Week 2, The Theory of Embodied Math

CS198–79 Philosophy of Computation Course Notes

Background reading: Chapter 1, Metaphors We Live By, Intro and Chapter 8, Where Mathematics Comes From; http://www.cabrillo.edu/~ewagner/WOK%20Eng%202/Lakoff%20&%20Johnson%20-%20Metaphors%20We%20Live%20By.pdf and https://www.scribd.com/document/339074367/Where-Mathematics-Comes-From

Supplementary Worksheet: https://d1b10bmlvqabco.cloudfront.net/attach/jdf1ibhhu4545d/hz5sgoa7845d1/je0gp43atkef/sp18_7_what_is_math.pdf

1.

We start with a quote from Einstein.

The words of the language [of mathematics] … do not seem to play any role in my mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined… The above mentioned elements are, in my case, of visual and some of a muscular type.

It reads that Einstein did not think about math in terms of mathematical symbols, but in sensations of a muscular nature. This quote is very germane to today’s topic: the theory of embodied math, outlined in the book Where Mathematics Comes From.

Did you like the book? I think this book is very polarizing. One of the reasons why it is polarizing is because it is an explicit rejection of mathematical Platonism. Briefly, mathematical Platonism is, as the author Lakoff puts it, a sort of “romance” of mathematics; that math is a set of eternal, unchanging, ethereal, human-independent truths. Mathematical Platonism gives a sort of timeless, sublime, perfect beauty to math. It is a very seductive idea, and also the default idea about what math is in this culture.

Here I should like to bring in a quote: “a great philosopher is someone who is wrong in a very interesting way.” (Corey Mohler, creator of Existential Comics) I think this quote applies very well to Lakoff in general. Parts of his theory seems just totally wrong, parts of it seem incoherent, and in any case one wants to reject many of its assertions, but overall, its core idea that math comes from, of all places, motor control, is so radical thatit is hard to honestly read this book and keep living your life as an unshaken mathematical Platonist.

To grossly summarize, one might say that mathematical Platonism says math is independent of human experience, and that the theory of embodied math says that math is dependent on human experience. As another gross summarization, let’s say one is the opposite of the other.

2.

So who has strong opinions against the theory of embodied math?

E: I think there’s a problem with it. If we say that math is dependent on humans, just constructed, invented by humans, then what about, for example, derivatives? The derivative of velocity will always be acceleration, even if someone doesn’t agree with it.

That’s a good point, but I’m not sure if that’s a problem with the theory of embodied math. I think Lakoff will agree with you. Lakoff might say: yes, math is a set of metaphors which everyone possesses, which everyone necessarily agrees with, because everyone has a body. Velocity and acceleration and derivatives are all embodied concepts that everyone necessarily has, because everyone has a body. Math is still dependent on human experience, namely the human body.

E: But if everyone agrees with it, in what sense does the theory of embodied math say that math is dependent on human experience? It seems that if something is necessarily agreed upon, it is independent of human experience. Like: if different people have different opinions about something, we might say that thing is dependent on human experience. But if everyone agrees about something, that thing must be independent of human experience.

You said: if x is something everyone necessarily agrees on, x must be independent of human experience. Here’s something I necessarily “agree” on: if I get shot in the face, I will die. Now I might believe: “Even if I get shot, I won’t die!” But then, if I in fact get shot, I will in fact die, with mathematical certainty. But the fact that I will die if I get shot in the face is certainly part of my experience. Another example: I have a hand. I will have this hand, insofar as I decide not to cut it off. In a way, I must agree with the fact that I have a hand. Now does this mean my hand is independent of my experience? I think this hand is very much a part of my experience.[1]

N: But Lakoff’s theory need not be contradictory to mathematical Platonism.

That’s right. Now Lakoff doesn’t deny mathematical Platonism. Rather, he says it’s a “matter of faith” whether transcendental, humanly inaccessible mathematical objects exist or not, and that the only sort of mathematics we should concern ourselves with, the sort which we can study scientifically, is a human-based mathematics. Lakoff doesn’t deny mathematical Platonism, he dismisses it. But this is a very different position from a bona fide mathematical Platonist, for whom the idea that there are transcendental mathematical truths is very important. Lakoff doesn’t care and doesn’t think anyone should care.

3.

Now, who strongly disagrees with mathematical Platonism?

S: Math might exist perfectly in the abstract. But that doesn’t mean it can necessarily explain the world, that it is a distillation of principles behind the workings of the universe. Mathematical laws may say one thing, but humans can do another thing.

