![[visualize a triangle]](triangle.png)
There is no doubt that the discovery, attributed to the school of Pythagoras, that the square root of two is irrational, was not only one of the most significant achievements of early Greek mathematics, but was also "the first crisis in the foundations of mathematics." I would like to consider the exact meaning of this crisis, to the extent that anyone can today, but some other issues must be dealt with first. Pythagoras was, in spite of the numerous legends that collected around his person, career, and school, as historical as any other early figure of early Greek mathematics or philosophy. What is remarkable is that the fabrication of legends did not stop in ancient times.
Here, for example, is a modern legend. This particular example is from William Everdell's The First Moderns , but I first heard it by word of mouth some 25 years ago:
...for a long time Pythagoras believed that every conceivable quantity could be expressed as a ratio (ratio in Latin is reason) of two of the available infinity of whole numbers - as for example 3/5 is the ratio of three and five or 119/120 is the ratio of one hundred and nineteen and one hundred and twenty. Then one day, one of Pythagoras's disciples pointed out to him that the diagonal of a square whose side was one unit could not be expressed that way. The two whole numbers needed to give the diagonal as their ratio did not exist; it was true and could be proved. Instead, one had to use the square root of 2, which is in this sense irrational and never "comes out even." Since they were all on a boat at the time, Pythagoras threw his student overboard and swore everyone else in his class to secrecy.
The truth, however, did not drown, and Greek mathematics was brought face to face with a brand new question... (Everdell 33)
Everdell doesn't cite any source for this, but he might have picked it up from Morris Kline's Mathematical Thought from Ancient to Modern Times :
The discovery of incommensurable ratios is attributed to Hippasus of Metapontum (5th cent. B.C). The Pythagoreans were supposed to have been at sea at the time and to have thrown Hippasus overboard for having produced an element in the universe which denied the Pythagorean doctrine that all phenomena in the universe can be reduce to whole numbers or their ratios. (Kline 32)
Kline doesn't mention any oath of secrecy being sworn on the spot; this detail may be Everdell's invention. It was of course well known, or at least widely asserted in ancient times, that the Pythagorean brotherhood was bound by an oath to keep secret the teachings of the founder, although, as Kline points out, "secrecy as to mathematics and physics is denied by some historians." In any case Kline gives no source for this account, although the mention of Hippasus at least gives a trail to follow. I will try to do this later.
Is this the truth about Pythagoras? A discovery that threatens his whole world-view is suppressed by murder, and then covered up? This tale will appeal to a certain kind of spirti, but I think what we have here is a sort of academic urban legend. It doesn't appear in any of the earlier modern histories of mathematics, such as Heath or van der Waerden, and I have been unable to trace it, in the form given above, to any earlier source that Kline, though I cannot say with certainty that it originates with him. I will try so say something about its ultimate sources, and then speculate on what this discovery might have meant to its discoverers.
Embarassments multiply when we turn to the ancient sources. There was an extensive hagiographic literature on Pythagoras, but it has relatively little to say about mathematics. Porphyry's Life of Pythagoras , for example, goes on for pages about the religious beliefs of the school, and the magical powers possessed by its founder: his "infallible" predictions of earthquakes, his ability to turn aside plagues, violent winds, hail, and to calm the sea, and things of this nature (c.f. Porphyry 49). There is little discussion of mathematics, and no mention of any story of the sort told by Kline and Everdell. Of course such an account might be expected to suppress unpleasant details, but they nonetheless mention the existence of critics of the school, and one would think, for example, that the early Christian polemicists would have picked up on it.
Worse yet, there seems to be no ancient evidence that Pythagoras even knew about the incommensurability of the side and diagonal of a square, and only the shakiest of evidence that this was known to later members of the school. Perhaps the earliest datable reference to irrationals is in Plato's Theaetetus, where it is said that Theodorus
...was proving to us a certain thing about square roots, I mean (the square roots, i.e. sides) of three square feet and of five square feet, namely that these roots are not commensurable in length with the foot-length, and he went on in this way, taking all the separate cases up to the root of 17 square feet, at which point, for some reason, he stopped" (Heath 203).
The evidence is negative: Theodorus shows that the square roots of 3, 5, 6, 7, etc., up to 17 are irrational; but as he does not show that the square root of 2 is irrational, this must have been known already. Iamblichus asserts, apparently without evidence, that Theodorus was a Pythagorean, so it must have been plausible that this subject was of interest to the Pythagorean school. The first direct evidence of any proof of the irrationality of the square root of two occurs in a passing comment in Aristotle's Prior Analytics:
For all who argue per impossibile infer by syllogism a false conclusion, and prove the original conclusion hypothetically when something impossible follows from a contradictory assumption, as, for example, that the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal to even if it is assumed to be commensurate" (Prior Analytics i.23, 41a26-27, tr. Thomas 111).
