Historically, what did mathematicians do before accepting the Completeness Axiom?

Question

In the textbook "Advanced Calculus" by Patrick Fitzpatrick on page 7 it says:

It has been "known since antiquity, there is no rational number x having the property $x^2=2$

This is in reference to the polynomial $p(x)=x^2-2$ and trying to solve for the roots.

On page 8 of the text,

if r is the length of the hypotenuse of a right-angled triangle whose other two sides have length 1, then $r^2=2$, and so the length of the hypotenuse is not a rational number.

I found this Wikipedia page indicating that the Completeness Axiom was recognized in 1817 by Bernard Bolzano.

So this axiom can be used to claim the geometric and algebraic problems above have answers that are irrational.

My question: What did mathematicians do before 1817? When they encountered the right-angled triangle or the polynomial above, did they simply say "there is no rational number that is the $\sqrt{2}$ and so we cannot move forward"?

Another way of putting the question: What lead Bolzano to the Completeness Axiom and why did we have to wait until 1817 for it to be recognized?

Answer 1

First of all, the Completeness Axiom is considerably more powerful than the statement "the square root of $2$ exists". The Completeness Axiom promises the existence of every irrational number. Before accepting the Axiom, it would be perfectly reasonable for a mathematician to accept that $\sqrt{2}$ exists but deny the existence of, say, $\pi$.

Second, "recognized the importance of" and "invented" are very different things. What Bolzano did was notice that everyone was using this axiom, but that no one was actually putting it into words. Long before Bolzano, people accepted the existence of irrationals; they just didn't have an axiom to point to and say "this is why".

If we want to go back far enough that irrational numbers actually weren't accepted, we have to go all the way back to ancient Greece - as soon as you have the Pythagorean Theorem, you have to accept the existence of $\sqrt{2}$ (unless you're okay with claiming that isosceles right triangles don't exist!). Before that, mathematicians certainly did believe that there was no such number - I can't find a resource specifically claiming "there is no square root of two", but the ancient Greeks did work under the assumption "every number is the ratio of two whole numbers".

Answer 2

It was known before 500 BC by those in the School of Pythagoras that $\sqrt{2}$ could not be expressed as a ratio of integers. And that motivated the need to deal with irrational numbers, i.e., with numbers that were not ratios of integers. However, general attempts at defining irrational numbers were unsuccessful for the next 24 centuries, until the late 1800's when Georg Cantor came up with his axioms for Set Theory. Earlier attempts at postulating completeness--such as the one you mentioned--did not offer a rigorous foundation for irrational numbers. So this axiom did not really change anything. Even before this axiom, people accepted that you could find a rational number that was as close as you would want to a number whose square was $2$. Finite decimal expansions were sufficient for this purpose, and infinite decimal expansions were conceived before this, even though a rigorous definition did not exist. The real breakthrough was Cantor's Set Theory, which ushered in the age of rigorous Mathematics and made infinite decimal expansions rigorous.

Answer 3

Before the late 19th century, mathematicians did not have the same attitudes about number systems that are today inculcated in mathematicians from an early age. There was no desire to define number systems in terms of other, more fundamental systems such as set theory. There was nothing like the modern consensus that the real number system was of central importance compared to systems that were either smaller (the rationals) or larger (e.g., the surreals or the hyperreals). A typical calculus textbook in the 19th century used a number system that included infinitesimals. People talked about "the line" as a philosophical entity, without any recognition that there could be a real line, a hyperreal line, or topologically exotic lines.

A mathematician in the era of Gauss, for example, would simply work with "numbers," and would not consider it worth worrying about whether $\sqrt{2}$ was a "number." People did want to work with consistent mathematical systems, but they didn't have anything like modern logic or model theory, so, e.g., it would not have occurred to them to try to prove that different number systems were equiconsistent.

There is a popular misconception that nobody explicitly constructed the reals before Dedekind. Actually, Simon Stevin (1548-1620) constructed them as infinite decimals. It's just that nobody in that era considered this sort of thing to be particularly important or fruitful. There is also an argument to be made that Euclid actually invented the real line, because there is a sense in which Euclidean geometry is exactly equivalent to the theory of the reals -- although Euclid's original formulation was not up to modern standards of rigor, unlike formulations such as Tarski's axioms.