book-recommendations

[marble solar]

For fcnl?

[frigid comet]

Brezis is good if you want to learn FA with a view towards PDE

[gray gazelle]

Anyone know of a category theory book that intersects logic?

[molten wave]

@gray gazelle

[gray gazelle]

@molten wave thanks

[steel viper]

topoi monkaS

[bold garnet]

What I don't get is that I buy a book "Elementary Topology" then I try and do the problems and like I can't do any of them. The threshold for doing the problems and actually understanding the material is so high that it seems almost unreachable for a regular person. I've been doing math for years and the difficulty of doing the problems is just too freaking high. Almost as if it was made this way on purpose to weed out people who can't do college or university.

[molten wave]

@bold garnet author?

[quick hornet]

one thing these textbooks assume is that you know how to "do math"

i.e. interpret definitions, follow and write proofs, that kind of thing

do you have a background in that kind of thing? say an intro to proofs course or a proof-based linear algebra course that reasoned on abstract vector spaces?

if not, introductory topology - and most other mathematical literature - will be a lot less approachable because it assumes you have that basic skill down

[molten wave]

I'm looking at

if this is the book you're talking about

then I question this part

I've been doing math for years

[quick hornet]

in the same way that, say, any English work assumes you know how to read and do critical analysis

any pure math text assumes you know how to read proofs, apply definitions, that kind of thing

i'd imagine every single intro topology in the world starts out with just a chapter or so of definition-pushing

which should be fairly rote if you have experience with that kinda thing

[molten wave]

I think it's important to refine

any "pure math" text

[quick hornet]

but if you dont have experience, it can absolutely be intimidating

fair mniip

"elementary" in this case does not mean "easy", it just means "introductory"

[molten wave]

maybe they just haven't done any pure math, which is fine, but there's a very clear line and perhaps a jump in difficulty or maybe a gap in the techniques used

[quick hornet]

yeah thats my point here

the materiral shouldnt be very scary if you have experience with pure math

at least in the introductory chapters

it'll take some time, sure, but everything in pure math takes time

you're not expected to be able to blitz through problems - they'll require thought, sometimes dozens of minutes of thought

the more core thing is that you need to know how "pure math" works (however ill-defined that might be)

definitions, logical relations, proofs, formalizations

again, this skill is as fundamental to pure math as reading is to english; it's just assumed you know it

if you're not practiced with this stuff, i'd caution that an introductory topology textbook probably isnt the best place to refine these skills

at least, not for a first introduction to them

this isn't me trying to be discouraging - it's being realistic

i'm not aware of any students whose first exposure to "pure mathematical thinking" was topology

not any successful students, at least

(i mean okay, i guess technically you could count chapter 2 of rudin, but even that is very much topology-motivated-by-analysis rather than topology-for-its-own-sake)

(which helps trim down the abstraction barrier a bit)

(and in any case, rudin is regarded as a very hard text for beginners and probably isnt suitable for many students unless they're being guided by a teacher/professor/whatever)

[this does bring up the related question of "why do topology texts bother with a chapter on basic notions of set theory, then?" I posit that this is more often used as review/practice and isn't intended to be a student's first introduction to the material - and perhaps there's a term which they havent seen before (maybe they had never seen a symmetric difference but the text uses the notion extensively) so it can help fill in gaps]

[that's a big digression from the topic at hand, though, so feel free to disregard]

[steel viper]

what are the problems

[gray gazelle]

starting serge Lang’s graduate text on algebra. Do I use this as a main text or a supplemental text? And if it’s supplemental what text should I use for my main?

[raw herald]

That depends on your ability, if it’s an undergraduate level ability, try summit and Foote or this gem: https://www.math.ucla.edu/~rse/algebra_book.pdf

also look at the pins

Really good algebra book guide in pins

[civic carbon]

ymmv, but I'm an algebraist and I get nothing out of Lang's algebra book. It really leans heavily into abstract nonsense.

[flint forge]

perfect

[worthy wigeon]

Hello, can anybody tell what would be a good diff eq book ?

[hearty steppe]

Use Paul’s online notes

[gray gazelle]

Petrovsky

Or if you need more application intensive stuff then ch8 in apostol

[worthy wigeon]

Thanks

[gray gazelle]

@worthy wigeon Differential Equations, Blanchard, 4th and Fundamentals of Differential Equations, Nagle-Saff-Snider, 8th

@worthy wigeon Advanced Engineering Mathematics, Kreyszig, 10th

[worthy wigeon]

@gray gazelle thank you

[gray gazelle]

what is a good beginner book for like dummies?

[restive raptor]

what topic?

[wooden sparrow]

Does Lang's basic mathematics cover all the math till highschool in more rigorous way? Or do I need other books to cover till all the highschool math?

[flint forge]

Well one issue with this

is that rigorously doing high school math doesnt make a ton of sense

i.e. its all computations

there is nothing informal per se about these computations

just they blackbox stuff

And then calculus happens and they ignore a lot of details

[wooden sparrow]

No, I needed to know where all the formulas and rules come from... Does the book help with that?

just they blackbox stuff
@flint forge I really don't want this

[flint forge]

To some extent there is no choice

as the justfication for some hs math

is the point of college math

Like we sort of just accept real numbers exist

until analysis

[wooden sparrow]

Like, AOPS book helped me understand why divisibility rules are the way they are

[flint forge]

Sorry I can only comment on the hs math part, i've not read lang's book

[velvet briar]

I'm thinking you mean "fully justified to a high schooler" in place of "rigorous" haha

That's a much harder question and I don't think any source will grab all

[wooden sparrow]

Yes, sorry for the mistake

I want rules and formulas to be justified, few of the axioms like commutative and associative properties are fine to be left out

So would Lang's basic mathematics be enough to cover till high school math where there's more than enough justification for the math till you get to college?

[velvet briar]

I mean Lang is good in that everything is there. If you don't know what you should be studying, just go further in Lang

If you feel something isn't justified, Google it. You'll get more that way

[wooden sparrow]

got it

Thanks for your time man

[weary nymph]

If even the reals are assumed to exist until early college, am I doing a risky move by learning analysis before calculus?

[flint forge]

not sure what you mean

it will be harder

but like it wont hurt you

understanding something better can never hurt

[weary nymph]

Makes sense

[hearty steppe]

Analysis is sort of like bridge between pure math and applied math

[gray gazelle]

I want rules and formulas to be justified, few of the axioms like commutative and associative properties are fine to be left out
@wooden sparrow those aren't axioms, if you are talking about commutativity and associativity of addition and multiplication. They have proofs. (And yes I know this is pedantic but it seems like something you would like to know given the material you are looking for.)

[hearty steppe]

But like studying analysis in depth I would say is pure math like if your spending a lot of time dealing with theory

[smoky surge]

out of curiosity what would yall say the preques to graph theory are?

[restive raptor]

some familiarity with basic sets, a little bit about bijections, and maybe a little combinatorics knowledge

[smoky surge]

gracias

i seems kinda like a neat field

[restive raptor]

@smoky surge to elaborate a little, most graphs are going to be expressed strictly as a set of edges and vertices, and proofs are done from that end, not the “this is what it looks like” end. Many proofs require constructing bijections, or at the very least, basic set fluency. Combinatorics is helpful for seeing those bijections, and combinatorial Problems naturally come up in graph theory

[bold garnet]

The book is called Elementary Topology 2nd ED by Michael C. Gemignani. I have worked at 3 different math tutoring centers over the course of around 10 years as well as privately tutored and have a BS and MS in math.

These problems are not easy. I don't see how a person can pick up this book and actually do the problems legitimately. Also the professor who was teaching the class was one of those professors who just sits in their chair and doesn't teach anything and expects you to figure it all out for yourself. It's harder to understand something when the professor is bad.

[quick hornet]

you have an MS in math without knowing introductory topology?

or are you referring to later chapters

i just grabbed a pdf and took a look and the problems dont look like anything special

but ive only looked at the first few chapters

but i mean, it's hard to address your specific... query without knowing exactly what youre asking

it sounds like youre just frustrated and venting

which is fair - math is hard - but doesnt give us much to work with

are you able to do the first few exercises? like even the ones given right after introducing metrics

[bold garnet]

Some proofs I understand and can do, but other proofs.......... its like how do I even know if I proved it or not. There are problems that I've done where I've solved something or proven something to only have a professor say "that was a good argument but the proof needed to be more rigorous".

