Best mathematical logic books according to redditors
We found 329 Reddit comments discussing the best mathematical logic books. We ranked the 107 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 329 Reddit comments discussing the best mathematical logic books. We ranked the 107 resulting products by number of redditors who mentioned them. Here are the top 20.
You need to develop an "intuition" for proofs, in a crude sense.
I would suggest these books to do that:
Proof, Logic, and Conjecture: The Mathematician's Toolbox by Robert Wolf. This was the book I used for my own proof class at Stony Brook - (edit: when I was a student.) This book goes down to the logic level. It is superbly well written and was of an immense use to me. It's one of those books I've actually re-read entirely, in a very Wax-on Wax-off Mr. Miyagi type way.
How to Read and Do Proofs by Daniel Slow. I bought this little book for my own self study. Slow wrote a really excellent, really concise, "this is how you do a proof" book. Teaching you when to look to try a certain technique of proof before another. This little book is a quick way to answer your TL:DR.
How to Solve it by G. Polya is a classic text in mathematical thinking. Another one I bought for personal collection.
Mathematics and Plausible Reasoning, Vol 1 and Mathematics and Plausible Reasoning, Vol 2 also by G. Polya, and equally classic, are two other books on my shelf of "proof and mathematical thinking."
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...
OCW
Single Variable Calculus
Multivariable Calculus
Differential Equasions
Linear Algebra
Helpful books
Mathematical Proofs: A Transition to Advanced Mathematics
Understanding Probability: Chance Rules in Everyday Life
Linear Algebra
Other Resources
Kahn Academy
Probability (keithpeterb on youtube)
Probability (Kahn)
Linear Algebra (Kahn)
There, Fixed up your links.
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.
Don't tell me I don't support a thing that I support so that you can criticize me for not supporting it, lmao.
You lot aren't too bright, eh? I guess it's easier to argue against us when you just make shit up on the spot. Here's a book for you, since you like reading so much.
And here's the free version:
https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX
> 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.
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
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.
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:
Both books use Haskell in an effective way to teach you discrete mathematics. (They don't assume prior knowledge of Haskell).
I've been working through the 2nd book and posting my notes and solutions here . You could compare solutions/notes and discuss stuff with me.
Pick up a book on proofs, and do as many exercises as you can. It's a big leap for a lot of students, going from plug and chug high school math to thinking creatively to prove results.
Also, learn some basic set theory (inclusion, union, intersection, cartesian product, etc).
Lay's Real Analysis book is very good for this purpose. It starts with basic logic and proofs, works in most of the set theory you need (though a bit light on Zorn's lemma), and then heads into the real analysis at a gentle pace. It also covers some basic topology along the way.
The Haskell Road to Logic, Maths and Programming:
http://www.amazon.com/Haskell-Logic-Maths-Programming-Computing/dp/0954300696
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.
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).
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.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
In response to the same question, my Logic professor suggested:
A few of my favorites:
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:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
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:
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:
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.
Would a very short one be of interest?
If we look at the "set" of all Cardinal Numbers, then that "set" can be indexed by the "set" of all Ordinal Numbers, call it V. Ordinal numbers are a standard way to talk about Well-Ordered sets. But it turns out that V has all the properties of a well ordered set, so it is a well ordered set and, additionally, it is strictly bigger than every other ordinal. But we can make the ordinal V+1, since V+1 != V we have V+1<V, but by definition of the successor ordinal V<V+1 which is a contradiction, V cannot be a set. Therefore, because the set of all ordinals cannot exist and the set of all cardinals is bijective with all the ordinals, it follows that the set of all cardinals cannot exist. Also, this does not depend on Choice or the GCH, if you don't assume them, then there are even more Cardinals so the result still holds.
This essentially says that the number of all infinites is too big for any infinity to count! Crazy, huh? Read about it here. Also if you want to start to learn more Set Theory and things I recommend looking for Suppes' book. It's a nice introduction to these ideas and will get you acquainted with the language and how things work. Once you have a bigger vocabulary and a bit more mathematical maturity I recommend Jech's book, it is gigantic, dense and tough but has everything that someone who isn't actively doing Set Theory research could ever need about Set Theory in it, if you can get through it...
