Where can I find an exceedingly gentle introduction to writing proofs? My math education is limited to non-AP high school classes many years ago. I’m now trying to relearn what I’ve forgotten (and then go beyond that) so I would like to grasp the underlying principles rather than only doing rote exercises to memorize techniques for getting answers without deeply understanding what’s behind those techniques. But I need to start with really, really simple proofs that explain the very basic techniques. For example, I know things like (a^b)^c = a^bc, and that I can solve for x in x/a = b/c by cross multiplying so that xc = ab, but I can’t prove either of those things. How do I learn to develop their formal proofs? Any of the texts I’ve seen so far are much too advanced for me.
Ask HN: How to learn mathematical proofs from scratch?
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Re: Ask HN: How to learn mathematical proofs from scratch?
#2Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair amount of what is called mathematical maturity.
The most basic subject to understand mathematical proofs is Euclidean geometry. There you will learn the basics of proofs and what it means to prove something.
Let’s look at x/a = b/c. You want to show that this equation has the same exact solution set as xc = ab. In order to prove this rigorously you’ll need to prove things about associativity. You’ll also need to prove that a unit isn’t a zero divisor in the real numbers. What we see is that to prove seemingly simple statements requires some machinery and to understand the necessity of this machinery requires mathematical maturity.
But maybe you don’t want to rigorously prove the above. Maybe you just want to understand why it is plausible that this is true. For that, pick up a beginning algebra book and actually read what it says and try to understand it. This is hard to do on your own.
Here’s a plausible explanation for why x/a = b/c has the same solution set as xc=ab. Note that a and c must be nonzero because we can’t divide by zero (this requires proof!). We note that
(x/a) times a
Is the same thing as x times (1/a times a). This is due to associativity. A nonzero number times it’s reciprocal is 1. And 1 times anything is itself. So x/a times a simplifies to x.
So,starting with
x/a = b/c
I can multiply both sides by a. I can do this since a is invertible and multiplying by an invertible element preserves equality (requires proof!). So what I get, after simplifying, is
x = (b/c) times a
I can rearrange things (by associativity) to write this as
x = (ab)/c
Now multiply both sides by c to get (I skipped a step by multiplying and simplifying at the same time)
xc = ab.
Re: Ask HN: How to learn mathematical proofs from scratch?
#3Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair…
Thank you for that. It’s helpful. I suppose my next question is how do I attain mathematical maturity most efficiently?
Re: Ask HN: How to learn mathematical proofs from scratch?
#4Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair…
+1 for Euclidian geometry. Once you “prove” that you can find angle B and C knowing angle A it’s a pretty eye opening experience. This is why this is/was? emphasized in middle school geometry.
Re: Ask HN: How to learn mathematical proofs from scratch?
#5Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair…
Thank you for that. It’s helpful. I suppose my next question is how do I attain mathematical maturity most efficiently?
Do you want something at the level of beginning algebra? Or something more advanced?
Re: Ask HN: How to learn mathematical proofs from scratch?
#6Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair…
Thank you for that. It’s helpful. I suppose my next question is how do I attain mathematical maturity most efficiently?
One thing that will help a lot is being thorough. Try to understand as well as you can, try to fill in missing steps and details, make sure the text itself is actually clear and correct and is not missing something. Try to guess how a proof might start before looking at it.
Broadening your field of view will also help a lot. Some introductory abstract algebra (groups, rings, vector spaces) would be a solid next step in my opinion, because there you will have sets of axioms and lots of proofs and you will learn about properties such as commutativity, associativity, inverses and identities in a more abstract and general way.
Re: Ask HN: How to learn mathematical proofs from scratch?
#7> For example, I know things like (ab)^c = abc
Is the formatting wrong on this? Because this isn't an identity.
Re: Ask HN: How to learn mathematical proofs from scratch?
#8Earlier quoted context omitted.
Thank you for that. It’s helpful. I suppose my next question is how do I attain mathematical maturity most efficiently?
Do you want something at the level of beginning algebra? Or something more advanced?
I know how to do beginning algebra and geometry but I don’t necessarily know why the various rote techniques work. So beginner as far as intuition. But also beginner for technique for things like linear algebra and calculus that I never learned at all.
Re: Ask HN: How to learn mathematical proofs from scratch?
#9Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair…
+1 for Euclidian geometry. Once you “prove” that you can find angle B and C knowing angle A it’s a pretty eye opening experience. This is why this is/was? emphasized in middle school geometry.
Any particular books you can recommend?
Re: Ask HN: How to learn mathematical proofs from scratch?
#10> For example, I know things like (ab)^c = abc Is the formatting wrong on this? Because this isn't an identity.
Yes - fixed now.