[restive raptor]

If you aren’t convinced a proof is airtight, it probably isnt. You should go through line by line, and justify every single claim and manipulation

Like obviously that doesn’t hold when you’re very familiar already, but when it’s new, justify everything. If you havent justified something, your proof is not rigorous enough or just straight up wrong

[gray gazelle]

@flint forge yo, there are views of the real numbers from different perspectives. learning about them is just entering the conversation. just because you can't see some infinitely small portion of the real number system to prove it to exist, doesn't mean it's not there. just like I am not really in the room with you, I am typing on a screen some where far away. but no matter how far, you have to consider my existence, real or not; just like the real numbers, real or not.

[valid moth]

this is a particular philosophy

[flint forge]

why was this in response to me

what did i say about this

im so confused

[quick hornet]

@flint forge yo, the gens Nautia was an old patrician family at Rome. The first of the gens to obtain the consulship was Spurius Nautius Rutilus in 488 BC, and from then until the Samnite Wars the Nautii regularly filled the highest offices of the Roman Republic. After that time, the Nautii all but disappear from the record, appearing only in a handful of inscriptions, mostly from Rome and Latium. A few Nautii occur in imperial times, including a number who appear to have been freedmen, and in the provinces.

[flint forge]

oh okay that makes sense then

[quick hornet]

im just joining the prattle-irrelevant-tangents-to-max club

[flint forge]

i cant tell if someone thinks that like

i dont believe in real numbers?

im so confused

@gray gazelle explain yourself

[quick hornet]

hyperconstructivism

[gray gazelle]

I thought that you didn't believe in the real numbers?

[flint forge]

why would you think that

[gray gazelle]

I scrolled up

[flint forge]

?

[valid moth]

I mean, have you ever seen one?

[gray gazelle]

to the thought about analysis before calc

[quick hornet]

are you misinterpreting this statement?

[gray gazelle]

yea

[quick hornet]

max is just saying that high school math classes dont formally define the reals

[flint forge]

note the 'until analysis' part lol

[quick hornet]

or what it means to be a "real number"

[gray gazelle]

ahh

[quick hornet]

and that an analysis class clarifies this

[gray gazelle]

yea

[quick hornet]

by giving a proper definition rather than a "just trust us, this works"

[gray gazelle]

I know

yea

sorry for the misinterpretation

[flint forge]

ur chillin

[flint forge]

hes a jaco alt

[wooden sparrow]

@gray gazelle thanks for the advice

Though I won't be able to find books that prove commutative and associative properties at high school level right?

[gray gazelle]

I have never really read a high school level textbook but no I wouldn't think so

you can find them on the internet pretty easily

at least for the natural numbers

[wooden sparrow]

Got it... Thanks again :D

[burnt vessel]

Does anyone know any good resources on differential geometry?

[flint forge]

i think do carmo is the standard intor

[burnt vessel]

Thank you!

[civic carbon]

I like lee

(Smooth Manifolds)

[marble solar]

I'm a fan of Spivak

Volumes 1 & 2

But I seem to be in the minority of such an opinion

[civic carbon]

my biggest complaint is that him using n and m for the dimensions of M and N respetively is unforgivable

[marble solar]

Does he?

Lemme pull it out

[civic carbon]

yes, grumble

I mean, everyone calls the dimension of M n

and then if you want a second manifold, N is a natural choice

but someone has to stop you when you say "Well, let's denote its dimension by m"

it's like a slow motion train crash

I would be very happy if they made a new edition where that was fixed. I agree the book has very nice floaty exposition if you want something readable [not a reference]

[marble solar]

I couldn't find an explicit example of this in volume 1

Third edition

[civic carbon]

huh, I don't have my copy of the book, so I can't look up the edition, but I do have a quote from it in my intro to proof packet

[marble solar]

LOL

[fast gull]

what is most amount of math textsbokks yall read concurrently?

[restive raptor]

if I'm using them to supplement something? like 5 (one per course). If i'm using them to learn on their own? tops 3, all same subjects (ie I'll look at several different books on the same subject)

[hearty steppe]

I’m reading 4 books for Analysis right now. I will probably be reading 2-3 for linear and abstract algebra

It’s not so bad if you enjoy math

Cuz I mean, there’s nothing like reading a good math book

Especially several good ones on a single subject

One of the books for analysis I’m reading isn’t an analysis book but a proofs book, Velleman

[restive raptor]

sure, but reading a math book requires consistently doing difficult problems. There's a maximum amount of time you can do that per day

[hearty steppe]

Oh yea I agree

I read the chapter then do the problems

[fast gull]

4 books for analysis?

for school purposes or casually?

[eager perch]

One of the books for analysis I’m reading isn’t an analysis book but a proofs book, Velleman
Is Velleman good?

[sage python]

I had a single class once with 4 textbooks we referenced at various times

It was an experience™️

[fast gull]

undergraduate?

[sage python]

Yee

[fast gull]

woah

what class

id understand if somewhat obscure subject

[sage python]

Analysis

[hearty steppe]

So my strategy is if I can do enough problem sets in a chapter in one or two books and understand it, I Probly won’t do the problem sets in the third book

And just move on to the next chapter

The idea is comparing perspective and learning from multiple angles

[sage python]

Used Baby Rudin, Sally's Fundamentals of Mathematical Analysis (nice coverage but the writing is ugh), Buck Advanced Calculus (baaaad), and Hoffman-Kunze Linear Algebra

[hearty steppe]

For now I’m sort of using Baby rudin on and off but it’s like the last book I look at

Right now I’m Juggling between Abbott, Schroder, and Apostol

Baby rudin is extremely condensed in each of the chapters

Not that it’s bad but it’s not really beginner friendly

I’m gona use Hoffman-Kunze and Janich for Linear

And Artin + Fraleigh for Abstract

[fast gull]

ah ok

i only read a lil bit of fraleigh

loads of exercises

[hearty steppe]

Yea that’s the thing you want to get good exposure to exercises and you may not have to do every single exercise

The main idea too is getting exposure to perspective

[fathom monolith]

im working through spivak and the problems are so hard, im wondering if i came to this book too soon

[marble solar]

Which Spivak?

Baby, little, or Comprehensive?

[limpid gazelle]

What are Baby, little, and Comprehensive Spivak?

[fast gull]

his books

Spivak Calculus is baby

Comphensive is probably the one with manifolds

im not sure what little is

or maybe little spivak is the manifolds one

[limpid gazelle]

That is exactly my guess

Ah

Comprehensive is probably Spivak's "A Comprehensive Introduction to Differential Geometry"

[fathom monolith]

I’m in calculus

So bb

[limpid gazelle]

Well Spivak does have a calculus book

[fathom monolith]

Literally the book

[limpid gazelle]

Nice

[hearty steppe]

Oh nice you got a physical copy

[fathom monolith]

i always perfer to have the physical copy whenever I can

[fast gull]

i hav physical of 4th

its just gray

[hearty steppe]

I don’t really care personally. If anything I prefer a digital copy so I don’t take up too much space

[fathom monolith]

they double as decorations

[hearty steppe]

Yea I just try not to collect too much stuff

[fathom monolith]

but yea idk the problems make me feel like I should be somewhere else first

[hearty steppe]

Maybe. Try Paul’s online notes

[fathom monolith]

kk ill igve it a look. thanks

[north spire]

but yea idk the problems make me feel like I should be somewhere else first
@fathom monolith You should struggle through them, in my opinion

Like, I don't personally like Spivak. But working through hard problems and thinking through them is the best way to learn this stuff.

[raw herald]

Spivak is a lot to handle if you’re a beginner

[pulsar aurora]

I switched over to apostol as I felt spivak assumed the reader had prerequisite for it.

