Animated Philosophy of Mathematics Discussion
As I am reading these math books, I'm reminded of an animated discussion I had with my Philosophy of Mathematics professor and a grad student years ago.
They kept saying that someone can't deny that the interior angles of a triangle add up to 180 degrees, that 2+2=4, and that it is impossible for there to be a square circle. I wouldn't grant any of those and I can deny those in general.
The interior angles of a triangle isn't 180 degrees for non-euclidean geometry. Once we replace Euclid's fifth postulate, it isn't necessarily that they add up that way. I added that the Universe is non-euclidean according to General Relativity.
2+2=4 for the integers, but for modular arithmetic it is not necessarily true. 2+2=1 while doing addition over the integers modulo 3.
They thought I was being overly pedantic. I should be able to say that the interior angles of a triangle is 180 degrees for euclidean geometry and 2+2=4 over the integers. I shouldn't be able to deny that.
My disagreement was stronger than that though. They kept adding that "the very meaning" of the terms imply those conclusions. I thought that mathematical concepts change. The meaning of triangle did once mean that interior angles were 180 degrees, but the meaning changed to expand it so that the idea of a triangle in non-euclidean space made sense. Addition was originally about objects (2 apples plus 2 apples) and lengths then got abstracted to general arithmetic. Later addition transformed to being about a binary operation with certain properties over elements of a set. In the future, the concept of addition will probably change again in ways that give more exceptions to the supposedly undeniable truths they were talking about.
For the square circle, they say that the very meaning of the terms were incompatible. I could think of an illustration that was both. I thought of a grid where the diagonals dilated so that closer to the origin the Square Circles centred on the origin looked like rotated squares and further out they looked like circles. Is that valid math? Probably not, but I just wanted to show I could have the concept no matter if it made sense or not.
What I was saying is that I rejected the idea of metaphysical mathematical objects and that concepts in math are instead human constructed abstractions whose meaning can change. The principle of non-contradiction applies to metaphysical objects. Aristotelianism tells us so and we are still under the shadow of Plato and Aristotle. Human concepts can be muddled, contradictory, or even crazy.
There was also discussion about Piaget ("do babies have the concept of person before object permanence?"), Thomas Kuhn (physicists didn't change their minds on Modern vs. Newtonian Physics, Newtonian physicists died off), Neils Bohr (they denied he was famous despite knowing of the Bohr model of the atom and the Copenhagen interpretation of Quantum Mechanics), Albert Einstein (his shock when the top-down general relativity explained the perihelion shift of mercury), the Heisenberg uncertainty principle (is temperature real? Since it is a macroscopic property, only in the sense that kinetic energy is real, and then only in the sense that position and momentum is real below Plank's constant), string theory and 10 dimensional space, materialism (if we know anything, it is that we make decisions. It might not happen as much as we think, but we make some decisions. Materialism is a very abstract construction far removed from the immediacy of making decisions and if it rejects will, then materialism is absurd), mind-body dualism (I don't believe in a spiritual substance), and those experiments that show decisions are made before it is thought (correct, the self is more than one's inner monologue).
They kept saying that someone can't deny that the interior angles of a triangle add up to 180 degrees, that 2+2=4, and that it is impossible for there to be a square circle. I wouldn't grant any of those and I can deny those in general.
The interior angles of a triangle isn't 180 degrees for non-euclidean geometry. Once we replace Euclid's fifth postulate, it isn't necessarily that they add up that way. I added that the Universe is non-euclidean according to General Relativity.
2+2=4 for the integers, but for modular arithmetic it is not necessarily true. 2+2=1 while doing addition over the integers modulo 3.
They thought I was being overly pedantic. I should be able to say that the interior angles of a triangle is 180 degrees for euclidean geometry and 2+2=4 over the integers. I shouldn't be able to deny that.
My disagreement was stronger than that though. They kept adding that "the very meaning" of the terms imply those conclusions. I thought that mathematical concepts change. The meaning of triangle did once mean that interior angles were 180 degrees, but the meaning changed to expand it so that the idea of a triangle in non-euclidean space made sense. Addition was originally about objects (2 apples plus 2 apples) and lengths then got abstracted to general arithmetic. Later addition transformed to being about a binary operation with certain properties over elements of a set. In the future, the concept of addition will probably change again in ways that give more exceptions to the supposedly undeniable truths they were talking about.
For the square circle, they say that the very meaning of the terms were incompatible. I could think of an illustration that was both. I thought of a grid where the diagonals dilated so that closer to the origin the Square Circles centred on the origin looked like rotated squares and further out they looked like circles. Is that valid math? Probably not, but I just wanted to show I could have the concept no matter if it made sense or not.
What I was saying is that I rejected the idea of metaphysical mathematical objects and that concepts in math are instead human constructed abstractions whose meaning can change. The principle of non-contradiction applies to metaphysical objects. Aristotelianism tells us so and we are still under the shadow of Plato and Aristotle. Human concepts can be muddled, contradictory, or even crazy.
There was also discussion about Piaget ("do babies have the concept of person before object permanence?"), Thomas Kuhn (physicists didn't change their minds on Modern vs. Newtonian Physics, Newtonian physicists died off), Neils Bohr (they denied he was famous despite knowing of the Bohr model of the atom and the Copenhagen interpretation of Quantum Mechanics), Albert Einstein (his shock when the top-down general relativity explained the perihelion shift of mercury), the Heisenberg uncertainty principle (is temperature real? Since it is a macroscopic property, only in the sense that kinetic energy is real, and then only in the sense that position and momentum is real below Plank's constant), string theory and 10 dimensional space, materialism (if we know anything, it is that we make decisions. It might not happen as much as we think, but we make some decisions. Materialism is a very abstract construction far removed from the immediacy of making decisions and if it rejects will, then materialism is absurd), mind-body dualism (I don't believe in a spiritual substance), and those experiments that show decisions are made before it is thought (correct, the self is more than one's inner monologue).