- lucasbfoley

# arithmetic & geometry, time & space

Updated: Aug 17, 2021

*****

**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**