For basic logic (first-order, classical) these are excellent textbooks...
I should also mention that there are some free, open source textbooks on standard logic (good books, written by scholars in the field) such as the following...
And if you want to go beyond classical logic, these are excellent textbooks...
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.
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:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
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
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:
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
I haven't read it myself, but I have heard Naive Set Theory recommended here several times before.
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.
I hope that helps!
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
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.
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.
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.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1394386195&amp;sr=8-1&amp;keywords=pinter+abstract+algebra)
This one is pretty good. I used it during undergrad:
http://www.amazon.com/dp/048669609X/ref=cm_sw_r_tw_s_awdm_IfxNxbFJS9ZQS
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.
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.
I heartily recommend Solow's How to Read and Do Proofs which would serve as a good foundation for thinking about 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.
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
I'm a fan of A Very Short Introduction series. Graham Priest did one for logic.
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
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.
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.
> 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.
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.
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!
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&amp;ie=UTF8&amp;qid=1474297774&amp;sr=1-4&amp;keywords=a+transition+to+advanced+mathematics
Seriously, that book is such a fucking good introductory text. It helped me so much.
Absolutely! There is a second book I would suggest to fellow proofy people. This one is aimed directly at proofs. These two combined were invaluable to me:
How to read and do proofs
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.
Two great introductions are:
both are short and very well written. Some books particularly close to my heart are:
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.
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.
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.
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.
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.
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.
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.
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:
If you're looking for mathematically-oriented logic puzzles, then you should check out Smullyan's series of puzzle books. You can find them on Amazon.
Both Quine and Tarski have great introductory texts.
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.
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&amp;qid=1450533541&amp;sr=8-2&amp;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/
In a quite literal sense:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
>What evidence do you have that it won't sink her campaign?
I never made that claim. I never made any statement of fact about the outcome of the campaign.
This is where you get yourself into trouble. You confuse facts and evidence and opinions.
You have made several subjective conclusions that are entirely based on opinion, and now you're confused and think they're statements of fact and therefore a certainty that requires an evidential rebuttal.
Here's a book that might help you.
Another very accessible book https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024
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.
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.
Halmos's Naive Set Theory is a classic.
http://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387900926/ref=sr_1_1?ie=UTF8&amp;qid=1320969231&amp;sr=8-1
Still the best math book cover ever
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.
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.
>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
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.
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.
>I mean its to late now to enroll...
Why wait? Pick up a few books on math and use your Google Fu to get yourself started.
I really like this book.
And instead of studying geometry (which I doubt you'd be using in college), study Logic instead. The way problems are constructed is similar to geometry. In geometry you have theorem and postulates, in logic you make proofs. You start out with two or three opening statements, and by using different combinations of OR, AND & IF-THEN statements, you can prove the final statement.
I'll give you this link about it but I'm hesitant to because it has lots of scary symbols and letters. Here. But save that for later. If you want to get started, take a look at truth tables .
Logic is so much more interesting than geometry because it'll help your Google Fu get even better. You can make Boolean statements when you enter a Google query. It also gets you on the path to learning SQL (which your brother may also be able to help you with). SQL is all about sets - sets of records, and how you can join them and select those that have certain values, etc.
You may even find this book a nice, gentle introduction to logic that doesn't require much math.
Basically, what I'm saying to you is this: you live in the most incredible time to be alive ever. The Internet is a super-powerful tool you can use to educate yourself and you should make full use of it.
I also want you to know that if you don't have a specialized skill, you're going to be treated like a virtual slave for the rest of your life. Working at WalMart is not a good career choice. That's just choosing a life of victimhood. Make full use of the Internet, and your lack of a car will seem less problematic.
(Note: I wrote this elsewhere)
Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:
When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
I don't know of any good online introduction to discrete math, but here are a few book ideas.
