On 9/16/2012 3:43 PM, Rex Allen wrote:

It seems to me that numbers are based on our ability to judge relative magnitudes:"Which is bigger, which is closer, which is heavier, etc." Many animals have this ability - called numeracy. Humans differ only in the degree to which it is developed, and in our ability to build higher level abstractions on top of this fundamental skill. SO - prime numbers, I think, emerge from a peculiar characteristic of our ability to judge relative magnitudes, and the way this feeds into the abstractions we build on top of that ability. =*= Let’s say you take a board and divide it into 3 sections of equal length (say, by drawing a line on it at the section boundaries). Having done so – is there a way that you could have divided the board into fewer sections of equal length so that every endpoint of a long section can be matched to the end of a shorter section? In other words – take two boards of equal length. Divide one into 3 sections. Divide the other into two sections. The dividing point of the two-section-board will fall right into the middle of the middle section of the three-section-board. There is no way to divide the second board into fewer sections so that all of its dividing points are matched against a dividing point on the longer board. Because of this – three is a prime. (Notice that I do not say: “this is because 3 is prime” – instead I reverse the causal arrow). =*= Let’s take two boards and divide the first one into 10 equally sized sections. Now – there are two ways that we can divide the second board into a smaller number of equally sized sections so that the end-points of every section on this second board are matched to a sectional dividing point on the first board (though the opposite will not be true): We can divide the second board into either 2 sections (in which case the dividing point will align with the end of the 5th section on the first board), OR We can divide the second board into 5 sections – each of which is the same size as two sections on the first board. Because of this, the number 10 is not prime. =*= The entire field of Number Theory grows out of this peculiar characteristic of how we judge relative magnitudes. Do you think?

HI Rex,

`Nice post! Could you riff a bit on what the number PHI tells us`

`about this characteristic. How is it that it seems that our perceptions`

`of the world find anything that is close to a PHI valued relationship to`

`be "beautiful"?`

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