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First off, go away and watch this video.

Mind blowing, eh? What especially gets to me is the chance it applies to real-world physics. To understand why, I have to blow your mind a second time.

While the proof used in this video is a bit dodgy (you can’t group together terms of an infinite series), you can accomplish something more robust via analytic continuity. Just like that first video, there’s real-world physics applications, one of which is the Casimir Effect.

Picture you have two giant plates partly submerged in water, parallel to each other and nearly touching. If any waves pop up or persist between the two, they’ll be forced to have a wavelength that divides the gap evenly: they’ll span the whole length, half the length, a third, and so on. What those waves are in the Casimir Effect is in dispute, as some argue they’re the virtual particles that form the quantum foam pervading our universe, while others think they’re boring old electromagnetic waves but with a Quantum interpretation. Whatever the case, the same types of waves are on the other side of the plates, but these have fewer limits on wavelength. Since wavelength governs energy, an energy difference is created between the two sides, and that balances itself by generating a force.

How much force is that? It partly depends on the sums of all the waves that fit in the gap.

1 + 2 + 3 + 4 + ... = sum i from 1 to infinityHere’s where it gets interesting. There are two ways this could pan out: either that sum is cut off after a finite point, or it carries on without limit. If it’s finite, the force should be a large positive number. If it’s infinite, it could be any number big or small (as it would probably be cancelled out by another infinity, since we’ve never observed infinite forces), but intuitively you’d guess it would still be positive.

Intuition does a really poor job of grasping infinity, though: the sum is -1/12, according to that analytic continuity. As barmy as that result sounds, it’s the one that best matches observation. Finite sums give entirely the wrong answer.

But if we need to assume an infinite number of waves exist in order to make sense of the world, doesn’t that imply an infinite number of waves exists? That, contrary to some theologians, there are actual infinities in the world? Let’s formalize this:

1. Theory A depends on assertion B.
2. No substitute for B is possible or plausible.
3. Theory A explains physical scenario C to the limits of our precision.
4. No other non-trivial theory we know of can explain C to the same level of detail without also depending on B, nor do we expect one to arise in future.
5. From the above, it follows that B is probably true.

It’s a weak inductive argument. At best, our belief in assertion B is no stronger than the evidence for Theory A. But when Theory A happens to be the most accurate theory devised by humans, and infinities form a core part of it, B can wind up very well supported.

Theologians may have been the first to realize the above, as the first time I’d heard this argument deployed was via presuppositionalism. That apologetic asserts the existence of the Christian God is a necessary assumption for basic logic to function, and by extension those that don’t assume God exists have at least one more contradiction in their worldview. Ergo, anyone who makes that assumption is on firmer intellectual ground.

There’s a number of deep flaws with this view, but more relevant is that it fails points 2 and 4 when plugged into the above argument. We have no explanation for how any god could be necessary for logic, let alone a specific one out of many; and we can pull logic out from the Anthropic Principle by pointing out that if physics didn’t follow some sort of logic, we wouldn’t exist. So the argument doesn’t allow you to assume a god’s existence.

But what about the existence of actual infinities? To this physics fan, that looks to be on solid ground. If the paradox from the first video goes on to be as well supported, then we’ve made it probable that multiple types of infinity exist in the real world.

Mind-blowing stuff, eh?

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