Introduction to Proofs Textbook Recommendation

[PManslaughter]

I'm planning on taking an Abstract Math and Linear Algebra II (essentially a more theoretical continuation to the first linear algebra course which all physics majors take) in order to later take courses like group theory, number theory, analysis, etc. But I have close to no knowledge with how to do proofs.
Could someone recommend an introduction to proofs textbook in which they found helpful when their knowledge of proofs was very limited?

 

[axmls]

I learned from Spivak's Calculus through self study as opposed to a book dedicated entirely to proofs. You can try "Basic concepts of mathematics" by Elias Zakon as well. It's available free from online (legally).

 

[PManslaughter]

I learned from Spivak's Calculus through self study as opposed to a book dedicated entirely to proofs. You can try "Basic concepts of mathematics" by Elias Zakon as well. It's available free from online (legally).

You were able to translate what you learned from Spivak to general mathematical proofs?

 

[verty]

There is truth to the saying, you should already know why something is true before you try to prove it. Most of the effort is to figure out if the claim is true or false, and why. And I think no one can teach you this, and certainly no book can teach you it.

For example, take Pythagoras's theorem. There are many proofs but if you just look at the theorem, it seems almost impossible to prove. Is a book like Velleman's How To Prove It going to help you to be able to prove Pythagoras's Theorem? Almost certainly not.

The best proof I've seen of that theorem is one that I saw in one of Polya's books, one that uses the proportionality of space. Any two similar figures, one being bigger than the other, will have an area that varies with the square of the change in size. For example, two similar triangles, one exactly twice as large, will have an area four times as large.

And looking at Pythagoras's theorem, it can be multiplied by any constant. If $c^2 = a^2 + b^2$, $kc^2 = ka^2 + kb^2$. So we can use any similar figures, they don't have to be squares, and the value for $k$ will just change. We could for example use circles where the sides of the triangle are the diameters ($k = {\pi \over 4}$). If we can prove it for any $k$, it is true for all $k$.

What figures should we use? Take the triangle itself and reflect it through the hypotenuse. Then drop a perpendicular from the apex to the hypotenuse and reflect those two triangles respectively through the other two sides. Now by proportionality of space, their areas are in proportion to the square of the side lengths. But we see that their areas do have the required relation.

Therefore, we have proved Pythagoras's Theorem. I know of no book that can teach one how to do this. Perhaps this is not yet a proof but converting it into a proof should be very easy because we know why it is true. So that's what I have to say about that, there are books that claim to be able to help but ultimately it is down to personal effort and creative thinking. There's no easy road, that is the point.

 

[Fredrik]

"How to prove it" by Daniel Velleman and "Book of proof" by Richard Hammack get a lot of recommendations here, and get very positive reviews at Amazon. I'm not familiar with the former, but I know the latter is very good.

 

[Fredrik]

So that's what I have to say about that, there are books that claim to be able to help but ultimately it is down to personal effort and creative thinking. There's no easy road, that is the point.

This is true for many proofs, but people who are just getting started with proofs are also getting stuck on things that are much more trivial and have nothing to do with creativity. For example, if the theorem starts with "for all positive integers n,...", it's obvious to someone with experience that the proof should start with "let n be a positive integer", but it's not obvious to them. There are plenty of "obvious" things like this that can be taught.

These books are also good introductions to set theory and logic. At the very least you need to understand truth tables and set theory notation.

The best proof I've seen of that theorem is one that I saw in one of Polya's books, one that uses the proportionality of space. Any two similar figures, one being bigger than the other, will have an area that varies with the square of the change in size. For example, two similar triangles, one exactly twice as large, will have an area four times as large.

I've been fascinated by this proof since I first saw it, but it's not just because the proof is so short. I'm fascinated by it because I still don't feel like I really understand it. In particular, I don't know how I would explain to a high school student that it's $ka^2=kb^2+kc^2$ rather than $ka^2=lb^2+mc^2$.

 

[verty]

I've been fascinated by this proof since I first saw it, but it's not just because the proof is so short. I'm fascinated by it because I still don't feel like I really understand it. In particular, I don't know how I would explain to a high school student that it's $ka^2=kb^2+kc^2$ rather than $ka^2=lb^2+mc^2$.

I had to look this up (in Olney's geometry book from 1870 or so). The three triangles are similar, this means $a:b:c\;::\;{ka \over 2}:{lb\over 2}:{mc \over 2}$ (bases and heights). But then $k = l = m$.

This is interesting actually, it seems to resist being put into words.

PS. $a : b :: d : e$ means "A is to B as D is to E".

 

[jack476]

I'm planning on taking an Abstract Math and Linear Algebra II (essentially a more theoretical continuation to the first linear algebra course which all physics majors take) in order to later take courses like group theory, number theory, analysis, etc. But I have close to no knowledge with how to do proofs.
Could someone recommend an introduction to proofs textbook in which they found helpful when their knowledge of proofs was very limited?

There is a wonderful introductory abstract algebra textbook that I recently found in my university library called "Introduction to Analysis and Abstract Algebra" by John Hafstrom. It's from 1967 and seems not to be too well known, but I cannot sing this book's praises enough for how well it explained the basics of mathematical proof to me despite me knowing almost nothing about proofs or proof-writing before I got to it. You do not need to be intimidated by the fact that it's an abstract algebra book, it explains almost everything from first principles (its later chapters assume some calculus knowledge, which if you're taking those courses shouldn't be a problem). It's more about teaching you proof and "mathematical maturity" (which is really what learning to do proofs is all about, mathematical reasoning and insight) in the context of abstract algebra and analysis than it is about those subjects directly.

Unfortunately, it's kind of rare and hasn't seen any reprints or anything, but a quick look at my state's online library sharing catalog shows that there are a couple dozen copies floating around university libraries throughout the state. I would very strongly recommend you get on your library network and see if you can either find a copy at your school or request a materials transfer for one.

 

[Fredrik]

I had to look this up (in Olney's geometry book from 1870 or so). The three triangles are similar, this means $a:b:c\;::\;{ka \over 2}:{lb\over 2}:{mc \over 2}$ (bases and heights). But then $k = l = m$.
[...]
PS. $a : b :: d : e$ means "A is to B as D is to E".

If I denote the height in the original triangle by $h$, I get $ka^2=\frac{ah}{2}$, and therefore $h=2ka$, not $h=\frac{ka}{2}$. Maybe your definitions are a bit different. That's OK though. I get $a:b:c:2ka:2lb:2mc$, and the result $k=m=l$ follows immediately from that too.

This method also explains why the area of the triangle can be expressed as a number times the hypotenuse squared in the first place. So I think it's easy to clean this up into a nice proof.

I was thinking that the fact that this proof relies on the validity of the standard statements about similar triangles makes it less intuitive than the proofs that only rely on the formulas for the area of a square and a triangle. But I just realized that there are no such proofs. The ones that rely on the area formulas use similarity as well, but you may not realize it because they use it in a step that feels like it doesn't require proof at all. (It's used to argue that if two triangles are such that their longest sides are the same and all the interior angles are the same, then the two shorter sides are the same as well).