Is it natural for self-learners to forget most proofs of the theorems they learn?

Question

When I read a theorem and read its proof and fully understand it, am I supposed to know the proof even after a long time or is it natural to forget the it?

I ask this question as I'm a self learner who forget many many proofs of the theorems I study even after reading it and understanding it (in many cases I rewrite the proof with full details entirely in my own words but even here I forget it after some time). Forgetting proofs annoys me really and I don't know if this is the natural thing or this is a something I've to deal with .

I also wonder, What should math student do with proofs? What is she expected to do with it? read it understand it and forget it or read it understand it and keep it forever? or just keeping the idea of the proof and reconstruct the details herself when needed?

To be concrete, in the last few weeks, I was studying Boolean algebra, ordinals and Godel first incompleteness theorem, Is it natural that I forget most of the proofs after sometime only keeping the main ideas and the main results from those topics? Is that what math student are supposed to do or am I facing a problem?

I'm going to major in math next September and I really want to know what I'm expected to do with proofs while learning for the coming years.

As a wider question, What are the outcomes math students should gain from studying some mathematical topic?

Answer 1

You will retain something as long as you practice it. It just so happens that for many, many theorems, it's the statement of the theorem that matters more than the proof.

I think a good example is the Fundamental Theorem of Algebra. The proofs for it are obnoxiously technical compared to how easy it is to state it. (You could explain the theorem to a talented high school student). The one standard proof requires half a semester's worth of complex analysis, while the other requires a whole semester's worth of Galois Theory.

So for a theorem like this, it's even seen sometimes where the theorem will appear without proof at the start of a book. And the result is then freely used.

This is perfectly practical thing to do. The main goal of any course in mathematics should not be to labor over the technical nuances, but to give the flavor and the style of the theory. And for so much of algebra and calculus, it's not really important why polynomials factor completely. It's just really useful that they do. If it hurts to work with an unproven assumption, the best I can offer would be to take something like this as an axiom -- albeit a redundant one.

Another example that comes to my mind in the study of algebraic curves is Hilbert's Nullstellensatz. (It is actually a kind of multivariable version of the fundamental theorem of algebra). The theorem, used as a black box, is the heart of the elementary theory of algebraic curves. But the proof detracts from the main content of curves and dives into commutative algebra instead. If you want to study commutative algebra, it's a great topic to cover. But if you want to study curves, take it on faith.

To backtrack a bit, I think most proofs aren't worth memorizing in full. Once you have enough experience in a particular area, you will know what techniques will generally work for what kinds of problems. You can simply forget all the details and work them out as you need them. At least, for the set of theorems which have so-called "follow your nose" proofs.

Some theorems have proofs with a particular trick, and it's best to remember a key phrase to remind you of the trick. In linear algebra, to prove the rank nullity theorem, all I ever keep in my head is "take a basis for the nullspace, then extend it". That phrase is enough for me to reconstruct the rest of the proof.

But all in all, don't fret over forgetting things. Focus on what you find to be important, read the rest once and forget it. You'll retain the spirit of the proof, even once all the details are gone.

Answer 2

As an undergraduate math major, there's a radical change in structure around late-sophomore to early-junior year (U.S. experience here). Prior to that point, most of the work is to calculate things with given formulas or algorithms. After that point, it switches to mostly proof-writing exercises. The big picture here is that the junior-senior years of the program are meant to introduce and build skills for possible graduate study, and someday, hopefully, discovering and proving novel theorems.

What should math student do with proofs? What is she expected to do with it?

To a large degree one is likely expected to use the tricks/techniques in those proofs to prove closely-related exercises, and expand those skills as time goes on. (I'd say secondarily you'll be expected to sketch and explain those proofs to hypothetical students or skeptical interlocutors, but that will be more the work of a teaching assistant than classwork.)

As a very simple example: Say you're shown the proof that $\sqrt{2}$ is irrational. The next exercise is probably for you to prove that $\sqrt{3}$ is irrational (which is almost the exact same piece of work, with a few numbers slightly modified). A test question might ask for a proof that $\sqrt{5}$ is irrational. A somewhat more advanced exercise would be to generalize the proof for the square root of any non-perfect-square integer $n$, or other roots above $2$ (which again are very similar, but veer off a bit more from the original each time).