This apparently refers to what is now usually given as the proof that root 2 is irrational, but he doesn't say who discovered it.
There is a passage from Proclus's Commentary on the First Book of Euclid's Elements which, in one version, is translated by Heath as asserting that Pythagoras discovered "the theory of irrationals" (ten ton alogon pragmateian). There is, however, a textual problem here, in that the word alogon, "irrational," appears in other manuscripts as analogon, in which case the assertion is that Pythagoras discovered the "theory of proportions." Heath remarks that the form ton analogon is not correct Greek; the right reading may be either ton analogion ("proportions") or ton ana logon ("proportionals"), and he opts for the latter. This reading seems to be the generally accepted one now, and is the one used in the translations of Proclus by Thomas, and by Morrow. All of these authors point out that it is quite unlikely that anyone as early as Pythagoras could have had a general "theory of irrationals"; at most, he or his disciples could have discovered the irrationality of the square root of 2, and possibly a few other ratios. One knows, in any case, how far the study had progressed by the time of the generation of Plato, for Theaetetus poses, in the dialogue named after him, precisely the problem of creating such a theory.
There is a scholium of uncertain date to the tenth book of Euclid, which asserts that "the Pythagoreans were the first to address themselves to the investigation of incommensurability." We will return to this later. Finally, Heath cites an Arabic translation of a commentary on Euclid, which asserts that the theory of irrationals "had its origin in the school of Pythagoras"; he thinks it plausible that this comes from the commentary of Pappus. Even if this is the case, it is five centuries removed from the heyday of the Pythagorean school.
We can conclude that it is quite possible that sometime in the fifth century B.C., a member of the Pythagorean school discovered the argument for the irrationality of the square root of 2 mentioned by Aristotle. Evidence that this was known to Pythagoras himself seems to be totally lacking.
What, then, about Hippasos and his fatal sea voyage? There are, so far as I know, only two references in ancient sources to anything like this. The first is the scholium to Euclid book X just mentioned. After a wordy discussion of irrationals and the infinite divisibility of geometric magnitudes, the scholiast adds,
There is a legend that the first of the Pythagoreans who made public the investigation of these matters perished in a shipwreck (Thomas 217).
The other is from a tract of Iamblichus, On the Pythagorean Life:
It is related of Hippasus that he was a Pythagorean, and that, owing to his being the first to publish and describe the sphere from the twelve pentagons, he perished at sea for his impiety, but he received credit for the discovery, though really it all belonged to HIM (for in this way they refer to Pythagoras, and they do not call him by name (Thomas 225).
If there is a common element in these two accounts, it is that of divine punishment for the violation of an oath. The equivocation over exactly which secret was revealed - the theory of irrationals or the construction of the dodecahedron - is paralleled in another legend about Pythagoras, the story that he sacrificed an ox in celebration of his discovery of the Pythagorean theorem, in some accounts, or his discovery of the theorem of "application of areas," in others (Plutarch tells the two stories in different works, c.f. Thomas 177). Whatever the original story was, it has evidently developed somewhat by the time of Iamblichus (4th cent. AD). In any case, we see that the story told by Kline and Everdell is pasted together from the two quotations above. The development of this legend did not end with Iamblichus.
The modern form of the legend is, in any case, implausible on its face: if it were true, it would be the unique case, not only of someone being murdered for a mathematical discovery (Hypatia was murdered for her religious and philosophical beliefs, and possibly her politics, not for any theorem she had proven), but also the unique case of a mathematical discovery taking place during a sea voyage. I know a few mathematicians with a hobby of sailing; none of them find it conducive to mathematical work. Finally, anyone who is inclined to think of Pythagoras as a murderer should recall Cicero's objection to the story of the ox sacrifice: a man who believed in the transmigration of souls into animals, and on that account abstained from eating meat, was not likely to sacrifice an ox. So - what is the appeal of the modern version?
It is probably only a coincidence that Kline's book was published at about the same time (1972) that coverups and conspiracy theories - the JFK assassination, Watergate - became a popular theme of media culture. Somewhat more relevant is the current academic penchant for ferreting out the "hidden agenda" of various scientific or philosophical theories, something which certainly did not originate then (one thinks, for example, of Nietzsche, or Horkheimer & Adorno) but rose to prominence shortly thereafter. Kline was quite probably innocent of all this, but these tendencies have doubtless contributed to the diffusion of our tale.
