After Finitude, Chapter 4: Hume’s Problem

Originally discussed on April 21, 2012

The subtitle of After Finitude is “An Essay on the Necessity of Contingency”.  This necessity was shown in chapter 3, and it can be briefly stated as follows: the principle of factiality teaches us that no thing exists necessarily.  Any and every thing could not exist, or exist, or exist differently.  There is no ultimate law governing the world: anything can happen, for no reason at all.

Recently, scientists at CERN announced they had observed a neutrino moving faster than the speed of light, an apparent impossibility.  It turns out this was the result of an equipment malfunction, but let’s imagine that the initial announcement was entirely correct.  After all, if anything can happen, then why not this?

There would be 3 ways of interpreting this.  First, we could say that there was a previously unknown element of physics at work:  under certain conditions, faster-than-light travel is entirely possible.  Second, we could say that an unknown law interrupted or altered the relativistic laws governing mass and energy; while faster-than-light travel was not possible before, it is now, thanks to the consequences of this unknown law.  A miracle from God would be a variation on this second possibility.

Both of these interpretations are still attached to the idea of causal necessity. or a fundamental uniformity of nature  Either we did not fully understand an existing law and mistakenly thought faster-than-light travel was impossible, or something caused a change in the laws prohibiting such travel.

Meillassoux offers the third, much more dramatic interpretation: our hypothetical violation of relativity took place for no reason.  His inspiration for this claim comes from the 17th century philosopher David Hume.

Hume outlined what he saw as a major problem: induction.  As an empiricist, he believed there were only two routes to knowledge: experience and the law of non-contradiction.  He pointed out that we cannot observe any necessary connection between one event and another; even if event Y has always had result Z in the past, we have no reason to believe that Y will always result in Z in the future.  Our experience cannot tell us that the future will be like the past.  Further, there is nothing inherently contradictory about the future being different; it is not a logical requirement that Z continue to follow Y.  There is no logical requirement that the Earth continue to circle the sun, because there is nothing contradictory in the dissolution or reversal of gravity.

There have been three basic methods of dealing with Hume’s problem.  First, the metaphysical solution: a necessary being, basically God, acts as a guaranteer of the future.  God instated and maintains all physical laws.  As we’ve seen, Meillassoux considers all necessary beings to be soundly refuted.

Second, there is the skeptical route, suggested by Hume himself.  Since experience and logic cannot prove that the future will be like the past, Hume alters the problem somewhat: instead, he asks “why do we believe that the future will be like the past?”  His answer is habit.  It is just a matter of humans believing ever more strongly in things they see happen over and over again.  There is a contemporary version of this; some philosophers of science, conceding that causal necessity or the uniformity of nature cannot be proven, suggest we simply leave the problem behind.  Instead of asking why the future should be like the past, we should be investigating the rules and assumptions scientists use when they make predictions.  All versions of the skeptical route essentially drop the ontological version of the problem in favor of an epistemological or methodological problem.  Instead of asking “Is it true?” they ask “why do we believe it?”

Meillassoux considers the skeptical dissolution of Hume’s problem to be something of a cheat; he wants to re-activate it as an ontological problem.  To do so, he needs to refute the strongest previously existing response to the problem: Kant’s transcendental solution.

Kant argued that if the laws of nature were contingent, that is, if they could change, they would do so – frequently.  The result of frequent changes in the laws of nature would be the complete destruction of our ability to think about the world, and since we obviously can think about the world, nature must be consistent.  Kant does not attempt to prove that the stability of nature is necessary – rather, he tries to show that we could not even begin to imagine or think about a world without it.

Take Hume’s primary example, that of a billiards table.  Hume asks why we should assume that when ball X strikes ball Y, ball Y will follow a particular path – bouncing off the sides or sinking into a pocket.  Why shouldn’t ball Y hover in the air or transform into a cell phone?  Kant acknowledges these possibilities, but he says that we can only imagine a ball transforming into a cell phone because the rest of the hypothetical context remains stable.  That is, in order for us to think about Hume’s problem with the billiard balls, we have to assume that the billiards table itself continues to exist, as well as the room, as well as the time it takes for ball X to travel to ball Y, and so on.  If nothing in the example remained stable at all, we would not be able to narrate it.

