Mathematics

/Σ/ To discover what was dearly missed in schooling, this general will serve as a playground. >What do you mean? Why talk of mathematics? >I hate mathematics. https://www.maa.org/external_archive/devlin/LockhartsLament.pdf >Okay, but I don't know where to start. I still hate what I was put through. I'm just as lost, but I hope this is what this general will serve as. Don't be afraid to ask any 'stupid' questions here. Recount of your earlier math experiences if you wish. >Books ( which I hope to expand, depending on interest ) http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf I realized I was able to enjoy math class not because of the rigidness but because they couldn't take away curiosity. >What do these functions mean? is what I would always ask. Only when we reach Calculus II do we see the importance of Taylor Series, but that's not how our calculators and ancestors did it, past 1715. Calculators use The CORDIC algorithm (first I've heard of it, still reading about it). Still, functions of the calculator, and how they're always implied to be known, but not the discovery of it, particularly frustrated me. How did ancients do it? From what I've researched, it seems that they've made tables. >https://en.wikipedia.org/wiki/Ulugh_Beg I don't know how exactly he got them. I think it was using a particular trigonometric theory and re-inputting new discovered values. Or if you precisely draw the angle you're looking for, you can get the ratio. What about you lainon? Do you remember having some mathematical inquisition, in spite of the system?

>What about you lainon? Do you remember having some mathematical inquisition, in spite of the system? Yes. Good reads, lainon.

Do you recommend any books for learning about electricity and magnetism for a math novice?

Taking calculus for the first time and really enjoying it. Currently a freshman of computer science but thinking of changing to math. Just worried that I'm not smart enough for real, proof based, math.

>>32959 This one: Concepts are explained in a practical relatable way.

>>32954 >Lockhart's Lament Severely misleading opinion piece. I prefer this: >Young man, in mathematics you don't understand things. You just get used to them.

>How did ancients do it? Typically they used some kind of sequence which approached the desired numbers, such as continued fractions. That combined with some identities and some tedious arithmetic is what made those kinds of tables.

are there different limits of number of posts for each thread in each board?, in humanity the limit is 300 and something, and here it seems to be 200 and something, I'm making this statement by assuming that the old math thread hit its limit(?)

>>32963 >Just worried that I'm not smart enough for real, proof based, math. Try it. There is a free book called Book Of Proof: >https://www.people.vcu.edu/~rhammack/BookOfProof/

In my opinion, there is some strange gap between syllogisms similar to the AAA about Socrates' mortality (such logic is often taught among various lawyers) and mathematical logic. That said, the same Gottlob Frege was aware of syllogisms when he created quantors. And I found this video, where the author seems to remove such a gap and at the same time it seems quite beautiful and simple with formulas and schemes. It would be interesting to know what you think. There are English subtitles built in and most of what's shown is also translated. https://youtu.be/w1Lm4OCoMdU

>>32963 Try this book man, I am currently learning math from scratch and this book is gold. I hope the book can help you as it did me.

>>33186 How is a discrete math textbook going to help him learn calculus? And discrete math is one of the areas of math that is already covered by a CS degree anyways. If he's interested in proofs and calculus he should read an analysis textbook. >>32963 Read pdf related >>32959 I feel like questions like this go into a thread about physics instead of one about math.

Is there any definition of the integral over R^n with n>1 that agrees with the Lebesgue (and hence Riemann) integral when they are both defined but does not do so through iterated integration? ie. a definition of integration for surfaces, volumes etc. which does not start as integration in one dimension?

>>32959 Griffiths Electrodynamics is pretty good.

In highschool, we each had to complete a personal exploration of mathematics. I still cherish this assignment and remember it as one of the highlights of my secondary education. I decided to explore Newton's Method of finding roots of a function. The teacher challenged me to solve this problem using my CS knowledge and without looking at any proofs. The iterative algorithm itself is quite simple, but 'discovering' it took weeks. He knew the ins and outs of the algorithm, so he was able to give me hints whenever I needed them. After proving it, I compiled my findings in a formal document amalgamating background information, the step-by-step proof and interesting cases. If you are looking to have fun with math, I highly recommend finding a topic like the Newton's Method and taking a stab at rediscovering it. Share your findings, discuss it with other people. This is real math; the painstaking process of coming up with new ideas and erasing them again and again because they are all soykaf, and nothing works until everything works.

>>33294 What do you mean by rediscover? If you already know how the formula looksblike, it's hard to rediscover it when you already know hownit should looks like. However, it's a nice idea. Trying to find solutions, formulas and etc for things mathematicians already know about. Who knows, maybe you can find a better proof, better aproximations, etc. Numerical methods are a good way because they don't assume a lot of knowledge. Sure you could try to find galois's proof of why there is no formula for the roots of polynonial of degree n > 4, but you do need all the knowledge galois had up to that point (he didn't invent groups out of nowhere, he had read a lot famous french mathematicians papers before).

>>33295 By 'rediscover', I meant figuring out the way to reach a conclusion. For instance, the foundation for my Newton's Method research consisted of only the formula. I learned what each of the variables meant, and everything else was weeks of mathematical trial and error. That process, for me, is 'rediscovering', but perhaps the word was too strong. More ambitious explorations (e.g. galois proof of quintic functions) could also be quite enjoyable, but as you mentioned, you need a lot of background info to actually make any type of progress. This may work if you have a solid understanding of mathematics, maybe even a degree in the field, but for laymen trying to reconnect with mathematics, this seems too advanced. When assigning the project, the math teacher told us to look for real-life problems you may solve/resolve with math. I remember a friend looked into the Elo rating system, and another looked into the transmission of disease. There are a lot of topics out there to 'rediscover' with mathematics.

>>33286 >ie. a definition of integration for surfaces, volumes etc. which does not start as integration in one dimension? The Lebesgue measure already satisfies this whenever the Fubini/Tonelli theorem does not apply.

Does anyone have a roadmap/chart of books, besides those created by /sci/, that show you the steps to learn math, with a focus on computer science?

>>33320 idk if im a good person for advice but pretty much all the math I know is in Rosen's Discrete Mathematics and It's Applications, Norman J Wildberger's Linear Algebra (covers Grassmans Vector Algebra), and high school maths (think Khan academy).

>>33320 Probably the most useful math a person can know, especially a computer scientist, is linear algebra. I highly recommend the book 'numerical linear algebra' by Lloyd N. Trefethan and David Bau III.

>>33322 What about discrete math? I'm pretty sure most CS majors don't take any linear algebra classes.

>>33348 They take it, but it's not as important as that anon is making it be. Combinatorics(which includes discrete mathematics and graph theory) is the most important math subfield for CS people.

