Anyone here into meta-mathematics? Any recommendations?

Bump

I made a physics major in his senior year look like a retard by pointing out aspects of this work I read on wikipedia. I unironically thought he would have answers instead I only got pissed.
I am quote certain no one there assumed me the fool (that I rightly was) since we had others in the round partaking in the discussion and no one really calling me out as possibly only have read the wikipedia summary. :)

The Wikipedia summary of what, exactly?

What on earth are you on about?

i think principia mathematica desu

My impression of Principia Mathematica is that philosophers care about it some, and mathematicians care about it not at all. I could be wrong though; I'm an outsider to both fields.

I see. So it's just an attempt to grasp the abstract nature of math? I'm a math major so I might actually look into this stuff.

It's very interesting stuff.

Have you heard of Gödel's incompleteness theorem?

Where did you get these impressions from?

b-ok.xyz/s/?q=Metamathematics

>just

>STEMfag

It's historically important because it motivated Gödel's incompleteness theorems. Though, it never became a real foundation like ZFC did.
Also, Russell & Whitehead developed type theory for the first time in PM, which became important in computer science and is currently of renewed interest in mathematics (in the form of homotopy type theory).

Thanks for this

I've heard of it, but I'm certainly not familiar with it by any means.

All I meant by the "just" was that that's a simple explanation of what it is.

So math majors don't get exposed to any of this?

I know mathematics Ph.D who don't even know what incompleteness theorems are.
Math major don't care about axioms, moreover any type of foundation of mathematics, and philosophy of mathematics.
... except their mathematics meet some "limitation", some contradiction or incoherence. Then, mathematicians just "fix", adjust their thing a little bit and go on. It should be look genuinely fucked up to philosophers, but that's how they do.

I have a YouTube channel where I cover a lot of ground in that direction

youtu.be/IR0GkYoRzeE
youtu.be/vI3wgxg3tYk

Attached: IMG_20190626_153605710_HDR.jpg (3072x4096, 2.04M)

You are the reason math threads belong on /sci/ and not Yea Forums

You are quite right; it is just sad.
Whitehead just wanted to show how aristotelian logic can be sufficiently improved; Russell gone full extreme and fell into it not realizing he didn't think about apodicticity. You can see this by how Russell reacted to axiom of reducibility.
Even if you do foundation of mathematics, they didn't care Russell's approach seriously because neologicism only go deep on Frege's research because they sufficiently can change to Hume's law, and don't care about Principia Mathematica(I mean, just see how long and needlessly complicated the book is)
Although type theory is getting a bit amount of good evaluation, it is fact that nearly nobody in mathematics cares about HoTT. It is just talked on computer science. Moreover it is not that good way to say it influenced by PM because of type theory, because it went very different route after that. They took out only the primitive element in concepts of PM.

Yeah i should change a word, not "Math major", but "a course in mathematics major".

You should look into the philosophical debates about which logical system we should use, ie classical logic vs things like constructivism, or dialethiesism. You'll learn a lot

Nice

Where do I start?

Nice recs, on the topic of
> books in fictionalism, formalism, platonism and intuitionism

Introduction to Philosophy of Mathematics, by Marco Panza and Andrea Sereni, is a relatively short, synthetic and accessible book.

As a PhD student in mathematics I'd agree about the latter part. Mathematicians, even the most philosophy inclined, don't talk all that much about Russell this day (although of course they know who he is).
Then again most mathematicians care jack shit about the foundations of maths. It's pretty interesting how philosophy has focused on it and all but ignored the meat of mathematical works in most subfields.

You can read Albert Lautman for more on that.

Was Russell and Whitehead's type theory anything like modern type theory though ? Sometimes I feel the idea almost goes back to Euler and Lagrange's first attempts at founding mathematics, but perhaps I'm just ignorant. Type theory is extremely removed from my field.

He's right though. I did a master degree in pure maths in one of the best uni in the world for mathematics. And I got a second master degree for teaching which is essentially an excuse to go over the basics (the first four year) again to gain a solid foundation in general mathematics. The only three times I really needed to think about foundational issues for class where a couple of two hours introduction in logic (in first and second year, respectively) that were there mostly because the teacher insisted on it, and on elective course in logic that only a few people took.

