how we can know there are things we cannot know
It is possible to know that there are things we cannot know.
This is quite surprising, and goes well beyond the so-called "zero knowledge" proofs of cryptography, although there are sufficient similarities that's it's probably worth starting with them.
The idea of a zero-knowledge proof is that you can prove that you know something without revealing what it is you know. This generally involves proving you can do something that would be impossible if you didn’t know something.
For example, consider a maze surrounded by high walls. In one wall there are two doors, side by side. If you want to prove to someone you know your way through the maze, all you have to do is run in one door, through the maze, and out the other fast enough that there would be no plausible way of searching for the path: you can only run so fast, and mazes take a long time to explore and find the way out of. By running it quickly you've dramatically increased the evidence that you've memorized its path without revealing anything about what the path is.
A different kind of zero-knowledge proof is interactive. For example, you can prove you're able to distinguish colours to a red/green colour-blind friend (everyone is always friendly in zero-knowledge proofs for some reason, which is a pleasant deviation from the usual cryptographic schemas involving malignant actors of one kind or another.)
In this proof, you start with two balls that are extremely similar except for colour. Your friend takes one in each hand and hides both hands behind his back, then shows you one, then puts that hand behind again and either does or does not swap them and shows you the same hand again, and asks if he has swapped them or not. Repeat as necessary: you will eventually convince your skeptical friend that you can indeed perceive something about his balls that he cannot (as it were).
So it is possible to know that information exists without knowing what the information is.
It is also possible to know that information cannot be known even though it must in some sense exist.
As promised last week, the key to this lies in thermodynamics.
Thermodynamics is the study of heat, which is the random motion of particles like atoms or molecules under Newtonian forces (or so we'd like to believe.)
Particles can move in various ways, and thermodynamics is grounded on the principle of equipartition, which says that the energy of random motion is divided up equally among all the ways a particle can move.
Each way a particle can move is called a state, as in "a state of motion". In quantum mechanical systems the possible states of motion are often discrete--the energy levels of an atom, for example--but they may also be effectively continuous for free particles. Because of the wave-like nature of particles, when when a particle is bound in box it can only occupy states where the value of the wavefunction is zero at the boundary. This restricts the bound states of motion to particular wavelengths that are integer fractions of the length of the box.
A great deal of thinking about the quantum world is focused on this kind of one-particle system. This is wrong, because reality is only manifest when there is more than one particle involved.
If we have more than one particle, the number of multi-particle states available to the system--which is the number of states over which the system's thermal energy will be divided up--depends on how the individual particles occupy the various single-particle states.
Consider two particles in a box. The states of motion might be left-ward and right-ward motion, and there would appear to be four two-particle states: both moving left (LL), both moving right (RR), one moving left and the other moving right (LR), one moving right and the other moving left (RL).
But those last two states assume we can distinguish the left-ward moving particle from the right-ward moving one.
What if we couldn't? Not because we aren't looking or have our eyes closed, but because from our perspective as children of time the particles are indistinguishable.
In that case, the number of multi-particle states changes dramatically: there is only one state of "a particle moving left a particle moving right", so instead of a total of four states there are only three.
This is the case because there is something fundamental that prevents us from distinguishing the particles. It is not that we do not distinguish them, but that we cannot: if we could, by any means whatsoever, tell the difference then there would be two separate states. In the ordinary case of classical mechanics there are frequently cases where we could distinguish two particles, but for whatever reason don't. They still obey classical statistics, and the states we don't bother to distinguish but could if we wanted to remain distinct. It does not matter if we do tell them apart: the fact that we are able to do so is enough to separate the single "one each way" state into two different states.
This is the point where philosophers sometimes get feisty, because there's a fairly important principle, sometimes called Leibniz's Law, that says "two things that have all the same properties are the same thing." This is also know as "the identity of indiscernibles", and it is false. Philosophers still teach it as if it was true, mind, even though we've known it to be false for almost a century now.
We know it's false because we have plenty of real systems that are made of indistinguishable particles. Elementary particles like electrons do not and cannot have labels that identify them. Every electron has the same mass, charge, and spin, and if we throw two of them in a box they become indistinguishable: there are two electrons in there, but there is no way at all, even in principle, to tell which is which.
Atoms of the same isotope are for the most part indistinguishable as well. When they are in a solid they can be distinguished by the position they occupy, but two helium atoms in a gas? Two oxygen or nitrogen or argon atoms in the air you're breathing right now? Two CO2 molecules? Again: there may be two of them, but they are not distinguishable by any means whatsoever.
And we know this because all kinds of thermodynamic properties depend on the number of multi-particle states motion that are available for the random energy of motion to be spread over. Mostly the big effects of particles being indistinguishable don't show up until things get very cold. For a gas at room temperature there are far more possible individual particle states available than there are particles, so it doesn't matter if the particles are distinguishable or not. But as the temperature of a gas approaches absolute zero the particles all try to get into the lowest energy states, and then the thermodynamics of the system becomes wildly different for distinguishable and indistinguishable particles.
One thing a particular type of indistinguishable particles does is form a superfluid at low temperatures, which means it has zero viscosity (resistance to flow). Helium does this at 2.7 K. If it consisted of distinguishable particles it would not. It does, so we know that no two helium atoms can be told apart by us. Full stop.
We sit on one side of a boundary that divides us from the Beyond. We live in the world of space and time, causality and non-contradiction. On our side of the boundary, where space and time rule the day, there is no way at all to distinguish two particles of the same kind. But on the other side, in reality, there are two particles, not one! There is something that makes them "not the same particle", but we cannot access it.
That is, in some way that is inherently and entirely inaccessible to us, the particles are not the same particle, because if they were the same particle there would only be one of them. There are two. Ergo, they are different, but in a way that we cannot ever access. If we could, the thermodynamics of cold gases--and many other things--would be different. So different that we would die.
So we can know there are things we cannot know. We know that two helium atoms are not the same atom, but we cannot know what distinguishes them.
And insofar as we want to identify that part of reality that is beyond with god--which I will layout the case for next week--this implies that god is (almost) silent: if god could communicate with us by any means whatsoever, the laws of physics would be different, and life would cease to exist in this universe.