>>33348 >I'm pretty sure most CS majors don't take any linear algebra classes. Most CS majors also don't know very much CS. Linear algebra is important because it allows you to grasp higher-level math concepts more easily. For instance, a common thing that one does in discrete math, specifically graph theory, is create an adjacency matrix. An understanding of linear algebra will greatly ease your introduction to this technique.

Almost 10 years ago i graduated with a math degree. Went onto programming. Never used ANY of the math, it was 100% useless.

>>33353 >I worked a job when I didn't had to use math as a programmer >therefore math is useless for programmers If all you do is webdev or similar stuff where you dn't even have to usee Bool algebra to simplify conditions - it is. If you want to create small ompany creating renderers or robots - it would be rather essential to develope in house software.

>>33357 I actively try to find fields that use math but you have to basically be PhD doing something very specific. Even with masters in machine learning, jobs using machine learning (like robots you mention) you just use libraries that abstract it all away. Graphics just has most baby linear algebra.

>>33348 I had both calculus and linear algebra during my licence and master (France).

>>33358 Certainly not the case. I just recently had to implement the gradient calculation step for a layer in a neural network because it used context-aware computation which couldn't be auto-calculated by tensorflow.

>>33375 implementing a basic calculation thats in every textbook isnt 'doing math'

>>33357 This represents it like 'if you do some stuff you wont need math, if you do some other stuff you will need it'. More accurately, doing 99% of stuff you wont need any math

>>33378 So the only math that counts is the stuff not in textbooks? Then sure! Almost no field uses math except math, physics theory, and cs theory. If you want to define things so your statement is just a tautology then why state it at all? Smh

>>33358 The people who heavily uses mathematics are creating libraries. If you are using any, you don't need to know mathematics in most cases. Want to use math in programming and it's not making some game? Go make some libraries, chances are you are going to need math there.

From what I've concluded after breaking out of my hatred of mathematics from schooling, is that mathematics is hilarious. After multiple iterations of decoupling mathematics from reality it somehow became a perfect descriptor of it and we found out that a lot of the things in earth (and in the universe in general) are mathematically perfect (bee honeycomb in the shape of hexagons being the perfect packing structure, sunflower seed distribution and the golden ratio, all that good stuff and more). There's a number called Chaitin's constant which is the probability of a randomly generated Turing machine program of a certain length halting and it's an uncomputable number so we have no idea what it is we just know it exists, it's transcendental, and we maybe have a lower bound of what it might be. Quaternions are pretty funny to learn about but frankly they can get fuarrrked when you have to use them in 3d rendering. Fourier transforms are fun - try them on images (imagemagick can do this). The math behind a neural network is very simple to understand, they're just weights and thresholds, and people have abstracted from that to what we have now. Honestly I don't even really care about the numbers, the theory is what I'm interested in and when I understand that I can apply it to various situations ranging from programming to art.

>>33394 >I don't even really care about the numbers No one does in math. I have read textbooks and papers that had no numbers at all.

>>33382 gradient descent is the fundamental operation in neural networks. Anyone that knows anything about neural networks, it will be first thing they learn. Typing out the equation is something anyone can do. It's baby CS student stuff.

I realize that my math foundation is very poor, and that's why I struggle so much in the math that I am learning now. I think I need to go back to elementary/intermediate algebra and start all over again.

>>33443 Take your time. It doesn't come over night. It took me about 3 years before I had the math skills I needed just to start what I wanted to do in terms of programming. Just keep putting yourself to the grindstone!

Why don't you learn mathematics from this playlist? https://www.youtube.com/playlist?list=PL5A714C94D40392AB

>>34171 Because it's a playlist of Wildberger?

Any sites to practice mathematics problems that require creativity (or, if you've read it, the ideal type of problems as described in Lockhart's Lament)? If it has problems that range in difficulty would be great. Essentialy, competitive programming sites but replace programming with math. The closest I know to such a place would be Project Euler, but even that just uses mathematics as a means to efficient/elegant code.

>>34575 I believe your best bet would be to find a compilation of Olympiad/Putnam style problems, assuming you are not looking to do research and simply hone your problem-solving skills. Otherwise if you are hoping to solve problems that are research level/unsolved, it might help to search for math blogs, use math stackexchange, etc. Some books you might find interesting: -the Mathematics via Problems series(currently has books on problem-solving in basic algebra and geometry) -The USSR Olympiad Problem book -Putnam and Beyond(has a ton of problems covering undergraduate level mathematics, emphasizing real analysis and algebra) all of which can be found on libgen ofc. Titu Andreescu(one of the authors of Putnam and Beyond) also has a lot of other books centered around a problem-solving approach to learning mathematics. The mathematics via problem series also takes a similar approach in that it teaches a subject through a collection of problems that may require some creativity.

>>34575 There are books sepcfiically for this purpose. I have one, "a collection of problems in complex analysis", it has hundreds of high-quality problems.

>>34577 These sites still sound kindof cyberhumped. Isn't the whole point of "A Mathematician's Lament" that math is an enjoyable activity of thinking. When I use lainchan.org/sec/ I'm not thinking about learning about security, when I think about a topic, such as the psyche, I'm not thinking about learning about the psyche. Same with math, its just for fun. >>34575 I like to draw shapes on paper and construct problems for myself to solve. Its kindof like deciding what you want to doodle, instead of a book telling you.

>>34190 >Because it's a playlist of Wildberger? So?

>>34650 Is this from a novice who genuinely doesn't know the answers to the questions Wildberger "covers" or am I browsing /sci/ somehow?

>>34660 The former option? I don't "do" /sci/ at all.