In practice you can make a whole career without running into an incompleteness theorem, because despite its general significance it's rather removed from most mathematical practice. Math is so fucking huge you can spend a decade studying a particular kind of interaction between multiplication and addition in a specific kind of algebraic structure.

Learn to use punctuation you annoying fuck

>Albert Lautman
Firstly, can I get some summary on what is his work?
Secondly, How do you think about Jean Cavaillès' works? I'm into some french epistemology works and this guy talks about mathematics.

Bump

Could any mathbros recommend me some good preliminary literature on the epistemology, phenomenology, and ontology of mathematics? Particularly mathematical or metamathematical foundationalism? My practical math skills are weak and I'm working on that (mostly for my own purposes), but my main problem is that I find it hard to jump right into foundationalist theoretical texts that presume you're familiar with the jargon already, when precisely what I'm trying to do is understand the tacit presuppositions underlying the jargon.

For example, when I tried to learn about the history of proof structure in logic, I kept wanting to ask, "what exactly does this author think the metaphysical status of 'proof' or 'self-evidence' even is?" But of course, if the nature of self-evidence is tacitly presupposed as itself self-evident, and not taken to be problematic, it's not going to be exhibited in the first place.

I'm interested in following things like this closely as they develop, for example following the tacit preconceptions of set theorists and intuitionists, paying close attention to the things they take to be primordially true, without even realising or asserting THAT they're "true" (because they're the unconscious ground[s] upon which conscious utterances have meaning in the first place).

Do any math people talk about this sort of thing, very self-consciously?

The stanford enclypedia of philosophy article of the material conditional is a good start.

Attached: list.png (2050x2508, 3.03M)

misread the op, sorry

>... except their mathematics meet some "limitation", some contradiction or incoherence.
That isn't fair and I don't think it is true either.

>Math major don't care about axioms, moreover any type of foundation of mathematics, and philosophy of mathematics.
They don't, until they have to. There are some points, eg. in functional analysis where you very explicitly have to invoke foundational axioms, see e.g. Hahn Banach, one if the more important Axioms in Functional Analysis and, by connection, in the study of ODEs, a more "applied" field.

>The only three times I really needed to think about foundational issues for class where a couple of two hours introduction in logic
You should have run into in the first hours of a Linear Algebra class.
You need AC, specifically, but equivalent, Lemma of Zorn to get that any Vectorspace has a Basis.

Some for some other important theorems, like Hahn Banach and I can't imagine they gave you a Master without having shown to you the proof of both.

Dont waist your time with Russell or Whitehead's math (there both great to read for Philosophy though, esp. Whitehead). Things have developed a lot sunce then. Notation, has changed, and most of the Pricipia Mathematica is unnecessarily cumbersome and not really illuminating. Once you've done your first formal PA proof of 1+1=2, you've seen enough.

At the moment I'm really into combinatorics on words, modal logic, and algebraic logic. I'll also say that if you coming at things from cognitive science, linguistics, philosophy, or computer science, please, please aquaint yourself with number theory and basic abstract algebra. It'll make learning stuff like model theory and non-standard models way easier to understand.

I'm only going into sophomore year, but I haven't yet. I don't know if any of the later classes explore it more though.

Is there anything that relates this to metaphysics?

Hello OP. Gödel's Proof is a short, substantive and excellent treatment of the Incompleteness Theorems themselves. The actual mechanics of the proof go a bit heaver toward the end, but the setup all makes sense.

Not that it matters but I have a complete set of PM in your pic related edition, mostly for reference. Since it's a reprint of the first edition, I also copied all the added stuff from the second edition in my old university library's 2nd ed. copy and stuck it in a binder, so I have a complete hard copy edition of PM available for personal reference. Russell's IMP is a prereq though, and worth a read.

Mathematicians do care about it somewhat for its historical context, and they'll name-drop it in their classes, but (for reasons that are clear) it is an abandoned project, more a historical curiosity than anything else.

