René Descartes is called “The Father of Modern Philosophy”, but he slept around; he also fathered the modern world.
Descartes straddles a great divide. Before Descartes, if you had a question, you asked your priest. Knowledge was in the Bible, and the church was the final word on everything. After Descartes, everything was different—his side runs to the modern age. He showed Europe something we now take for granted: knowledge is discovered, not revealed. He trusted his own intellect; he investigated the natural world and proved facts with his mind. Doing so, Descartes (and a few other people you will have heard of), forced religion’s retreat from the ground now owned by science. This took guts, not least because heretics were still occasionally being burned at the stake.
Descartes was more than just gutsy, though. He was a genius, world-traveller, friend of royalty, soldier, and dreamer. He loved to sleep and swordfight, indulged in mysticism, and knocked up his maid. He was so popular that he often had to hide from his friends to get work done—and boy, did he ever get work done. He was the world’s greatest philosopher and the world’s greatest mathematician. Cartesian geometry (the geometry of the game Battleship) is named for him.
Math is odd: everything in it is perfect and demonstrable. A rectangle can be divided with a single line into two triangles—that is not often or usually true: it’s always true. When Descartes turned his mind to philosophy, he kept a mathematical attitude: he figured all knowledge should be just as perfect and absolute. So, in the following selections from Meditations on First Philosophy, Descartes is trying to end the problem of skepticism, which is this: once we get a grain of uncertainty, it is hard to stop doubting.
I’ll give you stupid example: I used to study in bed. Philosophy is boring, so I fell asleep a lot with a book on my chest. Oddly, I often found myself dreaming about reading philosophy. I would transition very smoothly from awake to asleep, and I wouldn’t even notice. I would appear to myself to still be doing homework.
Right now, I am quite certain that I am awake, sitting in front of the computer. But that smooth transition I used to have while dreaming makes me worry a lot: there were times I didn’t notice falling asleep. So how do I know that I’m not snoozing, dreaming that I’m worried about dreaming?
In fact, how do I know I’ve ever even been awake? Wakeful life and vivid dreaming are very similar. Maybe everything is a dream! Maybe nothing is real! Maybe I am not real!
This is the problem of skepticism: doubt snowballs. A small moment of skepticism leads to big problems. Descartes wants to stop the snowballing. This might seem like a very minor thing, but it isn’t. Descartes wants to get rid of doubt altogether. He wants to remove any doubt, in any field, and make all knowledge perfect.
Think about that for a second. It should sound insane. Descartes wants to build knowledge on a completely doubt-free foundation, so that, forevermore, it will be certain. He wants to do away with error. He wants to make everything we know as certain as math.
Oh, and he wants to do it in a couple of days.
This may seem bonkers, but if anyone could do it, it’s Descartes. After all, he had already single-handedly reinvented one field, and he did it when he was only 23. You might remember from your grade-10 math class that shapes can be described with equations and equations with shapes. Descartes was the one who showed that, and he thus united geometry and algebra, two fields that had for all of history been separate. Fixing that took him a few weeks.
His philosophical strategy remains known as the Cartesian method. He says, roughly, he will try to doubt everything systematically, and see what remains. Instead of fighting the problem of skepticism, then, he flips it around; he won’t try to prove that his former ideas are certain. Rather, he will shake them to see if they tumble.
Descartes’ starts by doubting ‘a posteriori’ ideas, then he moves on to ‘a priori’ ones. It is impossible to overstate the importance of this distinction. It’s the fundamental discovery of epistemology—as important to philosophy as DNA is to biology. It is also one of those rare ideas that can change your life.
Take a step back from all the facts you know and consider them from a distance. Philosophers think that every one of them falls into one of two categories: facts you test through experience, and facts you can test using only your mind. (It is very important to see that the question is not how you learned these facts, but how you test them.)
The facts you test with experience are obvious. Is it dark outside right now? I check by looking. Is my father alive? I will call him. These are a posteriori facts. (For some reason, it’s always printed in italics.)
The facts you test with your mind are less obvious, but generally they have to do with math. How many degrees are in a triangle? The absolutely wrong way to find this out is to measure a few triangles. If you were to do that, you might find that your first triangle is 178º, say, and your second is 181º. Your third might be 177º. “They are all close to 180º, so close enough”, you might say, and you would be dead wrong.
You decided that you should average and round the measurements. Why average them? How did you know that 177º isn’t the right answer? Maybe all the other answers are wrong. Why round to 180º? What makes you think that nature works in round numbers, or even base 10 numbers? Why not round to a prime number? Why round at all?
