Under Construction

Paradox and Probability

The Red Queen shook her head. 'You may call it 'nonsense' if you like,' she said, 'but I've heard nonsense, compared with which that would be as sensible as a dictionary!' - Lewis Carroll - Through the Looking-Glass and What Alice Found There

Index

What is Paradox?

What Paradox Does To Truth

How Paradox Drives Emergence

Paradox and Prediction

Paradox and Probability

What is Paradox?

A paradox, writes Jim Al-Khalili in his book Paradox - The Nine Greatest Enigmas in Physics, is a statement that leads to a circular and self-contradictory argument, or describes a situation that is logically impossible. He also includes a category of 'perceived' paradoxes, for which there is a solution to the puzzle, a way out of the problem. The classic example of a paradox is the Liar Paradox, in which there are two contradictary statements - one claims the other is true, the other claims the first is false.

Well the solution to this problem, if you think it a problem, is that the words are just words, and so do not necessarily tell the truth, so only by taking them seriously do you even get the approximation of a paradox. If you knew that both sides were consistent, you could even identify which was which by interrogating just one side about what the other side says, and then choose the one that claims the other is lying as being the truth teller.

How does that work? Well the indirection solves the problem, and in the statements on the card there is, in the moment, a self-consistency for either side. Compare with a situation where you are faced with two doors, one leads to freedom, the other to certain death, each is guarded, one by a liar, and one by a truth teller. How can you choose which door to take? The answer is ask either guard what the other would say lay behind the other guard's door, and then choose the door that promises death.

If the liar guards the door to death, he will say that the other guard has the door to death, and the other would say it is the door to death.

If the liar guards the door to life, he will say that the other guard has the door to life, and the other would say it is the door to life.

If the truth teller guards the door to death, he will say that the other guard has the door to life, but would say it is the door to death.

If the truth teller guards the door to life, he will say that the other guard has the door to death, but would say it is the door to life.

So, that problem is solved, and we see that the promise of life or death can be seen through, but can we apply the same logic to the Liar paradox? Unfortunately not. In the liar paradox, as written on a card, A says B is true, while B says A is false, and because of the recursive logic, there cannot exist a situation where both statements fit one another.

Side A claims side B is true, and if side B is true, then side A is false, making side B false, but side B cannot be both true and false

Side B claims side A is false, and if side A is false, then side B is false, making side A true, but side A cannot be both true and false

And there is no clever way out of this conundrum, making it a proper paradox. Luckily, these are just words, tossed in a fine old salad, so all that gets forced to exist by the juxtaposition of incompatible A and B, is a neat and tidy example of words pointing to ideas that truly cannot mix. Words are trivial to jam together, because there are no enforceable rules to make them make sense.

What Paradox Does To Truth

What the words on the card in the Liar paradox are trying to emulate are actual truths, but if they were actual truths and non-truths, then the situation would be an actual paradox for real. However, you would tell it like this:

A is a truth, and because A claims B is true, A is in some factual manner the same as B being true. But B claims A is not true, so B is in a similarly factual manner, the same as A being not true.

If we assign boolean values to A and B:

A = 1 and A says B = 1

B = 1 and B includes A = 0, so B is truly 0 [false], but this implies A = 0, and that makes B = 1, and so on ad infinitum.

We properly require a better definition of truth. Philosophy talks of truth-makers, which are some facts that support whatever proposition is in question. However, while truth may emerge from truth-makers, the truth-makers in isolation are not truth.

How Paradox Drives Emergence

What we actually have under consideration is the nature of emergence, i.e. that whatever emerges from the juxtaposition of parts, is sometimes of a different nature entirely, from those parts.

Just as a whole is a different thing from constituent parts, in some way, it is also alike to each and every other whole object, in some way.

So we see the emergence of the number one as being from the parts of nothing-ness and whole-ness, there is no more to it than that. But when its paradox causes two-ness to emerge by being one divided as two parts, then we have the constituents of integers, which include a new emergent numeric property, that emerges because of the same-ness of kind of one and two, and it is that same-ness of kind that enables the mathematics of numbers. So two is both one divided and the sum of one plus one, and a number just like one, and a whole just like one. Thereafter the integers pop out, one by one, each alike in their number-ness, and whole-ness.

