arithmetic & geometry, time & space

*

arithmetic reveals patterns in counting

geometry reveals patterns in borders

arithmetic presupposes discreet entities to be counted

geometry presupposes discreet borders to be drawn

*

to mutually comprehend arithmetic, we must agree what the numbers and operators represent

we intuitively think, duh: 2 apples, 2 clouds, 2 grunts, 2 dreams, 2 universes - whatever it is, 2 its + 2 its = 4 its

this intuition overlooks, among other things:

- the cognitive processes underlying the objectification of that portion of reality from the remainder

- the cognitive processes underlying the conceptualization of that into apple

- the cognitive processes underlying the enumeration of apples into 2 apples

- the cognitive processes underlying the abstraction of 2 apples into 2

in other words, mutually comprehending arithmetic requires (1) comptabile cognitive processing (otherwise, we can't agree what "number" means) (2) stipulated values for figures (otherwise, your 2 may represent a value equivalent to my 7, and vice versa)

one might respond, take physical reality out of it, just use numbers & operators, agree on the numbers' values and the operators' functions, and calculate

two problems:

(1) numbers are physical: there can be no "number" without a physical representation, be it a pencil mark, pixels on a screen, figures projected onto a mental field by a wetware brain, bits in a microchip...

(2) numbers & operators are incorrigibly meaningless without a reality to which such symbols correspond; 2 + 2 = 4 begs the questions of what those symbols mean and why we should care; if we treat the arithmetic as a purely symbolic system, with no correspondence to a reality, arithmetic statements are either tautologies or nonsense

in sum: mutually comprehending arithmetic requires corresponding reality, compatible minds, stipulated meanings, and the time in which such phenomena emerge

*

the Pythagorean theorem describes the relationship between the lengths of these three lines

the Pythagorean theorem is not necessary for the relationship to exist, and we can comprehend the relationship without the theorem

the relationship is the shape; the shape is the relationship

the theorem symbolizes this relationship in an algebraic statement

*

arithmetic is time

geometry is space

2 (time 1) + (time 2) 3 (time 3) = (time 4) 5 (time 5)

2 (ok) + (ok) 3 (ok) = (ok) 5 (ok)

we move linearly through the arithmetic statement to comprehend it; each element of the statement is a discreet moment of comprehension; comprehending the statement requires a step-wise progression through its elements

O

we look at the shape and we see it

the shape is comprehended instantaneously

we can conduct operations in time to:

- measure elements of the shape (this angle's degree, that line's length)

- divide the shape into components

- categorize the shape in various ways (triangle, right triangle, ...)

- generate theorems to describe relationship patterns for shapes of this type (e.g. Pythagorean theorem)

but comprehending the shape occurs instantaneously; to see it is to comprehend it

when we see the triangle, there is no need for "ok...ok...ok…"

we just see it

O