Again, this argument: if I get shot in the face, it is mathematically inevitable that the bullet will pierce through my skull, mangle my brain tissues, and leave a hole at the back of my head. And when I say it’s mathematically inevitable I mean I can write out mathematical equations that describe how exactly the bullet will destroy my face. I might say that there’s a one-to-one-correspondence, in jargon an isomorphism, between mathematical laws and the physical workings of the bullet as it travels through my head. And I think a bullet traveling through my head is the least abstract thing in the universe.

4.

Something is off here. I asked for an argument for mathematical Platonism. The argument was: some mathematical fact, such as derivatives, for example, exist whether one wants them to or not. I asked for an argument against mathematical Platonism. The argument was: math might exist in the abstract but not have the all-knowing explanatory power that Platonism seems to bestow onto it. Then I answered both with the same example: a bullet bifurcating my face .

How come? Let’s recap. The argument for mathematical Platonism was basically an argument for the idea that math is independent of human experience. The argument against mathematical Platonism was basically an argument for the idea that math is dependent on human experience. So really what I’m going after is the dichotomy between “independent of human experience” and “dependent on human experience”. What seems to be straight in the middle of the dichotomy is the fact that humans have bodies. Now is a body independent of human experience, or dependent on human experience?

(1) If one says math is independent of human experience, then one should have no problem getting shot in the face.

(2) If one says math is dependent on human experience, again one should have no problem getting shot in the face.

Why? (1) because math is independent of human experience, but presumably math is what matters, and human experience doesn’t matter. So getting shot in the face can’t possibly affect math, and math is what matters. Maybe there will be a sort of math-heaven you’ll get to once you get shot in the face. (2) because, as a bullet approaches your face at bulletspeed, you can say to yourself, “I disagree with the mathematical idea that the bullet will obliterate my face.” And if you are right, you ought not worry about the bullet. The bullet’s not going to do anything. So you let the shooter, with his AR-15 positioned directly pointing at your face, pull his trigger without any complaints.

But obviously both are ridiculous, so this argument is a reductio ad absurdum of the dichotomy: math is neither dependent nor independent of human experience, just as the body is neither dependent nor independent of human experience. X is independent of human experience/X is dependent on human experience is just not the right syntax to think about these issues.

5.

Here’s a little thought experiment.

Math in Prison: Suppose you pick some guy/gal at random, let’s call them Bonnie, and stick them in an empty room. Bonnie has nothing to do, at all, but you provide them with a bunch of math books, let’s say, the entire Berkeley undergraduate and graduate sequence in math. You keep Bonnie there, locked in, for ten years. At the end of the ten years will Bonnie understand all the math? You might answer yes; Bonnie will be so bored they’ll turn their eyes toward the only complex piece of information in the room — the math books — and in ten years will absorb them all. Or you might answer no; maybe Bonnie just doesn’t have the aptitude.

This thought experiment is a sort of litmus test about whether you think mathematical ability is largely innate or largely a matter of circumstance. I’m inclined to believe that Bonnie will come out of that room as an expert in math, moreover that if we keep them in the room for ten more years they’ll discover hoards of new theorems.

M: I don’t think they will absorb all the math. This brings to mind the thought experiment about Mary and the colors.

Mary and the Colors Thought Experiment: Mary is a neuroscientist who studies colors all day, but she was born and raised in a black-and-white room and has never seen color. One day, Mary steps out of the room and sees color for the first time. Will she gain any new information?

So, in the Math in Prison thought experiment, Bonnie might understand the math in an abstract way, but maybe they’ll never understand it in the visceral way Einstein does. Similarly, Mary might understand color in an abstract way, but she’ll never truly understand color until she steps out of the room. The question here, though, is what constitutes “stepping out the room” in our math thought experiment? It should be some event where our locked up person suddenly understands all of math in a flash. But this seems much less plausible than understanding color, in a flash, after seeing color for the first time.

6.

Lakoff writes that the following argument is a part of the romance of mathematics, what I’ll call the Disembodied Thinking Computer Argument:

If math is disembodied, then computers can think.

Lakoff rejects this argument, and seems to think that if one rejects this argument, one must believe both that math is embodied, and computers can’t think. But does logic compell us to do so?

The above argument follows a P => Q structure, where P is “math is disembodied” and Q is “Computers can think.” The rejection of this argument is ~(P => Q):

It is not the case that: if math is disembodied, then computers can think.