As you proceed, you'll be expected to handle more sophisticated new proofs, with larger amounts of creative thinking involved, and a deeper toolkit of tricks/techniques from which to draw on.

Answer 3

I also wonder, What should math student do with proofs? [...] just keeping the idea of the proof and reconstruct the details herself when needed?

Yes, and not just a mathematics student. In my opinion, to learn mathematics properly we must not only make sure we fully understand the proofs of the theorems covered, but also ensure that we can reconstruct it on our own, and not from memory!

To be concrete, in the last few weeks, I was studying Boolean algebra, ordinals and Godel first incompleteness theorem, Is it natural that I forget most of the proofs after sometime only keeping the main ideas and the main results from those topics?

It is natural, and in fact I would say it is a good thing if you pinpoint the core bits. For example, I would say that the core behind all the basic facts about ordinals are:

1. We can perform transfinite induction/recursion along any well-ordering.

2. Any two well-orderings are comparable by embedding, and embedding implies isomorphism to an initial segment (or the whole thing), and any set of well-orderings have a minimal element under embedding.

3. Given any set $S$, the well-orderings on subsets of $S$ modulo isomorphism forms a well-ordering $W$ into which $S$ injects, and hence $S$ is well-orderable. This is the crucial theorem that enables transfinite iteration through any set. Choosing the minimum initial segment of $W$ that bijects with $S$ permits iterations relying on the 'number of steps' so far always being 'less than' (no surjection onto) the 'total number of steps'.

4. In ZFC, we can construct a canonical representative for any well-ordering $(W,<)$ by transfinite recursion along $(W,<)$, namely by $f(x) := \{ f(y) : y∈W ∧ y<x \}$ for every $x∈W$, and then $ord(W,<) := \{ f(x) : x∈W \}$. Note that both the transfinite recursion and the final step need the replacement schema. These canonical representatives are also called von Neumann ordinals, and we can define cardinals as ordinals that do not biject to any smaller ordinal.

Similarly for the first incompleteness theorem:

1. Godel's version requires ω-consistency and the β-lemma for encoding into PA, and translating the Y combinator into PA yields the fixed-point lemma, which can be applied to $( A ↦ ¬⬜A )$ to give the incompleteness theorem. Along the way you can check that the Godel sentence is (equivalent to) a $Σ_1$-sentence because $⬜A$ is a $Π_1$-sentence.

2. Rosser's version requires only consistency and can be obtained either by using "has a proof and no shorter proof of its negation" in place of "provable".

3. Arithmetic with addition and multiplication is in fact a red herring to the incompleteness theorem, and any computable system that can do basic reasoning about finite binary strings is already essentially incomplete. With ω-consistency we can reduce to the halting problem. With mere consistency we can reduce to a weaker problem which I call zero-guessing here. You can see that all I need to remember to recover the full proof of the generalized incompleteness theorem is the zero-guessing problem, as all the rest is quite easy to figure out.

4. Reasoning about provability within a theory that interprets arithmetic is cleanly represented in provability logic as the Hilbert-Bernays' provability conditions plus the modal fixed-point theorem. This representation makes it easier to grasp, and also makes it easy to figure out how to prove Lob's theorems (both internal and external forms) by simply 'translating' Curry's paradox, namely applying the modal fixed-point theorem to obtain $B ⇔ (⬜B ⇒ A)$ and then working from there.

Basically, since I know I can obtain all the basic facts from these core ideas, there is not much I need to keep around in my head for that purpose. Of course, the more we learn, the more core ideas we need to remember, but it should be much less than the amount we can reconstruct from them.

The hard part is in identifying these core ideas, since many textbooks do not do so. Sometimes, a textbook may also choose a longer route because it yields stronger results, but it may be better to understand the key mathematical structures before going deeper. For instance, arithmetization of syntax (via Godel's β-lemma) is essentially needed only to establish that PA can do basic reasoning about finite strings, so in my opinion it is not necessary to understand that at all if we simply want to understand the incompleteness phenomenon. Yet we should know it and isolate its relevance in the manner I just did.