[raw herald]

Yeah it’s like you must be mathematically mature

Maybe if you had a discrete math course before it you could try more easily but it’s hard, and also you really got to know your basic algebra chapter one even has binomial power series expansion proof problems

[north spire]

You can read it as a beginner haha but I think you'll need a bit of guidance. My professor assigned us equivalent readings from spivak's text, courant's text and apostol's text for our preparation for analysis this coming semester

I think if you don't enjoy reading it, though, there's not much point working through it tbh

[raw herald]

My picture of a beginner is a high schooler who just barely passed AP calc exams lol

[gray gazelle]

yeah you might want to learn some basic proof stuff before jumping into spivak, unless you're a god and can pick it up on the spot

[pulsar aurora]

I touched spivak not even knowing calculus. :p

[raw herald]

With guidance it makes sense

also the solution manual is helpful

[north spire]

I don't know what's on the AP calc exam so 🤷

[raw herald]

Business single variable calculus

Basically

[north spire]

Well, stackexchange and online forums are helpful for exactly that reason

You can just post your proofs/queries and get help whenever you need it

[raw herald]

They will critique your bad proofs?

I thought it was just like here is the answer lmao

I need to use stack more

[north spire]

For me, I post my proofs there and explicitly tell people to give hints towards a solution, assuming that I'm incorrect. I also usually tell them to place their solutions in spoilers so that I only look at them after I've given my best attempt.

It's good haha, they give good feedback.

[raw herald]

Ah that’s a good idea

You should ask them to leave most of the proof as an exercise

Lol

[north spire]

lol

I don't know, most of them are a helpful bunch. Extremely professional though. If you want feedback for your argument, then they'll just shove it straight in your face.

[quick hornet]

yeah, thats because of SE style guidelines

it's supposed to be like a "reference of questions and answers" much like a FAQ or whatever

rather than a proper forum

so personal remarks and whatnot are avoided where possible

[north spire]

They have a chat where you can be more informal. They also have decently high standards for what you should be asking and how it should be structured.

[quick hornet]

yeah, no fluff like "hello all" or "thanks in advance"

[north spire]

So, like, if you don't give your own attempt, your question is likely to be closed.

Oh nah, that's fine haha.

[quick hornet]

really? i thought they edited that out

[north spire]

Usually, they'll screw you over only if you're blatantly not putting in effort

Nah, it's usually okay. At least, whenever I've included a bit of fluff, it's fine

[raw herald]

Hello all, I am a very young student in America studying in Georgia. I have a challenge question and need to add 975+365 without using a calculator. Please suggest methods that I can use to solve this promptly. Thank you all.

Lol

[north spire]

That will just get deleted by the mods

The mods wouldn't even close it. It'd just get yeeted out of the site

[raw herald]

How long you bet it’d last?

[north spire]

It depends on which mod is looking at a given time. I believe some of them are super nice and just vote to close it. Some of them just delete this stuff immediately.

[raw herald]

You have no idea how shook I was when I realized that there were way less complex algorithms for multiplication than the third grade algorithm (in terms of computer complexity)

That and learning Pythagorean’s theorem works with my face and not just squares are the two shookethed moments

[north spire]

Must've been enlightening

[raw herald]

Yeah those r probably why I wanted to major in math tbh

[gray gazelle]

your face is a t r I a n g l e?

[raw herald]

Y e S

[gray gazelle]

n I c e

Mines a rhombicosahedron

[raw herald]

also btw there is a ham sandwich theorem

And it’s a lit name

[north spire]

but yea idk the problems make me feel like I should be somewhere else first
@fathom monolith Anyways, what I was saying before about Spivak wasn't meant to encourage you to beat your head against the wall. Like I said, I don't particularly like Spivak but I don't think you should give up just because the problems are hard.

If you need help at any point, just come to this server and ask questions. If you're writing proofs, then you may either just post them here or post them on stackexchange. You'll get lots of feedback and opportunities to clarify your doubts.

If you still feel like you can't handle Spivak, then perhaps choosing a more approachable text would be better. For instance, I've heard people recommend Serge Lang's A First Course in Calculus so that might be useful.

[potent jungle]

What do you guys think about:
https://www.amazon.com/Calculus-Made-Silvanus-Phillips-Thompson/dp/1456531980
for starting out calculus?

I want to really delve into it later on, is it a good start?

[marble solar]

@fathom monolith Helpful I finished my MS in pure math, and I still find some problems in Spivak very challenging. It's by no means meant to be an easy book, or a book that you read once and you get "calculus". Try to learn from it, because the way he presents the material/proofs is slick. It's probably my favorite calculus text

I keep coming back to it over the years to get his intuition for how things should go

[velvet briar]

@potent jungle
Rofl the book is from 1914

[hearty steppe]

Courant?

[wooden sparrow]

Anybody used Openstax books for their highschool or college?

What do you think of them?

[flint forge]

i used them in HS

i like the physics one well enough

[wooden sparrow]

Nice

@flint forge what about math?

[flint forge]

never tried it

to be honest all math books prior to calc are basically the same

and for calc the only standouts are like spivak

and apostol or smth

[wooden sparrow]

@flint forge I have a question about that

If all the books prior to calc are the same, then why do people say AOPS is better than some generic book? And why does it cost a shitton?

[muted hearth]

should I dive right into Spivak, or read How to Prove It first?

sorry for the interrupt

[wooden sparrow]

It's ok

[flint forge]

oh sorry

so

when i say pre-calc

i mean with respect to a high school curriculum

the AOPS books are kinda in their own bubble

tbh ive never read an AOPS

but afaik they are mostly comp math stuff

@muted hearth try spivak and go back to How To Prove It if needed would be my suggestion

[wooden sparrow]

So for let's say high school algebra, any book is fine?

[muted hearth]

aye, thanks

@wooden sparrow probably ~ I personally really like "Algebra and Trigonometry" from OpenStax, though

it's just an extended version of the precalc book

[flint forge]

yeah any book is probably fine

like they are all the same material plus minus a chapter or two

[muted hearth]

but Abramson explains stuff really well, I found

[wooden sparrow]

Ohh okay

So I noticed a thing

AOPS prealgebra has elementary number theory, where as openstax does not

[flint forge]

elementary number theory is not hs

[wooden sparrow]

It's prealgebra

I'm talking about

[flint forge]

I dont know what that entails exactly

but like i said

[muted hearth]

AOPS structure is a little unusual, it assumes that you want to work hard and progress quickly

IIRC

[flint forge]

AOPS has a lot of material outside the usual HS curricula

[muted hearth]

like

[wooden sparrow]

[muted hearth]

a standard text is for people who want to learn that bit of math, the AoPS series tries to cater towards "this is one step in my road to becoming a mathematician" people

[wooden sparrow]

I....umm want to become a mathematician 🙄

But their books are very expensive...
In my country they're unaffordable

[gray gazelle]

https://youtu.be/i8ju_10NkGY

[wooden sparrow]

@gray gazelle it's a book discussion channel bro

[gray gazelle]

But their books are very expensive...
In my country they're unaffordable
@wooden sparrow

[muted hearth]

@wooden sparrow to say that that's their target audience doesn't mean that they're that favourable of an option

i'm pretty sure openstax will do you fine if you're motivated

there's nothing any book has that can't be found in some form somewhere else

[wooden sparrow]

Got it

Thanks bro

[muted hearth]

AoPS are a little hard to find on libgen sometimes, N

[wooden sparrow]

AoPS are a little hard to find on libgen sometimes, N
@muted hearth I have been trying for a month to find an intermediate school collection

Couldn't find it

[muted hearth]

yeah

they're annoying

[gray gazelle]

Do you have access to some kind of opac?

[wooden sparrow]

What's that?

Opac?

