Annotations, Ch 18
“Aboard this ship, in the middle of space, I am free to define myself according to my will. I choose my axioms, both human and Vulcan, upon which I build my mathematics.”
- Spock
A note on mathematics
I don’t know as much about the history of mathematics as I’d like, but I do know this: everything you learned in grade school and university is really old. Geometry? Most of that is stuff that Euclid did back in Ancient Greece. Calculus? Newton and Leibniz made their big breakthroughs back in the 16/17th centuries. My point is that modern mathematics is very different from the stuff you were forced to learn in school.
To give you a taste of some more “modern” stuff, the following is a very important theorem in modern Algebra that’s actually pretty basic and about 200 years old (thus the quotation marks). It’s called the Fundamental Theorem of Galois Theory. There are several ways to state it, but I’m taking the formulation from Michael Artin’s Algebra: “Let K be a Galois extension of a field F, and let G = G(K/F) be its Galois group. The function H → K^(H) is a bijective map from the set of subgroups of G is to the set of intermediate fields F ⊆ L ⊆ K. Its inverse function is L → G(K/L). This correspondence has the property that if H = G(K/L), then [K:L] = | H |, hence [L:F] = [G:H].”
Confused? Don’t worry. It took a year for me to build up all the necessary vocabulary and theorems to understand the statement, let alone the proof. Once I did understand, it absolutely blew my mind. Galois theory is really cool. One of the things it explains is the reason why we can’t solve equations like this: x^5 - x + 1 = 0 using only the operations +, -, ×, ÷, √, and cube roots. Anyway, I digress.
So what does “geometry without the parallel postulate” mean, and what the hell does it have to do with Spock being introspective? (I’m not even going to go into set theory and the Axiom of Choice. It gets very abstract.) Well first, what is the parallel postulate?
The parallel postulate is one of Euclid’s axioms. Axioms are statements where their truth is taken for granted. You don’t prove an axiom-it’s the starting point for all other theorems. Now, someone might come along and decide that they’ll take as many arbitrary statements as they want, say these are axioms, and call that system a mathematics. Fine, you can technically do that-no one’s stopping you-but if your system ends up horribly contradicting itself, or if it’s just not very interesting, then no one will care about it and no one will study it. In math, you want to have as few axioms as possible and still get tons of interesting theorems. When you get into mathematical research (i.e. inventing/discovering math), striking this balance is important and hard to accomplish.
I’m actually going to cite Playfair’s axiom because it’s more intuitive (for me, at least) and it amounts to the same thing as the parallel postulate: “In a two dimensional space, given a line l and point x not on that line, there is only one line m that can be drawn through x such that m is parallel to l.” If you draw this on a sheet of paper, this statement seems pretty true. But it isn’t quite as obvious as the other four postulates, and for about two thousand years, mathematicians tried to use the other four postulates to prove the fifth postulate. Needless to say, they didn’t succeed.
In the 19th century, mathematicians decided to finally stop trying to prove that the parallel postulate was true, but to see what happened when it wasn’t. They ended up getting very new, different geometries, now referred to as non-Euclidean geometry. The most common examples cited of non-Euclidean geometries are hyperbolic geometry and elliptic (sometimes called spherical) geometry. In hyperbolic geometry, there are infinitely many lines m drawn through x that are “parallel” to l in a plane, while in elliptic geometry, there are no lines m through x that do not intersect with l. (There’s a nice diagram on the wikipedia page if you’re having trouble visualizing this- http://en.wikipedia.org/wiki/Non-Euclidean_geometry)
Which geometry is the “right” one? All of them, actually. All of these geometries have been proven to be logically consistent, and are equally valid ways of considering space. Consequently, you end up with some funky results that you thought couldn’t be true, like the fact that in hyperbolic space, the sum of all the angles in a triangle is less than 180 degrees. I mean, we’ve had that “sum of all angles in a triangle is 180 degrees” drilled into our heads since we were kids, and now I’m telling you that it’s not always true? And here we thought that math was so rigid and inflexible! Next thing I’ll tell you is that 2 + 2 = 5.
Well . . .
I’m kidding, right?
There is an object in “modern” mathematics called the zero ring. It’s more technical than this, but what it boils down to is that you define 0 = 1. Then 2 = 1 + 1 = 0, and 5 = 1 + 1 + 1 +1 +1 = 0, so 2 + 2 = 0 = 5. So yes, it is possible to have 2 + 2 = 5. In fact, it’s possible to have 2 + 2 = 0 (modular arithmetic, mod 2 or mod 4), 2 + 2 = 1 (mod 3).
. . . What? How can you define 0 as 1? Is math breaking down and turning into anarchy? Has everything gone to the dogs since Pythagoras and Euclid?
No. These are all perfectly accepted mathematical constructs, and every mathematician knows about them. In fact, he probably thinks they’re rather boring and he’d rather look at something more interesting, thank you very much.
Okay. Now that I’ve completely destroyed what you thought was addition, let’s cut right to the chase. What does this mind-bending math have to do with Spock?
There is a stereotype perpetuated in the general population that in math, there is only one correct answer, only one way of solving a problem, only one way of considering space, or integrating a function, or proving a theorem. This is not true. When you get into higher level math, there’s usually many different ways to prove a theorem (some more elegant than others). Mathematics is no longer about the One and Right and True answer like it might have been in your calculus class. Mathematics is about being logically consistent. Does this new theorem follow from our given assumptions, definitions and previous theorems, or does it lead to a contradiction? If the answer is yes, it’s logically consistent, then great! We have a new theorem. If the answer is no, then we go back and try to find where we made an error in our proof.
Spock is half Vulcan. He is a creature of extreme rationality, but it does not necessarily follow that he is completely rigid and eschews emotions. Like mathematics, you have to look at the underlying assumptions and this, more than logic, determines who he is.
Spock is half human. He is a creature who feels, but as with logic, it does not necessarily follow that those actions guided by emotions are illogical. We can draw another parallel to mathematics because emotions are guided by their own internal logic. That logic is highly individualized and the rules are difficult to discover, but they are there. In some ways, we can think of logic as Euclidean geometry, and emotion as non-Euclidean geometry. It doesn’t make sense to compare them, but each system is logically consistent.
So . . .
“Aboard this ship, in the middle of space, I am free to define myself according to my will. I choose my axioms, both human and Vulcan, upon which I build my mathematics.”
Jim is in a class of his own. Spock builds himself on axioms, but Jim-
“bends and breaks and believes the parallel postulate as he pleases, as it is useful, at his convenience. At his core, he burns with a drive to discover, push out into the dark of space, the thrill of the unknown, to live and thrive and fly and be free. To stand in the light and touch the darkness.”