Reddit mentions: The best mathematical logic books

We found 329 Reddit comments discussing the best mathematical logic books. We ran sentiment analysis on each of these comments to determine how redditors feel about different products. We found 107 products and ranked them based on the amount of positive reactions they received. Here are the top 20.

2. Learning to Reason: An Introduction to Logic, Sets, and Relations

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3. The Haskell Road to Logic, Maths and Programming. Second Edition (Texts in Computing)

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4. Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)

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7. Computability and Logic Fifth Edition

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8. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

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9. Foundations and Fundamental Concepts of Mathematics (Dover Books on Mathematics)

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10. An Introduction to Formal Logic

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11. Set Theory

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12. Mathematical Logic (Oxford Texts in Logic)

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13. Analysis With An Introduction to Proof, 5th Edition

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14. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (Source Books in History of Sciences)

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16. Model Theory: Third Edition (Dover Books on Mathematics)

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17. A Beginner's Guide to Mathematical Logic (Dover Books on Mathematics)

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18. Logic

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19. An Invitation to Applied Category Theory: Seven Sketches in Compositionality

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20. Elements of Set Theory

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🎓 Reddit experts on mathematical logic books

The comments and opinions expressed on this page are written exclusively by redditors. To provide you with the most relevant data, we sourced opinions from the most knowledgeable Reddit users based the total number of upvotes and downvotes received across comments on subreddits where mathematical logic books are discussed. For your reference and for the sake of transparency, here are the specialists whose opinions mattered the most in our ranking.
Total score: 36
Number of comments: 4
Relevant subreddits: 1
Total score: 34
Number of comments: 5
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Total score: 33
Number of comments: 5
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Total score: 16
Number of comments: 4
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Number of comments: 5
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Number of comments: 4
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Total score: 7
Number of comments: 4
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Number of comments: 4
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Total score: 6
Number of comments: 3
Relevant subreddits: 1
Total score: 5
Number of comments: 3
Relevant subreddits: 1

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Top Reddit comments about Mathematical Logic:

u/acetv · 14 pointsr/math

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

  • The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.

  • Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.

  • Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.

  • An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.

  • Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.

  • Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.

  • A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.

  • Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.

  • Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.

  • I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.

  • Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.

    Basically, don't limit yourself to the track you see before you. Explore and enjoy.
u/blaackholespace · 18 pointsr/math

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

u/pron98 · 1 pointr/programming

There's a 1966 book by Jean van Heijenoort that has many of the original talks and writings pertaining to the Hilbert/Brouwer debate. I think that the most interesting (philosophically) is a short note by Hermann Weyl (Hilbert's student who defected to intuitionism and then recanted), in 1927, that explains why both Brouwer and Hilbert are right.

Before posting Weyl's remarks, I'll quote some pertinent bits from Brouwer and Hilbert. As we'll see, Weyl said that all mathematicians were intuitionists, or thought they were, but it was Brouwer who discovered just how much of math was untenable from the intuitionistic point of view. He basically said that much of math was wrong:

Brouwer 1923:

> An incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is nonetheless incorrect, just as a criminal policy is nonetheless criminal even if it cannot be inhibited by any court that would curb it. … In view of the fact that the foundations of the logical theory of functions are indefensible according to what was said above, we need no be surprised that a large part of its results becomes untenable in the light of more precise critique.

It was Hilbert (the finitist!) who, according to Weyl had to make a radical philosophical jump in order to salvage mathematics, and he who had to defend his position. In 1927, in a talk where he personally lambasted Brouwer (and expressed surprise that he has a following), he explained that math contains "real propositions" with actual content as well as "ideal propositions". Hilbert first claims that his position is defensible in the tradition of math, but says it has two concrete advantages: it can save analysis, and its formal proofs are aesthetically more appealing as they're shorter, more elegant, and distill the essence of the idea of the proof.

> [E]ven elementary mathematics contains , first, formulas to which correspond contextual communications of finitely propositions (mainly numerical equations or inequalities, or more complex communications composed of these) and which we may call the real propositions of the theory, and, second, formulas that — just like the numerals of contextual number theory — in themselves mean nothing but are merely things that are governed by our rules and must be regarded as the ideal objects of the theory.

> These considerations show that, to arrive at the conception of formulas as ideal propositions, we need only pursue in a natural and consistent way the line of development that mathematical practice has already followed till now.

> ... [W]e cannot relinquish the use of either the principle of excluded middle or of any other law of Aristotelian logic expressed in our axioms, since the construction of analysis is impossible without them.

> ... [A] formalized proof, like a numeral, is a concrete and survivable object. It can be communicated from beginning to end.

> ... What, now, is the real state of affairs with respect to the reproach the mathematics would degenerate into a game?

> The source of pure existence theorems is the logical ε-axiom, upon which in turn the construction of ideal propositions depend. And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear. To make it a universal requirement that each individual formula then be interpretable by itself is by no means reasonable; on the contrary, a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument. What the physicist demands precisely of a theory is that particular propositions be derived from laws of nature or hypotheses solely by inferences, hence on the basis of a pure formula game, without extraneous considerations being adduced. Only certain combinations and consequences of physical laws can be checked by experiment — just as in my proof theory only the real propositions are directly capable of verification. The value of pure existence proofs consists precisely in the individual construction is eliminated by them and that many different construction are subsumed under one fundamental idea, so that only what is essential to the proof stands out clearly; brevity and economy of thought are the rasion d’être of existence proofs.

> … The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds.

> ... … Existence proofs carried out with the help of the principle of excluded middle usually are especially attractive because of their surprising brevity and elegance.

He also makes this remark, which turned out to be unfortunate in light of Gödel:

> From my presentation you will recognize that it is the consistency proof that determines the effective scope of my proof theory and in general constitutes its core.

Which brings us to Weyl in 1927 (in the comment below)

u/fgtrytgbfc · 11 pointsr/Thetruthishere

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

  1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,

u/jpredmann · 1 pointr/math

This is just my perspective, but . . .

I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.

​

Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.

​

I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:

​

How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics

Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.

Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.

​

As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.

​

Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:

​

A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )

​

Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).

​

If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.

u/angrycommie · 1 pointr/DebateaCommunist

Please read this and educate yourself. You are making yourself look like a fool.

Thinking Your Way to Freedom: http://www.temple.edu/tempress/titles/1982_reg_print.html This is an excellent textbook, and I studied under her.

The Logic Book: http://www.amazon.com/The-Logic-Book-Merrie-Bergmann/dp/007353563X

Perhaps after studying and educating yourself, you may see how your statements are in error. I would advise you stop throwing around "logical fallacies" everywhere. It makes you look like a fool, it is elitist and highly cringe-worthy. I used to be like you, but I acquired an education in philosophy (Kant is my main area of research, as is the philosophy of mind), and realized how foolish I had been. You're the embodiment of /r/atheism, grasping at the straws by (wrongly) insinuating the other person has committed logical fallacies by naming fancy ones- is at best cringe worthy. You are like a child who has been proven wrong by a professor who cannot accept his defeat, so he resorts to nit picking fictitious fallacies. It's funny how you must think you are a superb armchair internet intellectual, looking about a list of logical fallacies to use as your main driving point in arguments. I too, felt euphoric whenever I thought someone else committed a logical fallacy by going "ah ha! What you've just stated is a FALLACY! What now?" but I was young and foolish. This is epistemic poverty. I urge you to at least skim through the books I have suggested or any other logic book. They helped me immensely during my undergrad and grad years. I would recommend developing a strong background in logic and epistemology (mainly the JTB account and the Gettier Response to it). After having done that and you no longer want to grasp at the straws and act like a child, I will be more than willing to help.

u/let_me_count_the_way · 4 pointsr/HomeworkHelp

What this expressions says

First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.

This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)

What does ∀x∀yP(x,y) mean?

This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.

What does ∀xP(x,x) mean?

This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.

So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?

Proof by contradiction

We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.

So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)

not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.

So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.

not P(a,a)

Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.

Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)

This happens in two stages:

First we instantiate y
∀xP(x,a)

Then we instantiate x
P(a,a)

The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:

∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.

Hopefully that makes sense.

Recommended Resources

Wilfred Hodges - Logic

Peter Smith - An Introduction to Formal Logic

Chiswell and Hodges - Mathematical Logic

Velleman - How to Prove It

Solow - How to Read and Do Proofs

Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics

u/lamson12 · 2 pointsr/math

Here is an actual blog post that conveys the width of the text box better. Here is a Tufte-inspired LaTeX package that is nice for writing papers and displaying side-notes; it is not necessary for now but will be useful later on. To use it, create a tex file and type the following:

\documentclass{article}
\usepackage{tufte-latex}

\begin{document}
blah blah blah
\end{document}

But don't worry about it too much; for now, just look at the Sample handout to get a sense for what good design looks like.

I mention AoPS because they have good problem-solving books and will deepen your understanding of the material, plus there is an emphasis on proof-writing when solving USA(J)MO and harder problems. Their community and resources tabs have many useful things, including a LaTeX tutorial.

