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"?

--
Onward!

Stephen

http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html


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