Can a sphere be turned inside out, such that the outside becomes the inside, and vice versa, without creasing or tearing or poking a hole in the sphere?
Obviously not. Take any spherical object, such as a sealed balloon, and, without popping it, turn it inside out. Can’t be done.
Doesn’t have to be a real balloon, an imaginary one will do. Make it (in your mind) of some ultra thin material, with no real air inside to reduce the chance of popping. Then try to envision a way the thing can turn inside out.
You’ll quickly see it can do so only if the material of the “sphere” is not really solid, but has, at at least one point, a gap through which the material of the outside can pour into the inside, or vice versa. Of course, the material has to be thinner than the hole, else it won’t fit through the hole.
Now any material that is flexible enough and mesh-like, such that it contains at least one sizable hole, can perform this magic trick. But what if the material is solid; i.e., has no holes? Depends on what you mean by solid.
Infinitesimally solid? Meaning no matter how close you focus in on a patch of the surface it is everywhere connected with no holes? Can such a material exist in actuality? I have no idea. We know there are already “gaps” between atoms (or whatever) in ordinary surfaces, but that doesn’t mean anything can get through these gaps, especially if the objects are larger than the atoms (or whatever). This ignores all the forces acting on these surfaces, including those forces that could bring the eversion about; about that, wait a moment.
Even though such a continuous surface might not exist in actuality, it does exist in potential. Which is to say, we can write down equations for these kinds of surfaces which assume they are smooth.
How do you then evert surfaces like these, even if such an exercise is only in potential? It can be done. (Simplest proof I have seen: pdf.)
Spheres with continuous surfaces can indeed turn inside out—potentially. Which flies in the face of intuition, of what we know about real surfaces in actuality. Something has gone wrong with our intuitions then. But what?
Infinity. The trick, in principle, in potentia, involves infinite forces.
It is only when things can be pushed, stretched, and manipulated into infinite positions, which necessitates infinite forces, that the magic happens. Mathematicians have proved it can work; or rather, that it is possible in potential. They even have clever animations, but all of them are cheats because none can show infinite gradations of surfaces, and the power is all “from above”, in the minds of the programmers. None of these proofs say how, or rather by what mechanism, what cause, these eversions work. They just say (or rather imply logically) that if they do, infinite forces are needed.
There is only one infinite force.
What does this have to do with free will? Wilbur Salazar asks some excellent questions along those lines:
In your latest post, So-Called Random Numbers And Encryption, you defined randomness as something “where random merely means of unknown cause.” Later you imply that we might someday undo quantum uncertainty—certainly undoing Heisenberg’s principle. How does that affect your interpretation of free will? From what I’ve gathered binge-reading your posts, you seem to be a libertarian (dualist?). Do you think we will undo our notions of free will by undoing quantum uncertainty?
First, there is no proof quantum mechanics is strictly non-deterministic, only that if it is deterministic, (some? all?) causes are non-local. The papers linked in the post give illustrations of what some of these kind of non-local causes might look like. Whether these will be the only ones or there will be others, I don’t know. It may be that locality is itself false, in some or all cases.
However, this isn’t an article about quantum causality, but about our intuitions of free will. There is no proof that indeterminacy in QM causes free will, or is associated with it in any way. If indeterminacy is ontic (physically real), and our minds are totally caused by physical forces, then all our choices and thoughts are made wily nily. They therefore cannot be trusted, since trust is caused to happen “randomly.” Everything is thus nonsense. Even this conclusion.
Our minds, in the sense of our intellect and wills, are not, however, physical. So quantum mechanical forces, determinate or not, can only affect them indirectly. Still, our intellects and wills do operate, and operations have causes. The leading candidate for these causes is the same as the infinite cause in sphere eversion.
Free will exists, which is certainly true given the observation that we freely make some but not all choices. Should I shoot this guy in the neck and take the money in the till, or instead pay cash for the gas? Should I comment on this post and call Briggs a fool, or merely lurk?
Free will exists, but how is exists doesn’t make sense if you accept that, as Salazar asked, events like our choices have causes. If our choices are caused to happen—even by God—then choice isn’t free, but determined.
That kind of thinking leads to the argument, “I can’t see how the will is free if it is caused; therefore, the will is not free.” Which is equivalent to “I can’t see how spheres can turn inside out; therefore, spheres cannot turn inside out.”
I have no idea how our wills our free given that God is behind the causes of our wills and intellects. Neither do mathematicians have any idea how infinite forces must be employed to evert spheres. All mathematicians can demonstrate is the possibility, even if nobody has any idea how to bring such event about in practice.
We also have an enormous supply of positive demonstrations that our wills are free. It’s only that we have no idea how. But we have just seen that knowing how and knowing that are completely different, even though they are often conflated.
This is a weak proof, or rather a dim pointer to where a proof might lie. It must be, I believe, that a proof of free will will involve infinite causes and powers, just as with sphere eversion. Infinity is mind-busting; intuition fails. Even thinking of very large, not even infinite numbers, leads to dizziness. The mind rebels.
We’re at the point where we don’t even know how to frame a free will theorem in any proper way. We do know the infinite is utterly bizarre. We may be too stupid, in this life anyway, to figure it out. But no theory, in whatever shape or form, can disprove our observations.
To support this site using credit card or PayPal click here