Free intro to proofs books/course notes are a google search away and videos on youtube/etc too. You can also get a free library membership as a community member at a nearby university to check out books. Consider Aluffi's notes, Chartrand, Smith et al, etc.

You can also look into Analysis with intro to proof, a student-friendly approach to abstract algebra, an illustrated theory of numbers, visual group theory, and visual complex analysis to get some motivation. It is difficult to learn math on your own, but it is fulfilling once you get it. Read a proof, try to break it down into your own words, then connect it with what you already know.

Feel free to PM me v2 of your proof :)

u/lindset · 3 pointsr/math

During my sophomore year I took an "intro to proofs" course (known formally at the institution as Foundations of Advanced Mathematics) and I found it to be extremely beneficial in my development as a mathematician. We used Chartrand's "Mathematical Proofs" textbook (here's the link for those who are interested).

The text covered set theory, logic, the various proof methods, and then dug into stuff like elementary number theory, equivalence relations, functions, cardinality (culminating in Cantor's two main results), abstract algebra, and analysis. Obviously the book only scratched the surface on a lot of these topics, but I felt it accomplished its goal.

Part of my satisfaction with the course is likely due to the fact that we had a brilliant professor who taught the course in the spirit of what u/Rtalbert235 spoke of. He was able to clearly articulate the distinction between computation and theory. The way I like to say it is that he taught us the difference between pounding a bunch of nails into a 2X4 (computation) and building a house (proving theorems).

I don't mean to universally praise "intro to proofs" courses, however. I can definitely see how they can be horrible wastes of time if not done properly, and I can also appreciate the idea of "throwing" students into proof-based courses (analysis, algebra, and so on). For me though, I think it's worth the effort to try and optimize these sorts of classes, which will ultimately serve a LOT of math students who need to understand proofs, but don't necessarily have a desire to pursue the subject beyond the undergraduate level.

tl;dr - Given the right combination of textbook and professor, an "intro to proofs" course can be just what the doctor ordered for developing mathematicians.

u/phlummox · 1 pointr/learnmath

Oh, I'm terrible at calculus, haha. I teach discrete maths and logic, and never have to touch calculus at all, thank goodness :)

But a younger friend of mine is doing calculus just now, so I'll find out what he found useful and PM you. He did say that some of the books I'd recommended him were immensely useful for maths generally (not necessarily calculus in particular). In roughly ascending order of difficulty:


u/GeneralAydin · 10 pointsr/learnmath

There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.

To illustrate my point:

Linear Algebra:

  1. Here's Linear Algebra described in the sequence above: I'll just leave it blank because I hate pointing fingers.

  2. Here's a more serious intro to Linear Algebra:

    Linear Algebra Through Geometry by Banchoff and Wermer

    3. Here's more rigorous/abstract Linear Algebra for undergrads:

    Linear Algebra Done Right by Axler

    4. Here's more advanced grad level Linear Algebra:

    Advanced Linear Algebra by Steven Roman

    -----------------------------------------------------------

    Calculus:

  3. Here's non-serious Calculus described in the sequence above: I won't name names, but I assume a lot of people are familiar with these expensive door-stops from their freshman year.

  4. Here's an intro to proper, rigorous Calculus:

    Calulus by Spivak

    3. Full-blown undergrad level Analysis(proof-based):

    Analysis by Rudin

    4. More advanced Calculus for advance undergrads and grad students:

    Advanced Calculus by Sternberg and Loomis

    The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.

    Here's how you start studying real math NOW:

    Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into

    Discrete Math by Susanna Epp

    How To prove It by Velleman

    Intro To Category Theory by Lawvere and Schnauel

    There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.

    If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:

    Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.

    I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.

    Good Luck, buddyroo.
u/rrsmitto · 2 pointsr/learnmath

The only previous knowledge I really used when I took intro to proofs were some factoring methods that were helpful with proofs by induction, although they weren't necessary. That said, reviewing exponent/log laws, and certain methods of factoring couldn't hurt.

An intro to proofs course should be fairly self contained, meaning any necessary axioms and definitions should be covered in the course. Those examples that you gave are exactly the type of things that should be proven and not knowing them beforehand should be fine. The important thing is being able to understand and reproduce the proofs on your own, and with a bit of experience you will be able to intuitively reason whether a statement is true or false. This intuitive reasoning will also become much more important than memorizing later in the course when you come across statements you've never seen before that aren't immediately obvious.

I would recommend getting very comfortable with logic and basic set theory. I also highly recommend this book if you want some extra reading material (pdf). It's still one of my favorite math books. Hope that helps.

u/myfootinyourmouth · 1 pointr/math

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

u/c3261d3b8d1565dda639 · 2 pointsr/books

If you want a strong mathematical approach, check out Peter Smith's Teach Yourself Logic Guide. If you don't want to take as heavy of an approach, you can use the suggestions as a roadmap and pick-and-choose from the suggestions. Even the introductory logic book suggestions in that guide might be too math heavy, but you might at least read their reviews on Amazon. A lot of reviewers tend to link to books on either side: easier and harder approaches.

For what it's worth, while I was in University we used Computability and Logic in the second logic course, which is after the introductory course teaching basic propositional and predicate logic. It's not a book for learning logic, but it's an awesome book for tying together a lot of what you initially learn with computability, model and proof theory. In another course we used An Introduction to Non-Classical Logic. I really enjoyed both of these books, and they're relatively cheap, but as I said they are not introductory logic books.

I'll be happy to reply again if you have any further questions.

u/mathematicity · 6 pointsr/math

You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

Some user friendly books on Real Analysis:

  1. Understanding Analysis by Steve Abbot

  2. Yet Another Introduction to Analysis by Victor Bryant

  3. Elementary Analysis: The Theory of Calculus by Kenneth Ross

  4. Real Mathematical Analysis by Charles Pugh

  5. A Primer of Real Functions by Ralph Boas

  6. A Radical Approach to Real Analysis by David Bressoud

  7. The Way of Analysis by Robert Strichartz

  8. Foundations of Analysis by Edmund Landau

  9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi

  10. Calculus by Spivak

  11. Real Analysis: A Constructive Approach by Mark Bridger

  12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg

  13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.

    Some user friendly books on Linear/Abstract Algebra:

  14. A Book of Abstract Algebra by Charles Pinter

  15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer

  16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus

  17. Linear Algebra Done Wrong by Sergei Treil-FREE

  18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell

    Topology(even high school students can manage the first two titles):

  19. Intuitive Topology by V.V. Prasolov

  20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler

  21. Topology Without Tears by Sydney Morris- FREE

  22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov

    Some transitional books:

  23. Tools of the Trade by Paul Sally

  24. A Concise Introduction to Pure Mathematics by Martin Liebeck

  25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston

  26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith

  27. Elements of Logic via Numbers and Sets by D.L Johnson

    Plus many more- just scour your local library and the internet.

    Good Luck, Dude/Dudette.
u/rich1126 · 3 pointsr/learnmath

Any mathematical subject can be learned either as something applied, or something pure. That being said, if you're interested in pure math the main thing is learning proofs. That's the foundation of all higher level mathematics. If you can't fluently read (and eventually write) proofs, you aren't learning mathematics.

So, I'd suggest starting there. There are many books that are useful. In college, I used Lay's Analysis With an Introduction to Proof in a course, and found it very useful. Another thing I seen thrown around is the Book of Proof (pdf link there). I've never used the book personally, but it might give you a good place to start.

Then it's just a matter of going through various subjects. Discrete math (combinatorics, graph theory) is extremely accessible, and a pretty popular topic to begin learning proof with. You could also learn some abstract algebra (starting with group theory) as a more typical "standard" subject learned by math undergraduates. But really, if you want to learn math for its own sake, just find some books online and see what sticks. You have that freedom.

u/LADataJunkie · 3 pointsr/ucla

You will want to jump on 115A, but have a back up class in case you need to drop and realize it isn't going to work. I dropped 115A twice before I could finally commit and feel mature enough to do well in it.

One thing that really helped me was taking Combinatorics, a field that is fascinating to me. There was *some* proof writing in the class, but it was pretty basic (similar to proofs in statistics). I enjoyed writing those proofs and taught me the entire purpose of doing it. I was then able to do 115A with little difficulty.

I also got the following book, which is excellent (I used a much older edition) How to Read and Do Proofs by Daniel Solow.

https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024/ref=pd_sbs_14_1?_encoding=UTF8&pd_rd_i=1118164024&pd_rd_r=DRKKPQHM9KM7NF7XS7AJ&pd_rd_w=jAI7z&pd_rd_wg=TfejV&psc=1&refRID=DRKKPQHM9KM7NF7XS7AJ&dpID=51ljxm2YBEL&preST=_SY344_BO1,204,203,200_QL70_&dpSrc=detail

u/univalence · 1 pointr/learnmath

Yes, it's a very big guide. It's also intended for academics (mostly philosophers) with no training in logic who want to have a solid grasp of the topic for research work, so you definitely don't need to go through the whole thing; if you do, you'll probably be better equipped than most PhD students in logic!

