Can anyone explain these assertions on the number 3 to me?

[aldiboronti]

The following is from Plutarch’s Life of Fabius. I’ve underlined the sentence that’s relevant.

He likewise vowed to celebrate a musical and dramatic festival in honour of the gods, which should cost three hundred and thirty-three sestertia, plus three hundred and thirty-three denarii, plus one third of a denarius. This sum, in Greek money, amounts to eighty-three thousand five hundred and eighty-three drachmas, plus two obols. Now the reason for the exact prescription of this particular number is hard to give, unless it was thereby desired to laud the power of the number three, as being a perfect number by nature, the first of odd numbers, the beginning of quantity, and as containing in itself the first differences and the elements of every number mingled and blended together.

3 the first odd number? What happened to 1? The beginning of quantity? What happened to 2? I can’t make head or tail of the last bit "containing in itself the first differences and the elements of every number mingled, etc but if it helps my Dryden translation has that last bit as “the first number that contains in itself multiplication with all other properties whatsoever belonging to numbers in general,”

I’d have a new respect for 3 if I could work out what on earth Plutarch means!

[Senegoid]

Three is the third-lonliest number that you’ll ever do.

[Some_Call_Me.Tim]

Also, note this passage from the Book of Armaments.

[septimus]

It sounds absurd, but the ancient Greeks did not consider 1 to be a number. (The numbers were {2,3,4,…}.) As late as the 16th century Simon Stevin had to argue that 1 should be considered a number in his textbook!

I’m trying to figure out Plutarch’s money conversion. Assuming a denarius is valued at 200 drachmas, and a sestertius is 1/4 of a denarius, then the arithmetic works out with 338.3333 sestertia instead of 333. Is this a 1900-year old arithmetic error? Is the guy just expecting a 5+ sestertia discount for a party that big?

[Ignotus]

Classical Greek has three grammatical numbers: singular, dual and plural. So I guess in some sense three could have been seen as “the beginning of quantity”, as the smallest number taking the plural ending.

Just a guess though.

[Thudlow_Boink]

I have a number of things to say about this.

After that, you’d expect me to say more than one thing. Which may help to explain why, as septimus said, the Greeks did not consider 1 to be a number.

Another way of thinking of 3 as the first odd number, which may or may not be what Plutarch had in mind (though it’s not the way we moderns think of it), is that an even number can be divided into two equal parts, while an odd number can only be divided into two unequal parts. But you can’t do either of those things with 1.

Check out The Theology of Arithmetic, which includes statements like

The dyad would be the mid-point between plurality, which is regarded as falling under the triad, and that which is opposed to plurality, which falls under the monad. (p. 43)

and

The dyad is not number, nor even, because it is not actual; at any rate, every even number is divisible into both equal and unequal parts, but the dyad alone cannot be divided into unequal parts; and also, when it is divided into equal parts, it is completely unclear to which class its parts belong, as it is like a source. (p. 45)

and

For 3 is the first number to be actually odd, since in conformity with its descriptions it is ‘more than equal’ and has something more than the equal in another part; and it is special in respect of being successive to the two sources and a system of them both. (p. 50)

Does that clear things up? Yeah, I didn’t think so.

[Riemann]

I think I can get on board with the notion that “one” “two” and “many” are sort of distinct concepts, beyond just a sequence of numbers, and that three is in a sense the beginning of “many”.

Also:

“Just the place for a Snark!” the Bellman cried,
As he landed his crew with care;
Supporting each man on the top of the tide
By a finger entwined in his hair.

“Just the place for a Snark! I have said it twice:
That alone should encourage the crew.
Just the place for a Snark! I have said it thrice:
What I tell you three times is true.”

[Isilder]

@aldiboronti

The following is from Plutarch’s Life of Fabius.

Ah, well you see Plutarch rejected Aristotle… He rejected the peripatetic doctrine that
there is a logical reason for everything.

There’s little point in disecting Plutarch with regard to Platoism, Stoicism and Epicuraanism – all it would mean is that Plutarch supported waffling on with his own imaginations, and rejected Aristotle’s demand for supporting evidence for the ideas invented.
I think that what Plutarch says is that Fabian clearly locked onto the number 3, and Plutarch says that Plutarch too finds 3 beautiful, for being “odd” (ok, so he defines odd numbers start at 3.), and for being the first number (Maybe he means he can know he has “two” apples just by looking, but he has to count three apples to know its three ?).
That it is “perfect” or “contains the elements of other numbers” is nothing unique. Perfect might mean that its factors, apart from 1, add and multiply to the same thing, which is true of any prime. Or maybe he means prime… Its “differences and elements” are in other numbers, but so too other numbers have the same differencess and elements, and this was meaningless waffle.

So its all meaningless waffle and he could equally laud 7 , 11, 13, and 375.

… Do you have any respect for a anti-Aristotlean ?

[Thudlow_Boink]

@Isilder

and for being the first number (Maybe he means he can know he has “two” apples just by looking, but he has to count three apples to know its three ?)

Or maybe he agreed with OpalCat that a list has to have at least three items.