Not the theorem itself, but what the theorem describes.

Not the equation, a^{2} + b^{2} = c^{2}, but what the equation captures.

What sort of thing do they consider it to be? You can’t see, hear, taste, feel, or smell it. It is undetectable by scientific instruments of any kind. It’s not like a neutrino or gravity or the Higgs boson, where observing it (or at least observing its direct effects and thereby inferring its existence) is simply a matter of building the right detector.

The state affairs described by the Pythagorean Theorem is not subject to entropy or the law of conservation of energy or general relativity.

It is insolubly immaterial.

It’s not something that someone just made up, like unicorns. It’s not a tool that someone invented, like a hammer or a slide rule or a mnemonic device to help you remember the notes of the treble clef.

It is an objectively real thing that must be discovered to be known. In fact it *was* discovered, independently, by the Greeks, the Chinese, the Indians, and probably the Babylonians. This means that it existed prior to anyone knowing about it.

Moreover, there is nothing you can do to change it. It is an immutable fact of the cosmos. It does not evolve or grow old or wear down or collapse in on itself or expand over time like the physical stuff of the universe does.

It does not end with the death of the brain, and it would not end even with the deaths of all the brains. In a lifeless universe a squared plus b squared would still equal c squared. The state of affairs described by the PT would still obtain even if the theorem itself had never been developed.

But it was developed, and everyone can see exactly what it says. Everyone agrees completely that what the PT describes is indeed true of all right triangles.

Yet this peculiar thing is not itself an instance of a triangle. It has no material existence at all. It is everywhere and nowhere—available to anyone, yet existing in no specific place.

So in what sense can it be said to exist?

What is this eternal, objectively real, immutable thing that exhibits very specific properties yet has no material being and can only be experienced through a priori intuition?

Is it a spirit or a force or a magic spell? That would be difficult to square with the materialist metaphysics of naturalism. Is it an illusion created by specific neuro-physiological arrangements in the human nervous system? No, we have already seen that it existed prior to its discovery. It is objective and immutable, not illusory and subject to whim.

Where did it come from? What were its origins, by the naturalist’s account? Did it appear at the point of the Big Bang? Or later? What process created it? Where did this event occur?

Does the naturalist, committed as he is to materialist metaphysics, have to carve out a special ontological category for it? Is it some sort of quasi-materialist phenomenon? (That sounds rigorous-yet-vague enough to be useful in a debate.) Perhaps it inhabits some twilight region where the normal rules of naturalism don’t apply? Maybe we modify naturalism just a little. Maybe it’s “naturalism plus,” naturalism with a special carve-out for the state of affairs described by the Pythagorean Theorem?

But if you allow a carve-out for that, then why not for souls? Or for God?

What about the relationship between the ghostlike state of affairs described by the Pythagorean Theorem, hanging in the ether, and all of the right triangles we see every day and can point to all around us? Well, they all conform to the theorem. You could draw right triangles from now till the day you die, and if you square the two short sides, their combined areas will equal the square of the long side. Every time. This is how we know that the PT is a fact about the universe. It is objectively real, yet it is hard to say how naturalism’s materialist ontology can make room for it.

A thought experiment: say you get a little bored of the PT halfway through your lifetime of triangle-drawing and want to spice things up. You declare that, henceforth, the triangles you draw shall conform to a new theorem. Adding the square of the long side and one of the short sides shall equal the square of the other short side.

You will quickly discover that triangles are obstinate traditionalists. They positively insist on conforming to the old ways and have stubbornly refused to take the slightest interest in your new theorem.

That is because the PT describes something deeply and utterly true about right triangles. The state of affairs it describes is not merely an epiphenomenon of neuro-physiological states—an illusion created by your brain. It is a genuine facet of the underlying logic of the cosmos that you have perceived through a priori intuition. It is Truth, with a capital T, and is so quite independent of what you choose to think about it.

Nor is it true because you prefer to believe it to be true or because your brain has tricked you into believing it or because you happen to need a conceptual tool to help you do things with triangles. Understanding the PT is your mind perceiving the order of the universe, or at least a little piece of it. And this little facet of the cosmic order, which itself is not a triangle, will nonetheless govern every right triangle you ever encounter.

Yes, mysteriously, the idea maps to the world of mundane stuff so well that you can make very accurate predictions about all of the right triangles you will ever encounter—even across a lifetime spent drawing right triangles.

That’s because the form of the right triangle exists prior to any instance of a right triangle. Go ahead, create as many as you’d like. They will all conform. They will all instantiate. They will all participate. And the more precisely you draw them, the more precisely they will conform–or, if you prefer, instantiate the form or participate in the form.

But there is a twist. At some point in your lifetime of triangle drawing you might wonder just how accurate the PT is. You might get a ruler with finer gradations. You might get better instruments to ensure a “perfect” 90 degree angle. You might use a laser to make sure sure your sides are “perfectly” straight. And no matter what efforts you made along these lines, you would find that your tolerances would never be tight enough to produce measurements that *perfectly* match the PT. Zoom in close enough, and you’d find that all of your triangles have sinned and fallen short of the glory of the PT.

Should this cause a crisis of faith? No. It only shows that the physical world is messy at the margins and instantiating perfect forms using physical stuff is beyond what we can ever hope to do. (We can use computers to do it “perfectly” within a quantized virtual space, but that’s cheating—the physical world is analog until you get down to scales that are only relevant to humans in very specialized contexts.)

One thing we *can* say is that some physically instantiated triangles are more perfect than others. We know this because our intellects can perfectly grasp the form of a triangle. And we can see the gaps between what we know with absolute certainty to be a triangle and our attempts to instantiate that triangle. From this it necessarily follows that the state of affairs described by the Pythagorean Theorem is a real feature of the universe.

Of course, this meditation on the state of affairs described by the Pythagorean Theorem is illustrative of a much larger point. Namely, that naturalism, with its materialist metaphysics, struggles to account for the realm of a priori truths and our ability to grasp them. When pressed to provide such an account, naturalists typically resort to just-so stories about tigers lurking in the tall grass, and how our primitive ancestors must have developed a capacity for abstract brain illusions or we wouldn’t be here. Of course there is no direct evidence for these stories, nor could there ever be. They are exercises in speculation and are plausible in direct proportion to your belief in the premise that materialism is true. And that is all they are.

Objection: Earlier you said the PT was not affected by general relativity. But Einstein shows that real space is not Euclidean at all!

Reply: Yes, well, if you find yourself near a black hole and your triangles are behaving in a manner that would scandalize Euclid, rest assured there is a suitable non-Euclidean geometry for just such an occasion. But that is no help to the naturalist and his materialist metaphysics. The theorems posited by those geometries bear all of the properties relevant to our present discussion, insoluble immateriality, immutability, etc., and so pose the same problems for materialists. It’s worse, even. The geometries of curved space were discovered long before Einstein dared to believe that they describe features of the actual material universe, so they can’t be dismissed as the illusory product of brains generalizing from experience.

Again, the Pythagorean Theorem is not true because of any particular sequence of electrochemical impulses in the brain. It is true necessarily, all on its own. You see that it is true the moment you understand it. It is self-evidently and necessarily true. Recognizing this is a priori intuition at work. That is, the unique capacity of human beings to grasp such truths. But the Pythagorean Theorem doesn’t need you in order to be true. It is true quite independent of you or your electrochemical brain impulses.