How were irrational numbers accepted by mathematicians?

Question

What was behind accepting the existence of irrational numbers historically? Especially numbers that are not constructible on the real number line, say for example $\sqrt[3]{2}$. Was it a (somewhat) rigorous proof, or was it only an agreement or convention among mathematicians in those days?

Answer 1

Let me clarify a couple of things. No student of Pythagoras discovered irrational numbers, although this is a common misconception, Pythagoreans and even Euclid did not associate numbers with geometric points or segments, the only numbers available were positive integers. Instead they had magnitudes of different dimensions (segments, areas, volumes), and ratios of numbers and magnitudes. The latter produce what we call rational numbers, but also things like the ratio of circumference to diameter, which to them was not a number. But ratios could be compared, and what Pythagoreans discovered is that the ratio of the diagonal to the side of a square was not equal to any ratio of integers.

So the problem was in fact the converse, geometry and segments were the starting point, and integer ratios did not provide enough equipment to account for all geometric segments. Eudoxus then worked out a theory of ratios, presented in Euclid's Elements, that allowed to deal with ratios of magnitudes without relating them to any number ratios at all. This being said, if we modernize $\sqrt[3]{2}$ in the same vein that what Pythagoreans did is modernized in terms of $\sqrt{2}$ a construction of it was known since Archytas of Tarentum (c.428–347 BC), it just wasn't a straightedge and compass construction. Archytas constructed two segments that (we would say) have this ratio by intersecting a torus, a cone, and a cylinder. Greeks knew it as solving duplication of the cube problem. Later Menaechmus (c.380-320 BC) gave a simpler construction by intersecting two parabolas, or a parabola and a hyperbola, you can see details here Why did the ancient Greeks originally become interested in conic sections? Greeks and later mathematicians had no reason to doubt that conics intersect any more than that lines and circles do.

So the problem later mathematicians faced was not finding something like $\sqrt{2}$ and $\sqrt[3]{2}$ on a geometric line, but of accepting what they already knew produced such ratios as new kinds of "numbers". The process was indeed slow, medieval Islamic and European mathematicians called things like $\sqrt[3]{2}$ "absurd" (deaf-mute, unheard of), "irrational", "irregular", "inexplicable", etc., and even Cardano in 16th century uses them sparingly. Stevin's L'Arithmétique (1585) is considered the first work to advocate what we now call real numbers in their totality, by identifying them with infinite decimal fractions, and using iterated tenfold subdivision to locate the corresponding point on the line. According to van der Waerden this "general notion of a real number was accepted, tacitly or explicitly, by all later scientists". For details see Stevin Numbers and Reality by Katz and Katz, who add that "concerns about the reality of numbers generally preoccupy cognitive scientists and philosophers more than mathematicians".

Answer 2

For details, see Doubling the cube :

A significant development in finding a solution to the problem was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths $a$ and $2a$, the duplication of the cube is equivalent to finding segments of lengths $r$ and $s$ so that

$$a : r = r : s = s : 2a.$$

In turn, this means that : $r=a \sqrt[3]{2}$.

Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line.

See also : Thomas Heath, A History of Greek Mathematics. Volume I (1921), page 262-on for details and discussion.

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If I read correctly the solution, the construction (that obviously is not constructible by compass and straightedge only) ends up with two segments $RS$ and $RW$ that are in proportion $1 : \sqrt[3]{2}$.

Thus, subject to Conifold's remarks above about the fact that, for ancient Greek mathematics, $\sqrt[3]{2}$ is not a number, it seems evident to me that this is a "sufficient" basis for accepting the existence of a segment of that length.

Answer 3

I do not think there was any problem in "accepting the existence of irrational numbers" before the clarification of the axiomatic method (1910-1940).

The book X of Euclid's Elements looks more like an attempt to classify numbers (=size, magnitude) according to the complexity they were built. It was easier to add/subtract with the ruler, than to multiply/divide with the compass. And the famous problems like doubling the cube, squaring the circle, trisecting the angle were more related to finding the construction than to doubt about the existence of a solution.

I think mathematicians who were computing numbers on a board or abacus, like the Chinese and Indian, accepted rather quickly what we now call algebraic numbers, and invented algorithms to compute root of n-th degree equation long ago. Again, the complexity of the computation was much more of a concern than the existence of the solution.

The mathematicians who were still computing with geometric instruments, like medieval arabo-islamic mathematician, made some progress in solving fractions, quadric, cubic, bi-quadratic with the claim of inventing new methods, not new numbers.

With Al-Khwarizmi (Algorithm) and his book on computation with al-jabr (algebra) to compute geometric constructions with numbers, starts a language which will need 10 centuries for P. Wantzel to express (and to prove) that the duplication of the cube and trisection of the angle were impossible to solve with a straightedge and compass. Yet, Wantzel never doubted that $\sqrt[3]{2}$ and $\cos 10$ were otherwise perfectly standard numbers.

And the modern position is that, with the language of mathematical logic, you can rigorously classify numbers in set of increasing "complexity", and look back in history to see how these class of complexity did appear in time. You can also (mathematically) define what is an "undecidable" question, and then have yet a new class of complexity classes.

But, to come back to your question, I think that was about behind irrational numbers is not accepting their existence, but rather the complexity of computing with them, and first of all to define them.

Answer 4

Greek mathematicians (e.g. Euclid and followers) accepted points on a line that aren't constructible (or at least they didn't know to be constructible) as a matter of course. Lines were considered to be continuous, no "holes" in them. They didn't identify line lengths with numbers at all, as has been noted. That is a much later development, where initially the people starting to define the calculus assumed without second thought that "numbers" (real numbers we would say today) corresponded one to one with points on a line (e.g. for analytic geometry). That this has it's logical snares became known rather slowly, until we got today's definition of real numbers by Dedekind and colleagues to resolve the matter.