I think sticking to Epstein, or using Peter Smith's introduction (Or Paul Teller's book that Smith recommends and is free) is reasonable for a high-school course; If you go with Smith, you may want to cover a chapter or two from a more advanced book (for example, on the incompleteness theorems, the basics of computability, or the basics of model theory) if you move quickly.

Epstein's and Teller cover more ground, and Epstein presents a more "traditional" syllabus for mathematical logic. On the other hand, if you're interested in either philosophy or computer science (or any of the more exotic modern developments in logic), Smith's (And Teller's) presentation might translate more directly into the ideas you'll see there: the traditional syllabus on mathematical logic tends to better prepare students for model theory and set theory, which have less relevance outside of pure math.

Anyway, best of luck!

u/redditdsp · 12 pointsr/math

It's a fair point; applied category theory is really in its infancy. For a long time, it was considered pretty inaccessible and obscure. I think that's starting to change, e.g. with some new pedagogically oriented books (Cheng, Fong-Spivak), new international conferences, new journal, etc. But it might take time.


The most successful application so far is certainly Haskell, OCAML, and other similar functional programming languages. These were built entirely on category-theoretic principles, and have become quite popular (Haskell is used at AT&T, Amgen, Apple, Bank of America, Facebook, Google, Verizon, etc.).


There are control theory researchers such as Paulo Tabuada, robotics researchers such as Aaron Ames and Andrea Censi, and others who have explicitly used category theory in their work. For-profit companies such as Kestrel, Statebox, R-Chain, Conexus, etc. all use category theory more or less explicitly.


Whether or not electrical engineers—or others of that sort—will use CT depends on whether there are enough interested parties who can drive it more deeply into that domain. So far, the work has been at a very surface level because category theorists have to "go to them" instead of them "coming to us". As category theorists, we don't know enough about the depths of these fields to make a direct and immediate impact without preparing the ground. It takes time and effort, and we need more people on the case.


But if we continue—and I think we will—my guess is that in the future, people will use category theory to learn lots of different fields and connect their knowledge from one to another. A major value proposition of category theory is its ability to transfer information and problem specification from one field to another. I think that will eventually be broadly useful.

u/utmostoftopmost · 1 pointr/math

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!

u/softservepoobutt · 2 pointsr/TrueAskReddit

Honestly - through rigor. I would suggest studying logic, some philosophy (this is about the structure of arguments, and deduction in a general sense) and then something applied, like policy analysis or program evaluation. <- those last two are just related to my field so I know about them, plenty of others around.

Some suggested books that could be interesting for you:

Intro to Logic by Tarski

The Practice of Philosophy by Rosenberg

Thank you for Arguing by Heinrichs

Policy Analysis is instructive in that you have to define a problem, define its characteristics, identify the situation it exists in, plot possible solutions (alternatives), and create criteria for selecting the alternative you like most.

Program Evaluation is really just tons of fun and will teach a bunch about how to appraise things. Eval can get pretty muddy into social research but honestly you can skip a lot of that and just learn the principles.

The key to this is that you're either very smart and can learn this stuff through your own brains and force of will, or, more likely, you'll need people to help beat it into you WELCOME TO GRADSCHOOL.

u/Cezoone · 3 pointsr/learnmath

One thing I like to remind people, is that Linear Algebra is really cool and though it tends to come "after" calculus for some reason, it really has no explicit calc prerequisite.

I highly recommend Dr. Gilbert Strang's lectures on it, available on youtube and ocw.mit.edu (which has problems, solutions, etc, also)

I think it's a great topic for right around late HS, early college. And he stresses intuition and imo has the right balance of application and theory.

I'd also say that contrary to most peoples' perceptions, a student's understanding of a math topic will vary greatly depending on the teacher. And some teachers will be better for some students, others for others. That's just my opinion, but I firmly believe it. So if you find yourself struggling with a topic, find another teacher/resource and perhaps it will be more clear. Of course this shouldn't diminish the effort needed on your part, learning math isn't a passive activity, one really has to do problems and work with the material.

And finally, proofs are of course the backbone of mathematics. Here is an intro text I like on that.

Oh okay, one more thing, physics is a great companion to math. I highly recommend "Classical Mechanics" by Taylor, in that regard. It will be challenging right now, but it will provide some great accompaniment to what you'll learn in upcoming years.

u/functor7 · 7 pointsr/math

There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)

Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!

The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!

u/jhelpert · 1 pointr/learnmath

I graduated w/ degree in Math n' Physics but have been doing programming for startup for last 5+ years so many of my math skills got rusty.

While trying to get back into it went through several books and have found this to be the best if you're interested in more advanced mathematics: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094. It's not only been an excellent review but has fleshed out some areas I was weak (in higher level courses like complex analysis, topology, group theory the methodology of proofs was assumed and often not taught).

The explanations are solid, varied, and they go through each proof they present (often w/ exhaustive step-by-step details).

From there pick a domain you're interested in and pickup the relevant undergraduate (and maybe some graduate) level books/textbooks and see if you can pick it up.

u/willardthor · 2 pointsr/compsci

(Note: I wrote this elsewhere)

Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:

u/NeverACliche · 2 pointsr/math

>My first goal is to understand the beauty that is calculus.

There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.

There are some great intros for RA:

Numbers and Functions: Steps to Analysis by Burn

A First Course in Mathematical Analysis by Brannan

Inside Calculus by Exner

Mathematical Analysis and Proof by Stirling

Yet Another Introduction to Analysis by Bryant

Mathematical Analysis: A Straightforward Approach by Binmore

Introduction to Calculus and Classical Analysis by Hijab

Analysis I by Tao

Real Analysis: A Constructive Approach by Bridger

Understanding Analysis by Abbot.

Seriously, there are just too many more of these great intros

But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:

Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers

Discrete Mathematics with Applications by Epp

Mathematics: A Discrete Introduction by Scheinerman

u/schrodins · 3 pointsr/MachineLearning

What is the "Highest Level" of mathematics you have taken?
Math is substantially more like a foreign language than popular culture would lead you to believe. It takes practice and what I like to call 'settle time.'
If you feel like you have a strong grasp on the concepts of algebra I highly recommend starting from 'scratch' (first principals) and getting a book like http://www.amazon.com/gp/offer-listing/0321390539/ref=sr_1_2_twi_har_1_olp?ie=UTF8&qid=1450533541&sr=8-2&keywords=mathematical+proofs

It was the first textbook that made me really start to understand what is needed to think like a mathematician. Start at the beginning work though problems, set theory is so much more important than most people realize. It will be cloudy and frustrating but really try to work some problems, put it down for a week let it stew and come back to the problems you had trouble with. Do that over and over.
While you are doing that pick up any elementary Stats/Prob and/or Linear Algebra book and start flipping through from the beginning you will see all the tools you are learning in Mathematical Proofs in those books as well. Try to take what you are learning and see it applied in those books to add some extra hooks to attach things to in your brain.

For Numerical Analysis you are going to want to build a strong base in proofs, linear algebra, set theory, and calculus as you go forward. Don't let this stop you from starting to read up it is a great way to stay excited when you are learning things to know fun ways that they are applied but don't get discouraged. My Numerical Analysis class was a Sr level college course that started the semester with 24 Math and CS majors about half gave up before the mid term/

u/8975629345 · 3 pointsr/math

Hey mathit.

I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!

Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.

I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.

Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.

Do you agree with my choices? What else do you recommend?

I found online courses to be ineffective, I prefer books.

What's your opinion, mathit?

Cheers and many thanks in advance!

u/[deleted] · 4 pointsr/math

Add Coq Art: Interactive-Theorem-Proving-Program-Development in this venue. The book in the parent link is new to me but I like what I've seen so far. The Coq Art book is more about learning to use the Gallina langauge and techniques for writing and using tactics. It has a pragmatic feel, which is weird considering its subject is so close to pure theory. In all it makes a fun "lab" workbook for the autodidact and I am still working through it as I find time.

I also liked "Logic and Structure" by Dirk van Dalen because it gets on with symbols quickly but still has enough explaination for a beginner like me to follow.

I usually hate wordy logic books but I am enjoying Tarski's Introduction to Logic because his prose descriptions and explainations thoughtfully explain why instead of how. I have a few high-school/undergrad books that explain the how in boring and long winded prose. Barf. Tarski's Intro to Logic is also expanding my vocabulary. I wish I would have read this one first, followed by Dirk van Valen's, and finally the top one on constructive logic.

u/speakwithaccent · 2 pointsr/math

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

u/meshuggggga · 2 pointsr/math

For discrete math I like Discrete Mathematics with Applications by Suzanna Epp.

It's my opinion, but Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers is much better structured and more in depth than How To Prove It by Velleman. If you follow everything she says, proofs will jump out at you. It's all around great intro to proofs, sets, relations.