>>34668 Oh, good. Give me a second to breathe a sigh of relief. Wildberger is a parody of ultrafinitism come to life, what we might call a "ultra strict hyperfinitist", neither of which themselves exist among credible mathematics. Not only does he advocate throwing out and redoing at this point what it seems will be every field of modern math's current formalisms to make *some* parts of classical trigonometry simpler and more prominent, but he also, knows, and does not seem to care, that his proposals make many other parts of math, even of planar geometry, much weaker, just because he doesn't like the typical *informal* descriptions of orthodox math given to students and/or wants to be known as the guy who wrote them this century. Unfortunately, he's no Cantor. His most rigorous work, the "rational trigonometry", produces only a handful of very basic new discoveries, and does so at the cost of, for example, circles being continuous. Seriously; here's him explaining in his own words that, in the 21st centruy's geometry, it makes perfect sense for a line to both have points inside a circle and outside a circle but not intersect that circle (and not just any line and circle, but the unit circle and the line y=x !!) : https://youtu.be/kcirwIwRIUw?t=484 And it doesn't get better from there. I haven't even watched a majority of his videos (I always get to irritated with what a waste of time most of them are), but you'll also have your "foundational course in mathematics" regularly interrupted with digressions about counting things, rants about the "logical problems" of just using a different ontology to justify extensions than he does, explanations that it's actually okay to make those extensions when he needs them for his projects (ie geometric figures), rants about how "the pi number" exists only up to how much of it has been physically written down (until you add more, then *that* becomes both a number that wasn't before and the new pi, yes really: https://youtu.be/lcIbCZR0HbU?t=1860 ), or that angles don't exist because they have no finite trisection construction but that his replacement for them do despite having no linear comparisons at all (this is also relevant to quadrances but I *think* he obliquely permits distance so it's much worse here), and more and more. I've not even skimmed half of Wildberger's "Foundations" but it's already clear we're on alternatively useless and shaky ground. I have to say, this criticism should carry its quadrature of the weight above coming from me personally. I'm a strictly-all-formalisms kind of philosophy of math guy who prefers proofs without the axiom of choice and even has a lot of sympathy for the general concept of finitism specifically that most mathematicians just don't bother with, and I see the work of real finitists, like say the mad sexy attempts at the project like the recreation of peano arithmetic within finitism, as the sort of herculean efforts much of today's research mathematicians ought to get back to aspiring to. So, in that light, I kind of despise Wildberger for making them stand next to such pompous, self-centered quackery in the popular mind. Even within the university system Wildberger made it his mission for far too long to embarrass as many of his colleagues as possible by association through his rather craven methods of discussion and self-promotion. Like I said, it should be clear from the great number of windmills he's tilting at over the years that he quite wants to be a legendary mathematician, but from the little preview of his work I've given here hopefully you see both why I described him as disingenuously vapid because he lacks the chops to back it up before and that you'd be wasting far too much study time trying to dredge up some educational value from him instead of, I dunno, just reading Lang and Spivak followed by an assortment of springer cheese singles. I intentionally went for breadth instead of depth here because, like I said, spending too much time on Wildberger for myself irritates me, but if you've already seen any of his videos and have any specific questions about why their acutal math counterparts are really just fine I'd be happy to help out. To save us all time and frustration, just do yourself a favor and don't watch more until after you've thoroughly studied the real math at hand in them, because then you'll have the good sense to quit early and go study more real math instead of getting your reasoning skills polluted by the mathematical equivalent of logic-free assembly without mnemonics.

>>34814 Hmmm, I see. But what do you recommend instead? The playlist seems very convenient since it starts with the very basics of mathematics and takes it to advanced level. What would be your replacement of that playlist? >t. not that anon but someone who wants to learn maths long years after graduating from high school with no further mathematical experience

>>34864 If you're still pretty confident in your ability to catch up quickly, you could do a lot worse than Serge Lang's "Basic Mathematics". Here's a playlist of what looks like overview videos I found; I haven't watched it but it looks like he covers it all: https://www.youtube.com/watch?v=WybeKK6oMh0&list=PLMcpDl1Pr-viA25VUkHNmcUkWx9usPgyb&index=74 If you're not, probably the most cost-effective resource is Khan Academy. Just hop in wherever you need to and ride through to at least College Algebra. I'm not in love with their teaching style, but they just about make up for it by taking things very slow through lots of detail; I don't know of another resource that goes into as much detail without asking for a pile of money (I've never been very good at piracy but I find that elementary resources are fuarrrking impossible to find because no one both cares about math generally and thinks they're worth getting and distributing over uni textbooks and publications). What sort of things are you looking to learn, where do you think you need to start, etc? I can give some more specific advice if you'd like.

>>34872 >What sort of things are you looking to learn Everything from middle school mathematics to college level mathematics (till the abstract algebra parts in a normal university curriculum). In high school I studied calculus a little bit and still remember a little part of the curriculum but I believe that I lack understanding of concepts. I can operate with numbers, but I can not visualise things iniside my head. for example proving the -(-x) = +x was impossible to me a couple of days ago. I memorised it during my school years, but never questioned the very basic maths or understood the concepts. I mindlessly went through the formulas as I progressed with solving more and more questions. I would like to make a fresh new start and make it right this time. The reason why I was hopeful about the playlist of Wildberger was it started from very basics and with 200 videos, moved on to harder topics. I wish to do the same. Quickly grasping the basics of mathematics, being able to understand and visualise concepts and then moving on to advanced topics. >khan academy I hate their playlist orderings. They are not in order and the same concept is explained in 3 different playlist in separate courses. Could you provide with a good course order to take?

the school system fuarrrked me bad, i used to be able to solve 3rd degree polynomials by head but because of a soykafty math teacher that forced me to use his dumb methods i completly lost that ability, fuarrrk the school system, it has only taken from me and never brought anything.

is there alot of overlap of coding and maths? Im a math major and have been going the K&R for the last few days

>>34928 Yes. Programming is basically an intersection of math and logic. An understanding of math will greatly benefit you when coding.

>>34928 https://en.wikipedia.org/wiki/Curry–Howard_correspondence

>>34884 Math should be taught through developing your own computer algebra system

>>34938 Could you perhaps expand on that a little bit more? What articles or books do you recommend to proceed in this direction?

>>34962 Maybe they mean writing a text based equations solver. Start with something that just does operations on numbers, then something that can solve for 'x'.

What is your opinion on Algebraic Calculus: A Radical New Approach to Higher Mathematics for Students of Electronics and Computer Graphics by Dr Roderick Lumsden http://www.rodericklumsden.com/algebraic-calculus/ Algebraic Calculus puts forward a rather radical new proposal for the teaching of basic undergraduate mathematics aimed especially at students of electronics and computer graphics. The text draws on methods I worked out in my Ph.D. work several years ago which in turn drew on the several books on Differential Geometry that came out largely in the sixties. The central thesis of the book is that basic Calculus can be developed much more easily and painlessly if the Exterior Derivative is used from the start. This leads naturally to Differential Forms and these in turn can easily be shown to give Integration simply by defining it as a product between an n-form and an n-simplex. So I’m doing two things: I’m showing that the idea of differential forms, which crystallised around a hundred years ago, allied to the concept of simplexes, suffices as a foundation to develop the entire body of the calculus easily and quickly, and gives a much more coherent line of development. I’m putting it in a way that is clear, readable and, hopefully, entertaining. So I have preferred English readability to mathematical formality wherever reasonably possible. Along the way, I cover in some depth various supporting fields such as Vector Algebra, with an introduction to the up-and-coming area of Geometric Algebra, and I also give a good, but more critical, introduction to the subject of Generalised Functions. The book ends with a brief introduction to Projective Geometry. The salient original features of the book can be summarised as these: I introduce differentiation using the exterior derivative on a scalar function to generate a 1-form, so making it multivariate from the start. I define integration as a product between a differential form and a simplex. I use the axioms of a group to show that the addition of angles in the circle leads naturally to the idea of complex numbers. The book incorporates geometric algebra into the presentation of vector algebra and analysis from an early stage. Generalised Functions are introduced fully based on differential forms, and this treatment prepares the way for an advanced coverage of Fourier and Laplace transforms. In the account of this first edition, I said: “I make no bones about the fact that I’m not happy with any of the existing presentations of Generalised Functions and that I have deep reservations about the Fourier Inversion Theorem. A natural result should eventually have a natural proof. There’s no question that this has been achieved for Stokes’s Theorem, which initially required a somewhat tortuous proof, but when developed in the recast framework of Differential Geometry falls out automatically as the general extension of the Fundamental Theorem of Calculus. In the present treatment I regard both of these as so basic that they are effectively definitions of different grades of Integration and to all intents and purposes need no proof.”

