On 9/16/2012 3:43 PM, Rex Allen wrote:
It seems to me that numbers are based on our ability to judge relative

"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),


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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