By Jeff Michaels
Disclaimer, these corrections are not an affirmation of any particular person’s position. These are just clarifications for the sake of mathematical accuracy.
- 21:30 “Let’s just take something everybody knows like multiplicative commutativity. A times B = B times A. … There are situations where this doesn’t apply, such as matrices.”
Matrix products are NOT multiplication, they are a shorthand for function composition akin to
. This is because from Linear Algebra, we know that any finite dimensional linear transformation can be represented by a matrix, and it turns out that we have discovered a cool pattern for composing these linear transformations in the form of matrices, and we call this pattern “matrix multiplication”.
This is in NO WAY related to the operation “multiplication” and no one should ever expect that it follows the same rules of multiplication proper. What Crocoduck has done here is taken an operation that ISN’T multiplication, interpreted literally an analogous linguistic device (a bad one, pedagogically, if this is how people will interpret it), declared that the analogous device doesn’t behave literally like the thing it’s analogous to, and confirmed that there must be a contradiction afoot and so the rules aren’t “really” rules and so everything is just a linguistic device all along.
- 23:57 *Discussion about Linear Algebra and Quantum Mechanics
I’d just like to point out here that Linear Algebra was not invented TO DESCRIBE quantum mechanics. Linear Algebra was its own field of study, proving theorems in its own right, for its own reasons, and it JUST SO HAPPENED to be a good fit for Quantum Mechanics, as did other branches of math. There’s more to Lie Algebras than just Linear Algebra, let me assure you.
- 24:25 *Discussion about Non-Euclidean Geometry / Differential Geometry.
Here’s a nontechnical description of Differential Geometry. We live on Earth, an Earth is a sphere. When we get really zoomed in on the world, it looks flat. We have flat ground, flat tables, and so on. Due to this “flatness” we can do Euclidean-Geometry, the parallel postulate holds and so on. But if we zoom way far out, the surface of the Earth is curved (as captains knew for centuries) and so Geometry behaves very differently. The sum of the angles in a triangle are more than 180 degrees and so on.
Differential Geometry contextualizes Euclidean and Non-Euclidean Geometries in a coherent system that explains their application by describing objects called Manifolds. Manifolds are collections of points who, when zoomed out, are governed by a Non-Euclidean Geometry, such as a Sphere. When zoomed in, in Math we say “there exists a local neighborhood for each point such that…”, then the Geometry local to that point in that neighborhood becomes Euclidean. As you can see, Euclidean and Non-Euclidean Geometries aren’t contradictory. They’re different contexts for examining a set of points and can be contained in a coherent theory. Contradictions cannot be contained in any theory.
His discussion of “you have to change the underlying axioms” is patently false. What you have to do, is come up with a more universal set of axioms which admit both sets of axioms as special cases, as Differential Geometry does very well. These rules are not mutually contradictory.
To anyone who understands these details at a technical level, Crocoduck’s argument looks something like this:
- Sometimes doors are open.
- Sometimes doors are closed.
- A door being open is inconsistent with a door being closed.
- Therefore there really is no such thing as a “door” but rather a linguistic device which we use to describe some sort of reality that has nothing to do with a door.
His bit about Non-Euclidean Geometry really exactly as retarded as this argument here. Logics, I might be swayed to believe are linguistic devices. But objects constructed from them end up having immutable properties that impose on the wielder in a way the wielder cannot get out of unless they fundamentally change the logic. In math terms there’s a great example of this.
We have this branch of mathematics called Galois Theory which discusses the underlying structure of the solutions to polynomial equations in terms of Group Theory. From this (And the Abel-Ruffini Theorem prior) we have that general polynomials of degree 5 or higher do not have algebraic formulas for general solutions in the vein of, say, the quadratic formula. Not only have we not found one, we know IN PRINCIPLE they CANNOT have one.
How does this make sense if math is a language? If polynomials are linguistic artifacts, we could change them as malleably as we change our language to say whatever our language needs them to say. But we cannot. We can build something else to do what needs to be done, but that something else will not be a polynomial.
Given this example of polynomials, I conclude with one final analogy:
We invented this thing to hold together wood called a “nail” but had a hell of a time pounding them into the wood. So we invented this thing called a hammer. The hammer was great at pounding in the nail but was terrible at being used to eat soup. This is because the hammer we made, despite being invented as a tool originally, was really a hammer. Being a real hammer, it had a set of mind independent facts about what it can and cannot do in the world which come with a set of mind independent consequences about what we can use the hammer for.
Analogously with mathematics, we invent things called operations or sets to describe some real world thing. Maybe our motivation for investigating them in the first place IS to make them our tool for some problem about physical reality. But then we investigate the tools we derive further and find out that they yield more abstract and complex objects such as polynomials or manifolds. Those objects, it turns out, have invariant and defining properties that we can’t muck with while keeping the structure of the object intact in the same way that disassembling the hammer makes it no longer a hammer.
Those properties which impose restrictions on us, the users, in a way language proper does not and cannot in principle, reflect the realness of the objects under construction. It’s no secret that we spend decades discovering the properties of mathematical objects IN THEMSELVES before they end up becoming useful to physicists later. This is because using an object is different from knowing what an object IS.