As I have looked at more and more complicated mathematics I have often found formulas and theorems and so on where I understand the proofs for them but I still don't feel like I understand how they really work. Unless I am given a somewhat intuitive explanation, I tend to feel alienated. While I don't doubt the validity of the proofs, to me it often feels unsatisfying if I cannot explain concepts intuitively. Of course, some proofs go hand-in-hand with explanations (they are merely a way of setting out formally what we already know). More often though, it feels like I have to accept they 'just work' because they have been shown to be mathematically rigorous. Am I being too demanding, or is there a way to carry on this desire for intuitive explanations as I progress further?
To illustrate, here is an example of a proof that there are infinitely many primes (Euclid's Theorem) which also makes intuitive sense to me:
- Consider any finite list of prime numbers $p_1,p_2,...,p_n$
- Let $P$ equal the product of the list. $P = p_1p_2...p_n$
- Let $q = P+1$
- If $q$ is prime, then the list is incomplete
- If $q$ is composite and there were a number $p_x$ on the list that could divide it evenly, then $p_x$ would have to be able to divide $P$ and $P+1$, meaning it would also have to divide $(P+1)-P=1$. Since no prime goes into 1, no number $p_x$ meets this requirement. Therefore, we can conclude that if $q$ is composite, it must be divisible by a prime not on the list. Hence, no list is complete
I understand this in layman terms to mean if you multiply the list together and add 1, then every prime number in the list would be 'one off'. It seems the proof formalises this insight.
However, other proofs that involve lots of rearrangement don't seem to provide any idea of 'why' the theorems/formulas work. It just seems like with a lot of manipulation of an equation, the theorem/formula springs out. Example (proof of the Cosine Rule from brilliant.org):
Here, it almost seems as if we got lucky and because we ended up with the identity $sin^2x + cos^2x \equiv 1$, it all simplifies down and we end up with the Cosine Rule.