Also, knowing some Linear Algebra is great for Multivariate Calculus.

u/ase1590 · 1 pointr/AskComputerScience

Bitcoin mining programs are pretty advanced pieces of software, especially due to the rise of ASICs and other esoteric hardware that must be supported. You might want to consider an easier project for the time being.

Fundamentally, you need to have a good grasp on data structures before you can really get going with Merkle trees and cryptocurrency mining in general.

Here are some free resources to get started

If you want to consider grabbing a book, Algorithms (4th ed) is generally regarded as a decent book to get started on thinking with and using algorithms and data structures.

edit: depending on the time you have, you may also want to consider reading this on set theory. It tends to make later algorithms a bit more clear as a lot of it draws from set theory.

u/TheElderQuizzard · 3 pointsr/philosophy

A book on Real Analysis or Discrete Mathematics would be good. You’ll be able to practice proving things with familiar topics like calculus and real numbers.

Very nice discrete book
Good intro to analysis of the real number line

From the preface of the first book
> Until this point in your education, mathematics has probably been presented as a primarily computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions about the real world. In this setting, your primary goal in using mathematics has been to compute answers.
But there is another side of mathematics that is more theoretical than computational. Here the primary goal is to understand mathematical structures, to prove mathematical statements, and even to invent or discover new mathematical theorems and theories. The mathematical techniques and procedures that you have learned and used up until now are founded on this theoretical side of mathematics. For example, in computing the area under a curve, you use the fundamental theorem of calculus. It is because this theorem is true that your answer is correct. However, in learning calculus you were probably far more concerned with how that theorem could be applied than in understanding why it is true. But how do we know it is true? How can we convince ourselves or others of its validity? Questions of this nature belong to the theoretical realm of mathematics. This book is an introduction to that realm.

u/467fb7c8e76cb885c289 · 2 pointsr/btc

> Never read it, will google them after this reply.

It's so fucking cool it's unreal. Not up to date with recent developments but wanna check it out again properly soon.

>Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.

That'd explain why it was so dense lol - I dived from no mathematical logic (apart from like basic predicate calculus) and using first order symbols sparingly.

>There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommendthat you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626
>
>Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.

Thanks :D I'll check it out. Given your breadth of knowledge on it I imagine your background is pure mathematics?

u/sgoldkin · 2 pointsr/logic

The best introductory logic text you will ever find: Logic: Techniques of Formal Reasoning, 2nd Edition Donald Kalish, Richard Montague.
This book is especially good if you have done any programming. The structure of main and sub-proofs corresponds to main program and subroutine calls. You can pick up a used copy for around $23 here: https://www.abebooks.com/book-search/author/kalish-montague-mar/ and you can see the table of contents here: https://www.powells.com/book/logic-techniques-of-formal-reasoning-9780195155044 (but, obviously, don't buy it for $133!)

For meta-theory, take a look at: Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560. This book explains things in a clear way using ordinary English, before setting out the proofs.
And, if you are interested in model theory, take a look at Model Theory by C.C. Chang and H. Jerome Keisler, https://www.amazon.com/Model-Theory-Third-Dover-Mathematics/dp/0486488217 and you should get a good idea of what additional mathematics you might want to pursue.

u/bitchymelodrama · 0 pointsr/math

Possible path:

Learn to think like mathematicians because you'll need it. For example, Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al is a good book for that. When you got the basics of math argumentation down, it's time for abstract algebra with emphasis on vector spaces(you really need good working knowledge of linear algebra). People like Axler's Linear Algebra Done Right. Maybe, study that. Or maybe work through Maclane's Algebra or Chapter 0 by Aluffi.

After that you want to get familiar with more or less rigorous calculus. One possibility is to study Spivak's Calculus, then pick up Munkres Analysis on Manifolds.

Up next: differential geometry which is your main goal. At this point your mathematical sophistication will have matured to the level of a grad student of math.

Good luck.

u/GOD_Over_Djinn · 1 pointr/math

I don't want to say that it's impossible for you to get through Spivak, but I think it will be frustrating. Spivak is, I think, most useful for someone who already knows a little bit about what calculus is about. You might be better suited going with a gentler introduction to calculus like Stewart (pdfs exist on torrent sites if you don't want to drop $200), or even a proofs book like Chartrand, Polimeni, Zhang.

u/ADefiniteDescription · 2 pointsr/math

> and i got the impression that may be i should also learn about the history, the context, the development and the goals of mathematical logic in order to appreciate it fully

I don't know that this is really necessary to be honest. It's certainly interesting, but isn't necessary to having a firm grasp on logic as done in mathematics departments.

That being said, here are some suggestions. You should start by consulting Peter Smith's Teach Yourself Logic, which is a huge document listing any and every source you'll ever need to learn logic.

As for history - that's a bit trickier. Frege's a tough place to start. He's a clear writer and the origin of analytic philosophy, but his Begriffsschrift notation is a pain in the ass to read (it doesn't resemble anything you will be familiar with). Also his logic is second-order, which isn't exactly standard (although it's more accepted nowadays).

If you want to start with Frege, I recommend reading his Begriffsschrift (Concept Script, a formal language of pure thought modelled upon that of arithmetic) and his Grundlagen (The Foundations of Arithmetic). The latter is much more important in my opinion. If you decide you want to get to the real nitty-gritty, you can consult the brand new, first ever full translation of the Grundgesetze, which includes a huge appendix which teaches you the notation.

For a broader perspective Kneale & Kneale's The Development of Logic is a great (secondary) source. If you want primary sources, van Heijenoort's From Frege to Gödel is the place to go.

> apparently mathematical logic is supposed to be a rigorous analysis of logic?

I wouldn't really characterise it this way. In my experience, most people split the study of logic into three main camps: mathematical logic, philosophy of logic and philosophical logic. The first camp is the proving of theorems in various logical systems, usually with an eye towards mathematical application. The last camp is the application of logic and logical tools to philosophical problems. And the philosophy of logic is the study of the nature of logic itself. That's a bit closer to the types of things you touch on at the end there.

Many, many people just study one of the branches I list, and you can too, without worrying about the philosophy.

u/autoditactics · 14 pointsr/math

Hartshorne's Geometry: Euclid and Beyond is a much more readable book compared to his other well-known work.

In addition to Needham, I've heard very good things about Remmert's Theory of Complex Functions for its use of history and Wegert's Visual Complex Functions for its visual approach to complex analysis, similar to but perhaps more rigorous than Needham. Kenji Ueno's three-volume A Mathematical Gift is similar in its intuitive explanations, but it covers various topics in mathematics as opposed to just complex analysis and can act as a nice introduction or as light reading (yes, he has another three-volume work on AG). I can also recommend Foundations and Fundamental Concepts of Mathematics by Howard Eves for its breezy overview of the foundations of mathematics, for anyone interested in that.

Edit: Links

There are also some nice books on calculus, such as Excursions in Calculus by Robert M. Young and New Horizons in Geometry by Mamikon A. Mnatsakanian and Tom M. Apostol (of Calculus and Analytic Number theory fame).

u/CopOnTheRun · 1 pointr/learnmath

I was researching this topic a while ago and, Eves' Foundations and Fundamental Concepts of Mathematics came up as a popular choice. I bought the book, but I can't say I've ever gotten around to reading it so maybe someone else can vouch for it.

On a related note, now that you've reminded me of the book I'll definitely have to read it over break. Thanks stranger =)

u/SnailHunter · 3 pointsr/learnmath

Mathematical Proofs: A Transition to Advanced Mathematics was the book for my college proofs class. I found it to be a good resource and easy to follow. It covers some introductory set theory as well. Just be prepared to work through the proof exercises if you really want a good intuition on the topic.

u/rdar1999 · 3 pointsr/btc

I'm glad prof. faux decided to randomly cite an elementary introduction to logic such as hunter (without citing any particular page).

Don't get me wrong, it is a pretty decent introduction afaict, especially for undergrad students, but I'd be really delighted if he could mention what he meant in something a bit more used, if not slightly more rigorous, like boolos: https://www.amazon.com/Computability-Logic-Fifth-George-Boolos/dp/0521701465

And, of course, I'd feel real joy if he could cite a particular page to back up what he meant.

u/Dr_Frank_Baby · 2 pointsr/math

I've found Alfred Tarski's Introduction to Logic: and to the Methodology of Deductive Sciences to be a great primer on sentential (predicate) logic. Tarski was a good jumping off point from mathematics to mathematical logic and analytical philosophy. Discovering this book was a turning point in my life; it galvanized my interest in mathematics and lead me to study the foundations of mathematics and philosophy of language in my free time.

u/yggdrasilly · 3 pointsr/learnmath

Two great introductions are:

u/boterkoeken · 8 pointsr/logic

For basic logic (first-order, classical) these are excellent textbooks...

u/Klaark15 · 3 pointsr/logic

Hey there.

