I'm reading "Professor Stewart's Incredible Numbers," by Ian Stewart and in there it is claimed that

In early Greece, the list of numbers started $2, 3, 4,$ and so on: $1$ was special, not "really" a number. Later, when this convention started to look really silly, $1$ was deemed to be a number as well.

This is new to me. What evidence is there of this and when did the change occur?

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    $\begingroup$ Much the same question here: hsm.stackexchange.com/questions/5326/… $\endgroup$ – fdb Sep 19 '17 at 14:52
  • $\begingroup$ Well, 1 is not prime either :-) . It's always been a sort of special number, so it depends on what "number" means (like what the meaning of 'is' is) $\endgroup$ – Carl Witthoft Sep 20 '17 at 11:45

The opening words of the Platonic dialogue Timaeus are:


εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων, τὰ νῦν δὲ ἑστιατόρων;


One, two, three, — but where, my dear Timaeus, is the fourth of our guests of yesterday, and our hosts of today?

clearly showing that the word εἷς "one" was used for counting in the same way as all other positive integers.

If at some point later on (some) Greek mathematicians accorded a special status to the number $1$, I would be tempted to compare this to a modern discussion of whether $0$ is or is not to be included in the natural numbers -- or whether the zero ring is an integral domain or not. It is a technicality, of the sort that only becomes meaningful once one has run into the necessity of "constructing" objects that have already been understood on a more intuitive level.

To be sure, there seems to be a certain attraction to the statement as presented by Ian Stewart: it is intriguing to consider the possibility that the Greeks somehow thought "differently" about numbers than we do. It is a statement that borders on the unprovable. Heidegger is even more outrageous, writing somewhere (in "Modern Science, Metaphysics, and Mathematics", which is reprinted in the "Basic Writings" collection) that the Greeks didn't even consider 2 to be a number: since 1 is the unit, and 2 is only ever a "pair", and not a collection of "two" separate objects. Only when the number three is first encountered, does the concept of number really start to shine forth -- or so the the little magician from Messkirch would have it. It sounds good, but I have yet to be convinced that there is more to it than idle speculation.

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  • $\begingroup$ I do not see how a passage with 1 and 2 separated by a comma "clearly shows" anything at all, let alone what Greek mathematicians (rather than the general populace, or Plato) considered 1 to be. Or how a direct consequence of Euclid's definition from the Elements (explicitly ascribed to Pythagoreans by Aristotle) "is a statement that borders on the unprovable". $\endgroup$ – Conifold Nov 27 '18 at 6:05
  • $\begingroup$ @Conifold You disagree with my construal of the passage "one, two, three, but where is the fourth?" as a 'clear example of counting'? I mean, you are fully entitled to your scepticism, but there is such a thing as 'taking it too far'... $\endgroup$ – RP_ Nov 27 '18 at 9:48
  • $\begingroup$ By the way I do not mean the statement by Stewart is itself unprovable. What I am saying is that I can see that it could exert a certain type of fascination, since it seems to convey the idea that the Greeks had a different conception of what a 'number' really is. It is this latter idea that I called unprovable. (One's concept of number is not unlike Wittgenstein's beetle in the box.) It is germane to the ideas of Jacob Klein, for example, who busied himself with exactly this type of metaphysical arguments. $\endgroup$ – RP_ Nov 27 '18 at 9:51
  • $\begingroup$ Badiou wrote books using what he considers to be set theory, and I wouldn't take that as reflecting modern mathematical practice either. But nevermind that. Greek populace at large happily used letter based arithmetic, and yet there is not a trace of it in the mathematics of Euclidean tradition, derived from Pythagoreans. "Tempted to compare this to a modern discussion of whether 0..."? Don't be. $\endgroup$ – Conifold Nov 27 '18 at 9:58
  • $\begingroup$ Well, I won't if you could offer any good arguments against doing so. Right now you just seem to rely on your dismissive tone for doing the job, if you will pardon me saying so. What I'm claiming is that the whole fact that Greek mathematicians rejected 1 as a number is an artifact of their attempts at formalization, just like arguing over whether the zero ring is an integral domain is an artifact of our way of doing number theory (i.e. partly via modern algebra). Since the Greeks saw a number as a multitude of monads, the monad itself couldn't be a number, if they wanted to avoid circularity. $\endgroup$ – RP_ Nov 27 '18 at 10:03

Euclid's Elements Book VII:

Definition 1: A unit is that by virtue of which each of the things that exist is called one.

Definition 2: A number is a multitude composed of units.

See also: Aristotle on unit (monas) and number (arithmos) and Metaphysics, Book N, 1088:

"One" evidently means a measure. And in every case it is some underlying thing with a distinct nature of its own [...] And this is reasonable; for the one means the measure of some plurality, and number means a measured plurality and a plurality of measures. Thus it is natural that one is not a number; for the measure is not measures, but both the measure and the one are starting-points.

These speculations affect the "theory of numbers" of Pythagorean origin and not the everyday counting practice; see Plato, Republic, Book 7, 525a, with the distinction between the "science of arithmetic" (ἀριθμητική) and the "art of calculation" (λογιστική).

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    $\begingroup$ I don't think this addresses the claims made by the OP. A "multitude" could be composed of a single unit. $\endgroup$ – Carl Witthoft Sep 20 '17 at 11:41
  • $\begingroup$ @Carl Witthoft. No. The old Greek had healthy opinions. $\endgroup$ – Wilhelm Sep 20 '17 at 14:47
  • $\begingroup$ @K.B. Absolutely, if you consider the acceptance of slavery a healthy opinion. But I agree that Aristotle's text quite explicitly contradicts Carl Witthoft's point: "the measure is not measures" really means that a single individual can never be regarded as a "group". (This somehow reminds me of the insistence of Euclid that the whole be always greater than the part -- distinctions such as these were more rigidly enforced in Greek mathematical thought.) $\endgroup$ – RP_ Sep 21 '17 at 13:49
  • $\begingroup$ @René: In their time the acceptance of slavery was a healthy opinion. You musr not judge everything from the modern standpoint. $\endgroup$ – Wilhelm Oct 1 '17 at 19:54

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