Non-foundationalist foundational thinking
Anthony Coulter
fom at anthonycoulter.name
Sun Jun 5 00:55:08 EDT 2022
[Note to moderator: if it's not too late to reject my previous
message, please do so and send this one instead. In my last
sentence I garbled the word "foundational" in a way that is
confusing.]
Another example of the distinction between "foundational" and
"foundationalist" thinking can be seen in a proof of the Law of
Cosines using vectors. Define the Euclidean plane as an affine
space with an inner product. (Here "affine" means "no distinguished
origin" rather than "no notion of length or angle.") Given a triangle
with vertices A,B,C, we introduce [lowercase] vectors
a = C-B, b = C-A, c = A-B, and compute:
|c|^2 = |a-b|^2 = (a-b)\cdot (a-b) = a\cdot a - 2*a\cdot b + b\cdot b
= |a|^2 + |b|^2 - 2*|a|*|b|*cos(theta)
QED. This would be a valid "foundationalist" proof of the Law of
Cosines but, in my opinion, a poor "foundational" one. The eponymous
cosine enters the formula with the application of this identity, which
is justified in the proof because it is the definition of the angle
between two vectors:
(a\cdot b) = |a|*|b|*cos(theta)
or
theta = arccos( (a\cdot b)/sqrt(a\cdot a)*sqrt(a\cdot b) )
Why in the world would someone define an angle in terms of three
dot products, two square roots, and an arccosine? And why does space
come equipped with this bilinear operation mapping two vectors to a
scalar? The formula only makes sense if you already understand the
Pythagorean Theorem and the Law of Cosines. No human civilization
invented dot products before discovering those two theorems, and if
someone were to invent dot products for non-geometric purposes (e.g.
linear algebra) it would be unlikely that they would come up with the
idea of taking an arccosine of that particular algebraic expression
unless they were already using cosines elsewhere in geometry.
Thus from a "foundational" standpoint, lengths and angles are
conceptually prior to inner products, and my proof of the Law of
Cosines contains no geometric insight---all the real insight is hidden
in the choice of the definition of angle.
But from a "foundationalist" standpoint, taking lengths and angles as
primitive and defining the dot product as |a|*|b|*cos(theta) turns out
to be complicated. The dot product exists even if one or both of the
argument vectors are zero, but the angle is undefined in this case.
Inner products can also be defined over complex Hilbert spaces (where
the angle exists but might be complex and thus unintuitive) or fields
where neither cosines nor square roots (needed to define |a| and |b|)
are available. And even in the familiar Euclidean plane it turns out
to be a huge pain to define angles numerically: you have to construct
a number system, introduce proportions, and (if you want to define the
radian the usual way) find a way to represent the length of a circular
arc.
If your goal is to efficiently build mathematical theories, the
convenience and generality of vectors, dot products, and analytic
geometry are ideal. If your goal is to establish the initial properties
of fundamental concepts, synthetic geometry is superior. I would accept
the proof I gave of the Law of Cosines when I am wearing my
"foundationalist," theory-building hat. But I would not accept it when
I wear my "foundational," theory-discovering hat.
This might not be the distinction that Tim Chow is going for but it
seems at least related.
Anthony
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