>>35100 I click on the hyperlinks but nothing happens.

>>35100 >basic Calculus can be developed much more easily and painlessly if the Exterior Derivative is used from the start. I completely agree. However the formatting of the book, with bolded words everywhere, is not very good

Seriously considering self studying math. I was in college and took up to real analysis II, but dropped out and joined the military. It's been a few years but I really miss it to be honest, I'm not sure how fast I'd pick it back up.

>>32984 Sorry I don't understand. Syllogisms are used both in hilbertian axiomatic schemes and natural deduction systems. Whatever logic you use, it appears to be just a regular form of modus ponens.

i lllvoing calculus of vvariation

>>36436 Same! I just picked up Griffiths "Exterior Differential Systems and The Calculus of Variations" at the library today after reading a paper that confused me, looking forward to learning some calc. of variations properly. Every time I go to the library and look at the math section I realize that there's so much cool soykaf I could read, it sometimes feels overwhelming, but I also feel a bit like a kid in the candy store. Often I leaf through a few titles that seem interesting even when I came for something specific.

>>32959 How much of a novice are you? If you know basic multivariate calculus, Purcell & Morin or Wangsness is good. Otherwise, I would recommend instead Classical Field Theory and Relativity from Susskind's Theoretical Minimum series.

>>33320 It's a process I'm doing myself at the moment. If you're restarting your studies, it's fair to to start with precalculus, which is a retreading of high school algebra and geometry but oriented in a way to prepare you for studying calculus.

The derivative is simply the inverse of boundary.

>>32954 here is my contribution to the thread I don't know how useful this book might be for programming, but the quality of problems and it's beginner approach for mathematical rigidity makes it worth while for anyone who wants to improve their problem solving skills but don't know where to start.

>>32954 >>34872 Hello, for the past little while I've realized my math skills and basics are pretty poor and was hoping someone could direct me towards some good resources to relearn/review. This playlist seems like a good resource but I know myself and without any practice questions or ways to review none of it will stick well enough for it to be worth it to me. So, if any of you wish to help out a retard don't be afraid to reply.

>>36698 roughly whats your current level? can you do alegbra, trigonometry, matrices, logs, ect.

>>36516 dual, not inverse. The boundary map doesn't have an inverse in general.

>>36707 I'm somewhat confident in my algebra skills but in school I was the type of kid to forget everything after the test so nothing really stuck.

Question for anyone who studies math: How do you find balance between problem solving independently, and learning new material? if you had to give a ratio for these activities, how would you split it? I personally feel most comfortable spending more time learning new material and choosing a few problems to work on, but I'm not sure if that's the best route.

>>36993 Ideally while you're learning new material you're doing problems related to that material, and while you're working on a problem you're learning/reviewing things related to that problem. It definitely helps to periodically review things you've already learned or re-do problems you've already solved. Often the added perspective of a few more months of study can really make the old material more clear and sit in your head more permanently. Plus it can make you feel more competent to tackle new problems/subjects.

>>32954 >Lockhart's Lament After reading this it does reinforce some of my doubts I had while I was studying math in high school. I won't say that all of my failings in math are due to the failings of the education system since I am a bit of a retard, but it does explain why a lot of concepts that I learned never made sense especially calculus.

>found out that the smallest factor of a number n is smaller than sqrt(n) + 1 thanks to math Thank u math by primality tests now run in sqrt(n) instead of n/2 lie the dum dum I used to be What other math do you know is used in programming?

>>37083 I hope you take into account that sqrt(n) may be a divisor of n(which is why it's smaller than sqrt(n) + 1). You can make it faster by taking into account that all prime numbers greater than 3 are of the form 6k +/- 1.

I learned how to produce NFAs and DFAs of a mathematically defined language. What can I do with these?

>>37291 You can write tools like grep, awk, sed, flex fuzzy finders... etc

>>37083 You mean largest factor right? >>37291 Same here. Pretty fun stuff. Im not really sure what you can use them for tho lol

Kinda have a weird question I took Calculus 1-3 in college, got A’s etc, but I dropped out of college. I have also worked a little bit in Spivak a long time ago, but gave up because I got stuck. I know all of my basic geometry, trig, algebra, etc. If I want to restart my math journey, should I return to Spivak’s Calculus, or do something like Zorich’s Analysis?

>>34814 >ultrafinitism come to life When I read the Wikipedia page for that, I thought it is surely not real. Yet having watched a talk of his, he seems to believe it. I am sceptical of infinity, but that is just stupid. That numbers too big to represent don't exist is just an arbitrary practical limitation. We can think about many things that don't exist, so the meaning of such a statement about existence as it relates to mathematics is not clear. My view is that it's only logical to talk about things which could be brought into existence, and hence not infinity. In all of the maths I know, I have yet to see anything resembling infinity, or any reason why such a thing should be possible.

>>37291 >>37393 If you need automaton to do some stuff, you can write program that generates table for it.

Any recommendations on an (easier) intro to geometric deep learning besides Bronstein? He dives in immediately with group theory and differential topology, which I'm pretty unfamiliar with. I figure I'll need to learn group theory or topology eventually, but I'd rather not get caught in the weeds.

>>38581 i'm afraid the resources you're looking for likely don't exist, and even if something does it wont be much good. geometric deep learning relies very heavily on these areas and you'd be better off learning them. i'd recommend familiarizing yourself with topology and group theory (any undergrad book or lecture series will do), then reading something to familiarize yourself with manifolds and some basic differential geometry. for this, both "An Introduction to manifolds" by Tu and "Introduction to smooth manifolds" by Lee are good choices.

How to write a simple prover from scratch?

