Basics of Proofs (2017) [pdf] (opens in new tab)

Some universities now have a Lean proof assistant logic course that teaches natural deduction http://leanprover.github.io/logic_and_proof/

Whenever I have asked professors in the past how they first learned proofs the answer was always from doing Euclid in highschool which is no longer taught

I studied CS and electric engineering for 6-7 years total, and both degrees included some pretty math-intense courses, proofs and all. I remember how I really, really wanted an explanation of the underlying proof system but all I got was just more intuitive explanations, all of which felt out of place.

What is unusual is that I kept looking. "How to Prove It" was something I really wish I read ahead of calculus. It is only after this book it clicked for me, all of it. Having understood the rules of the game it became possible to play with math..! Which I still do from time to time.

I think that this material should be taught at high school. Bits of it even earlier.

Philosophy was the only field where I learned formal logic at all. I went into higher math, did some CS, and also got a law degree[1], and am now doing a Ph.D. in informatics, and still the only place I have really been exposed to formal logic was my undergrad degree in philosophy.

[1] It is amazing (or not, depending on your view of lawyers) that lawyers are basically not taught anything about making and proving formal arguments, at least in my experience.

> But I think it would be helpful for many people to first learn logic and deduction "the precise way", and then do actual mathematical proof where you can jump over more obvious parts.

Absolutely. And I can not imagine a mathematics degree which doesn't place formal logic front and center in the first semester. When you are starting out you absolutely need to suffer through the rigourous formal arguments.

>Ironically, the one subject which often teaches formal logic as an early introductory class for undergraduates isn't mathematics, computer science, or physics, it's philosophy.

I can not believe that this is the case. Even engineering undergrads have to learn formal logic in their first semester.

> When I was first introduced to mathematical proofs, I was perplexed by how fuzzy and intuition-based the notion of proof was.

This was precisely my issue when studying proofs in school. Do you have any suggestions for resources to get started down the right path?

> Moreover, mathematicians rarely prove anything from axioms, they start from other statements which are considered more trivial.

Not more trivial but from the ones already proved and those closer to the thing you are proving. There is no need to go back to the axioms if you know, and can reference, proof of step N-1, just go from there.

There is also this 'misconception' that mathematical theorems follow from the axioms. They do, of course, but the axioms were choosen just right to make things that were working to still work, with some weird consequences like axiom of choice

Our discrete mathematics professor leaned heavy on proofs, but also introduced pop quizzes for proofs of algorithms and other mathematical constructs as an experiment that term which sent my anxiety ablaze. You either had the intuition for that particular problem or you didn’t. Studying would not help you that much. The success rate on the quizzes was very low surprising no one.

As someone who tutored a Formal Reasoning class for a top philosophy department, I don't think we should be teaching anyone logic. It's rigid, its made up, and it doesn't help much for most things. I think the way proofs are written in math is cool, and I think the fact that mathematicians are, in the end, aware that it is impossible to rigidly demonstrate anything is perfectly fine and it makes their proofs a lot easier to read and interpret. Formal logic should never be treated as first order or ever used as such, it will only hamper mathematics. I understand the need for a rigorous language to describe mathematics, I understand why people like it so much, and why it has found so much use. It is still not the groundwork (because there is none).

Isn't (classical) geometry basically about proving properties from a few theorems? I.e., logic, proof, and deduction.

A lot of people are exposed to Geometry in High School. It might not be universal, but it's pretty close. IIRC, Eucid's Elements was the second most read book, after the Bible. Many cities even have a street named after the guy.

I know, it's not the same as a formal logic course. But it's not that far off, either. There is some preparation in earlier years of schooling.

I like your analogy with riding a bike: If you want to ride a bike, what is more important, learning how to ACTUALLY DO IT, or to learn the physical and mechanical rules underlying it? Furthermore, technical knowledge will always be incomplete (and that applies also to logic in a very technical sense), while once you learnt how to ride a bike, your knowledge, in a way, is complete.

This is fantastic. Perhaps similarly, I personally found it much easier to complete math problem sets after I began to write out an explicit list of steps of what to do.

For example, I broke down problems with to-dos, such as:

1. Find the definition for what math_term_X means in a particular problem.

2. (For breaking down part of the problem): Figure out how to show that a particular object is lesser than or equal to another project.

3. Write down headings for each case I need to prove.

...and so on.

Writing down explicit steps was far more practically helpful to me, than my previous conception of problem-solving from the quote about how Feynman solves problems (that is: "Write down the problem, think real hard, write down the solution"). Some people may not need to write down steps, but I was personally able to learn a lot more with a specific, more verbalized approach.

It's very neat and helpful to have a flowchart suited to any general problem, which I'll try out in addition to my current approach of writing down a list of to-dos for solving specific problems. Thanks a lot for sharing.

Maybe it's worth one more section near "Any Ideas On Why It Is True?", something like:

Are you beginning to doubt whether it's true?

Try to think of a counterexample.

Is there something that keeps getting in the way of a counterexample working? Can you prove that that always happens?

I think the problem a lot of people have (even math majors) is expecting the lecture to be the first introduction to the topic. It's like taking a literature class and going to lecture without doing the reading. Reading (or even just skimming) the next section/chapter of the book before the lecture allows you to focus and ask questions on the parts of the material you didn't understand during the lecture. I very rarely wrote down full proofs in my notes during the lecture. I focused on the lecture itself and wrote down the pieces that I wanted to remember.

>which we all dutifully copied into our notebooks. Very uninspiring.

I had 5 years of that, a very enjoyable time.

And another fun and powerful technique is "proof by intimidation" (https://en.wikipedia.org/wiki/Proof_by_intimidation), in which you don't have to know what you're doing, but other people have to think you do.

Ironically when rigorous proofs were invented by the greeks, Euclids "proof by picture" was no less rigorous than modern formal logic.

Over the years I’ve noticed it too, the time lag between similar posts ranging from a few hours to a day or so. My simple explanation is that a person reading the original post either remembers a related nugget of Internet that they think people would also find interesting or else they find it while googling, inspired by the initial post.

I was not aware of the other post when posting this. I stumbled upon this PDF coincidentally and thought it would be valuable for HN.

>Is this something that HN does on purpose?

No, the HN algorithm is extremely simple and does not consider syntactic similarities of the titles.

It makes some sense to have a leveling-of-expectations course for freshmen that both inflates their GPA and says "Okay, whatever you were told in high school, we expect you to use this."

Affirmative action and many related criteria mean that students aren't just admitted by considerations of ability.