how mathematicians “prove” that mathematical proofs, such as the Pythagorean Theorem, are true for any input?
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In a proof, we use previously known true statements to go from assumption to the thing which we need to prove. It basically boils down to several steps where we show if A is true, then B is has to be true also. Simplest examples to check out are probably some basic proofs in algebra/number theory, like the proof that square of every even number is even.
There is also ‘proof by contradiction’ where you start off by assuming that the thing you want to prove is NOT true.
For example, √ 2 is an irrational number. To prove this by contradiction assume that it is rational, i.e. it equals a/b then there are a series of logical steps that ultimately show that a contradiction occurs, for example that ‘a’ is both odd and even.
Using the same argument it is possible to show that √ n is an irrational whenever n is not a square number.
They do so by using variables for their proofs rather than discrete numbers. If you can prove a specific algebraic relationship to be true, then by definition it must be true for all numbers that you could input no matter what.
The idea is to show that even if you don’t know the inputs, you can still know a certain thing for certain. Usually we do this by assigning the unknown inputs variable names, like A B and C for the side lengths of the triangle in the case of the Pythagorean theorem. Then we write down some initial observations about how those variables are related, and do some math to show that the result we wanted to prove must follow from those initial observations. The Pythagorean Theorem is a geometry theorem so it’s much easier to prove using visuals. If you want a good example I suggest just looking it up, there are lots of great accessible proofs online.
Proofs are a series of agreements you can make based on logic and previous agreements.
For example.
People should agree that A = A, and B = B. Silly as it is, its important because I’m about to start including B’s and A’s left and right.
If we agree on that then we might agree A + B = AB
We might also agree on how x and y are sorted out on a graph.
We can then also agree what a slope is, Go over A times, and up B times….we’re on to something.
I’m not going to go over the whole proof, but eventually we’ll find going over A and up B in a consistent way makes consistent patterns; patterns that are the Pythagorean theorem.
The wiki has many proofs by construction as well as formal proofs.
They prove it by showing what rules don’t say, but that rules still mean. For example, if balloons aren’t allowed in the house, we can also say balloons aren’t allowed in the living room. The rules don’t say balloons aren’t allowed in the living room, but since the living room is in the house, we can prove that balloons also aren’t allowed in the living room.
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