The need for rigorous proofs in Sets

Question

I am a $10$th grade student who is studying Elementary Set Theory. In a lot of questions, I come across, the solution or rather the proof requires being proved by identities or by assuming an element of a set, i.e $a \in \{A\}$ and so on. What I am unable to understand is why we require rigorous proofs for Sets Theory and to broaden the scope of the question why do we require rigorous proofs at all in Mathematics?

Answer 1

Because we don't really know something is true until we prove it. Here's my favorite example of this:

Question: Given a circle, what is the number of regions determined by n points on its perimeter?$^{[1]}$

We begin by looking at different cases, starting from when $n=2$:

The pattern seems clear, doesn’t it? The number of regions appears to be doubling every time we add an extra point. So we suspect that with $n = 5$, there will be $16$ regions. Let's check to make sure:

That was tedious to do, but clearly we're right!

Okay, so for $n\geq2$, we have that the number of regions with $n$ points, $R(n)$, is given by $2^{n-1}$. Why bother looking for a rigorous proof of this when we can just accept that it's true and get on to more interesting results in mathematics? We have papers to publish and 80 years to live, so we shouldn't waste time proving minutiae that are obviously true.

So we conclude with that with $n = 6$ the number of different regions will be $R(6) = 2^{6-1} = 32$.

But it isn’t. It’s 31:

And if you're wondering what the general formula for the number of regions is, look below—it isn’t the simple one we had in mind at all. It’s $$R(n) = \frac1{24}(n^4 - 6n^3 + 23n^2 - 18n + 24)$$

And that’s why mathematicians need proof. We require proof because things aren't always as they seem. Mathematicians won't accept something as true just because it's true for all numbers less than $2\times10^6$. We need cold, hard, logical proof that something is always true, because for all we know, a claim might turn out to be false when $n = 2\times10^6+1$. Or we might think that a general idea is true, and say so with a condescending "obviously," when in fact, it turns out that sometimes the math isn't what we expect.

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$[1]$ Problem from David Acheson's "1089 and All That: A Journey Into Mathematics."