You mention that your brother is bright -- how bright exactly? First of all, Computability and Logic is quite an advanced book that is typically aimed towards 2nd year logic students, and is usually for students who have taken a rigorous discrete mathematics course in their first year.

It delves quite deeply into the theory of logic and the philosophy of mathematics and would not be suited as a light exercise book for someone unless they have taken a math-heavy first-year logic course and are planning on taking up electrical engineering or something of the sort.

As for Hurley's book, a Concise Intro to Logic, well, this is on the other side of the spectrum -- it is very watered down compared to other logic readings, and pales in comparison (to most other introductory logic books) with regard to depth and breadth on formal logic.

It's usually aimed at first-year philosophy students who are taking introductory courses in logic or critical thinking, and most of it is simply rote-learning certain forms of argument as well as a lot of "quick and dirty" techniques which mimic that of a dry maths textbook. If you're looking for an interesting exposition into logic, then this book is certainly not it -- it would serve better as a high-school introduction for logic, and if prescribed to anyone older, would be very lackluster.

Here are some suggestions for you:

u/ShowMeHowThisWorks · 17 pointsr/math

I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.

The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.

You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.

But it's probably best to get a good foundation before embarking on Real Analysis and the like:

Discrete Mathematics with Applications by Susanna Epp

How to Prove It: A Structured Approach Daniel Velleman

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Book of Proof by Richard Hammock

That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.

u/JonnJonzz87 · 1 pointr/math

In a math course I recently took that was basically an introduction to math proofs we used Mathematical Proofs: A Transition to Advanced Mathematics which I found to be a great text. It begins by going through the language and syntax used in proofs and slowly progresses through theory, different types of proofs, and eventually proofs from advanced calculus. There are so many examples that are very well laid out and explained. I would highly recommend it for learning proofs from scratch.

u/antonivs · 1 pointr/philosophy

> The distinction is that in math, all foundational meta-theories are require to get the right answers on simple object-level questions like "What's 1 + 1?". If your mathematical metatheory answers, "-3.7" rather than "2", then it is not "different", it is simply wrong. We can thus say that Foundations of Mathematics is always done with a realist view.

The natural numbers are an interesting example, which goes back to ADefiniteDescription's point about a privileged model. The basic axioms of arithmetic are categorical, i.e. have only one model, up to isomorphism. Not all theories have this property, though.

If it could be shown that some moral theory similarly has only one correct interpretation - that all alternative interpretations end up being isomorphic - then that could support a kind of realism, at least in the context of that theory. A lot would depend on the nature and scope of the theory in question, and its interpretation.

So perhaps Parfit's position would be better captured by saying that he believes there are unique true answers to moral questions, as there are for questions in categorical mathematical theories.

> What's a good textbook for that field, anyway?

The books I studied are quite outdated now, but a classic modern text is Model Theory by Chang & Keisler. That might be more comprehensive than you're looking for. You could try Model Theory: An Introduction - its first chapter is quite a concise basic intro. There's also A Shorter Model Theory.

u/themarxvolta · 4 pointsr/logic

If you're interested in non classical logics I'd recommend "An introduction to non classical logic" by Graham Priest (it has modal logic and other very interesting non-classical logics). It's a good overview of the field.

For denser subjects in classical logic like computability, Turing machines, Gödel theorems, proofs for compactness, correctness, completeness, etc.; I'd go for a classical work by now: "Computability and logic" by Boolos, Burgess & Jeffrey. It's not an easy reading though.

u/Lhopital_rules · 64 pointsr/AskScienceDiscussion

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

u/Buttons840 · 2 pointsr/haskell

The Haskell Road to Logic, Maths and Programming

http://amzn.com/0954300696

I read only the first chapter or two a long time ago. I don't remember much, but I do remember I was able to progress through the book and learn new things about both math and Haskell from the text.

I didn't have any trouble getting the outdated examples to work. I had read LYAH previously though, so I wasn't a complete beginner.

I would really enjoy hearing what others have thought about this book.

u/Banach-Tarski · 2 pointsr/math

I learned all the set theory I need in Analysis with an Introduction to Proof, which features a good intro to set theory before getting into the analysis. It's a bit light on cardinal numbers, Zorn's lemma and the Axiom of Choice, but you can read up on those in a more advanced source. Personally, I think it's best to learn about those later when they come up in other subjects (like functional analysis) because otherwise it can be hard to see how to use them.

u/BMammaJamma · 18 pointsr/learnmath

I highly recommend reading "Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand et. al. It helped me get a better understanding of how to write a proof as well as organize my own thoughts.

Here's the Amazon link: Mathematical Proofs: https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX

u/funnythingaboutmybak · 2 pointsr/learnmath

I got my bachelors in Spanish and I have one more semester left to finish my masters in mathematics. As someone coming from a liberal arts degree, proofs were foreign to me, and handling anything with more than one variable was just asking too much. When I took Linear Algebra (my first proofs class), I had peers like the ones you mentioned, who just "got it" without taking notes while missing a third of the class lectures. And here I was slaving away, lost in a web of confusion. That class almost broke me. But over time, I learned a few things which were catalysts for my math competency:

  • I learned the framework of proofs and logic from this awesome book so that whenever I saw a theorem, I already had an idea of how to tackle it
  • I gave no shits about people thinking I'm stupid and thus asked A LOT of questions in class
  • I showed up to office hours religiously; it's insane not to utilize this one-on-one time with a PhD with tuition rates being as they are
  • I drew everything I could to help me understand concepts; if you can see the forest from the trees, the details will fall out
  • Related to the above, I tried building a visual intuition of things I was learning which helped me see past the slew of variables and greek letters
  • I memorized all definitions and stuck them in Anki; you're screwed trying to do proofs without definitions
  • Those other smart guys had to do everything I was doing at
    some point in time, they just got a head start; so I put in the hours and caught up (even surpassed in some areas)
  • I immersed myself in math: gave talks at conferences, got a job related to math, talked to other students about it, blogged about it, etc...

    Anyway, the struggle is real, but after all those focused hours of engaged studying, the intuition will finally be there and your brain will then "compress" that information so you have room to learn more. Ad infinitum.

    Hope that helps.
u/GapOutThere · 6 pointsr/math

You need a good foundation: a little logic, intro to proofs, a taste of sets, a bit on relations and functions, some counting(combinatorics/graph theory) etc. The best way to get started with all this is an introductory discrete math course. Check these books out:

Mathematics: A Discrete Introduction by Edward A. Scheinerman

Discrete Mathematics with Applications by Susanna S. Epp

How to Prove It: A Structured Approach Daniel J. Velleman

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Combinatorics: A Guided Tour by David R. Mazur

u/yerdos2030 · 1 pointr/logic

I can recommend two books which I have read recently.

  1. An Introduction to Mathematical Logic is more structured and formal description of logic.
  2. [Introduction to Logic] (http://www.amazon.com/Introduction-Logic-Methodology-Deductive-Mathematics/dp/048628462X/ref=sr_1_7?ie=UTF8&qid=1449702263&sr=8-7&keywords=mathematical+logic) gives more insights and helps to get a big picture of logic.
    I enjoyed both of them a lot and going to read them again.
u/thedude42 · 1 pointr/cheatatmathhomework

This book refocused my life. It gave me the recognition of the value of my CS degree I did not have while I was doing my degree.

After I read GEB I read this:

https://www.amazon.com/gp/aw/d/0674324498/ref=mp_s_a_1_1?ie=UTF8&qid=1524603692&sr=8-1&pi=AC_SX236_SY340_QL65&keywords=frege+to+godel

I didn’t understand most of it and I didn’t follow most of the proofs, but reading the words of these men was quite a wild ride because I knew where the story would end, and reading the arguments between these brilliant people and seeing how each was so convinced of their view, and how wrong they were, and how some had grace and others lacked it... really fascinating thing to bear whiteness to.

u/bediger4000 · 1 pointr/logic

Consider Raymond Smullyan's A Beginner's Guide to Mathematical Logic. It has some history of logic mixed in with pretty good coverage of propositional and first order logic, as near as I can tell. Lots of exercizes, which helps me personally.

u/thebrokenlight · 1 pointr/math

If you need an introductory text into Set Theory and Logic, you should try Kunen's Set Theory: An Introduction to Independence Proofs or Jech's Set Theory.

Then I would recommend reading Aczel's paper on Non-well-founded sets (1988).

For some historical context, I would urge you to read the amazing graphic novel Logicomix.

All of these books can be found online.

u/shamrock-frost · 1 pointr/math

The Haskell Road to Logic, Maths and Programming. I had already fallen in love with programming, and with Haskell, and this book showed me how well math, logic, and computer science play together. Shoutout to my aunt Trisha for giving me this book as a Christmas present in my junior year of high school

u/sillymath22 · 6 pointsr/math

I would say book of proof is the easiest to get you started and its free online with solutions!

I would start with the above its the easiest to read for an introduction to proofs books I have come across yet it still presents everything you need.

