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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?

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    $\begingroup$ In mathematics, nothing is based on agreement, but only on a rigorous proof. See "Doubling of the cube" in Wikipedia, where this proof is explained. $\endgroup$ – Alexandre Eremenko Dec 12 '15 at 13:58
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    $\begingroup$ I'm voting to close this question as off-topic because this question is trivial: the answer is easily found in Wikipedia. $\endgroup$ – Alexandre Eremenko Dec 12 '15 at 13:59
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    $\begingroup$ Please define what you mean by the term "construct a number." It does not have a single meaning. For example, using origami a length equal to $\sqrt[3]{2}$ can be made, so your premise would be wrong in that setting. $\endgroup$ – KCd Dec 12 '15 at 15:47
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    $\begingroup$ After Dedekind gave a rigorous definition of real numbers that allowed for a rigorous proof of the intermediate value theorem, such basic existence questions like for the number $\sqrt[3]{2}$ could be settled. $\endgroup$ – KCd Dec 12 '15 at 15:49
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    $\begingroup$ I think Dedekind came after the term was already established and accepted, see solution of cubic equation was much befor, a rigorous proof would certainly imply the construction of cube root of (2), exactly similar to pythagoras proof of not only discovering the irrational numbers but also CONSTRUCTING IT EXACTLY $\endgroup$ – Bassam Karzeddin Dec 12 '15 at 16:42
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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".

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    $\begingroup$ Very useful & interesting historical description you rarely find it from the net, I shall review it in details to clear out this issue, since I may have reasons of being so skeptical about the real existence of cube root of (2) on the real number line, (say X - axis), even with intermediate theorem & the endless digits representation of its representation $\endgroup$ – Bassam Karzeddin Dec 14 '15 at 6:45
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    $\begingroup$ User Conifold, thanks for a nice answer. It should be pointed out that the relevant work by Stevin is his L'Arithmetique rather than De Thiende (the latter is a practical guide to using finite decimals). $\endgroup$ – Mikhail Katz Dec 14 '15 at 16:05
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    $\begingroup$ @bassam karzeddin In this you are not alone. Philosophy of mathematics called intuitionism rejects actual existence of most real numbers, they exist only "potentially", they are "becoming". Brouwer:"For us the point is the sequence [of nested intervals] itself, not something like 'the limiting point to which according to the classical conception the intervals converge'"dspace.library.uu.nl/handle/1874/26962 They do accept cube root of 2 however, or even pi, because there is a finite law to determine the whole sequence for them, they are "finished" in this way, unlike "lawless" others. $\endgroup$ – Conifold Dec 15 '15 at 19:50
  • $\begingroup$ @Conifold I went through those references (not in details), which seems to me they intuitively CONCLUDE, If we accept things without rigorous proofs (specially in mathematics) then it is only assumptions concluded intuitively, which would create continuously a huge volume of baseless mathematics that one day may come to an end, especially we have the contradiction as rigorous proof of impossibility of such constructions, I think there must be a valid reason somewhere!? $\endgroup$ – Bassam Karzeddin Dec 19 '15 at 12:41
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    $\begingroup$ @bassam karzeddin We do accept things without rigorous proofs, namely axioms of any theory, both for intuitive and pragmatic reasons. Occasionally they do lead to contradictions, like the set of all sets in Frege's original logic, but "baselessness" is a minor technical problem that affects most of mathematics little if at all. E.g. details of Zermelo's fix to Frege is barely noticed by working mathematicians. It is not the body of mathematics that gets adjusted to some valid base, but a suitable base is adjusted to the existing body of mathematics. $\endgroup$ – Conifold Dec 22 '15 at 3:38
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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.


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.