But that’s clunky, so let’s simplify things a bit. P => Q can be written as ~P or Q, so ~(P => Q) = ~(~P or Q) = P and ~Q:

Math is disembodied, and computers can’t think.

So the rejection of the Disembodied Thinking Computer Argument compells us to believe both that Math is disembodied and that computers can’t think. (Which seems strange; one explanation is that logical rejection of the argument is different from an intuitive recoil from the argument.) Now this is a far cry from what Lakoff wants to say: math is embodied, and computers can’t think. So Lakoff isn’t logically rejecting the Disembodied Thinking Computer Argument: if he did, he would have to say math is disembodied. So there is plenty of room to believe both that math is embodied and computers can think; it turns out our intuitive recoil from the Disembodied Thinking Computer Argument isn’t what it seems.

I point this out because Lakoff’s sleight of hand is somewhat hard to notice, and one might fall into the trap of believing that if math is embodied then computers can’t think. But I don’t think there’s a basis to that argument. I want to make this foundation precise, because the rest of the semester I want to argue both that math is embodied, and that computers can think.

7.

One last argument: if you reject faith in mathematical Platonism, you must reject faith in God. Theology dictates that mere humans want to know Him, but can never fully understand Him. Similarly, mathematical Platonism says, mere humans want to know math, but can never fully understand math.

A: There’s a problem here. The theological component has a “Him”. But obviously math doesn’t have a gender.

Are you sure about that?

I think math is very male! It is construed in a very masculine way, which partly explains the sexism in STEM disciplines; also the idea of mathematical authority, and authority in general being sexist in a more obvious way. Later, we’ll delve deeper into the theology and sexism of math when we read a book called The Religion of Technology by David F. Noble.

8.

N: About your point in (section 6 of these notes), I think there’s a problem. If P is “math is disembodied” it doesn’t necessarily follow that ~P is “math is embodied.” You would have to say “it is not the case that math is disembodied”, and that’s different from saying that math is embodied. Math might be something else not disembodied, but it doesn’t have to be embodied.

That’s a fair point, and an important one generally. To say “math is embodied” is stronger than to say “it is not the case that math is disembodied”. But in this case I don’t believe it matters: it is certainly the case the “math is embodied” and “math is disembodied” are mutually exclusive, and that’s all we need for the argument. To draw a diagram:

( Math ) | ( ) Math

Math is either in the circle or not in the circle. If it is in the circle this represents the sentence “math is embodied”, if it isn’t, it represents “math is disembodied.”

N: But Lakoff still leaves room for mathematical Platonism. So Math could be both inside and outside the circle.

In that case you would have to be talking about two different kinds of math, which we could write like this:

(Math1) Math2

Where Math2 is presumably the Platonic conception of math, and Math1 is the math the humans can know or ever hope to know.

[1] This might be confusing, and I think it’s because there are several senses in the phrase, “independent of human experience”. In one sense, I’ll call it the judgment-ready sense, a thing T is independent of human experience if and only if everyone ought to agree that T is true. We have a strong moral sentiment that everyone ought to agree that climate change, for example, is true, or at least that the Earth is round, for example, is true. That is, we might say climate change is a fact “independent of human experience”, and that the Earth is round is a fact “independent of human experience”. If someone denies this, we can confidently brush them aside, and say to ourselves, “they’re denying a fact that’s independent of human experience; they’re being irrational, I’m being rational, therefore I have no reason to talk to them.”

But the sense in which I say, “independent of human experience”, is not a judgment-ready sense. I’ll call it a necessarily-agreeing sense: again, if I get shot in the face, I die, whether I agree or disagree. It’s not clear if there’s a sharp distinction between the two: it may be the case that whether I believe in climate change or not, climate change will eventually come and maybe Berkeley will then be submerged underwater and I will drown. The difference is more in immediacy: while someone can disagree with a fact that is independent of human experience in a judgement-ready sense, and go on living their life perfectly well (for now), nobody can disagree with a fact that is independent of human experience in a necessarily-agreeing sense and go on living their life — to “disagree” with the existence of my hand would mean I would have to cut it off, and that affects my life significantly. One might say: a fact that is independent of human experience in a necessarily-agreeing sense is true only insofar as humans exist, but a fact that is independent of human experience in a judgment-ready sense is true regardless of human existence.

Why humans in particular? What’s so special about humans? Later we should like to build up a framework to be able to say “complex-enough being” instead of “human”. But let’s stick with “human” for now.

Facilitator: Jongmin Jerome Baek (jjbaek.com)