How the square root of 2 became a number

It's funny that everybody remembers Pythagoras for the right-triangle theorem, but that wasn't what made him important at the time. The right-triangle square equality had been known empirically for centuries. What he proved that was so completely earth-shaking is that for some right triangles A, B, C there is no rational Q for which QA = C. That was what was so important about his proof.

According to Van der Warden "Science awakening"[1] Ancient Greeks treated numbers as some kind of "dirty" (Real) model of a platonic Ideal of a quantity. Numbers were invented by filthy traders and accountants while wise philosophers used geometry to reason about quantities.

This attitude to numbers can be felt even now when we are taught to solve straightedge and compass construction problems. Greeks had no issues dealing with square root of 2 with geometry or "geometric algebra" how Van der Warden names it.

[1] https://archive.org/details/scienceawakening0000waer

I never learned of these formal definitions in high school mathematics. Nor in the lower level college ones that I took. There’s a beauty to this perspective— irrational numbers are what rationals are not.

I happen to be reading Poincare's Science and Hypothesis right now and he introduced a way of defining the square root of two that I found enlightening. In less articulate form: consider two sets, one of which contains all numbers whose square is less than two and one of which contains all numbers whose square is greater than two. The square root of two is then the symbolic name for the element that divides those two sets.

Poincare says it better in the book though.

Irrational numbers have become a constant nuisance for me in Isabelle. I really wish that the Greeks' initial hypothesis that everything could be expressed in rationals was actually correct.

Something I always love to ponder: Would aliens who perceive reality vastly differently than humans come up with different number systems?

For example, for the longest time we thought of numbers as 1 dimensional and refused to consider 2-dimensional numbers. Even know we try to shunt them off to the side as much as possible ("complex", "imaginary") even though they model reality more closely.

Might a hypothetical alien race begin with 2D numbers? or something entirely different?

Interesting timing for this video that talks about bias in stem history (for example how naming discoveries/inventions is done in different cultures)

https://www.tiktok.com/t/ZTNLLvDYm/

How do you prove irrational numbers doesn’t repeat down the line?

I'm with Dedekind on this one -- Cantor's work, by and large, was hot garbage and led and continues to power some of the most naval-gazing mathematics ever invented.

Dedekind had a great idea that at its heart was a constructive notion of what a "number" was in terms of our ability to approximate it. That's the core of intuitionism, and after a long dark interval has finally come back into prominence in modern mathematics.

Why is it better to invent a weird new class of numbers (irrationals) rather than just identify that there is something wrong with how we think about this issue?

Put another way, why don’t we reject out-of-hand the notion that sqrt(2) cannot be calculated, given that right isosceles triangles do exist in reality and their hypotenuse has a definite length?

Put yet another way, why not just say sqrt(2) equals 1.41 (or however much precision you need) + some infinitesimal amount?

lol finally a quantamag math article where I can follow the math

> The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction.

Let me try it! If I make a square with sides 1 inch long, then measure the diagonal with a tape measure I get... 1 and 6/16ths! Only whole numbers and the ratios between them involved there, so I guess that's all the Greeks needed after all.