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I am currently in High School, possibly interested in pursuing a career in Mathematics or the Philosophy/Logic behind Mathematics. I don't have a lot of money for books or videos about Higher Mathematics, but I have been using Wikipedia to learn Maths.

I don't understand why nothing can be as concise and beautiful as Wikipedia. When we learned about functions in school, my teacher took an entire class just to explain them to us. He kept trying to use visuals and all these different tools to explain what a function is, and then, mildly confused, I looked up "function" on Wikipedia, and here is the definition I get:

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

I would much rather use Wikipedia to learn Mathematics, because of how concise it is. Sometimes I don't want to know every single thing about a function, I just want a sufficient definition.

Is there any encyclopedia or book that will have definitions and teach you Mathematics in this way? I'm not trying to sound egotistic. I am just a very fast learner and I would much rather blaze through a paragraph that explains things very concisely than spend hours learning the same material in different ways. Is Wikipedia good for learning Mathematics, or is there an alternative way to concisely learn what I want to know?

user340812
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    No, wikipedia is not good for learn mathematics. Wikipedia is a compendium of facts and theorems and sometimes some proofs. The information is minimized and compressed, it is not good for learn something but it is good for searching information. –  May 19 '16 at 17:55
  • Wikipedia is fine for some things. I think the critical thing is to think hard about what you are learning. The key to that is doing plenty of problems, the harder the better. – almagest May 19 '16 at 17:57
  • Most math textbooks past calculus and "discrete mathematics" take this approach to definitions. A lot of people who are not so mathematically inclined find such definitions baffling, so teachers often need to give motivation. (I remember my frustration was in the supposedly "advanced" intro physics class in college, where we spent a class and a half talking about how to add vectors.) – Thomas Andrews May 19 '16 at 17:58
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    Possible duplicate of Are mathematical articles on Wikipedia reliable? – Wojowu May 19 '16 at 17:58
  • The information in Wikipedia is mostly reliable. It is not a good learning tool, always - the same way that a dictionary should not be the primary way you learn a language. – Thomas Andrews May 19 '16 at 17:59
  • everyone on this forum use mostly internet for studying maths (and science in general) : wikipedia + pdf + online courses of teachers + forums + ebooks, of course it is the best and most reliable source of knowledge of all time. – reuns May 19 '16 at 18:02
  • user1952009 Can you direct me to some of these resources you are speaking of? – user340812 May 19 '16 at 18:03
  • Thomas Andrews This is what I am talking about! If I were to teach how to add vectors, It would literally require 2 minutes. I would just say the sum of two vectors $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $(x_{1} + x_{2}, y_{1} + y_{2})$ . It's literally that simple, and I feel like teachers really overcomplicate it when they use 50 different examples! – user340812 May 19 '16 at 18:04
  • see what I obtain after googling 'derivative + integral + calculus + pdf' : 50 different courses on the subject, all of different levels, with different explanations. – reuns May 19 '16 at 18:06
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    The accuracy of a wikipedia article on math is inversely proportional to its level. I'd be leery of the introductory articles on basic calculus and so on; but several of the K-theory pages,for example, were written by Andrew Ranicki himself. – anomaly May 19 '16 at 18:07
  • @anomaly: Why "inversely", then? – hmakholm left over Monica May 19 '16 at 18:30
  • @HenningMakholm: Because I was writing quickly and apparently not paying much attention. :) – anomaly May 19 '16 at 18:35
  • I agree with Masacroso and anomaly. Wikipedia is a rather good place to start searching for sources, because of the citation policy. But it is a very bad place to start learning mathematics, because of the hyperlink rabbit hole, and because proofs are like second-class citizens in that realm. One also has to be aware that Wikipedia is full of errors, especially in low-level articles (for mathematics say below graduate level) because they can be easily edited by people who succumb to the Dunning-Kruger effect. High-level articles generally escape this because hardly anyone reads them. – user21820 Mar 28 '18 at 03:27
  • But your real question is about learning mathematics well, not about Wikipedia. Since you're at high-school level, you may be not aware that real mathematics is all about theorems and their proofs, about abstract structures, and cannot at all be learned in the same way as memorizing facts from an encyclopedia. To get an idea of what it is really like, I recommend Spivak's Calculus as an excellent rigorous textbook that will give you a glimpse of a mathematical field called real analysis. Also try "How to Prove It" by Daniel Velleman. – user21820 Mar 28 '18 at 03:32

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Wikipedia is a good source for reminding yourself of definitions, and is generally accurate. However, learning math from Wikipedia is probably not a very effective strategy because it does not have a "path" for you to take and it is an extremely terse regurgitation of facts.

I would recommend the book "A Transition to Advanced Mathematics". Its pdf is freely available online and it provides a great introduction to proofs and basic set theory as well as some of the other foundational concepts in more advanced math.

Ken Duna
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  • I'm only familiar with everything up to about an Intermediate Algebra class. I do have a book around the house called "What is Mathematics? Ideas and Methods". Do you think I am ready to transition to Advanced Mathematics right now? – user340812 May 19 '16 at 18:02
  • I think you should be ok. The book is relatively self-contained (it defines pretty much every piece of terminology it uses), and any algebra you would need is extremely basic stuff like how to FOIL, etc... – Ken Duna May 19 '16 at 19:53
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The others explained quite well how to use wikipedia. Good for you if you can get a good understanding of concepts only from definitions, it means you have good visualisation abilities. High school lessons are indeed full of drawings and examples (in some countries more than others), but the more advanced your classes are, the less explanations you will have. There will always be a point, though, where definitions will not be enough.

PS : I think you would be very interested in metamathematics (look up a little first order logic on wikipedia).

Vincent
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