Related but not exactly overlapping book recommendation: The Art of Writing Reasonable Organic Reaction Mechanisms.
Ask HN: How to learn mathematical proofs from scratch?
61–70 of 125 posts
Re: Ask HN: How to learn mathematical proofs from scratch?
#622 books I recommend: how to prove it, and the book of proof, get a real analysis book, and if possible take a course. it took me around 5 years to be able to prove stuff... you need relaxation, fluid thinking, and a breadth of knowledge of facts you can use to prove stuff we can try your example we want to show that (a^b)^c = a^bc let's work on the LHS by the definition of an exponent, we know that a^b is just a * a…
side note; there are some subtleties here for the rationals and serious difficulties with this method for irrationals. I agree this is a good start for someone thinking about how to prove some simple math ’rules’ though, so please don’t take this as discouragement; rather encouragement to keep opening up minds when discussing math!
Re: Ask HN: How to learn mathematical proofs from scratch?
#63Just skip pen and paper proofs and go play around with Agda or Lean or something.
You'll learn something much more rigorous and concrete.
Re: Ask HN: How to learn mathematical proofs from scratch?
#64Boy, you have a long way to go.
Master precalculus, with emphasis on conceptual understanding and computation. It's the base.
You may want to read There’s more to mathematics than rigour and proofs by Terry Tao.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
Re: Ask HN: How to learn mathematical proofs from scratch?
#65Many people here are recommending texts like How to Prove It by Velleman. From this book: "Many students get their first exposure to mathematical proofs in a high school course on geometry. Unfortunately, students in high school geometry are usually taught to think of a proof as a numbered list of statements and reasons, a view of proofs that is too restrictive to be very useful."
One solution to this problem is to learn from a book purely on proofs, as Velleman suggests. It seems to me, however, that given your background and your wish for an 'exceedingly gentle introduction', this method might not be the best for you.
While I agree with the view that the attachment between geometry and proofs is detrimental to students' learning of both topics, I would like to make the argument that in a case like yours, learning proofs through geometry is actually a great place to start.
Proofs are not traditionally linked to geometry without reason; in school, geometry is the closest thing you get to "real" mathematical thinking. Since you are already familiar with geometry, revisiting it, this time through a lens focused on proofs, would be an effective way to bridge the gap between traditional school mathematics and proof-based thinking.
At this point, it comes down to finding the right geometry book. I highly recommend Introduction to Geometry by Richard Rusczyk: https://artofproblemsolving.com/store/book/intro-geometry. While it is designed for the advanced high school student seeking to learn geometry in a different way than what is taught in school, it just so happens that this makes it a great book for your purpose as well.
Having already learned geometry, you will be able to focus more exclusively on the proof aspect of the book. Take a look at some of the excerpts listed in the link above to see if the style of the book suits what you are looking for. After learning geometric proofs, you will then be able to easily extend the same ideas to proofs in other subjects.
Re: Ask HN: How to learn mathematical proofs from scratch?
#66At Imperial College London there's an intro to proofs course taught with interactive exercises supported by a proof assistant, Lean. The exercises (and the proof assistant) are freely available online at https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g... . Disclaimer: I didn't fully work through this game, I never studied at ICL and I can't vouch for its effectiveness, I simply heard about it and thought…
This might not be the best introduction for someone who hasn't proved "a = b iff a - c = b - c", and e.g. may not be familiar with "if and only if," or the fact that P => Q is considered true if P is false, even if Q is also false.
> P => Q is considered true if P is false, even if Q is also false
It's a good thing then that this framing is rather unlike the intuitionistic mathematics actually suggested.
Re: Ask HN: How to learn mathematical proofs from scratch?
#67Undergrad discrete mathematics and symbolic logic made proof writing click for me. You have a set of things known to be true as handed down from on high, you have a set of operations to transform those true things into equivalent statements, you then go about the work of using those transformations to connect true statements together into a massive graph and extend that graph through speculation if possible. With tha…
I studied CS and it never clicked for me.
For me, a proof is essentially transforming one formular into another one until someone who understands math says "yes, now you have proven it!"
For me, any step is as good as the others.
I wish, I could understand what's happening.
Re: Ask HN: How to learn mathematical proofs from scratch?
#68When you're cross-multiplying, you are relying on a basic identity that if ab = ac, then either b = c, or else a = 0.
So you want to prove that this multiplying both sides is valid?
This is something that is very close to the fundamental axioms of arithmetic (Peano axioms).
To prove it you have to show that the basic rules like associative and distributive property will derive ab = ac from b = c.
x(yz) = (xy)z # associative law
x(y + z) = xy + zy # distributive law
b = c
b - c = 0
For instance:
ab - ac ≠ 0 # subtract ac from both sides
a(b - c) ≠ 0 # distributive law
a ≠ 0 AND (b - c) ≠ 0 # Follows from 0x = 0
b - c ≠ 0 # right branch of AND above
b ≠ c # add c
We relied on some existing rules, like being able to add the same quantity to both sides, but we didn't multiply both sides of the inequality by the same factor; we relied on inferring something by using the distributive property to rearrange the difference of products ab - ac into a product form a(b - c). If a product XY is nonzero, Y must be nonzer, and so must X; if either is zero, then it falls victim to the 0x = 0 rule: zero times anything is zero.
Re: Ask HN: How to learn mathematical proofs from scratch?
#69At Imperial College London there's an intro to proofs course taught with interactive exercises supported by a proof assistant, Lean. The exercises (and the proof assistant) are freely available online at https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g... . Disclaimer: I didn't fully work through this game, I never studied at ICL and I can't vouch for its effectiveness, I simply heard about it and thought…
This might not be the best introduction for someone who hasn't proved "a = b iff a - c = b - c", and e.g. may not be familiar with "if and only if," or the fact that P => Q is considered true if P is false, even if Q is also false.
That may be the case. Otoh, this is being used as an intro course for first year college students, who don't have any background in logic afaik. So perhaps it could work. I don't have enough information to say either way.
Re: Ask HN: How to learn mathematical proofs from scratch?
#70At Imperial College London there's an intro to proofs course taught with interactive exercises supported by a proof assistant, Lean. The exercises (and the proof assistant) are freely available online at https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g... . Disclaimer: I didn't fully work through this game, I never studied at ICL and I can't vouch for its effectiveness, I simply heard about it and thought…
This might not be the best introduction for someone who hasn't proved "a = b iff a - c = b - c", and e.g. may not be familiar with "if and only if," or the fact that P => Q is considered true if P is false, even if Q is also false.
Tangent: ever since highschool I've wondered about the schism between (a) the elements of "showing one's work" for algebra / calculus problems and (b) formal syllogisms to justify each step.
I have to wonder if highschool math would make more sense to some students if (a) and (b) were taught together.