Greetings lains, I promise this post is about math, please bear with me. I am a software reverse engineer and master's student in computer science. I have found myself taking a liking to theoretical computer science. In particular, I am interested in learning more about programming language theory (type theory, program analysis, semantics, compilers, etc.), cryptography, formal verification, complexity theory, and computability theory. That said, my mathematical background is a bit lacking for these areas of study. Unfortunately I cannot take any math classes to bolster my background, so I will have to self-learn any material needed. What mathematical preliminaries should I cover in order to have the proper background to pursue my interests?

>>38736 current researcher in theoretical compsci here, luckily the areas that you've expressed interest in have fairly light prerequisites. complexity theory and computability theory don't really require any prerequisites at all, just mathematical maturity. Michael Sipser's book and accompanying mit course lectures are a great starting point. If you like the type of material covered, "Computational Complexity: A Modern Approach" is a great choice for further reading. I'm not an expert on cryptography and formal verification, but make sure you have a strong background in elementary number theory, Richard Borcherds has a lecture series on this, with accompanying textbook "An Introduction to the Theory of Numbers". Some abstract algebra knowledge is also good to have here. For programming language theory the only real prerequisite is an understanding of some well-known programming langs and a background in mathematical logic. Some set theory and basic category theory would be useful here. I'd also urge you to familiarise yourself with graph theory, most introductory books on the subject would suffice. a solid foundation in graph theory is required to access many deep and beautiful areas of compsci. I'd also advise you to make sure you have a solid understanding of probability theory and statistics. Maybe even some basic real analysis is good to have. good luck!

>>38708 Depends on what "from scratch" means for you. Do you only have vim and a C compiler? Or are you starting from a batteries-included environment such as Python or Haskell? Also what kind of prover? Do you want to automatically derive proofs for propositional statements? Do you want to be able to write statements and proofs in first-order (or higher-order) logic and have them checked for correctness? Do you want to prove theorems in Presburger arithmetic or recursive functions?

>>38738 Thanks for the thorough response friend. I know that I am unlikely to be able to pursue all of what I mentioned, I just wanted to make sure I have the proper background to dabble in these things to narrow down my interests. I will take your advice into consideration.

Does math not scare you guys? It's an absolute confrontation with the absurd and I find it hard to keep going with this stuff sometimes. But I couldn't get myself to really enjoy programming since it's bound to whoever is in charge of the digital electronics supply chain

remember that compshit undergrad is not math

I was a loser NEET who's going back to school later in life for a CS degree, and I can't pass the fuarrrking math exam to get into a lot of the required classes. I scored just barely below the threshold twice, which means I have only one more try or I have to take some other math class for no credits whatsoever just to get into those other classes, mostly coding-related. I've been putting it off for over a year now. I'm only taking 2 classes this semester because I've already exhausted all of the other classes that don't need the math prerequisite. This is so stupid. In real life, if I needed to do some math equations, I'd have a reference sheet by my side listing the way to do them, the same way I have little Bash loops and Perl syntax open in a .txt file at all times on my computer. In the exam room you only get a piece of scratch paper. If I do take that optional math class over the summer I might miss out on signing up for classes for the following semester. I don't even like thinking about this, why can't they just let me take a coding class? I'm running out of time and I have fuarrrking homework today too. I also have to schedule another meeting with an advisor who I haven't seen in a year after telling her I would pass the math exam. Ah man this is like kicking dead whales down the beach. I just don't like math very much, maybe the software development path was a bad idea. I was thinking of pivoting to systems administrator, hopefully that doesn't render any of the classes I've taken thus far useless

>>38749 Just take the extra math class and stop procrastinating. It's one class.

>>38749 Stop looking for easy-ways-out and learn the material.

>>38570 >In all of the maths I know, I have yet to see anything resembling infinity, or any reason why such a thing should be possible. Rejecting the notion of an infinite continuum makes modern analysis, topology, and geometry basically impossible. For instance, all functions are continuous if the continuum is finite. You can probably work around these limitations and do some useful math, but accepting the existence of a continuum is one of the foundational assumptions of 20th century analysis and topology and you'll have great trouble recovering even the most basic results. >>38745 Mathematics is the opposite of absurd

>>38741 > Do you only have vim and a C compiler? Probably yes. I would prefer this. I don't like to be dependent on some huge ocaml package manager or something. > Also what kind of prover? I don't want to have some fancy autoderivation, maybe just creating some tactics like they say "ring" for bruteforcing axioms. And yes I would like to write statements and proofs for simple theorems like "(a+b)^2 = a^2+2ab+b^2". I need to write it from scratch to convince myself that it has a strong foundation. In future I would to experiment with an alternative logic system but maybe I need to write a prototype in lean or something before. Currently I don't have much experience.

>>38754 I recommend Type Theory and Formal Proof, by Nederpelt and Geuvers. There are also a couple of books by Barendregt on lambda calculus if you need a stronger foundation. You don't have to depend on package managers, you are never forced to do so; if you need a library just make sure you have it and your header files. It is more important that you use a language of your choosing, preferably one that is closer to your goal, such as those in the ML, Lisp, and Prolog families. If you want to go for scratch you can implement your dialect in C, otherwise, if your problem is one of bootstrapping, you can choose an implementation you like and keep the source around. Actually, to better your chances to be provably correct, you would be better served doing everything on assembly, and perhaps implement Standard ML or Scheme as they are formally specified. There's a point where you have to compromise, meet the machine halfway. You might want to start reading the little prover and the little typer.

Could everything be explained by combining set and type theory using something like anonymous functions, assuming that one's consciousness and perception is what defines all reality, including physics? I was thinking something along the lines that existence is a gradient of ∞ in size, and that one's perspective is equal to λ ∈ R² (∞) (R² being a euclidean plane where at minimum four points would be plotted via an anonymous function, and that being an element of infinity) A simple way of putting it is, you have a plane of infinity consisting of a linear gradient that's black and white. Life exists by plotting a square on this gradient of infinity. That would be said life's perception of reality. Decision would take place within the square by thinking "which side do I think is prettier, this shade of dark gray, or this shade of lighter gray" (or maybe assign one side to 0 and 1), and equally dividing this defined cube to get closer to a perfect, ideal, singular color/end goal. The more complex the being, the better it's ability to divide. Getting the perfect color never happen of course since this gradient would go on forever, but this way of thought would eventually lead to the creation of lainchan in theory lol. When death occurs, maybe this recursively thinking cube is removed from the gradient of infinity where that now lost energy I guess manifests elsewhere. Idk if I should continue to entertain this, I talked to a few friends about this but felt pretty embarrassed, let alone someone with a phd since I'm not very sure on this yet. But if it sounds interesting I'll gladly elaborate in detail.

>>38820 Sounds dumb lol. It's an unprovable theory with no tangible consequences on reality. All you have is theology with a mathematical veneer. It also sounds a little schizo. Like you just heard a bunch of words and now you're throwing them together randomly. One can infer through logic perhaps two things about the soul, anything more is just mad speculation.