[gray gazelle]

Online Public Access Catalogue

Maybe you know someone who is university who has access

They often have links to springer books

[wooden sparrow]

I... Don't have any access to any university or friends from any

[gray gazelle]

Oh ok

[gray gazelle]

a less known transition to mathematics book, Introduction to Mathematical Structures and Proofs; Gerstein, I laude over How to Prove it; Vellemen

The Foundations of Mathematics; Stewart and Tall, is also a great addition to anyone's collection

[marble rock]

what do you guys think of Abel’s Theorem in Problems and Solutions

for galois theory

https://www.springer.com/gp/book/9781402021862

[dense sphinx]

anybody got the solutions to spivak calculus 4th edtition?

cant find it

[pulsar aurora]

The solutions are found within. (also I don't)

[dense sphinx]

they only have slected ones not all of them

[gray gazelle]

Try to prove your solutions

Makes you infinitely better at math

[quick hornet]

i mean yeah, spivak is a proof-based book

not sure how youre supposed to solve spivak problems without proofs

when it literally asks you to prove things

[marble solar]

You just kinda jump in and do them

If you get stuck, look at Spivak's solutions

That's how I was taught proofs

[eager perch]

i mean yeah, spivak is a proof-based book
@quick hornet What is this book?

[quick hornet]

spivak's calculus

is what is being discussed

(well there are other books written by guys named spivak, but spivak's calculus is the most famous "spivak")

[eager perch]

Thank you, I will look that up!

[gray gazelle]

How about non-proof based calculus texts. Thomas, Anton, Stewart. What are people's opinions and experiences with these?

[weary citrus]

Used Stewart in high school, terrible design and color scheme. Used Thomas in uni, better design with the light blue. Both are wordy so I never read too much, but I heard Stewart can be a bit lazy at times,
perhaps Thomas is that way as well. Both books can be found for download with some google searching, so doesn’t hurt to check both of them out.

[shy shore]

Obligatory warning: Spivak's "Calculus on Manifolds" is a different book by the same Spivak, and has stuff like his "Calculus" as a prereq

[fathom monolith]

I used stewart for Calc 1, 2, and 3. I liked it alot, but the text and the classes in hindsight were more about training engineers than building a foundation

[hearty steppe]

Stewart has a good amount of exercises but I don’t remember the chapters going over concepts that well. Use Paul’s online notes

[marble solar]

Thomas is the best of the three

[broken meadow]

never heard of anton but yes thomas is better than stewart

[marble solar]

I learned calculus from a combination of Spivak + Thomas

Spivak was for the theoretical training, Thomas was for the more practical hands on

[marble solar]

@civic carbon I found out where spivak uses M,N for n, m dim'l manifolds respectively

It's a bit painful

[dense sphinx]

I just want to verify my proofs

[flint forge]

no

[dense sphinx]

wahh

[trim narwhal]

What are the books you would suggest to me to study from algebra to linear algebra?

[limpid gazelle]

See pinned messages in this channel

I believe Daminark gave a pretty good list

[valid moth]

do you mean hs algebra

[trim narwhal]

Yes

[valid moth]

maybe try lang's Basic Mathematics

[trim narwhal]

I'll take a look at it, thanks

[muted hearth]

@trim narwhal Algebra & Trigonometry is one of the couple of OpenStax books that I think is actually really good, too

also on the calc note, I randomly stumbled upon Differential and Integral Calculus by Piskunov

and I really like the way it's written

[trim narwhal]

@muted hearth Thanks for your suggestions :)

[potent jungle]

Hey, there are two book recommendations for calculus in #books-old and I wanted to ask you if I should read both, choose one, and after finishing, what's next?

[gray gazelle]

@potent jungle after calculus people usually branch off to linear algebra

And after that real analysis

You can start real analysis right after finishing calculus too

The best thing to do would be to download the course plan of a reputed college

And see which topics are taught in what sequence

And try to follow that

[potent jungle]

Are the books suggested in #books-old are enough (for calculus)? I mean, I see courses of calculus that has I II & II parts

I just don't know if these book really go into calculus the way I want to before branching off

@gray gazelle I really want to a deep understanding of calculus before moving on do you think these 2 books enough? (or as I asked, should I go with one of them instead of two?)

[gray gazelle]

I'd say complete a single book and then move on to a more rigorous treatment of calculus

Like spivak

Spivak is usually quite difficult to do as a first book for calculus

So having a considerable background will be of great help

[gray gazelle]

spivak is a good bridge text from calculus to elementary analysis but not a good first book. your first calculus course should be about doing calculus; a deeper understanding comes later. just my view.

[gray gazelle]

Hey guys wondering if you guys have any Calculus 1 book suggestions?

[main flax]

i like thomas early transcendentals

[long anchor]

never heard of that book

[restive raptor]

@gray gazelle are you looking for something more on the rigorous or more on the computational side

@potent jungle if you want to do calculus in depth, I would suggest spivak

[acoustic pelican]

So I have heard many varing opinions on using spivac to learn calculus. I just wanted to ask what spivak does and does not cover from a proper first real analysis course?

[velvet briar]

It's a light analysis course, good for people entering pure math and becoming familiar with proofs

Let me take a look and see if there's anything specific it's missing

[north spire]

@acoustic pelican What you'll find in a proper first real analysis course will differ from place to place

So, like, I know some places might use spivak as an introduction to analysis

[quick hornet]

it doesnt cover topology at all

its introduction to series somewhat neglects to define formal power series or justify why this notion is actually necessary

[gray gazelle]

It's not even an introduction to analysis imo

[quick hornet]

and just uses the semi-formal definition

[gray gazelle]

It's just calculus but difficult

[quick hornet]

the main difference though is in the "feel" of its problems

like its sequence-based questions are far more well-behaved than you'd expect from an analysis course

it also doesnt cover stuff like the lebesgue criterion, or countability arguments in general

[gray gazelle]

Towards the end of the book he covers fields and stuff so maybe that can be regarded as an intro of sorts

The child to baby rudin

[velvet briar]

Oof haha, that's pretty light

[gray gazelle]

Yep

[quick hornet]

i mean i dont think an intro analysis course absolutely needs to cover fields but yeah

lacking stuff like "all complete ordered fields are R"

feels pretty sad

admittedly my intro analysis course didnt prove this

but it discussed it

[velvet briar]

But I like fields

[gray gazelle]

On the other hand baby rudin feels like a summary of sorts lol

It needs a teacher or smthn

[civic carbon]

I open real analysis with talking about the axiomatic description of the reals. Not with proofs or anything, but I think understanding that leads to understanding the somewhat unique structure a lot of analysis proofs end up having.

I also think it is pretty helpful to let go of the idea that real numbers are decimals

[quick hornet]

well less "let go" and more "formalize what that actually means" i feel

[velvet briar]

I like thinking about points on a line

Or open sets I guess haha

[civic carbon]

To me, analysis is less about formalizing, and more telling the rather bizarre story of how people formalized what mathematicians had been doing for 200 years.

(but I'm not a formal/rigor person)

[quick hornet]

hm

perhaps its better to say

"develop a context where we can actually make sense of what 'decimal' means and how they express real numbers"

[velvet briar]

Any good book that is as easy as necessary to teach proper post-calc analysis?

I'm just wondering what to recommend haha

[flint forge]

Rigor is a crutch for people witnout divine intuition

[civic carbon]

yeah, I woudl not do that, really, I just want to never mention decimals at all when thinking about real numbers

[quick hornet]

eh fair

[gray gazelle]

You don't even need that adjective

[quick hornet]

but i think its still good to discuss

at least at some point

for one, decimal expansions can provide nice (counter)examples

[flint forge]

The day i write a proof that isnt morally correct ill cut off my hand

[quick hornet]

(as long as you handle the 0.9 repeating issue)

[civic carbon]

like, that's a special case of thinking about the reals as the completion of the rationals. And it is a theorem that every real number is the supremum of a set of rational numbers all of whose denominators are a power of 10. But it is not, to my mind, a useful or important theorem.

[quick hornet]

i think its "culturally" important though

like students go into the analysis course only really knowing real numbers as decimals

so saying "this isnt really wrong - it totally fits our definition and is a valid way to present a real number - but perhaps isnt the most useful or nicely-behaved thing to work with"

is IMO good to bring up once or twice

obviously you shouldnt spend entire lectures on decimal expansions, theres not much interesting stuff to say on them

[acoustic pelican]

the main difference though is in the "feel" of its problems
what do you mean by that? Are the questions throughout the book more like the ones near the beginning where the way to solve them is not obvious from the get go. That would be quite interesting.