But, when we get to four, and nine, etc, the potential for square-ness, and so on, emerge. These are a different quality from linear integers, even taking account of multiple-ness.

So, getting back to truth; truth-ness is undoubtedly akin to same-ness, but is it identical with same-ness?

The truth of 1 + 1 = 2 stands or falls by the partial identity implied by 'equals', and in this case 'equals' implies numeric same-ness and quantitive same-ness; we are not dealing with apples and pears, and they are indeed the same. But truth itself can be a truth of non-same-ness, such as 1 + 1 not = 3, so clearly truth emerges from comparing, and is a virtual quality, just as is whole-ness, same-ness, and number-ness, and every other thing.

The point is that all these emergent virtual things come to exist on their own terms. So a thing is true when it embodies, among whatever else it embodies, the virtual object Truth, irrespective of truth-makers, though these may be useful in identifying truth.

Paradox is at once true and not true. So we would do better by stating:

A has truth = 1 and A includes the fact B has truth = 1.

B has truth = 1 and B includes the fact A has truth = 0, so B truly has a truth value = 0, but this implies A is false about B, so A has a truth value = 0, and that makes B correct, and so B truly has a truth value = 1, and so on ad infinitum.

We could say A's truth = 1 iff B's truth = 1. But B's truth = 1 iff A's truth = 0. It all amounts to the same thing.

The paradox is that A and B have contradictory truth values, that are bound together 'as if' they were quantum entanglements, but which also undo one another. The twist is that they cannot be untwisted, so the fact is that the truth states of both A and B are neither true, nor false, but rather they are paradoxical - a state of affairs that logic usually does not allow for, because there is the Rule of the Excluded Middle. However, excluded or not, we have seen that some new third way must come to be whenever logic fails and paradox reigns.

Paradox and Prediction

C.D.Broad (1925), in reference to Emergence, claims that emergent properties cannot be predicted from lower-level features of the parts that form the new whole - 'cannot, even in theory, be deduced from the most complete knowledge of the properties...'.

However, without a complete theory of Emergence, perhaps Broad's claim is a little rash. Without very good reasons to show otherwise, I would be inclined towards the view that full knowledge of the parts should, in principle, provide at least a very good idea of what might be expected to emerge. After all, it is only through such an exercise of the imagination that anyone is able to match the parts to the wholes, and thereby explain Emergence with any coherence at all.

Paradox and Probability

Paradox, as we have seen, creates necessary emergence, some of which is claimed to be hard/strong, and some of which is claimed to be soft/weak. However, I am not so sure of the distinction. As with the creation of integers, driven by paradox, after the first couple, there is not much novelty, yet there is exactly the same level of paradox driving the process. When we consider notions of geometry, we also must conclude that some values cannot ever fully emerge in sufficient detail to fulfil the undoubted completeness of the whole - just consider π and the circle.

While paradox is clearly the engine of necessary emergence, and demergence for that matter; and while novelty is often seen whenever things do emerge, I believe the case could be made that some forms of emergence are not strictly necessary, but are strictly possible, and as a consequence of these factors may occur with a specific probability.

The integers, or linear numbers as I would describe them, as we have seen are strictly necessary, due to paradox, and are still emerging, so I claim. By contrast, squares, roots, and some factors, while associated with non-primes, and being what I'd call flat numbers - having area, are not [or were not] necessary to the same extent, but they were, are, or will be, possible. The same holds true for numbers with more factors, and the potential for volumes, hyper-volumes, etc.

Somewhere along the line of integers, probability allows the emergence of new forms of numbers, and very occasionally these will have the internal structure necessary for defining the properties of elementary particles. Whether those particles are quarks, or the constituents of quarks, I don't know, but the general rule would seem to be that objects with more dimensions can contain objects with fewer dimensions, though the reverse is also the case - that objects with fewer dimensions can contain objects with more dimensions.

Of course, the objects were talking about are virtual mathematical objects, each of which may be instantiated as many times as possible, or as required, to form material objects in a real Universe. And it is this feature that allows all photons to be photons, and the same for gluons, electrons, protons, etc. etc. Each type of particle is a particular combination of internal numeric properties, and each instance of a particle is a particular combination of internal and external numeric properties. The numbers are fundamental, and from them emerges everything.