In the mathematics of the ancient Greek 1 has not been considered to be a number. Nevertheless 1 counted as a divisor of perfect numbers like 6 = 1 + 2 + 3. Is there any explanation for this inconsistency?
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1$\begingroup$ Why do you think that 1 was not considered a number by Greeks, is there a source? Greeks had a numeral for it since the time immemorial, and Pythagoreans distinguished it as "the number of reason" in their numerology. $\endgroup$– ConifoldCommented Oct 28, 2016 at 21:41
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$\begingroup$ @Conifold: Euclid excludes 1 as a number. $\endgroup$– Franz KurzCommented Oct 29, 2016 at 7:32
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$\begingroup$ @Conifold. I think you are confusing "numerals" and "numbers". $\endgroup$– fdbCommented Oct 29, 2016 at 11:07
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2$\begingroup$ I suspect that Euclid was influenced (through Plato) by the refined Parmenidian concerns that multiplicity and change are illusions and only One is (he strives to keep motion out of geometric arguments for similar reasons). But it had little effect on common usage by Greeks, or even mathematical usage, Euclid himself does not maintain the distinction consistently in the Elements. $\endgroup$– ConifoldCommented Oct 30, 2016 at 21:20
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3$\begingroup$ The Treviso arithmetic of 1453 also excludes 1as a number. $\endgroup$– Mark DominusCommented Nov 13, 2016 at 4:23
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2 Answers
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Euclid (and this is not "the Greeks", although the same definition can be found in other places as well) defined a number as a multitude of ones. Since 1 is not a multitude of ones, 1 was not a number. 6, on the other hand, is a multitude of ones, and for this reason has 1 as a divisor. And since 6 is not a multiple of 6, its only divisors were 1, 2 and 3. In particular, there is no inconsistency whatsoever in your example.
Occasionally Euclid includes 1 as a number, in particular in a couple of propositions that are believed to go back to the Pythagoreans.
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I am not convinced that there is any inconsistency. In El. VII, def. 2 Euclid writes “A number is a multitude composed of units.” ( ̓Αριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.) In Def. 22 he says: “A perfect number is that which is equal to its own parts.” (Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν.) He does not say that the “parts” of perfect numbers are themselves “numbers”. The parts are a “unit” (that is: 1) and one or more “numbers”.
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$\begingroup$ In VII, Def. 3, Euclid defines "part" only in terms of numbers: "μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ...." But clearly one is considered a part in the text. $\endgroup$ Commented Oct 30, 2016 at 1:51
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$\begingroup$ @MichaelE2. I see your point. I think in Def. 3 he is defining what it means if "a number is part of a number". Apparently it is not meant as a exhaustive definition of "meros". $\endgroup$– fdbCommented Oct 30, 2016 at 10:30
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$\begingroup$ We seem to agree. Euclid's definitions rest on a common understanding of some terms, like "measures." It could be that a number being a multitude of units, the unit is understood to be part of a number. Defs. 3/4 would extend the notion to the relation of "a number of/to a number" being either part or parts. My understanding is that there is scant evidence to decide such issues, but I'm not familiar with what there is. $\endgroup$ Commented Oct 30, 2016 at 12:40