For people like me who have a serious interest in the HISTORY of mathematics, as opposed to proper cutting-edge mathematics, it holds interest. Recently I've recognized the need to hold my History of Math library section apart from the math-as-such (I have all these books):

Euclid's Elements (Heath's English standard in the proper 3-volume Dover set, not the BN trash)
Gauss, Disquisitiones Arithmeticae
Descartes, Geometry
Cardano, Ars Magna, Games of Chance, Book of my Life
Whitehead/Russell, PM
Gödel, the actual statement-paper of the Incompleteness Theorems (has a proper title equally as tedious)
Russell, IMP,
Hardy, Mathematician's Apology
Carroll, Euclid and his Modern Rivals
Fibonacci, Liber Abaci
Bourbaki, multiple volumes
Robins/Shute, Rhind Papyrus (a fair summary but the ancient Chace text is the English standard exposition I think; Chace is an amazing work of scholarship that I've have the pleasure to completely read)
I also have a binder with print copies of all of the Unabomber's professional mathematical work. Interestingly, he dabbles with a digit-problem in number theory which is tossed off by Hardy in the above as an example of a trivial, uninteresing, "not properly mathematical" problem.

Desired additions: Conics, more Bourbaki, Russell Philosophy of Mathematics, Whitehead's Process and Philosophy (context + Yea Forums meme)

Attached: logicguide.png (1766x4567, 2.65M)

>try Gelfand’s Algebra
>get stumped on a problem and get depressed because I can’t solve it
>drop the book for weeks until I forget how much I hate myself and then the cycle starts back again
Is there a book that actually teaches instead of gives you a bunch of problems? I need a book for brainlets that lays it all out for me so I can study and complete exercises.

Attached: D4722B9B-3AA0-40FB-8304-53978E76FC4F.jpg (254x199, 14K)

Stop sinning against St Gödel and just read the Bourbaki already
b-ok.cc/book/2304020/370ad3
There's book one

vocaroo.com/i/s1qb1hHWaAmV

If you want to understand how a mathematician thinks, then you should read math but not journal articles or textbooks. You should read monographs and lecture notes, which tend to go a lot more into "motivation" and "intuition". This sort of discussion also sems to be more common in applied math, than pure math. For example, stuff like Paul Halmos and John H Conway really expose how mathematicians think. In more philosophical areas, Noam Chomsky's early work, as well as much philosophical logic tends to explicate the intuition and motivation of the mathematical theory under construction.

>self-evidence
No such thing.

There are only axioms and rules of inference, none of which have to be thought of as "self-evident". "Proof" is a syntactic concept not much different than the grammar rules governing any natural language. Nothing mysterious or metaphysical about it. Of course, whether such a "proof" actually proves something relevant in the real world is a different issue, but whatever we might think it proves cannot possibly be anything more than a well-corroborated model, as it happens in the natural sciences.

>(((Israel M. Gelfand)))
>Morris Kline
>no (((Rudin)))
>no Apostol
>no (((Halmos)))
Not gonna make it.

there was a thread on /sci/ recently about pm and whitehead in general. most thought he was a crank but some on there really held him to high esteem.

Maybe we see different thread of whitehead, I saw /sci/ seems to agree whitehead's later metaphysical term do some important works

That's perfectly normal. I recall problems that took me a couple of days, with one problem in the 200s being incredibly hard, I believe in was in the Vieta's theorem section.
The exercises are all about tinkering and struggling with them until you pull through, be it in an hour, or in a week from now, after you go through some sections and return. Other beginner algebra books may teach you, but they won't help develop your mathematical maturity and overall problem-solving skills quite like this one. If you blew through all the problems effortessly, the book would be a waste of time, wouldn't it?

But yes, it's more of a supplemental text. You should read it with any precalc book like Axler or Lang's.

Imagine attempting to rigorously study logic without including algebraic logic and boolean algebra.

Theology of Arithmetic my Iamblichus

Attached: 51d1TURbyTL.jpg (317x500, 34K)

looks like a crank book

its theology in the rational Metaphysical sense, not religious doctrinaire

Skip the 'analytic' bullshit and go straight for Hilbert and his school. Tarski is a must obviously.
Also the early sections of Bourbaki's set, just to realize how most of what is written on the subject is and will forever be useless to mathematics no matter how advanced. I cite Bourbaki because his is a very foundational work while being very skeptical of metamathematics.

If you are going for old classics you can't keep Archimedes out. His works are pure esthetics.
You won't have a problem finding a good translation in whatever your language is. Though if you want to meme deeper, i've heard that his dialect is very pure and is not to bad way to get into ancient Greek.

Thanks everyone.