Harristotelian Logic

Talking about logic got me to thinking about how easily it is to prove that mathematics is consistent, as that is equivalent to saying that every valid mathematical statement reduces to a tautology.

And it is so obvious it's weird, like, of course every valid mathematical statement reduces to a tautology, so all valid mathematics is consistent.

My key example that came up talking about logic and equals being equal, is, with the circle:

Given x^2 + y^2 = z^2, note that by substituting in numbers you always get to a tautology.

e.g. x=3, y=4, z = 5

4 + 16 = 25

25 = 25

and letting the equals mean equal requires that EVERY valid mathematical statement reduces to a tautology, or as mathematicians like to say, an identity, when you plug in the numbers.

Otherwise, like if you end up with 3=5 then hey, it wasn't a valid mathematical statement!

Then by definition, mathematics is consistent.

So, mathematics is consistent as everything reduces to a tautology.

Oddly enough, everything does. And if you figure that out then you understand one of the most remarkable things about logic, mathematics and reality itself.

Everything reduces to a tautology.