Pythagoras, Athanasius, Lord Russell, and Monty Hall

Anything approaching a fully satisfactory explanation of the phenomena of knowledge requires the co-operative efforts of all those who believe that there is a world of real existence independent of human minds and that this real existence can be truly known as it really is.
— Francis Parker, “Realistic Epistemology
July 6, 2018
Once two subatomic particles interact, “they lose their separate existence. No matter how far they move apart, if one is tweaked, measured, observed, the other seems to instantly respond, even if the whole world now lies between them. And no one knows why.” (Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn [New York; Alfred A. Knopf, 2008], page 3.)

The problem lies in those words “instantly respond,” which suggest an influence that is able to travel much faster than the speed of light, which, according to Einstein, is not possible. How does the tweaking or measuring or observing of the one affect the other?

Right triangles

Consider a Euclidean right triangle whose base is 3 units and whose upright is 4 units. The Pythagorean Theorem tells us that the hypotenuse is 5 units long. Always. No matter what the units are.

Now suppose that I say of this triangle “Oh wait! I’ve made a mistake. The base is 2 units.” In that case, the hypotenuse is approximately 4.4721 units (actually,  20  units).

The 5-unit hypotenuse, does not say to itself, “Dear me, I am much too long,” and become shorter. No. The very instant that I said, “Suppose the base is 2 units,” the hypotenuse “became” 4.4721 units.

But “became” is perhaps not the correct word to use, for the hypotenuse did not transform from the longer to the shorter. It was a change that occurred instantaneously. Once I said, “Suppose the base is 2 units,” there was no intervening unit of time — not a nanosecond — when the hypotenuse was anything but 4.4721 units long.

Something similar will happen if we tinker with the upright, or even the hypotenuse. Change the upright, and the hypotenuse will instantly change. Change the hypotenuse and either one of the two legs will change or they both will — instantaneously. Indeed, effortlessly.

Material implication

Consider the basic form of material implication as developed in Russell and Whitehead’s Principia Mathematica: p ⊃ q, which we read, “If p then q.” Or, more explicitly, “If the proposition p is true, then the proposition q is true.”

If that implicative proposition, made up of two separate propositions (p and q), is true — leave aside how we may know this — then certain interesting things follow.

Suppose I can prove to you — again, never mind how — that the proposition q is false (formally, not-q, or ~q), it follows that p is false (formally, ~p).1 Again, p does not gradually transform into ~p. And again, we do not say that p becomes false. It just is false. There is no waiting for the falsehood of q to work its magic on p. If q is false, so is p. Instantaneously. Effortlessly.


Suppose I have two kittens in a box, and I tell you (truthfully) that one is all black and the other one is black and white. If I ask you to reach in and take one out without looking, what is the likelihood that the one you take out will be a female?

Setting aside the technicalities of genetics and other biological mysteries, you would say, “50 percent.” And you would be right.

But, if — before you take one out — I tell you that the all-black kitten is male, and I ask you, What now is the likelihood that you will take out a female, I cannot say for sure what you would answer. But the correct answer is that the probability is 33.33 33  percent, or 1/3, that the kitten you take out is female. And the probability that the kitten you take out is a male is now 66.66 66  percent, or 2/3.2

These probabilities changed the instant I told you that the all-black kitten was male. They did not have to hear what I said. Having no physical connection to me, unaffected by the strength of my voice or its timbre, and not needing so much as a nanosecond to calculate, they become 1/3 and 2/3, respectively. Instantaneously. Effortlessly.

It is important that we not try to visualize this as if we were thinking of cells in an Excel spreadsheet. Even there, when you have a series of equations and you change a value in one of the terms in a cell, there is a momentary delay while the signal from one keystroke affects the existing displays of previous keystrokes. Because those effects occur very fast, you may not even be aware of it (though it can become noticeable when you are making changes to much more complicated equations, as when you are preparing the check digit of an ISBN). The Excel spreadsheet is merely displaying the correct answer. In our example, the 1/3 likelihood that you will take out a female kitten was not a display of the correct answer; it was the correct answer. More precisely, it is the correct answer.


This “instantaneity” is a familiar concept in Trinitarian theology. The Christian theologian is explicit on this matter: concerning the Son, he will say that the Father begets the Son, but “there was not when he was not.”

The begetting is not a temporal event. It does not even follow upon anything that the Father does. The Father does not say, “I shall have a Son,” and lo! there is a Son.

It is not to be thought of as a kind of creation, where God says, “Let there be light,” and there is light where there was none before. The Son, say these theologians, like the Father, has no beginning.

Similarly with the Holy Spirit. The procession of the Holy Spirit (whether a single procession from the Father or a double procession from the Father and the Son does not matter for our purposes here) does not occur after the begetting of the Son. One wants to say that it is contemporaneous with the begetting, but even that characterization is not to be admitted, because it would mean that the begetting and the proceeding occurred at the same time. And by this account, there was no time for them to occur at the same. “There was not when they were not.”

My point in bringing Christian theology into this discussion is merely to show that the idea of one thing’s being in some sense dependent (the word here is not precise) on another, yet not following it in time is one familiar to the theologians. The Father does not cause the Son; he begets him.

For because the Father, the Son, and the Holy Spirit do not exist in time, no relationship that depends on the existence of time can describe their relationship. And the relationship “A is the cause of B” suggests that there is something temporal involved.

Cause and effect

Where there is no time, there is no relationship of cause and effect. The theologians can say that God caused the world to come into being because by their account, the world has a beginning, that is, it exists as a temporal entity.

In my examples I have shown that geometrical relationships, logical relationships, and probability relationships are similarly atemporal. The laws of cause and effect (whatever they may be) do not apply to them because they are not temporal relationships.

I do not say that the law of cause and effect has been overthrown. I do not say that it is invalid or a superstition.

Ayn Rand defined the law of cause and effect as “the law of identity applied to action.” Where there is no action, there can be no cause or effect.

What I do say is that the concept of timelessness is not so unfamiliar to us as it may perhaps seem at first. Whether the theologians are right about God is not relevant to this discussion. All that is relevant is that the concept of timelessness is not so unfamiliar. It has a place in theology, the laws of probability, logic, and geometry.

And yet, in all these cases, a change in one thing results in a change in another, without any intervening influence or force.

My speculation, then, is this: Can it be that entanglement is a similar relationship? that the reason a change in one particle is reflected by the same change in another is that they are connected in a way that does not admit of forces or influences, perhaps connected in the way that a hypotenuse is connected to the legs of a right triangle? or the way that a probability “changes”?

If so, then it is not true to say, as some popularizers of quantum mechanics say, that the law of cause and effect has been invalidated. If I am correct, all that has happened is that the physicists, like the geometers, logicians, statisticians, and theologians before them, have discovered relationships that are not temporal.

1 This is the argument form modus tollendo tollens, the method of denying by denying; that is, we deny p by denying q, also known as “denying the consequent.”
2 This is a special case of the “Monty Hall” problem.
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