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&amp;qid=1494805054&amp;sr=8-69&amp;keywords=proofs+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.
Smullyan's Beginner's Guide to Mathematical Logic is incredible and quite accessible: Link
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
> 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?
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.
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.
http://www.amazon.com/Q-E-D-Beauty-Mathematical-Proof-Wooden/dp/0802714315/ref=cm_cr_pr_product_top
It's a nice little book, very small, notebook sized and only 60 or so pages, but it's quite enjoyable. :)
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 :)
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.
>Basic. Ass. Film. Theory. Read a mutherfuckin book nigga. Movie's are >not just flat images.
Read lots of books. Maybe you should try one as well. How about http://www.amazon.com/Logic-For-Dummies-Mark-Zegarelli/dp/0471799416/ref=pd_sim_b_1
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.
I really like Nancy Rodgers' "Learning to Reason".
Keith Devlin has a course called Introduction to Mathematical Thinking that covers a subset of the material in Nancy's book.
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.
http://www.amazon.com/Computability-Logic-George-S-Boolos/dp/0521701465
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.
The Stanford Encyclopedia of Philosophy is a gem. It contains articles on any topic of relevance to philosophers, typically with a great deal of attention paid to the history.
You will likely want to look at modal logic. Apparently this is a good introduction.
As for history, this book and this book will be very, very good.
There are a number of excellent (scholarly) survey articles on certain subjects within the philosophy and history of mathematics, but I would need more specific guidelines on what you'd like to learn, and what you know.
Finally, if you dig through the archives of the n-category cafe you can find some interesting posts and discussion from working mathematicians and philosophers of math... You will have to do some digging, as most of what's there is pure math or mathematical physics, but the more philosophical posts have wonderful discussions.
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.
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:
some point in time, they just got a head start; so I put in the hours and caught up (even surpassed in some areas)
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.
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.
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:
So you could pick one of those, and see if it meets the level you're after. There are also some standard books on "how to write proofs" which often get recommended: Velleman, How to Prove It: A Structured Approach <https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-ebook/dp/B009XBOBL6&gt;, and
Solow, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes <https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024&gt;, and they might be useful. I've never read either, shamefully, tho I probably should.
Also: something that can be handy once you're past the basics is the amazing (but large and expensive) Princeton Companion to Mathematics. Which has something useful and interesting to say about pretty much every major area of modern mathematics.
I hope that helps!
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.
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)
read this: http://www.foxnews.com/us/2014/04/17/man-facing-525-federal-fine-after-not-paying-for-soda-refill-at-va-hospital/
Your statement either assume that people can not be trusted and/or follow basic directions.
My 8 year old niece and I both feel that you should also read this: http://www.amazon.com/Logic-For-Dummies-Mark-Zegarelli/dp/0471799416
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.
That's a great book. There's a good section about triangle numbers relating to squares as well.
Another good one with lots of visual proofs is QED, which features diagrams showing geometric formulas (such as the volume of a pyramid).
The Haskell Road to Logic, Maths, and Programming.
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.
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.
If you'll read this, I'll buy it for you.
I am a fan of The Logic Book
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!
This is a good place to start: http://www.amazon.com/Logic-Book-Merrie-Bergmann/dp/007353563X/ref=pd_sim_b_2
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.
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!
You can also check Chartrand's et al book:
https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094
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&amp;qid=1486754571&amp;sr=8-1&amp;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!
What? The fact that you BOUGHT a stolen laptop shows that they ARE NOT worthless. Maybe this would help you out?
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.
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.
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.
> The Church as a whole is guilty until proven innocent.
Ah, so you've graduated to an even more ridiculous claim -- EVERY Catholic is guilty without evidence. Okay.
> what the FUCK are you doing to protect those children being abused by priests?
What are you doing? Bad mouthing innocent people anonymously on the Internet? Great job.
> I'm not bound by any of that
Correct, you are free to make assertions based on no evidence.