If you are really honest, you’ll admit that you were making up the rules as you went.
The correct way to find the degrees inside a triangle follows. Do it with me. Really! Grab a pencil. It’s worth it.
Start by drawing a tipi. Then draw another line parallel to the base that intersects the crossed tipi sticks at the top. Draw a half circle on top of that line. Clearly, that arc is 180º, because it covers the straight line. (If it were less or more than 180º, the line would be curved or bent.)
The angles at A, B, and C are the same as at angles D, E, and F, respectively, because the lines are parallel. So if D, E, and F are 180º, A, B, C must add up to 180º too.
And that’s it. That’s how you prove that the interior angles of a triangle add up to 180º. Notice something though. We didn’t prove it for any particular triangle. We proved for all triangles. That, if you think about it, is amazing. We proved something that will always be true about every triangle everywhere, and it’s not trivial at all: a triangle’s interior angles are 180º, the same as the angles on one side of a line or half of a square. Neat!
Even deeper, though, is that we proved this using only our minds. This is really amazing: we can prove an eternal truth about the universe with our eyes closed. Meditate on that. It should be astonishing. What are the chances that we, one little species on one little rock in one little solar system, should have access to eternal mathematical truths? This is one of the deepest, most mystical, most amazing things I can imagine.
Finally, you should have felt something when you proved this. Did you? Did you feel a fleeting sense of clarity, of surprise, of finality? This one thing is finished, once and for all: you have demonstrated a complete, perfect fact. It is delightful.
That was an a priori proof. There are other a priori truths, some trivial (bachelors are unmarried men) and some profound (some infinities are bigger than other infinities). All of them, though, are proved using only the mind. Your senses cannot not help at all; in fact, your senses get in the way.
For much of history, a priori truths were truth. People genuinely thought that we could best prove things about the world using our minds alone. Actually, some people still do think this, if only a little bit, now and then, and mostly for God and physics. Ancient Greek philosophers thought they could prove that the world is made up of atoms and void using only their minds. Now, I hear, physicists think the universe is round in all dimensions, as if that makes sense.
Despite the nutty uses to which a priori arguments have sometimes been put, they are very deep. I think—and I’m not alone—that a priori truths are more certain than a posteriori ones, and that they are more permanent, more discoverable, and even, maybe, in some way, more true. Whether the atmosphere is 72% or 68% nitrogen (a very important a posteriori fact) seems somehow fleeting compared to the proof that there is an infinite number of infinite numbers.
Descartes agrees about the relative profundity of a priori truths. In Meditations he first questions a posteriori facts. Then he questions a priori knowledge. He rejects both as being untrustworthy, though the difficulty he has in ridding himself of a priori facts shows how certain mathematics really is.
Doing this, he knocks down the foundation of his knowledge. Without his mind or his senses, it seems like nothing certain remains. He cannot trust his eyes. He cannot trust his mind. What is left? You can almost feel the despair and the sense that he has gone too far, to the edge of sanity—because as well as being a brilliant philosopher, mathematician and swordsman, Descartes was also a great writer.
It turns out that the solution is the problem itself. He doubts everything, and doubt, paradoxically, is the answer. You’ll see. Trust me, though; it’s amazing. He does judo and uses the force of uncertainty against itself. It’s astonishing. It’s beautiful. It’s brilliant. The one small fact left over is called ‘the cogito’, the most important sentence in all of philosophy, the one bit of philosophy that everyone knows.
From that one perfectly certain—and ridiculously small—point, Descartes wants to rebuild all of science. I think it is safe to say that he failed in his efforts, and that it doesn’t matter he couldn’t succeed.
He showed that the one certain and true thing—“I think”—is only found within, in personal reflection. It is not revealed by God and still less by the church. When he laid out his system of doubt, established the most rigorous criterion for truth, and found the answer within, Descartes drew the line; he was on one side of the watershed, and everyone before him was on the other.
Descartes died of sleep deprivation, unless the Pope poisoned him. He was so beloved, though that the French tore his corpse apart for souven-ears. Really.
 Yes. Remember when you fought with your brother and you said “I hate you” and he said “I hate you +1” and you said “I hate you infinity” and he said “I hate you infinity + infinity!”? He was actually right.
Some infinities are bigger than other infinities. How much bigger? Infinitely bigger, of course. There are actually an infinite number of infinite numbers infinitely larger than infinity, and someone (not me!) can prove this perfectly. Just trust me, though. The first person to prove it went insane. No, seriously.