[quick hornet]

but its worth at least mentioning

[civic carbon]

oh yeah, I mean, my day one discussion is that I write "pi = 3.141592..." on the board and ask if it's true... and if so in what sense...

[quick hornet]

ah sure

okay thats reasonable then

@acoustic pelican not exactly sure what you mean here

i think "the way to solve them is not obvious from the get go" characterizes most higher mathematics textbooks

including spivak

[civic carbon]

but yeah, if you use decimals in proofs you almost always end up stuck having to do more work because of the .99=1 thing]

[quick hornet]

(at least the non-computational content of spivak)

my point is more that, like

i wouldnt expect "prove stone-weierstrass" to be a problem in spivak

but IIRC it is in rudin

[civic carbon]

I mean I'd say even Calc II a lot of stuff is not obvious where to start, even in e.g. Stewart. To me, that's almost the central learning goal of Calc II.

[quick hornet]

thats an interesting take

a lot of students are certainly put off by the fact that the path to tackle an integral isnt "automatic"

like its not just "memorize a list of rules and then use the one that it looks like"

unlike differential calculus

[flint forge]

in fairness isnt hat problem r ight after he cites a way stronger version of stone weier

[civic carbon]

then in calc 3, the learning goal is slightly more nuanced: "There are many correct solutions to a problem, but some of them are heinous"

[quick hornet]

oh yeah max he proves that like

given a continuous function on some complex interval

[flint forge]

algebras of functions approximate

suitably interpetted

[quick hornet]

theres a sequence of polynomials with the limit that function uniformly on that interval

[flint forge]

no i think thats not what is in rudin

his is entirely general

[quick hornet]

really? lemme check

[flint forge]

using certain sets of functions

with distinguisung properties

[civic carbon]

tangential, but I don't think I've ever even looked at a proof that all meromorphic functions on the Reimann Sphere are rational

[flint forge]

and stuff

he calls them algebras iir

ive seen one actually

its cute

[civic carbon]

(which I think of as spritually related to Stone-Weierstrass)

ohhhhhh

it's "holomorphic functions are constnat"

so you just clear out the zeroes and poles

and go "now my thing is constant"

[sage python]

Yeah

[civic carbon]

right?

[flint forge]

yeah

that bad boy

[quick hornet]

fair

[flint forge]

one of the few thms in rudin i liked

[sage python]

To be fair I think Rudin does the polynomial proof first since that actually is part of the general proof?

[quick hornet]

in this edition at least he gives a fairly detailed sketch

[dapper root]

Wait is Stone-Weierstrass not just that continuous functions are estimated by polynomials on compact sets or w/e

Is Stone-Weierstrass theorem 7.32 from that screenshot?

[flint forge]

yes

[young surge]

terrence tao or rudin? which analysis book do i get?

pros / cons to each?

[flint forge]

I would just go thru both for a bit

And stick with whichever you prefer

Its kinda just a personal preference at the end of the day

[young surge]

ok i will do so. just from reviews and browsing, it seems like the tao books are more approachable somehow

[flint forge]

Ive never read them

[young surge]

they're just called "analysis I and analysis II" and the rudin book i'm looking at is "Real and Complex Analysis"

[hollow current]

m

why wont you go through both books

like they prolly complement each other

[young surge]

that's probably what I should do huh

like reference the other book if I get stuck on a concept or something

[hollow current]

but isnt tao like introductory analysis?

like Rudin's real analysis is advanced one

[young surge]

yeah i think the tao books will be easier for me

i'll start there

[hollow current]

rudin's introductory analysis is Principles of mathematical analysis

[flint forge]

No

Rudin doesnt have a book called real analysis

Yosh is refering to principles

Oh wait

[young surge]

this book

[flint forge]

Sorry messages didnt load for me

Yeah you want principles

Not that book

[civic carbon]

yeah, that book gives me nightmare

ss

[hollow current]

Rudin doesnt have a book called real analysis
@flint forge he has

just above lol

[flint forge]

That is not called real analysis

The second half of the title matters here lol

Anyway

[young surge]

oooh you know what, the table of contents to this principles book is much more similar to tao's books

like nearly the same structure

this is the one i want for sure

[flint forge]

Basically every undergrad class in analysis

[young surge]

thanks guys

[flint forge]

Has the exact same structure

Lol

[hollow current]

i havent read real and complex analysis but by contents it just implies that it is after principles

[young surge]

oh ok. I don't come from a math background. I'm basically looking into this to flesh it out better

[flint forge]

Be warned yosh

Once you get to the part titled differential forms

Its time to switch booms

Books*

[hollow current]

because principles is ending with lebesgue measure and real and complex begins with it

[flint forge]

You should learn measure somewhere else

Anyway

Second half of rudin is just bad

[hollow current]

i just switched to Zorich for a moment

[young surge]

my original plan was to follow some lecture notes on measure theory, but then I was thinking, I don't really have a strong background in advanced calculus / analysis so why don't I just follow these books that include measure theory at the end?

is that a bad plan do you think?

[flint forge]

I think you should switch to a different book for measure theory

But you should start w rudin

[hollow current]

for measure theory Kolmogorov is nice

[flint forge]

U want a strong foundation in analysis

Friendship ended w lebesgue

[young surge]

ok thanks for the tips guys. gotta head out soon

[flint forge]

Now Haar is my best friend

[civic carbon]

I liked Royden for measure theory.

I wish I had any intuitive sense of what a differential form is.

[flint forge]

Its a tensor product of the exterior algebra w C^infty

Ez

This is more or less my intuition unironically

[gray gazelle]

Does anyone know any good into mathematical logic.
Is reading about Gödel undecidabilty theorm alright?

[molten wave]

no

first study the mathematical logic

then study godel's theorems in that framework

[civic carbon]

I like Godel, Escher, Bach for an informal introduction to it

[cyan jasper]

does anyone have the aops number theory book here

if so, i want to continue my study of number theory with a different book

does anyone have any recommendations?

im seeking to enhance my skills in competition math

[gray gazelle]

problem solving strategies by engel and putnam and beyond should be fine imo

[restive raptor]

putnam and beyond is very difficult but yeah it's good

[gray gazelle]

@molten wave thanks for answer
Btw I made typo as I meant 'intro' not 'into'

[limpid gazelle]

$e^n=\sum_{n=0}^\infty\frac{n^n}{n!}$

[hasty eagle]

Whoever:

[marble solar]

no no no no

Not allowed

[mystic lark]

lol

whys this in book disc tho

[hasty eagle]

N/𝔄:

[heavy barn]

$e^e=\sum_{e=0}^\infty\frac{e^e}{e!}$

[heavy barn]

[gray gazelle]

Any recommendations for biology textbooks? I really like the way math and physics textbooks have an axiomatic basis and progress by providing a motivation and a logical extension of whatever came before. I've been having trouble finding books like that for the other natural sciences, mainly biology.

Ik there are mathematical biology books, but those are more about modeling than they are about general biology from a more mathematical perspective.

[hearty steppe]

You are correct most math bio books are for modeling (computational).

Also like me, you may also be interested in eventually getting into the physics aspect of biology which is soft condensed matter. I got a lot of biophysics book recommendations but haven’t really checked them out. Need to work on a biophysics background.

One book series I am aiming to check out soon is Murray’s Mathematical Biology. Also check out the book Mathematics for Neuroscientists and Intro to Computational Biology (Waterman). Be open to computational biology as well as computational neuroscience.

A lot of the material you will be expected to learn which is generally relying on modeling methods will be ODEs and PDEs, numerical methods, linear algebra, graph theory, some abstract algebra to get into group theory, stochastic processes, some real and complex analysis exposure as well.

I will say however, keep your head up. There is definitely going to be room for the marriage between theoretical mathematics and biology. It is not very accessible to most people to make that jump in the same fashion as quantum biology is a new area with much uncertainty and room for exploration.

It’s really only a matter of time before we can better crack the code of biology into mathematical translation. This will be required for innovations in quantum biology and biotechnology initiatives involving nanotechnology to name but a few possibilities.