If you want some more challenging problems I would recommend

A transition to advanced mathematics

u/polp4a · 2 pointsr/UBC

Both 215 and 220 need plenty of practice. So as long as you set time aside for that you should be well on your way.

For 220 I would review some basic proof techniques (contradiction, contra-positive, induction) but not worry too much about knowing the details. In general we were never ask to prove anything where we couldn't apply the basics from a proof we had already learned.

We used Mathematical Proofs: A Transition to Advanced Mathematics (https://www.amazon.com/dp/0321797094), which was a very clear text with plenty of practice problems. If you have time I would recommend reading chapter 2 and 3.

u/HigherMathHelp · 5 pointsr/math

It's common to have some difficulty adjusting from lower-level courses with a computational emphasis to upper-level courses with an emphasis on proof. Fortunately, this phenomenon is well known, and there are a number of books aimed at bridging the gap between the two types of courses. A few such books are listed below.

u/mixed_massages · 1 pointr/math

Thank you very much for your reply and explanation. It's starting to make more sense now - I think I just need to study Model Theory on its own. (I saw that you praised this book in another thread. Would you recommend it for self study?)

u/knestleknox · 44 pointsr/math

I'm in literally the same boat with you. Full time software engineer and I graduated almost a year ago.

I think two things motivate me to self-study:

  1. It's something I enjoy

  2. It's someting I'm excited to apply to some project (personal or professional)

    To a degree, 2 is kind of a subset of 1. Anyway, when I come home I just find an hour to set aside and work through a book, ebook, or video to develop the skill in question. Recently it's been this book. And someday's I just wanna play video games and be lazy. So I think a large part of it is understanding that it's a slow process for the most part.
u/drunkentune · 1 pointr/PhilosophyofScience

As someone that didn't start off in math, I've always heard that Godel and Tarski are formalizing Russell's set paradox. Have I got it all wrong?

P.S.

I always love reading about famous people in philosophy that are also immensely important in other fields. Tarski's T-schema is an excellent correspondence theory of truth in philosophy; he's even bigger in logic (by the way, his Introduction To Logic is a great read).

The same goes for Kant - I sat in on a sociology class years ago that started off discussing Kant. Later, when discussing the class with the professor he admitted that he didn't know Kant was Serious Business outside of sociology.

P.P.S.

You said, "I spent lots of time as a youth thinking about the "deeper" meaning to the world we inhabit of the theorems (which ultimately is very little)." If you deny that we discover a deeper meaning to the world we inhabit when we discover the connection between the falling of an apple and the rotation of the planets, or between table salt and sodium, we've got a serious dispute.

u/clqrvy · 3 pointsr/askphilosophy

For the development of modern mathematic logic, a great volume with primary sources is From Frege to Godel.

http://www.amazon.com/From-Frege-Godel-Mathematical-1879-1931/dp/0674324498

Kneale & Kneale is definitely the go-to source for a broader history.

u/arbn · 4 pointsr/AcademicPhilosophy

That depends on why you're studying Logic.

Do you plan to use Logic as a tool for doing Philosophy? If so, I recommend studying Logic for Philosophy by Theodore Sider. You will get a more rigorous, formal treatment of propositional and predicate logic than what your introductory textbook likely contained. You will be exposed to basic proof theory and model theory. You will also learn, in depth, about several useful extensions to predicate logic, including various modal logics.

Do you want to become a logician, in some capacity? If so, the classic text would be Computability and Logic by Boolos and Jeffrey. This is an extremely rigorous and intensive introduction to metalogical proof. If you want to learn to reason about logics, and gain a basis upon which to go on to study the foundations of mathematics, proof theory, model theory, or computability, then this is probably for you.

Also, perhaps you could tell us what textbook you've just finished? That would give us a better idea of what you've already learned.

u/UsesBigWords · 1 pointr/askphilosophy

I recommend this to all beginners -- I like the Barwise & Etchemendy book because it's aimed at people with no background at all in logic or upper-level math, it's restricted to propositional and first-order logic (which I think logicians of all stripes should know), and it comes with proof-checker software so that you can check your own understanding instead of needing to find someone to give you feedback.

After that, you'll have some familiarity with the topic and can decide where you want to go. For a more mathematical route, I think Enderton (mentioned previously) or Boolos are good follow-ups. For a more philosophical route, I think Sider or Priest are good next steps.

u/CoqPyret · 2 pointsr/askphilosophy

Category theory is an overkill. If you think you're gonna have an easier time with it, you're mistaken. Category Theory is an extreme generalization of abstract math. Although, there's a very nice intro that you can get started with: Conceptual Mathematics: A First Introduction to Categories by Schanuel and Lawvere. It's accessible to most high school students.

What you are trying to understand is trivial. Most any intro to proofs/higher math book has an explanation of the subject.

In general, you need to learn how to think logically because the way you're going right now won't get you anywhere.

Again, read a book on the very basics of logic and sets. It would contain everything you need to know. For example,

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

> ... relation between finite and infinite.

...relation between finite sets and infinite sets. Just about everything in math is a set. There are many different types of relations. Some are functions, some are equivalence relations, some are isomorphisms.

> Just because something is an adjective or property does not mean it can't be negated.

Ok. Opposite of infinite is finite. In fact, we can say that a set is finite if it is not infinite. But limit is a number and infinity is not. You can't compare apples to oranges.

> In fact almost everything has an inverse.

Relations and special kind of relations called functions have an inverse. Also, operations can be inverse.

u/mniam · 8 pointsr/math

When I took a graduate set theory course, the book used was Kunen's Set Theory (Amazon), which I enjoyed. I've also read through some parts of Jech's Set Theory (Amazon, SpringerLink) and liked what I read.

u/sylviecerise · 1 pointr/TwoXChromosomes

I would just dive into it to see if it makes more sense! Here is a guide about delta epsilon proofs, which is one of the most common basic proofs you learn about in pure mathematics. Real Mathematical Analysis is a great textbook about real analysis. Also, if you're worried about the math, I would look into philosophical logic—Logic by Hodges is a good text for that and it won't involve any necessary background in math.

u/Cialla · 1 pointr/askphilosophy

The Logic Book is a good text for FOL and the early theorems of meta-logic (soundness and completeness of propositional and first-order logics). It's somewhat slow going though.

A more mathematically inclined text is Herbert Enderton's Introduction to Mathematical Logic. Enderton goes into more of the meta-logic, including incompleteness, Lowenheim-Skolem, and computability. He also touches on second-order logic toward the end.

Along the lines of meta-logic, Boolos and Jeffrey's Computability and Logic is very good as well. (Er, and Burgess. I can only vouch for the 3rd edition, which is pre-Burgess.)

Given that you're already familiar with FOL, I'd lean toward Enderton or Boolos and Jeffrey with the caveat that The Logic Book has endless practice problems and, iirc, answers to many of them in the back of the book (the others have fewer (but more interesting) problems).

If you want to go beyond FOL, I second stoic9's suggestion of Priest's book.

u/Abstract__Nonsense · 1 pointr/learnmath

The Haskell Road to Logic, Maths and Programming takes you through a lot of the basic “essential” math for CS, much of what would be covered in a typical discrete math course, but taught along side Haskell which is fun!

u/BraqueDeWeimar · 1 pointr/math

I posted this in /learnmath but didn't get any response so I'll give it a try here.

I'm a senior high school student and I'm learning linear algebra using Pavel Grinfeld's videos and programming in Haskell with this book.

What can I do to practice and apply concepts of linear algebra and programming?

Any recommended textbooks to complement the LA course?

Is it a good idea to solve project Euler problems in order to acquire programming/math skills?

u/Silvaticus08 · 3 pointsr/mathbooks

I think "Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)" is a solid book.

It starts off with what I would expect in a discrete math course (which is generally a first proofs course) and ends with a few chapters that would begin a second step writing intensive proofs course: number theory, calculus (real analysis), and group theory (algebra).

There are also many resources online that will help you once you've gotten through the basic notions in the book.

u/Edmond_cristo · 3 pointsr/math

What's up, man. I failed geometry twice (sophomore and junior year) in high school. I barely graduated high school with a 2.0 gpa. I am now a senior studying math and computer science (going to be getting masters in math). I am at the top of my class, and I will be graduating with a ~3.91 GPA.

Math, just like anything else, is about practice and perseverance. I thought I sucked at math (and basically everyone told me I was more of an "english" kind of guy). But when I got to college, I found that I really enjoyed the challenge, and I found the material interesting as hell. So I worked my ass off at it.

If you work hard (some may need to work harder than others!) and persevere, then you will be fine. There will definitely be challenges, but that's what makes math so fun.

edit: Also, unless you are a math major, I can't imagine you will be getting into too much rigorous theory. You will likely continue mostly just be doing calculations (Calc 1, Calc 2 and Calc 3). That is how it is at my university, at least. However, if you are a math major, it can't hurt to get a head start on writing simple proofs. For that, I recommend the following book: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094/ref=sr_1_4?s=books&ie=UTF8&qid=1474297774&sr=1-4&keywords=a+transition+to+advanced+mathematics

Seriously, that book is such a fucking good introductory text. It helped me so much.

u/kanak · 17 pointsr/compsci

If you're planning on learning haskell (you should :D), why not do a book that teaches you both discrete maths and haskell at the same time?