>>38736 I would recommend Practical foundations of mathematics by Paul Taylor. It truly is a remarkable book which essentially tries to build foundation from a general perspective using type/category theory. That being said, it is also quite difficult to follow, I'd recommend to take a look at the first two chapters to get a general idea >>38820 >Could everything be explained by combining set and type theory using something like anonymous functions, assuming that one's consciousness and perception is what defines all reality, including physics? First of all, it's not clear at all what variant of type theory you're referring to. In particular, intensional type theory is incompatible with set theory, while other variants may not consider types as h-sets (mainly because identity is considered to be an extensional property). One may also consider variants where univalence and/or function extensionality holds. Now, I suppose what you call anonymous functions are essentially lambda terms, and lambda calculus is indeed a valid computation model. However, there is no proof or empirical evidence as to whether the universe is turing complete. This is way outside of my field of expertise, but I believe EPR and other paradoxes that suggest non-locality may be a problem to such perspective. The last past is quite vague. It seems to be a solipsistic/theological argument.

>>38823 This looks like an interesting book, but it might be a bit out of reach with my background. It might be good for me to ease myself into it with something a little easier. Any recommendations on that front?

>>32954 >LockhartsLament This was a great read thank you! To help others check it out I looked for a different link as this one has 404'd, here it is: https://maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf

>>38993 There are a few key differences between yourself and the likes of Galileo and Einstein (I don't think Einstein ever told someone to kill themselves). Addressing the matter of your "revolutionary theory" vs. theirs: they made concrete predictions about the world that could be tested and proven. This is what makes something science (describing Galileo and Einstein as mathematicians was an interesting choice). You make unsubstantiated claims that cannot be proved or disproved. You might as well speculate that the universe is a simulation. Your ideas do not even meet the bare minimum requirement of being well formed. >Could everything be explained by combining set and type theory using something like anonymous functions is essentially the maths equivalent to an engineer saying: >Could we achieve perpetual motion by combining cars and trains using something like tempered glass It doesn't make sense to say that, and it makes me wonder if you understand the things you are talking about.

>>38994 I'm not a famous mathematician, I'm a random engineer who likes reading papers in their freetime, nor do I think my theory is revolutionary. I was looking for people to give me input so I can tell where to go. Either way, the way you speak is why people stray away from this subject. Failure and soykafty ideas are inevitable no matter your gifts because no one is perfect. After all, this thread is meant to be for aspiring mathematicians of all levels.

>>38995 >the way you speak It's great that you pivot from telling me to kill myself to tone policing. >why people stray away from this subject Probably better off without them. I don't think Einstein or Galileo would have been put off if someone called them dumb. >nor do I think my theory is revolutionary Yet you compare it to some of man's greatest discoveries, and if true, it would disprove almost all major religions and change the way we look at reality itself. I don't see how you could propose such a thing and see it as anything but revolutionary.

>>38996 this is getting silly, i wish you luck on whatever you study

>>33294 here in the east we have this concept in our philosophy called "experience" which means that reading or discovering some phenomena is much lesser in quality to actually experiencing it like deriving it/thinking about it for long time. we have this story, which I think is fake but still is a good example. that when scientists in the west found about quantum phenomena and sub atomic particles, people who have been meditating for years have also observed the same particles when they meditate for long times, so they went to meet the western scientists. but when they arrived they found the scientist had gone crazy because he was not able to understand what he was seeing in the instrument.

>>38993 Galileo got famous for criticizing Aristotle, and any competent person knew that Aristitle's physics was bunk. Opposition to him was purely ideological. Einstein's theory of special relativity was immediately obviously correct to anybody that has studied electromagnetism. His theory of general relativity took years to develop, during which time he had correspondence and support from the world's best and most eminent mathematicians.

>>38708 Take a look at "John Harrison, Handbook of Practical Logic and Automated Reasoning". Everything written is accompanied by examples on a limited subset of the OCaml. There is also a great amount of explanation and history.

>>39008 some french frog apparently had the idea earlier. Read about it years ago but didnt care enough to remember details. When it comes to math & coding what im really missing is a relation of cost. Like traditionally sin cos is very expensive but on gpu its apparently hardwired and can be cheap. In other words algorithm design can fail due ignorance of architecture changing cost of functions.

I'm looking for a good read on λμ-calculus ("lambda mu calculus") - any reading tips? In particular I'm looking to formulate a version of λ-calculus fit to describe hypercomputation and programs that have cycles. λμ-calculus is fit for formulating cyclic programs and I'd like to know if LAST just-so-happens to be equivalent to λμ-calculus because it describes λ-calculus + an extension not defined by the author - due to its syntax it is quite natural to evaluate it as-if the program is stored on a cyclic tape, but I can't analyze it as λ-calculus for this exact reason. Hypercomputation... I'll worry about next. I've attached two papers I don't understand yet lol. They're about explicitly cyclic λ-calculi.

>>32954 Are there any books that teach math through programming? I don't know any APL (or J) but I thought it would be pretty good choice. I found some books on the subject: >https://github.com/EvansWinner/List-of-array-language-math-books >https://code.jsoftware.com/wiki/Books >https://aplwiki.com/wiki/Books >archive.org/details/APL_books I was looking for a book that uses some other language because I don't know any APL. The only other book that was close was the new NoStarch book "Learn Physics with Functional Programming" that uses Haskell.

>>42168 Not a book or programming but you may enjoy MetaMath. It's a very simple language for proofs that can be computer-verified.

>>42168 I found 2 books that use Haskell: >Discrete Mathematics Using a Computer (https://www.dcs.gla.ac.uk/~jtod/discrete-mathematics/) >The Haskell Road to Logic, Maths and Programming (https://staff.fnwi.uva.nl/d.j.n.vaneijck2/HR/)

>>42168 APL probably will not be good for your goal unless you just happen to be one of those rare people who quickly take to APL. It was a language designed to be more like human language and like human language it generally does not throw errors, it just produces a result and if you don't already know what that result should be you will have no idea that it is wrong. Also, with the exception of Dyalog and Mastering Dyalog, every APL book will have code which will probably not run in any of the APLs you will be able to run and much of the code will go about things in antiquated ways. You will be learning a dead dialect for the most part, like learning English from a book on Old English. If you do attempt to work from one of these old books you will want to use GnuAPL but you would be better off learning APL with Dyalog.

>>35100 do anyone has this book as pdf?

>>42037 Maybe you can find something in Lambda Calculus and Combinators and its references.