> You are enabling the abuse by trying to silence me for no good reason other than that you're offended
I'm enabling abuse by pointing out that it's ridiculous to accuse millions of people that are innocent of wrong doing without any evidence? Whatever you say buddy.
> people who are innocent don't care what people say about them
Ah, I see. So you can accuse millions of innocent people of doing something horrible with no evidence, and if anyone objects, that confirms your evidence-free claim? Very nice logic.
> Maybe had some suspicions you've tucked down deep
Devoid of any chance of actually defending your point, you've resorted to cursing in bold capital letters and ad hominem attacks. "Here's a solid fact of life" people who abandon reason and substitute it with these desperate measures are (1) losing the argument (2) realize they are losing and (3) everyone else realizes it too.
> Or perhaps you've been a victim of abuse yourself
Or perhapppsss.... maybe... just maybe... I think it's wrong to accuse millions of innocent people of heinous things with no evidence? I found a good book for you https://www.amazon.com/dp/0192893203/
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?
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.
Q.E.D.: Beauty in Mathematical Proofs by Burkard Polster.
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
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.
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.
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.
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
https://www.amazon.com/Logic-Dummies-Mark-Zegarelli/dp/0471799416
And here you go! Since you failed to explain yourself, it is much needed, I suppose. Lets play this shitpost game together then. It seems that I won't have any kind of reasonable answer from you, so why not engage in some tomfoolery?
Oh! I have even better book for you:
http://www.hpmor.com
It has something about giving proper arguments and explanations, I'm sure. Cannot recall it though - its been a while since I read it last time.
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!
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.
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
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.
I can recommend two books which I have read recently.
I enjoyed both of them a lot and going to read them again.
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.
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.
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).
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.
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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.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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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.
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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.
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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:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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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.
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.
Certainly recommend getting a copy of Rosenlicht if you're in this situation. This is what I'm currently reading in preparation for an honors-level analysis class. Should be pretty easy to follow even without much of a strong proof-based background.
As far as proof mechanics and mathematical foundations go, I recommend "How to Read and Do Proofs". It's not a bad reference to have at your side, just in case.
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
S t'en as un un si gros, fais-lui plaisir et achète ce bouquin: https://www.amazon.fr/gp/product/0192893203/ref=oh_aui_search_detailpage?ie=UTF8&amp;psc=1
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&amp;qid=1524603692&amp;sr=8-1&amp;pi=AC_SX236_SY340_QL65&amp;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.
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 =)
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.
Bhai, thoda paisa kharch karke apne liye kuch achha kar lo. 1 2
>EDIT: Wink? ;)
Yup. (I was trying to be facetiously "cute"... so many Q.E.D. proofs end up as geometric drawings! )
But on the serious side, yes. Keynes liked to "play around" with math-like "quasi-proofs" (and to expropriate and abuse physics concepts) and succeeded in baffling many with what was basically a lot of literary "fancy footwork" -- but his assumptions are so many (and so questionable) and he piles on so many things as "givens" (when they in fact are nothing of the sort -- falsely dismissing the inevitable failure of his equations with "in the long run we're all dead" and similar faux-witty rhetoric) -- that nothing of his really qualifies as a Q.E.D. He simply ignores that anything really NEEDS to be demonstrated.
>Thinking that holding a purse makes you less of a man indicates that you view purses negatively because they are associated with women. That makes you sexist.
Whoa, whoa, whoa. Learn some logic, pal. Here's Logic For Dummies, which will suit you perfectly.
This is not sound logic:
John views purses negatively.
Women use purses.
Therefore, John views women negatively.
I mean come the fuck on. Personally, purses don't bother me because they're associated with women; purses bother me because I think purses are fucking stupid. Women use purses to carry around a bunch shit they don't need, but society tells them they need. For example: make-up, lipstick, hair-brushes, combs, nail-polish, and so much other shit. That's why I hate purses. It has nothing to do with what gender is associated with purses, it has to do with the fact that purses are fucking stupid. In conclusion, you're a jackass that needs to learn some basic fucking logic.