physics-String-Theory-Science-ebook/dp/B003WUYP56

They were both a bit ostracized for being a decade or two ahead of everyone else in their criticism of string theory and philosophical manias.

I think at some point physics got hijacked by pure mathematicians.

The many-worlds-interpretation and string "theory" are completely un-related (and note where I put the quotes)

I think you meant:

not even wrong

I might suggest the references & bibliography from: http://www.amazon.com/Not-Even-Wrong-Failure-Physical/dp/0465092764/ref=sr_1_1?ie=UTF8&qid=1320808955&sr=8-1

I'm in the same position you are, but I think it depends on algebraic geometry as developed by the Grothendieck school, among other things. I'm trying to study some of the prerequisites for tackling Hartshorne's Algebraic Geometry. Maybe this would help you get started: Invitation to the Mathematics of Fermat-Wiles. I haven't looked at it, though.

It's essentially impossible as an early undergrad to understand the proof. If you're really desperate to be able to understand it within the next 5 years, here is a 400-page book aimed at helping undergrads understand the proof.

That was the premise of "Not Even Wrong", that string theory remains outside the scope of science due to its complete lack of testability.

So that leaves the string theorist with "ad hominem" attacks like this post essentially calling everyone who disagrees with them "stupid" i.e. "non-specialist".

I read The Trouble With Physics about when it came out, so quite a while ago. In trying to find that reference I stumbled on the Not Even Wrong book / blog, which seems a slightly more up to date version of the same thing.

My understanding of the point of the criticism - and this isn't at all my field, so take all of this with that in mind - is stronger than we don't currently have a way to test string theory. The argument from the Trouble With Physics was, if I recall it correctly, that string theory was not so much a theory as a class of theories, and a sufficiently broad class of theories that with the right constants inserted, they could be made to model any result and consequently were unfalsifiable, regardless of any improvements that may come in experimental physics. How much truth do you see in that criticism?

>Hossenfelder’s argument, in brief: There’s no reason to think nature cares what we find beautiful

I'm not string theory supporter anyway and I pointed to its conceptual problems in the time, when Dr. Hossenfelder posted on article about extradimensions after another (see bellow) - but a bit more sanity and less ideology would be useful even when judging the string theory fiasco:

Reality check 1: Dr. Hossenfelder pursuits “ugly” bottom-up phenomenological approach to physics rather than up-bottom “pretty math based” stringy/susy theories – but even uglier fact is, that this (her?) phenomenological approach failed as well. There is no beautiful but failed and ugly but successful approach to theoretical physics: only failed theoretical physics of all kinds thinkable during last four decades.


Reality check 2: At least Lee Smolin or Peter Woit wrote their insightful books well before string theory fiasco – but where Dr. Hossenfelder was, when they pointed to its problems? After battle everyone is general, after wit is everyone’s wit… ;-)

Reality check 3: Her hypocrisy and opportunism goes even deeper: When string theory was still hyped, Dr. Hossenfelder also jumped into its bandwagon for example by many studies involving extradimensions – but now she bravely pretends, she was never involved into this hype.


Dr. Hossenfelder popularity solely depends on short memory of laymen public i.e. that people forgot, she was herself a great promoter of extra dimensional stuffs and black holes and that she made money and scientific "credit" with writing about them (Observables from Large Extra_Dimensions, Signatures_of_Large_Extra_Dimensions, Black hole relics in large extra dimensions, Black Hole Production in Large Extra Dimensions at the Tevatron, Observables of Extra Dimensions Approaching the Planck Scale, [Suppression of High-P_T Jets as a Signal for Large Extra Dimensions](https://www.researchgate.net/publication/2001593_Suppression_of_High-P_T_Jets_as_a_Signal_for_Large_Extra_Dimensions and New_Estimates_of_Lifetimes_for_Meta_stable_Micro_BlackHoles-From_the_Early_Universe_to_Future_Colliders), Schwarze Löcher in Extra-Dimensionen, Black hole production in large extra dimensions at the Tevatron) just before ten years.

I think Hawking and Green both are string theorists? I just started reading Peter Woit's book about theory of everything/quantum mechanics. He argues that string theory is not able to be proved right or wrong scientifically, and is basically not valid science.

Yes. Absolutely. I hated math until grade 12. Then I worked really really hard and had 3 tutors (really) and still barely passed. I am bad at math. But now I fucking love it. I really love math puns.

Take a look at a book called Here's Looking at Euclid. I don't understand a lot of what's in it, but it gives me way more of a drive to figure shit out than "you should know this" and "you may need this some day".

I mean, clearly I'm a huge nerd. But I'm a huge nerd who spent all of my primary and secondary education loathing math to the very core of my being. It made me cry myself to sleep at night. The way we teach math really really needs to be revised.