>>32954 >To discover what was dearly missed in schooling this hits home for me! my mathematics knowledge has been really spotty over my education and i want to firm it up, and working my way through intermediate linear algebra and college calculus. i find im unlearning some bad thought habits/processes as im moving through that i picked up in high school that were previously clouding the eureka moments and the beauty of mathematics that im now starting to bathe in. im finding it difficult to think mathematically. when i encounter a difficult problem, i get into a state of mind where im just rabidly trying to find an answer and i end up feeling like deer in the headlights, and all im doing is just testing a bunch of different ways i can do the problem in my head, but i find i get so stuck on the possible solutions i have, i miss the actual solution. my question is this, how do i get into the problem solving mindset? or are there alsny resources that you would recommend regarding problem solving? thansk for reading my blog

>>42609 school teaches you to just memorize formulas to fill in different numbers which is only good enough to train mediocre workers. exact formula pattern matching is not sustainable when you go beyond basic math and wish to touch novel knowledge. good math students build a mental map how formulas inter connect and transform into each other. That reduces the knowledge base to a fraction and makes operating within it much faster. Understanding the meaning also allows you to pattern match on meaning instead of syntax. This allows to find connections you wouldn't be able to notice if you merely memorized solutions.

>>42610 One way I'd go about adopting this is by recognizing that mathematics is formal logic applied through an informal language. When you learn new mathematical concepts, remember that they hold true because they are built on other formula or rules that have been established as true through the same process, all the way down to the axioms of the specific mathematical branch you're working in.

Has anyone here read ‘How to Prove It’? Seems like a good introductory maths book.

>>43312 If you want to learn to write proofs, it's not. If you know how to write proofs, it is.

>>42611 >One way I'd go about adopting this is by recognizing that mathematics is formal logic applied through an informal language. When you learn new mathematical concepts, remember that they hold true because they are built on other formula or rules that have been established as true through the same process, all the way down to the axioms of the specific mathematical branch you're working in. The idea that all of mathematics is reducible to logic and a fixed set of fundamental axioms is what Russell and Whitehead thought when they wrote Principia Mathematica. Then Gödel came along and proved it untrue.

>https://www.maa.org/external_archive/devlin/LockhartsLament.pdf 404'ed doe

>>43315 >Goedel's incompleteness theorem Biggest urban myth in mathematics today, it's not a real thing. Goedel wrote a sensational philosophical paper, not a valid theorem. https://jamesrmeyer.com/ffgit/godel_flaw

>>43315 That isn't really true. Goedel just proved that a set of axioms cannot prove itself consistent while being consistent. If you think about it, a set of axioms being able to prove itself consistent would not give a very strong justification that it actually is consistent. The axiom "true" proves itself consistent, yet that tells you nothing. >>43624 What is the meat here? The author believes that the function Z is inconsistently defined? >This is because there can be infinitely many expressions that have the same value as any given number So? What does this imply? >But the sequences of symbols “fff 0 + ff 0” and “fffff 0” are not the same value when substituted for the free variable of Φ So? There was no claim that \Phi must be some sort of homomorphism. As I understand it, Goedel's claim is simply that Z and \Phi agree when n = "ffff...0" is a natural number, and no further. "fff 0 + ff 0" is not a natural number, it is a string of symbols. Goedel's proof is also not the only proof of the theorem, in fact his numbering scheme is one of the more complicated ones. This author also seems to think that ad hominem attacks are valid, so I'll do it as well. He also has some hilariously bad argumentation about other topics. Here's one https://jamesrmeyer.com/infinite/lebesgue-measure . Aside from the atrocious writing style that makes it so hard to understand what he's even saying (a hallmark of crankery), he makes a fundamental mistake: he assumes that every point in B is an endpoint of a "complete interval" in A. I can come up with unions of intervals for which it is not true. Let u_1 < u_2 < ... and v_1 > v_2 > ... be increasing/decreasing sequences of irrational numbers respectively, having the property that sup_i u_i = u = inf_i v_i. Then set J_1 = (u_1,u_2), J_2 = (v_2,v_1), J_3 = (u_2,u_3), J_4 = (v_3,v_2), ... . u is clearly not in the union of all these complete intervals, and it is also not an endpoint of any of the intervals J_i. His entire torturous argument completely falls apart because he doesn't produce any reason to believe that every element of B must be an endpoint of a complete interval in A (which would imply that B is countable, which he seems to not realize would also be a much shorter way to conclude his "proof"). Oh, and he addresses it! https://jamesrmeyer.com/infinite/lebesgue-measure#ConvergingPlea >Applying a limit means that the intervals of Bn decrease to degenerate intervals, giving a resultant set B which consists of single points that are separated by the complete intervals of a set A. And for this scenario it is the case that within the interval [0, 1] every point is either an endpoint of an interval of A or an interior point of an interval of A. By not addressing it! He literally is just restating what he said earlier again without proof. "It's obvious". Someone even gave him a very nice proof that the inverse of \phi is not in a complete interval for a specific type of enumeration (one in which a rational number p/q is labeled with the integer at least q-1) https://jamesrmeyer.com/infinite/lebesgue-cranks. Here's his response: >It’s immediately obvious that there must be something wrong with the above “proof”, since the result is saying that the irrational φ is an interior point of a non-degenerate interval within which there cannot be any interval of the set A Not so! It just proves that there is an irrational number that is not an endpoint of any complete interval. That does not make it an interior point of the complement of A. His other refutations are meaningless. He seems to not really understand that being a boundary point of A is not the same as being an endpoint of an interval in A. I think this is where is entire misunderstanding stems from. I'm much more familiar with Lebesgue theory and not so familiar with foundations, yet his argumentation on Lebesgue theory is blatantly incorrect, I have low confidence in the correctness of his argumentation on Goedel's theorem. I do thank you for linking this website though, his writing is so hard to parse and contains so many flaws, it's good practice for grading first-year analysis students.

hey lainons is "The Linear Algebra a Beginning Graduate Student Ought to Know " a based book or is there a better one

>>43624 if godel flaw, where paper showing? Is Big Math afraid?

>>43625 One unskippable proof step equates two objects from two non-compatible languages. Basically, according to Meyer, Goedel's paper conjects that a complete proof may exist if these two are equivalent, but leaves it an exercise to the reader. So the original paper is then an elaborate conjecture with a suggestion for how a theorem may be constructable, but not a theorem. Then Meyer attempts to demonstrate that this equivalence is not constructable, and as such the suggested pseudo-theorem is not realizable. There is way too much jargon both by Goedel and less but still too much by Meyer for me to rigorously evaluate it because I can only be bothered to learn so much about mathematicians' conlangs. >>43690 https://jamesrmeyer.com/ffgit/incomplete-formal

>>43693 tank u body too short or empty

>>43695 wait! You fooled me! >Swierczkowski claims that for this reason his proof his superior to most other proofs of incompleteness. However, the author makes an elementary error in his proof that renders the proof invalid. Those are not other proofs of incompletness being wrong, this is the same guy! They aren't published in any journal! Why the fuarrrk should I bother, if guy doesn't publish! So it is after all that Big Math doesn't want You to know about incompletness theorem being wrong, and they don't publish mind-breaking result and wants you to not bother with proving ZFC consistent!