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    $\begingroup$ I would appreciate your answer without being able to upvote it for the history it contains, but please note that my question is too tricky (at least for me), it is like that "how then can we accept its existence somewhere on the real number line even we know for sure it can't be located exactly, you may consider me so skeptical to the limit that I deny completely its existence on the real number line, further I assume its existence is only hypothesis & so unlike $\sqrt[2]{2}$ which has rigorous proof of existence like any rational number on the real line number, noting rational are too dense $\endgroup$ – Bassam Karzeddin Dec 13 '15 at 6:54
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    $\begingroup$ I'm so sorry for wasting your time, I couldn't convey the idea, but you may give me the last chance, One day (few thousands years back), One student of pythagoras announced a DISCOVERY of irrational numbers, it was forbidden, then he was KILLED, later mathematicians found his discovery was very true and ADDED those constructible irrational numbers to their rational numbers to form a set of all real numbers, it was simply starting by CONSTRUCTING EXACTLY sqrt(2), so my question, WHO & HOW (in HISTORY of math), Discovered the cube root of (2) FIRST since it was not added to real numbers yet.. ? $\endgroup$ – Bassam Karzeddin Dec 13 '15 at 13:35
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    $\begingroup$ Nice, you are about to get into my mind, I'm asking about the first proof of cube root of 2 being EXISTING on the real number line, every one almost understands the irrationality of both, but the first Sqrt(2) has exact location, whereas the second cube root of (2) doesn't have location on the real number line, with rigorous proof, (only approximation to whichever number of digits you desire (no limit), I wish you get the HUGE difference between them, let me claim then the first is a real existing being on the real number whereas the second is legendary existing most likely a fallacious $\endgroup$ – Bassam Karzeddin Dec 13 '15 at 14:26
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    $\begingroup$ @Mauro No, the question is about the first proof of the existence of $\root 3\of 2$. The point is that there is no obvious (classical) geometric construction that starting with a segment of length $1$ allows us to produce a segment of that length, and therefore it is not entirely clear on what basis a pre-XIX century mathematician would accept its existence. $\endgroup$ – Andrés E. Caicedo Dec 13 '15 at 15:05
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    $\begingroup$ @Mauro I went roughly throw these references, I don't think they are acceptable or a rigorous proof of existence of cube root of (2) on the number line, they mentioned it clearly that Philo line is impossible construction, specially we know that there is a rigorous proof of impossibility of such construction, I wonder if there is any valid reason? $\endgroup$ – Bassam Karzeddin Dec 19 '15 at 12:23
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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.

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    $\begingroup$ The aim of the question is not to make confusion Ruther than arriving at the fact, and one can't argue on unequal ground, there were quite useful comments on my answer here quora.com/… that had been deleted by a moderator Logan M, asking about history and saying this is modern maths. Actually I was expecting some very experience historian would provide such convincing historical background based on rigor than conclusion, I simply looked on the information provided, but found it unconvincing..? $\endgroup$ – Bassam Karzeddin Dec 31 '15 at 14:51
  • $\begingroup$ Sorry for the above mentioned link, I meant this link: quora.com/profile/Bassam-Karzeddin-1/answers?sort=recency $\endgroup$ – Bassam Karzeddin Dec 31 '15 at 14:55
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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.

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    $\begingroup$ If you mean arbitrary points on a number line, then it is obviously constructible (its length from (0, 0) coordinates, How: remember that rationals are very dense, also constructible numbers that are not rationals are also very dense, so where can you choose your arbitrary point outside this field of continuous points, the definitions used on later stages are not rigorously proved, since they keep hiding behind ambiguous concept of alleged non existing infinity, which leads all other sciences to be astray ...? $\endgroup$ – Bassam Karzeddin Jan 4 '16 at 16:47
  • $\begingroup$ @bassamkarzeddin, "arbitrary" as in $ \pi$ or $\sqrt [3]{2}$? You keep contradicting yourself... $\endgroup$ – vonbrand Jan 4 '16 at 21:47
  • $\begingroup$ Please note that those numbers you mentioned are all the question about, I simply deny their existence on the real number line, so there is exactly zero probability to choose an arbitrary point that happen to be your suggested number as pi or cube root of 2, a number would be redefined as a finite sequence of digits "no matter where you place the decimal notation" (between/before/after) the digits or any number that can be constructed exactly, (no approximation, limits, convergence, intermediate theorem, infinity, ....) $\endgroup$ – Bassam Karzeddin Jan 5 '16 at 14:48
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    $\begingroup$ @bassamkarzeddin These numbers are irrational numbers, but irrational numbers are a subset of the reals. You can't deny that they exist in the real number line. $\endgroup$ – HDE 226868 Jan 14 '16 at 0:04
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    $\begingroup$ Why my original question is being continuously updated? wonder! $\endgroup$ – Bassam Karzeddin Oct 30 at 11:01

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