[gray gazelle]

This has been really helpful, my area of study is pure math and physics, when I started my lower division studies I got really into quantum programs and picked my classes around that. But recently I've been hearing about biological computing which is exactly the sort of thing I'm interested in, unfortunately I always took the easy way out of bio because I found the method of learning tedious and unmotivated.

Thanks!

[hearty steppe]

I’m basically interested in pure math and bio myself tbh

[solemn mantle]

@heavy barn

What

[shadow nebula]

[gray gazelle]

[shadow nebula]

[broken meadow]

[shadow nebula]

[white cradle]

[valid moth]

[white cradle]

[hollow current]

good books on topology?

[north spire]

Introduction to Topology by Bert Mendelson is what we're using for our reading course and it's nice.

@hollow current

[hollow current]

fok epub on libgen

@north spire do u have pdf

[north spire]

yeap, i'll send it to you in DMs

[gray gazelle]

abhi can you send me one too? im curious to see how it fares against munkres

at least in terms of topics covered

[north spire]

Sure, but it's only like 200 pages and doesn't aim to cover a lot from what i can see.

We're using Klaus Janich's Topology as supplemental material

(uh give me a second, i'll have to get onto my computer)

[gray gazelle]

lol dont force yourself, i can find a pdf

djvu
epub
mobi

libgen please just gimme a pdf

[north spire]

Uh okay shit lol, it isn't sending because the files are too big

[gray gazelle]

post it here and delete it instantly

jk that might be against the rules

[north spire]

https://djvu2pdf.com/

[gray gazelle]

tru

[north spire]

Download the djvu version and just convert it

Janich's topology should be easily available as a pdf

[gray gazelle]

this will be a nice break from my diffgeo homework's index hell

calculating lie brackets will never be fun

[north spire]

Janich's book actually goes quite far but is quite memey so idk if it'll be in your taste to read it. I like his writing so I just read it. It's not really ideal as a main text because it lacks problems but actually, you could just get those in problem books.

[gray gazelle]

quite memey
🤔

seems like mendelson starts on metric spaces and then goes to abstract topological spaces. i dont think that's a bad approach to it, but im a bit biased since i jumped right into abstract spaces

[north spire]

Basically, imagine living in a medieval town as a poor farm boy and just going about your daily routine till, one day, your Knight friend returns from his adventure that he set out on several years ago. That friend kind of sits you down and tells you all about his adventures and it's a fantastic story. But sometimes, he goes off on weird tangents and talks about things that happened in the downtime between each adventure and those don't dampen the mood of the story but kind of lift it up and sound extremely interesting.

That's kind of what it's like reading his book.

[gray gazelle]

that sounds lit what the fuck

[hollow current]

in medieval knight friend will first take my food

then tell stroies

[gray gazelle]

ok i've been skimming the exercises for mendelson and they look pretty standard

[north spire]

seems like mendelson starts on metric spaces and then goes to abstract topological spaces. i dont think that's a bad approach to it, but im a bit biased since i jumped right into abstract spaces
It's a way to give motivation? Idk but the metric space approach works for us. We haven't formally learnt a lot of math (only analysis I and linear algebra I) so it makes sense.

[gray gazelle]

nothing too difficult

i took topology at about the same level (only one year of "rigorous calc" and linalg 1&2) and we went right into topological spaces

but the first lecture was metric spaces

mendelson discusses homotopy
:)

[north spire]

Haha we haven't actually started school so idk. We've done, like, several lectures already for analysis I but basically, we've only just finished the construction of R and are now moving into all the actual stuff with analysis

[gray gazelle]

which construction of R did you guys do

dedekind, cauchy, or some other thing

(those are the only two im aware of)

[north spire]

We did dedekind cuts, I believe we'll do the construction via Cauchy sequences later.

[gray gazelle]

oh nice, i did dedekind cuts as well

i would like to never go through it again

but it was pretty cool seeing it

[marble solar]

Are you computing explicit lie brackets

or is it more of an abstract thing

Like bashing them out to get a proof

[hollow current]

hm

[north spire]

It's actually pretty nice. My friends are somewhat tired of it but I find it very cool. I learnt this stuff beforehand, though, so it was a great refresher and he changed a few things around so proofs were simpler.

[gray gazelle]

i sat down and did one lie bracket just to check that two vector fields' flows didnt commute, thats it

[hollow current]

What Zorich does is Cauchy, isn't it?

[gray gazelle]

actually computing lie brackets is kinda...

🤮

[marble solar]

It's a little long sometimes, but you do it a few times and it's not too bad

[gray gazelle]

that's true

[marble solar]

Now computing jones polynomial for some knot?

[north spire]

What Zorich does is Cauchy, isn't it?
Yeap, he works with sequences mostly. But idk, I wouldn't say that what he has written is a construction of R. It's getting there in some parts but it's also lacking.

[gray gazelle]

i havent actually gotten much computation practice with lie brackets, we've been working purely theoretically with them so far

the homework has a few computation qs though

[marble solar]

I don't see much of a need to work with them directly that much if you're not going to be a riemannian geometer

diff. geometer

[gray gazelle]

i plan to take RG in the fall so it might be good to know

[marble solar]

Ahh, yeah that'd be good then

What book they use? Petersen?

[gray gazelle]

for the RG class?

[marble solar]

yA

[gray gazelle]

id have to check the course page, one sec

[marble solar]

I took RG from Petersen, and he didn't even use his own book

[gray gazelle]

"It will cover chapters 0-9 of the "Riemannian Geometry" book by Do Carmo."

[marble solar]

Ahh the classic

I never got into Do Carmo. I've been preferential to Spivak and Petersen

[gray gazelle]

i've heard good things about do carmo's curves and surfaces book, so hopefully his RG book is also up there

maybe i'll check out petersen

[marble solar]

Lee's is really short

Petersen is a monster of a text, hope you like PDEs

[gray gazelle]

i have never worked with a pde in my life

lol

[marble solar]

It's ok, apparently many analysts have never taken PDEs

[gray gazelle]

PDEs isnt even a required course here

it might be for the grad students but im not sure

undergrad? optional

[marble solar]

for grad courses generally very little is required

[north spire]

It's an undergraduate course over here

But comes in a finite sequence

[marble solar]

The only real requirement is to pass your qualifying exam, and if you do that you can normally get out of the first year courses

[gray gazelle]

one of my ta's did that lol

[marble solar]

That's my entire plan

[gray gazelle]

studied the first year grad real analysis and skipped it

[marble solar]

IDK first year courses are kinda mehh

You use books that are meant to get you up to speed quickly, so they don't spend a whole lot of time building intuition. Then you go bash out as many problems as you can. Lots of homework and tests

[gray gazelle]

that doesn't sound very fun

[marble solar]

It can be, if you have a good group. I just prefer smaller courses with like 2 or 3 people and you meet in the office hours working on obtuse problems

With everybody equally lost

Wait is RG undergrad @gray gazelle

[gray gazelle]

it's a "crosslisted" course

so grads and undergrads take it

if you take it as a undergrad, it counts as an undergrad course, and if you take it as a grad, it counts as a grad course

why do you ask?

(also i realize that probably doesn't answer your question, but im hesitant to call it an undergrad course definitively)

[marble solar]

It sounded as if grad courses were distant from you, but by discussion it's very close i.e. RG

Just curious, it was very much a graduate course where I was

[gray gazelle]

what do you mean by distant? (ill take no offense if you mean mathematical maturity/ability wise)

taking grad courses apparently isnt an uncommon thing for third/fourth years here (and it seems to be encouraged in some ways)

a lot of the 4th year courses are crosslisted similarly so you end up taking some anyways

[marble solar]

It's general knowledge that first year grad courses suck so much because of all the focus on very tough weekly problem sets, exams, finals, etc.

You don't get any cool projects

2nd/3rd year courses usually have much less in the way of evaluation

i.e. everyone shows up and gets an A

[gray gazelle]

i'd assume that's mainly for the core first year courses (e.g. for quals)

i had a professor tell me that the core courses are hard, but the other grad courses are slightly less so

[marble solar]

The other courses are hard in a different way

You have to read it, and find a way to apply it to the research that you're working on

That's not always the case, but it's the hope

[gray gazelle]

this has been pretty interesting to hear about

thanks for the insight!

ill keep it in mind for when my fourth year rolls around and grad school applications start happening

[marble solar]

What year are you?