>>43685 I took a look at it and seems to be a good book as it goes straight to the point (something rare), has plenty of examples and even more exercises. It's not a book for anyone but for those who really want to learn the topic. >>42609 >when i encounter a difficult problem, i get into a state of mind where im just rabidly trying to find an answer and i end up feeling like deer in the headlights It reminds me of myself: for years I thought my efforts were useless as there was no way to measure the quantity of results of such efforts but, a few weeks ago I was reading a logic book (classical, not mathematical) from the late 70's and in the introduction the author gave some advices about good studying practices, one of wich was writing down the basic concepts and results of each chapter; in our case, it would be writing down every definition, every theorem and every result worth to be remembered. It has worked! That's a really confortable way of consulting the most useful information per chapter and, to my surprise, solving the problems has become less difficult. Apart from this I've been writing down my thoughts (not only observations) when solving each problem, like "whow can I justify this equality?", "This 'proof" does not seem to be one to me, but for the time being I will leave as it were such since I have no time to invest on formalities" or "this concept arises from a question like this...", etc. These practices makes you realize wich are your shortcomings, and thus, what you have to study; how may attempts have you tried, what approaches have you tried (this leads you to realize which one works on on which case), what are the concepts you master, the concepts you don't fully understand, the concept or reason that prevens you from solving a problem and, most importantly, writing down all of this helps you understand everything in the long term. But here is the trick: you need to understand that this takes time, and you must study several times the same topic.

>>43693 The more I read from this guy, the more fallacies I find. Every refutation I've looked at, he basically says "you can't have a number-theoretic expression of the form A = GN(B) where A is an expression with a free variable, GN is a godel numbering, and B is an expression in the meta-language, because A and B aren't the same type!" This is such an obviously incorrect line of reasoning that I can't help but feel that everything he's writing is in bad faith. He hides behind jargon and convoluted sentences in natural language, which perhaps obscures to a layman how often he makes assertions with no justification whatsoever.

Does anyone know any math forums (the more obscure the better)?

>>43702 He supports that one approach to refutation in detail in this paper: https://jamesrmeyer.com/pdfs/ff_representability.pdf

My calculus-based physics course taught me so much about trigonometry, stuff I really should have learned in high school. Like, angle sum/differences, double-angle, writing each function in terms of all the others, etc. That calc-based physics course only allowed one 'granted' expression without proof or derivation: ΣF = ma. All other expressions had to be derived in proofs, whether on exams or labs. It had an over 85% failure rate for first attempts. Yes I failed, but man what an awesome math boot-camp that course was! :DDD

>>43717 this paper essentially just repeats the same fallacious argument. >Since gna(x) and gnb(y) are formulas of a formal system, the actual format by which they are represented is immaterial; that is, the value of the function is the same for any given value of the free variable, regardless of whatever format the formula is represented in. x and y are integers. If you choose a different "format" (goedel numbering) of the input, then of course the resulting value will change, because the resulting input will change. In fact, I believe gn_a(x) = x suffices totally for his purposes. He doesn't actually describe what he means by "contains the information of GN_A[X]" though. Does he mean that, from gn_a, you should be able to reconstruct GN_A? Where do people rely on this? He claims this fact is implicit in Boolos's proof (https://topologicalmedialab.net/xinwei/classes/readings/Go%CC%88del/Boolos%20proof%20Godel%20Incompleteness%201989.pdf), but I see nowhere the need for a representation of GN_A as a formula in the formal system. All that is needed is a one-to-one goedel numbering and a way to rephrase expressions and an algorithms acting on expressions into this goedel numbering. For example, you could use the ASCII character codes and an interpreter coded using arithmetic expressions on those ASCII character codes. Nowhere do you need the ability to recover the original expressions without knowing the goedel numbering itself, which is implicitly fixed a priori anyway. >On the one hand, the numerical value of the formula gna(x), for any given value of x, must be independent of the actual symbols used for its own representation. Indeed, x is essentially encoding an ordered list of integers. If we agree on how those integers correspond to symbols, we're golden. >On the other hand, since the assumption is that it can refer unambiguously to a formula of its own formal system, the numerical value of the formula gna(x) for any given value of x must be dependent on whatever actual representation is chosen for the formal system Indeed, again, we have to agree how numbers correspond to symbols for anything to proceed. If you pick a different way to assign numbers to symbols, you have picked a different goedel numbering. >so that for any given value of an x that corresponds to some X, when the formula gna(x) is represented in representation (i) the formula gna(x) has one specific value, but when it is represented in representation (ii) it has a different value for precisely the same value of its variable x. These "representations" amount to assigning a different index -> alphabet mapping. Of course you get different strings, because you've decided to use a different goedel numbering. Here's the relevant paragraph explaining what he means by "representations", and I assume also "format": >Given a particular set of symbols, there are several distinct fundamental representations of any formal system that uses those symbols. Given a specific formal system, there can be one representation which we refer to as (i), and which includes the symbols 0, S, ), (, ¬, ∨, ∀ (as above). There can be another representation, which we refer to as (ii), and which is identical except that the positions of the symbols 0 and S are reversed. That is, wherever a 0 occurs in representation (i), a S appears in that position in representation (ii), and wherever a S occurs in representation (i), a 0 appears in that position in representation (ii). This means that for any sequence X that is a formula of the formal system in representation (i), there is a sequence in representation (ii), which we denote as X′, that represents the same formula. >Clearly the function GNA[X] will not have the same value as GNA[X′]. Note also that GN_B and gn_b don't actually appear in the proof. They could be removed completely without changing the argument. Another sign of crankery: overcomplicated proofs, poorly described variables, mixing up notation.

Hello lainons I am 19, and since I finished my first year of high school I haven't interacted with mathematics at all. At the time, I was completely uninterested in mathematics, and frankly I don't remember anything. The average 12 years old probably knows more about mathematics than me. With time, I realized mathematics are extremely important. My illiteracy in this field becomes each day more and more of a shame, which has prompted me to "get into it". I have numerous things to study, and adding something to the list is almost destined to fail. However, in a brief moment of motivation, I looked up where to start, and Serge Lang's book Basic Mathematics came up. Do you think it's appropriate for someone who can't differentiate algebra from calculus (although in my country, they aren't taught separately nor named like that) ? Please feel free to recommend me other works to start with, just no pop science soykaf. Thank you for your insights.