There are atleast two books that do this:

u/aleph-naught · 3 pointsr/math

I'll second the Halmos text, it's short and sweet; if you're looking for something more comprehensive I'd suggest Set Theory by Thomas Jech.

u/edcba54321 · 1 pointr/math

Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos

u/Untrained_Monkey · 2 pointsr/math

Introduction to Logic: and to the Methodology of Deductive Sciences by Alfred Tarski really helped me understand the key concepts of mathematical logic when I was young. Dover has republished the book and you can find used copies in great shape for $5 USD.

u/GeneralEbisu · 6 pointsr/math

I'm also planning on doing a Masters in Math or CS. What do you plan to write for your masters?


> Anybody else feels like this?

I think its natural to doubt yourself, sometimes. I dont know what else to say, but just try to be objective and emotionless about it (when you get stuck in a problem).

The following books that helped me improve my math problem solving skills when I was an undergrad:

u/hrtfthmttr · 1 pointr/politics

No. That's not quite there. But at this point, I've handed you the keys, you have to take the drivers ed yourself. This is a pretty good book to start with

u/oneguy2008 · 1 pointr/askphilosophy

Great to hear! Some good texts to consider, which will take you through the end of the main syllabus in an introduction to first-order logic are Chiswell and Hodges and Smith. If you ever get the chance to look back through this material I'd recommend taking a look at Goldfarb, but I don't think that's a great place to start in your situation.

u/brandoh2099 · 1 pointr/math

IMHO if you don't understand AC and its equivalents, then Jech is not the book you should be reading. That book is pretty heavy and is used (as far as I know - I have a handful of friends who work in set theory) as a research reference. Maybe read Naive Set Theory first. Despite its name, it's reasonably advanced but way more readable.

u/inducing · 4 pointsr/math

Right now I am studying Proofs from "Learning to Reason: An Introduction to Logic, Sets, and Relations" by Nancy Rodgers. Prior to getting started I looked at tons of "Intro to Proofs/Transition" books and the vast majority of them (including the popular darlings) are, frankly, just mostly doorstops - there's no way you could come out being able to do proofs by studying them.

Rodgers starts out with prop. logic and builds everything on top of that. Everytime she introduces a new topic, she gives logical justification (chapter 1 explores the logic extensively) that makes the proof structure work (very satisfying and makes the concepts stick around longer e. i. you are not just monkeying around with mish-mash of various tools, but actually know what you are doing)- never seen that in Real Analysis/Linear Algebra books that are, supposedly, designed to teach you proofs.

For example, in an intro to Real Anal, they just throw you the structure of Induction Proof and expect you to prove away - unrealistic. They dont show you why the proof works (logic and intuition behind the proof), wont let you explore the syntax of the proof before you get more comfortable with it and since one doesnt have a firm foundation made out of prop. logic, one's on a very shaky ground ready to break down whenever something serious comes on. With Rodgers, whenever something big and scary shows up, you just take everything apart into its logical building blocks like she teaches you in chapter 1 and it will make perfect sense.


But the worst part of RA books is they assume you are intimately familiar with Deduction and wont spend a half a page on it and that's 99% of math Induction Proof structure. Rodgers spends half the book exploring the intricacies of Deduction arguments. Basically, Rodgers' book explores math grammar in all its gory detail, is sort of a very revealing math porn.

If you ever studied a foreign language, you know there are 2 types of books. The ones that spell out all the grammar and give all the necessary vocabulary with an intention that you'll read some real literature in your target language in the future and those that skip the grammar or are very skimpy on it and give you pre-determined phrases and various random knowledge bites instead. The first category of books take the tougher road, but it pays off the at the end. Rodgers' book is one such book.

All in all, I just cant imagine learning proofs from Linear Algebra/Real Analysis books. Because, they are mostly about concepts inherent in these subjects and not proofs. Proofs are there to prove the said concepts, so there wont be enough time/space to explore proofs in-depth which will make your life tougher.

u/PeteBunny · 2 pointsr/math

Don't think of your abilities as fixed. The number of proofs you encounter grows from where you are now. You did not know algebra when you started. You will be increasingly exposed to proofs as you go along. Spend time on them. I recommend you get a tutor, or at least read some extra material. https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_69?ie=UTF8&qid=1494805054&sr=8-69&keywords=proofs+math

u/IAmScience · 5 pointsr/IWantToLearn

My symbolic logic course used The Logic Book by Bergmann, Moor, and Nelson. It was a solid introduction to formal logic. I've found the knowledge to be useful across a wide variety of disciplines.

u/paul_f_snively · 2 pointsr/programming

You're welcome! But upon reflection, "Category Theory in Context," I think, presupposes too much background in abstract algebra to really qualify as "introductory." And I criminally overlooked An Invitation to Applied Category Theory: Seven Sketches in Compositionality! BTW, it's even available online.

u/PsychRabbit · 1 pointr/math

Goedel's Incompleteness Theorems, by Raymond Smullyan.

From the preface:
> [intended] for the general mathematician, philosopher, computer scientist and any other curious reader who has at least a nodding acquaintance with the symbolism of first-order logic..and who can recognize the logical validity of a few elementary formulas.

I'm guessing most of the people on /r/math meet that description and more. If you want a general introduction to mathematical logic and computation, you should read Computability and Logic by George Boolos. If you can read Boolos, you can probably read Smullyan, and if you read them both you should emerge with some understanding of incompleteness.

u/bstamour · 1 pointr/programming

Have you seen The Haskell Road to Logic, Maths, and Programming? It's a pretty decent intro to higher math, and each chapter has a Haskell module.

u/sellphone · 2 pointsr/math

Naive Set Theory if you want a more textbook approach, the book mentioned in my other response if you're looking for something more like a story with proofs.

u/mlitchard · 31 pointsr/AskReddit

I was once a teacher's aide for an autistic teen. He seemed very bored with the 3rd grade arithmetic the teacher thought was his limit. One day, we had some extra time. I asked him if he wanted to read my set theory book. It's difficult to assess consent and comprehension, but we have our ways. I figured out that not only did he like this book, but he could follow along. It took about 3 months, but he was able to learn basics of sets. What makes me sad is the hard truth that people who know about this kind of math, generally don't find themselves being an educator for special needs students. His higher math education ended when I left. That's not right.

u/TimeSpaces · 2 pointsr/math

Try these books(the authors will hold your hand tight while walking you through interesting math landscapes):

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

A Friendly Introduction to Number Theory Joseph Silverman

A First Course in Mathematical Analysis by David Brannan

The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and Numbers by K. G. Binmore

The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas by K. G. Binmore

Introductory Modern Algebra: A Historical Approach by Saul Stahl


An Introduction to Abstract Algebra VOLUME 1(very elementary)
by F. M. Hall


There is a wealth of phenomenally well-written books and as many books written by people who have no business writing math books. Also, Dover books are, as cheap as they are, usually hit or miss.

One more thing:

Suppose your chosen author sets the goal of learning a, b, c, d. Expect to be told about a and possibly c explicitly. You're expected to figure out b and d on your own. The books listed above are an exception, but still be prepared to work your ass off.

u/tntenson · 3 pointsr/math

Halmos is good.

Others at the undergraduate level:

[Enderton's Elements of Set Theory] (http://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407)


Moschavakis' Notes on Set Theory



If you are just looking at books on sigma algebras, you should look at measure theory books. Royden, Halmos are the standard texts.

u/Recursionist · 3 pointsr/math

How do you learn proofs? Do you just memorize them straight up? Can you prove simple things in Set Theory and Point Set Topology on your own? There are only so many techniques for proving things. You absolutely need to master them. After that all you have to remember is a few definitions/theorems/lemmas and an odd trick here and there.

It could also be notation/symbols constantly throwing you off.

Anyway, I like these 2 books below:

Mathematical Proofs by Gary Chartrand et al.

Mathematical Writing by Franco Vivaldi.

u/tgallant · 2 pointsr/philosophy

Quine's Methods of Logic and Mathematical Logic (in that order) have been my favorites, and I've heard good things about Tarski's Introducion to Logic: and the the Methodology of Deductive Sciences but have yet to get around to it.

u/erisson · 5 pointsr/compsci

You may also want to check out The Haskell Road to Logic, Maths and Programming.
This book focusses on logic and how to use it, so you get to learn proofs. It even hits corecursion and combinatorics. If you think math is pretty but you want to use it interactively as source code, this could be the book for you.

u/bhldev · 2 pointsr/learnprogramming

Hello,

You can take a step back and learn about the philosophy and history of math. Once you learn some famous mathematicians and what their discoveries mean you will have more interest. One book is "What is Mathematics"

Then you can pickup some difficult texts. Doing a million problems mechanically is useless, except perhaps to pass tests. Get some idea of how to construct and read and appreciate a mathematical proof. Learn how to write proofs and prove common theorems and what those theorems mean. I recommend this https://www.amazon.ca/gp/aw/d/0321797094 to give you an overview.

