# Why do some proof method feel more satisfying than others?

Let's say we are asked to show that 1+2+3.. =n(n+1)/2, then a very simple way to prove this is to use induction. The proof is simple, consider P(1) and show P(n+1) from P(n). However, it feels quite unsatisfying.

On the contrary, I think one would appreciate many of the proofs mentioned in this se thread much more. For instance, I would imagine that many people would appreciate and say that the geometrical proofs are more interesting than the induction proof.

Why is this the case?

• "Satisfying" in the eye of the beholder ... Also, when the going gets tougher, the proof method may not always be intuitive. Apr 18 at 20:35
• Besides the fact that how satisfying something is is subjective, this sounds like more of a Psychology & Neuroscience question than a philosophy one. Apr 19 at 1:52
• "use induction". Is that what you meant to say? Did you mean "deduction," arriving at a conclusion by starting from fundamental axioms? Apr 19 at 2:10
• @MarkAndrews Not OP but that was probably intentional: en.wikipedia.org/wiki/Mathematical_induction and the term induction is fitting as you move from a special cast to proving a universal principle. Apr 19 at 9:08
• Proof by induction was always my favorite. I find it quite satisfying. Apr 19 at 9:10

I think what is satisfying lies in the eye of the beholder. Though two versions of what a beholder might find satisfying are exemplified in this answer:

https://math.stackexchange.com/a/34400/979158

The first is the beauty of simplicity. Like without any (specialized) prior knowledge like geometrical series, graph theories, combinatorics and so on, you're able to visually see, how the proof works and that it works. So from the standpoint of a teacher or anybody trying to exemplify and explain that relation to someone else that's absolutely beautiful.

The thing is, that is practically equivalent to those:

But you could get away without knowing any of the mathematical notation or other levels of prior knowledge and abstraction, yet you wouldn't have to cut on the essentials of the proof itself.

The problem, or rather the beauty, of mathematical and complete induction is that it is not problem specific, but can be generalized to a whole range of relations. The idea is essentially that under the assumption that the relation holds for n (or even for all elements m ∈ N, where m >=0) then it also holds for n+1. So what's left is to prove that it works at all, by additionally confirming a base case, usually the start of the natural numbers by 0 or 1.

Yet as said if you follow this algorithm you'd have proven that it works, but you wouldn't have needed to understand what it does it in the first place, how people came up with that relation, how and where you could apply it or why you wanted to prove that in the first place. It's something like being a car mechanic without a drivers license that has never driven a car in their entire life. You know what you need to do to fix it, but you would only know in the abstract what "fixed" even means.

Now again eye of the beholder, for some the journey is what is satisfying rather than the destination, so that could be a nice proof, but practically you'd need prior knowledge of what it does and if you don't have that it makes you feel like having applied something without fully understanding what it does (and you don't have to, in order to apply the algorithm) and now having your fingers crossed hoping that it will hold.

And the other source of satisfaction is the second proof in that initial post, that is a proof that you came up with yourself, or that put your prior knowledge to good use, that was challenging and complicated but in the end all parts fell in place perfectly. There's just a beauty in seeing something working as expected, acquiring a complicated skill, being able to use your knowledge and the like.

But again that leaves you with the unsatisfying answer from the beginning, what is or isn't satisfying lies in the eye of the beholder and what scares of some might be the source of satisfaction for someone else and what has a simplistic beauty to some might be "too obvious" for others. Also some just care for the fact that it works, while others like it when they understand how it works and how they can make use of that.

• As a programmer, I often build programs that I will never use. Sort of like your car mechanic example. Apr 22 at 2:47