Origin of Numbers: Pythagorean Theory

[arivero]

In the "origin of zero" thread, I saw the following remark:

the thought I'm playing with for the moment is that every rational number has its origin in 1.

I had heard a different story, in a pythagorean mood: The first thing we can say is if a number, a magnitude, a rythm, a length, etc is odd or even, ie if it can be divided or not in equal parts. From this we get the smallest odd quantity, 3, and the smallest even, 2. The unity is not got because it can not be divided after all. So after getting the odd and even numbers, we get the unity, as the difference 3-2.

 

[matt grime]

Do you mean mathematical origin or philosophical origin?

 

[Zurtex]

If you can try and find a book called 'Surreal Numbers' by Donald Knuth. Very enlightening about the importance of what some refer to as 'trivial' properties of numbers.

 

[mathwonk]

or cultural origin, or geographical origin, or temporal origin?

 

[honestrosewater]

I had heard a different story, in a pythagorean mood: The first thing we can say is if a number, a magnitude, a rythm, a length, etc is odd or even, ie if it can be divided or not in equal parts. From this we get the smallest odd quantity, 3, and the smallest even, 2. The unity is not got because it can not be divided after all. So after getting the odd and even numbers, we get the unity, as the difference 3-2.

How do you define the smallest even number without using the concept of unity? Wouldn't the smallest even number be the even number whose parts cannot be further divided (i.e. whose parts are unity)? The concept of the smallest even number would then be derived from the concept of unity.
Also, if an even number is a number which can be divided into equal parts, and an odd number is a number which is not even, then unity is an odd number, as it cannot be divided into equal parts. I can't think of a definition of oddness which excludes unity without using it. Let x and y be individual variables ranging over set S. Say x is even if there exists some y such that (y + y = x). How will you define oddness?

 

[HallsofIvy]

I would define "smallest even number" (of a given set of integers) as the even number, x, in the set such that if y is any even number in the set, then x is less than or equal to y. That doesn't use "unity".

The problem I have with arivero's "...ie if it can be divided or not in equal parts. From this we get the smallest odd quantity, 3, and the smallest even, 2. The unity is not got because it can not be divided after all." (I assume he meant "at all") is that saying "a number is odd if it cannot be divided in equal parts" certainly does apply to 1. If it cannot be divided at all, then it certainly cannot be divided in equal parts and so is odd.

 

[neurocomp2003]

Math is devised into quantities...and from a psychological standpoint the first quantity you recognize is 1.