The secrecy surrounding Pythagorean teachings was nothing unusual for the ancient mystery religions, all of which had their oaths of secrecy, and legends of divine punishment for their violation. To see evidence, in this particular case, of a "coverup" is just a clumsy anachronism. It is based on a failure to see just what this secrecy amounted to, as well as a failure to appreciate just what the discovery meant.
It is probably true that the early Pythagoreans were shocked by the discovery of irrational magnitudes - if indeed they were the ones that made it. For it is quite true that the discovery invalidates a naive understanding of the Pythagorean belief that, as Aristotle puts it, the principles of mathematics "must be the principles of all existing things" (Metaphysics A 5, 985b, tr. Thomas 173). In fact, this is one argument against the notion that the incommensurability of the side and diagonal of a square was known to the early Pythagoreans. On the other hand, the existence of irrational ratios did not prevent Plato from assimilating a good deal of Pythagorean number-mysticism, as the Timaeus, for example, shows. The participants in theTheaetetus betray no particular anxiety about the existence of irrational ratios; it is for them simply one mathematical question among many. So the existence of irrationals cannot be viewed as immediately fatal to the philosophical (or religious) project of the Pythagoreans. On the other hand, it might have posed a severe challenge to their mathematics.
The case for this is largely speculative, but I think it is at least plausible enough to bother stating it. The argument begins with the curious circumstance that there are not one but two proofs of the Pythagorean theorem in Euclid. The first, in Book I, uses only the most elementary properties of triangles and their areas. The second, in Book VI, is a shorter argument, but it relies on properties of similar triangles, which in turn depends on the theory of proportion developed in Book V. Heath makes a case (Heath 148) that the proof in Book VI is the Pythagorean one, whereas the elementary but longer proof in Book I is Euclid's.
The argument of Book IV, in brief, is as follows: suppose that ABC is a right triangle, with the vertex of the right angle at C, and draw CD perpendicular to AB. Then the triangles ADC, CDB are similar to ACB, so that
AD/AC=AC/AB and DB/CB=CB/AB
from which it follows that
AD.AB=AC2, DB.AB=CB2.
Since AB=AD+DB, adding these equations yields
AB2=AC2+CB2
which is the Pythagorean theorem. The question is, how does one justify, for example, the inference that DB/CB=CB/AB implies DB.AB=CB2? This would seem to be a matter of elementary algebra: if a/b=c/d, then ad=bc. But here we must remember that AB, AC, etc. are geometrical magnitudes, and the Greeks had no concept of "number" which included arbitrary magnitudes. If the sides of the triangle have a common measure, i.e. are all integral multiples of the same length, then the proof is easy, but if they are not, then it is not at all obvious how to proceed. Once the Pythagoreans learned that two lengths need not have a common measure, one of their prize proofs fell apart.
It is probably a significant fact, then, that the Pythagoreans were among the first to study ten ton ana logon pragmateia, the theory of proportions. Of course this probably referred to the discovery, traditionally accorded to Pythagoras, of the relation between musical overtones, and also to a collection of elementary results on arithmetic, geometric, and harmonic means. But after the discovery of irrational ratios, a more acute problem was on the table: that of providing a foundation for the proof of the "Pythagorean theorem." This, as one knows, they did not achieve; that honor belongs to Eudoxus, whose work on proportions is substantially Book V of Euclid, as we are told in a scholium to that book. Euclid, finally, removed the original motivation for this work, if indeed it was he who came up with the proof in Book I, but by then there was a powerful new theory of ratios which could then be applied to other problems. This is certainly not the only time mathematics has developed in this fashion.
If this account is true - and Heath makes a plausible case for it - then the discovery of irrational ratios was not so much of a philosophical or metaphysical crisis as a purely mathematical one. This might be construed as a disappointment for the philosphers, but not for mathematicians.
Richard Crew, August 2000.
[1] William R. Everdell, The First Moderns, University of Chicago Press 1997.
[2] Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford Univ. Press 1972 (references are to the 3-volume edition of 1990).
[3] Sir Thomas Heath, A History of Greek Mathematics vol. 1, Clarendon Press 1921, reprinted Dover 1981.
[4] Porphyry, Vie de Pythagore - Lettre a Marcelle, tr. Edouard des Places, Les Belles Lettres 1982
[5] Ivor Thomas, Selections Illustrating the History of Greek Mathematics, Harvard Univ. Press 1939.
[6] Proclus, Commentary on the First Book of Euclid's Elements, tr. Glen Morrow, Princeton Univ. Press 1970.