Finally nothing beats taking advanced classes in university.

If this all seems a bit too much then maybe you can pick up something specifically for your purpose like 3D Math Primer for Graphics and Game Development, 2nd Edition 3D Math Primer for Graphics and Game Development, 2nd Edition https://www.amazon.ca/dp/1568817231/

Get yourself to high school math level first (understand the unit circle, exponents, algebra, trigonometry) and you can move up from there.

u/schmendrick · 5 pointsr/AcademicPhilosophy

Computability and Logic by Boolos, Burgess and Jeffrey is good but seems to cover much of the stuff in Hunter. You may want to dig deeper into set theory, model theory, proof theory or recursion theory and look at some references specific to those topics.

u/shimei · 15 pointsr/compsci

Depends on what kind of math you are looking for. For example, there is a middle school outreach program called Bootstrap World which is about teaching algebra using functional programming. You could take a look at their materials.

If you're looking for university-level math, there are some books like The Haskell Road to Logic, Maths, and Programming. I haven't read it, but I think it covers discrete math sort of topics.

u/simism66 · 5 pointsr/math

I think you might be looking for formal logic? The reasoning in mathematical proofs is based on systems of logic that can be formalized and treated as mathematical objects in and of themselves so that we can study the properties of the reasoning we employ in mathematics.

Though formal logic is used in mathematics, it is not limited to it. Formal logic can be used in thinking about anything. Philosophers, for instance, use it all the time in trying to think rigorously and carefully about difficult philosophical questions.

For an introductory book on formal logic, I like Peter Smith's Introduction to Formal Logic.

Since you also said that you're looking to understand the world in a probabilistic way, it also might be worth taking a look into Bayesian Epistemology

u/origin415 · 6 pointsr/math

I haven't read it myself, but I have heard Naive Set Theory recommended here several times before.

u/zitterbewegung · 42 pointsr/math

The rate of your learning is defined by your determination. If you don't give up then you will learn the material.

Look at the book that is required and only learn what you need in the class. Don't learn everything in the book either. Just learn what you need to do well and refer to the books when you get confused.

Note don't try to learn everything that's below. Only use it to learn what you actually need. This can be overwhelming at first but just set aside a set time to study this.
EDIT I added more books and courses.
OCW
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Helpful books
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_3?s=books&ie=UTF8&qid=1312542911&sr=1-3
http://www.amazon.com/Understanding-Probability-Chance-Rules-Everyday/dp/0521540364
http://www.amazon.com/gp/product/048663518X/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0155510053&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0YXJR9EVHCH9PCBDN372

Khan Academy
http://khan-academy.appspot.com/#calculus
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/probability--part-1?playlist=Old%20Algebra
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/linear-algebra--introduction-to-vectors?playlist=Linear%20Algebra

EDIT: I knew nothing about topological quantum computation about 1.5 years ago but then I took a independent study in college and I was assigned 1-3 papers a week to read. Eventually I got it a few months ago. What got me through it was not giving up...

u/hoijarvi · 1 pointr/science

You might be interested in The Haskell Road to Logic, Maths and Programming or just google haskell+math. Formal work seems to be navigating towards haskell now. My background is in power engineering, so I'm very familiar with numerical stuff, but lacking in discrete math. That's what I'm trying to patch.

u/mechtonia · 1 pointr/AskReddit

I know just the book that you need but its name escapes me. This is a placeholder for when I get home and can check the bookshelf.

EDIT: Here you are: Foundations and Fundamental Concepts of Mathematics

u/surement · 1 pointr/learnmath

A course I took previously used this book; it has a chapter on introductory real analysis, which is what you want to get at. I would not suggest going directly to a book like Rudin, as he (in my opinion) tends to amplify the "general route" problem that you mention.

u/Proclamation11 · 1 pointr/UMD

That's not the standard 310 textbook. This is. Also, in 310 you go over the first 3 chapters of the 410 book. I'm not disagreeing with your comment otherwise though.

u/I_regret_my_name · 3 pointsr/learnmath

Not really, sorry. The only analysis textbook I'm familiar with is this one I used for my class, so I don't know how it compares to any others.

u/faydaletraction · 1 pointr/math

Having a proof explained to you isn't even close to the thrill of proving something yourself, IMO. My advice would be to get your hands on an intro to proofs text and work through some of it. If you don't like writing proofs or think it's boring, your time at university is probably going to bore you to tears.

If you want an intro proofs book, you might start here. The text is very clearly written and chapters 9-13 will give you a very basic notion of what ideas will be at the core of some of your upper-level math classes (abstract algebra, real analysis, etc).

u/den_of_sins · 1 pointr/math

My department uses Enderton's Elements of Set Theory in our introductory Set Theory course. It starts at the basics and has some large cardinal and forcing stuff towards the end, I believe.

http://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407

u/that-cosmonaut · 2 pointsr/math

Smullyan's Beginner's Guide to Mathematical Logic is incredible and quite accessible: Link

u/hspecial · 1 pointr/math

Set theory. http://www.amazon.com/Set-Theory-Thomas-Jech/dp/3540440852

I did out all the proofs that are left out in the first few chapters and developed an undergraduate level paper on ordinal numbers.

u/KontraMantra · 12 pointsr/AcademicPhilosophy

In response to the same question, my Logic professor suggested:

u/adollopoftrollop · 1 pointr/AskReddit

I believe that this is the standard for Logic classes. It's the one I have, and you can find a solutions guide online to check your answers. It's a tad bit expensive, unfortunately!

u/Dhush · 2 pointsr/statistics

No problem. FWIW my intro to proofs class used this book and I thought it served its purpose well

https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094

u/iopha · 1 pointr/philosophy

If you are interested in studying the development of mathematical logic and the philosophical disputes surrounding the foundational questions (viz., the disputes between logicism, formalism, intuitionism, etc.) then Russell should not be studied in isolation from Frege, Hilbert, Brower, et al. Perhaps: http://www.amazon.com/From-Frege-Godel-Mathematical-1879-1931/dp/0674324498

This might be too technical, I don't know!

u/topoi · 3 pointsr/askphilosophy

It depends what you're trying to get out of it.

There are literally hundreds of introductory texts for first-order logic. Other posters can cover them. There's so much variety here that I would feel a bit silly recommending one.

For formal tools for philosophy, I would say David Papineau's Philosophical Devices. There's also Ted Sider's Logic for Philosophy but something about his style when it comes to formalism rubs me the wrong way, personally.

For a more mathematical approach to first-order logic, Peter Hinman's Fundamentals of Mathematical Logic springs to mind.

For a semi-mathematical text that is intermediate rather than introductory, Boolos, Burgess, and Jeffrey's Computability and Logic is the gold standard.

Finally, if you want to see some different ways of doing things, check out Graham Priest's An Introduction to Non-Classical Logic.

u/krypton86 · 3 pointsr/math

> Is there any good book with problems/examples that I could work through in order to thoroughly prepare myself to be able to write proofs for a Real Analysis I course?

Besides Velleman's "How to Prove it," try Mathematical Proofs: A Transition to Advanced Mathematics or maybe How to Read and Do Proofs: An Introduction to Mathematical Thought Processes.

The book I used in my "Intro to Proofs" course was A Transition to Advanced Mathematics. It was pretty good, but the edition that I used had several mistakes in it. Also, it's waaaay too expensive.

Now for the unpleasantries —

Suggestions aside, the main problem here is your "thoroughly prepare myself to be able to write proofs for Real Analysis" goal. Working through a proofs book on your own will be seriously challenging, but the thought of taking Real Analysis without at least two other proofs courses under your belt is terrifying to me. I had to take "An intro to mathematical proofs" followed almost immediately by a proof-based Linear Algebra course before I was even allowed to contemplate a Real Analysis course.

Come to think of it, how in the hell are you even allowed to do this if you haven't taken a proofs course before? Are you sure this is even possible? Are prerequisites not enforced at your school? No one, and I mean no one was permitted to take Abstract Algebra or Real Analysis without the required prerequisites at my university. The only way you could get around it was by being the next Andrew Wiles.

Just to drive all this home, I was a straight-A Physics/Math major with the exception of two courses: Thermodynamics and my first proofs course. I've never worked so damn hard for a B in my life. Come to think of it, I actually recall quantum mechanics being easier than my proofs course.

I'm being sincere when I ask you to reconsider this plan. You are asking for a world of pain followed by the very real possibility of failure if you do this.

TL;DR: Unless you are remarkably sharp and have loads of time on your hands, this is probably a mistake. You should take a more elementary proofs course before tackling Real Analysis. Good luck, whatever you choose to do.