[gray gazelle]

technically third right now because of the summer courses

in the fall i'll be starting the actual third year that all the math majors do

and ill be taking most of the third year courses that they all have to take

A Path to Combinatorics for Undergraduates - Andreescu, Feng.
A Course in Combinatorics - Lint and Wilson.

If someone knows about these two , which one to pick ? ( These were suggested by the course)
If there is any other intro combinatorics book please advise .

[brisk pelican]

Hey, I wanted to ask you guys, I want to learn applied math, and I don't know where to go, I started Calculus but I feel like I am missing a lot of the basics, how can I start and build my knowledge?

I don't think I am missing a lot of knowledge from the basics, but I do feel I need to refine it

I really feel lost when starting to think how to start, like there no "roadmap" or a series of books that is recommended for my case

[pulsar aurora]

For anything before calculus, any text book regarding HS algebra and precalc will do. Some people even recommend Kahn academy

[fierce marten]

A Path to Combinatorics for Undergraduates - Andreescu, Feng.
A Course in Combinatorics - Lint and Wilson.

If someone knows about these two , which one to pick ? ( These were suggested by the course)
If there is any other intro combinatorics book please advise .
@gray gazelle You can try a Walk Through Combinatorics by Miklos Bonas . This book is sweet . I mean it has all sorts of problems ranging from easy to hard .

As for the both books you asked I have read A path to Combinatorics book also but I won't suggest it for the first read . But if you are familiar with basic stuffs like recursion ,generating function(like say snake oil methods and stuff ... ) and basic counting and a bit about graphs then sure you can also try the both books you mentioned all I wanted to say was those books you mentioned is not for a intro to combi but only for those who are already familiar with a bit of combi as I mentioned ....

[gray gazelle]

As for the both books you asked I have read A path to Combinatorics book also but I won't suggest it for the first read . But if you are familiar with basic st...
@fierce marten oh so what you would recommend as a intro book . I know tiny bit basics like AP-MP , inclusion-exclusion and p&c a bit .

So what should i read

I tried bona , it has great examples

but I didn't like it

any other recommendation ?

[gray gazelle]

kenneth bogart

@gray gazelle

[patent walrus]

what is a good book on "decision theory"

[obsidian burrow]

Are you the croatian spider man?

[wooden sparrow]

No he doesn't look like that

[echo kiln]

has anyone read thomas calc

or james stewart calc

which one do you recommend

and why

what are the advantages of each one of them

[hollow current]

@echo kiln Thomas Calc is not bad

Thomas Finney i mean

i haven't read Stewarts' one so i cannot compare

but like Thomas provides you with most of the needed info for calc

[echo kiln]

does it have a lot of high quality exercises?

but really a lot

[hollow current]

well, it have exercises

i wont say that there are a lot of them, like 40-50 per section

and supplementary for each chapter

they are not so hard

[echo kiln]

not even for calc 2 or 3?

[hollow current]

tbh, i didn't look in his exercises much

[echo kiln]

lol

[marble solar]

I like Thomas, he has exercises that require less calculator work, and a better development of theory and proofs than stewart

[echo kiln]

oof thank goodness it requires less calculator work

[civic carbon]

I feel like Thomas has a lot of exercises that are numerically incredibly messy.

[flint forge]

oh my GOD

i had a really aggro Stormzy song playing in my headphones (which i wasnt wearing)

and i hit play on that and put the headphones on

and i thought it was like a grime song about thomas the train and was so hype

[broken meadow]

thomas is good

https://youtu.be/xtrcbkYx82U

wait this is the wrong channel

fuck

[valid moth]

i mean jan also sent that in this channel so

you're not getting banned before him

[broken meadow]

possibly

[broken meadow]

nice

do you like my video

[solemn mantle]

Welcome

[cobalt arch]

Which branch of foundations in math is complete and consistent? If such an approximation is to be found is there a book that starts from the most rudimentary theorems and lemmas according to the axioms it has and slowly builds up into other branches of math?

[flint forge]

euclidean geometry

completeness and consistency is a lot to ask for

or at least, being provably complete and consistent

oh really

thats what i meant i just forgot the name

thank u

learn it from the top down

ever heard the name Lurie?

jan being a lurie alt

would be the real twist

lurie is like

an honorary zoomer

he has such zoomer energy

[hearty steppe]

Max you are barely a millenial if your 23 yrs old

xD

[flint forge]

im 21

im not a zoomer or millenial

[hearty steppe]

Oh shit you are a zoomer

Wdym

I classify zoomers by age

[cobalt arch]

Why would it be a waste of time?

[hearty steppe]

Ugh damn it I thought this was chill

[cobalt arch]

I am not arguing with you just curious to know

I see

[flint forge]

channel classifications apply like yield signs in traffic

all channels are discussion general until someone needs to do math

wait thats the opposite

ofa yield sign

i swear im a good driver

[cobalt arch]

Is ZFC a better place to start?

Haha

Naive set theory?

[flint forge]

i think he means

halmos

[cobalt arch]

Yeah

[flint forge]

which is ironically not naive set theory

[cobalt arch]

After foundations where do I head towards?

Oh

What kind of algebra? Abstract?

Seems fair

But doesn't real analysis and aa take sets in a naive way?

Or more intuitionally rather than logically?

After RA and AA where do I head to?

[pulsar aurora]

Seems a bit jumping the gun trying to roadmap this shit when you haven't, assumingly, studied any of it.

[cobalt arch]

Does endertons book cover propositional calculus only?

Yeah that is true neveza

Oh

I have many books on all of these fields I just need to map it out.

[sage python]

You probably don't benefit too much yet from looking especially long term

Like you might just do analysis and be like you know what actually biology is dope and math is for nerds

[cobalt arch]

So RA + AA -> Logic -> ST?

[valid moth]

no dami the best method is to completely plan out your path in math from khanacademy to Hartshorne (in one year)

[sage python]

And then those long term mappings didn't do much for you

[pulsar aurora]

Basically. Plus, one's person path might be different from another persons

[valid moth]

honestly the only thing I ever got out of long term planning was pleasure out of fantasizing about it

[sage python]

Pretty much lmao

Like it's a high to imagine yourself in however long and being ahead of where you are

Just focus on what's in front of your nose. It seems right now algebra and/or analysis is your upcoming step, pick one and go with it

[valid moth]

I mean it can be interesting, if you like record it somehow

then you compare how you're doing

but I mean once it becomes useful it's basically just goalsetting

which is a good thing

there's a difference though

[pulsar aurora]

Then, at least, for me, a single book takes me a while to chew through. Like a year or two. So, at most, plan out maybe the next subject or book and leave it at that

[valid moth]

I wouldn't recommend eating your books

[pulsar aurora]

Oh... no wonder I"m not learning. 😦

[sage python]

Yeah I mean eventually goals are good but priorities can change over time, look at JohnDS

[valid moth]

personally I learn best by osmosis

[sage python]

Lmao

[valid moth]

smh dami this is just a training arc, he will be back soon

[sage python]

Having a tab open with the book

[valid moth]

or not

[sage python]

I love how that emote is named "pensive" but it mostly looks sad lol

[cobalt arch]

The thing is why would I take AA if it's about sets with different axioms?

Like fields and rings

Groups

[valid moth]

lol dami true
I just realize my emotional association with the word pensive had changed due to the pensive emotes

I associate it more with sadness now

interesting

[pulsar aurora]

@cobalt arch Perspective and new tools, I imagine. Learning a bit more rigorous calculus in Apostol, you'll operate on a few different basis to prove the same thing to show other alternatives or perspective to a problem, as well, to show consistency in the 'rules'

[cobalt arch]

Hm I see

I have so many RA books

[pulsar aurora]

Have you even studied any of them?

Or do you just hoard books, but never read them? Like people who buy games, but never play them

[cobalt arch]

Yeah I am studying one in greek

It is calculus but still haha

[gray gazelle]

Anyone got any recommendations for a book on proofs and logic?

Ping when respond ima be offline for a bit

[main flax]

@gray gazelle

[gray gazelle]

Thank you

Thanks very much

[hearty steppe]

Velleman

[civic carbon]

I've read dozens, and I like Galovich best

[velvet briar]

https://www.google.com/url?sa=t&source=web&rct=j&url=http://discrete.openmathbooks.org/pdfs/dmoi-tablet.pdf&ved=2ahUKEwjEqNXMrdDqAhXHgnIEHRsCA0QQFjAAegQIAhAB&usg=AOvVaw3qckwD1F6JlR6GOQUarnb3

[civic carbon]

but it is, in my mind, an area where all the textbooks are utterly terrible.

[velvet briar]

Sorry for the awful url. That's a free discrete book that does logic really well

[civic carbon]

I think a lot of the problem is that most books don't have a sense of audience, and there is certainly a lot of pedagogical disagreement on what the purpose of the course even is.

[velvet briar]

Yes try a few books and find what level of difficulty you're looking for

[civic carbon]

my undergrad course used Lay's Analysis text, and I'm not sure I've seen anything better? But you get the content wrapped around Analysis content.

the course is particularly difficult to self-teach because it is a writing course.

a lot of the goal of the course is to go from understanding why the product of two odd integers is odd to being able to write a tight, readable argument that that statement is true.

and you can self-assess "do I understand why this is true" a lot better than you can self-assess "Is my writing clear, concise, and readable"

[hearty steppe]

This one? @civic carbon https://books.google.com/books/about/Doing_mathematics.html?id=5PglAQAAIAAJ

[civic carbon]

yeah, though I've only taught out of the second edition.

but I no longer require a textbook, but this is what I list as my optional text

my biggest complaint is that it is too fastiduous and technical for its own good.

an undergrad intro to proof student does not need to think about the fact that formally the ordered pair (x,y) is syntactic sugar for the set {x, {x,y}}

[For that matter, I also do not need to think about that]

I'm definitely not claiming it isn't important for someone to think about.

[marble solar]

Do these proof books help?

From what I've talked to with professors, they prefer just jumping in on linear algebra or analysis, take your time at first and give the students lots of models

[civic carbon]

I really find myself thinking "wow, there are 30 intro to proof books and they all suck, I should write an intro to proof book" often. But then I realize that 30 people have already thought that 😛

[marble solar]

I was just thrown in the deep end with proof based Calculus, with no real prior experience for either proofs or Calculus

[civic carbon]

I really, really like my intro to proof course. I think it is incredibly useful, though I have very specific pedagogical goals in it, which I think most people do not. I'm also very aware that most of the students in the course will not go to grad school, so I'd aim differently if I expected that to be the case.

[marble solar]

I think Linear Algebra/Discrete math books usually do a good job of going over proofs

[civic carbon]

The "sink or swim" attitude towards teaching math has a lot of huge downsides to math accessibility. It strongly privileges certain students.

[marble solar]

I mean this was at a Community College, most of the students weren't exactly from priveleged backgrounds or had prior exposure to these things

[sage python]

Lmao that's a hot take Jan

I like it

[civic carbon]

Point set topology was perfect for me where I was at when I did this, but I also went on to get a PhD. I take a lot of inspiration from point set topology when I teach intro to proof.

[marble solar]

I can understand the worry, though. In general, I haven't seen these intro to proof courses done that well as independent courses

I think it can be great as like a supplemental 1-2 unit thing as someone takes linear algebra or analysis

[sage python]

Hot doesn't imply wrong or ironic

Just hot

How is that booing? I said "I like it"

[civic carbon]

I do a lot of questions that are "Here is a made up definition. Read it, understand, answer these questions about it"

[sage python]

But yeah I started with Spivak Calc (I guess I had probably seen induction in high school to find the sum of the first n numbers)

[civic carbon]

Because the skill students will most need in future math courses is the ability to read and internalize definitions, and that is very very far from automatic.

[sage python]

I also had epsilon calculus but I was able to place out of like, one quarter of intro calc on the placement. Didn't even know the proper definition of the integral lmao

[marble solar]

As you said Zeta, most of these books are poorly written; and most math instructors aren't exactly amazing at teaching intro to proofs for the most part

[valid moth]

why define it as {x, {x, y}}? I assume there were earlier attempts to rigorously define it that didn't pan out?

[civic carbon]

But I think in Intro to Proof there is a lot of incentive for professors to make a bunch of template questions because it is what the students want. And then you're just forcefeeding students algorithms for writing very narrow types of proofs, and that does absolutely no good.

[restive raptor]

my intro to proofs class was in Coq!

it was inquiry based

[valid moth]

nice lol

[restive raptor]

we needed to make our own axioms

it was not an easy class

[valid moth]

how's ibl stuff

is it fun

[restive raptor]

really fucking hard

but fun

[valid moth]

oof

nice lol

[restive raptor]

I mean our assignments didn't even have explicit questions. They were just "prove everything you can about this"

[valid moth]

Lol

prove everything you can about groups

go

[marble solar]

Like my spivak's calculus course was basically like "Here's 12 proofs that will be on the exam, you better know them" he didn't really have us try to make up our own proofs until the next course

[valid moth]

okay brb

[restive raptor]

one of my friends tried to prove the twin prime conjecture for many hours, not knowing it was open. I did the same with unique prime factorization in Z(sqrt(-d))

[civic carbon]

I find memorizing proofs to be worse than useless. But clearly someone thinks it is useful because al ot of professors do it.

[marble solar]

I think before you ask students to come up with their own ideas, you give them a model of how it's done ~ and from what I've seen of intro to proofs is they want you to be able to produce your own proofs without having that ingrain structure

[civic carbon]

I remember vividly the frist time I taught number theory I told the students to learn two particular proofs for the final. And then I had questions about them, but I did not "label" the questions as being about them. And basically no one got it.

[limpid gazelle]

Ah, Naive Set Theory book is pretty good

[marble solar]

I encourage people to memorize proofs if it's beyond your understanding

[civic carbon]

e.g. I told them to the proof that there are infinitely many primes, and then had "Prove or Disprove: If p is prime, then p!+! is dividsible by a prime greater than p."

[main flax]

what's p!+!?

[marble solar]

p! + 1

[main flax]

oh

[civic carbon]

oh whoops, yeah p!+1 sorry 😛

[valid moth]

!!!

[main flax]

close enough

[civic carbon]

I should have made the next question the same thing about (p+1)!

[valid moth]

it's just (p+1)!

lol

[marble solar]

Right, but I think just getting beginning students in the rhythm of what the expectations are at first; getting them to read the structure and spit it back out is very useful

It's what you do when you learn a language

[limpid gazelle]

So we have a bound p_{n+1} ≤ p_n! + 1

[valid moth]

whoever I'm going to bound you

[limpid gazelle]

I mean it's better than 2^{2^n}+1

[civic carbon]

I think a lot of the problem is that many students take a long time to "get" quantifiers. It takes a lot of time for students to be able to correctly identify whether a statement about quantifiers is right or not, nevr mind non-trivial proofs about it.

[marble solar]

I've mainly been through, and worked with Community College students who have little background, little experience in learning/studying things more abstract than they could usually process

[civic carbon]

I think you can blend "not memorization" and "following a template" by making students fill in parts of a proof at first.

[marble solar]

Yeah, the first course we had was "Just learn the basic structure" then in the second course we had to come up with our own proofs

[civic carbon]

in all teaching, the correct question for the professor to ask is the easiest one that will challenge some students. As mathematicians we are pretty terrible at judging that for intro to proof.

And it is especially difficult because you will always have, say, 20% of students for whom you could cover the first month of the class in one day and they'd get it all pretty well.

[marble solar]

Math is a creative endeavor, but before you've earned the right to be creative you must first understand what the current existing structures are, why they are the way they are, and what needs improving

[civic carbon]

So there is a very delicate balance, even more so than in most classes, of "How do I help the students who are struggling while not boring the students who are exceling to death"