Kant’s point is that if nature were wholly chaotic, then laws would have changed so often that we would have noticed them changing – or rather, they would have changed so often that we would never notice anything at all, since constant changes in gravity would mean the Earth would never have had a chance to form in the first place.

Everything we’ve seen so far, from the first two responses to the faster than light neutrino (an unknown aspect of relativity, or a higher law changing relativity) to the metaphysical, skeptical and transcendental responses to Hume’s problem, all depend on some version of necessity.  They can all share a common response to the idea of complete contingency: laws must be necessarily stable, because if they were not, they would change.

Imagine we have been rolling a die for the past hour, and every time, the die has landed on a three.  We would begin to assume the die was loaded, that someone was cheating us.  Now imagine millions of dice being rolled every second of every day for billions of years – and every single one of them has always come up as a three.  This is what a contingent stability of laws looks like to all the above responses: a ridiculous, completely absurd statistical marvel.  This is the problem Meillassoux must solve: if laws are contingent, they would change, frequently.  He will call this the frequentialist implication, and it is what he seeks to refute.  Doing so would show that Kant is wrong: thought is still possible in a contingent world, and stability is thinkable and not a statistical marvel at all.

Think about the die again.  When you roll it, there are six possibilities given in experience: we have a 1:6 chance of landing on a three.  When we assign a probability to any given event, we say it has a 1:X probability of taking place.  By the end of this argument, it will hopefully be clear that the X in 1:X does not actually exist in the context of the contingency of laws – the changing of a given law is not a matter of probability.

So there is a 1:6 chance of a die landing on a given number.  Now, begin to add in all the other logical possibilities: because the laws governing the dice could change, the die could, without contradiction, transform into a hungry tiger, or hover in the air, or change its mass and crash through the table.  Now, there is a 1:9 chance of the die turning up as a three.  We can count these possibilities as long as we like – in fact, it seems that we can count them into infinity.  We seem to be trapped into believing that there is a 1:∞ probability of any one of these things happening.

Meillassoux’s crucial question is: which infinity are we talking about?  Following his teacher Alain Badiou, he will say that the mathematician Georg Cantor proved a de-totalization of the infinite.  There are, in effect, multiple infinities, and they can be differentiated from one another.  (This section will also be a useful introduction to Badiou, our next subject)

Take infinite set A:  A is the set of (1, 2, 3…).  1, 2 and 3 are some of the elements of set A.  We can rearrange these elements to show that the elements of A can form a larger infinity than A:  ((1, 2), (1,3), (2, 3)).  The further combinations of the elements of set A are larger than set A – this larger set is called the power set of A.  This same procedure can be carried out with any infinity.  Further, element (1) is itself an infinity.  We can say that element (1) is itself a set: the set of 1 is (X, Y, Z).  And we can see how those elements can form a different infinity, larger than set 1: ((X, Y) (X, Z), (Y, Z)).  (Q6)

Every infinity contains both smaller and larger infinities.  Now we have an obvious question: is there a largest infinity?  In other words, is there one set that contains set A, the power set of A, set 1 and all the other sets we can form?  Let’s call this largest set the “set of all sets”. The answer is no, and this is the key to eliminating the X in our 1:X probability of laws changing.

We have already seen the first argument against a set of all sets.  Any and every infinite set can produce a larger infinite set, and itself contains other infinite set.

The second argument is similar to Bertrand Russell’s barber paradox.  There are two, and only two, kinds of sets: reflexive and non-reflexive.  A reflexive set includes itself within it. So reflexive set A, including itself within it, is A(A, B, C).  Non-reflexive sets do not include themselves, so non-reflexive set 1 would be 1(2, 3, 4).  An example of a non-reflexive set is a bowl of peaches: a peach is not itself a bowl of peaches.

So what would a set of all sets be?  Reflexive or non reflexive?  Let’s call the hypothetical set of all sets C.  If C is reflexive, we could write it as C(C, D, E).  C is an element in set C.  A set of all sets would have to include itself, otherwise one set (itself) would escape it.  The set of all sets must be reflexive.

Now, consider the non-reflexive sets.  Is there a complete set of non-reflexive sets?  In order for our set of all sets, C, to truly include all sets, then there must be a complete set of non-reflexive sets.  Let’s call the complete set of non-reflexive sets 1.  Now, 1 itself can either be reflexive, or non-reflexive.

If1 is reflexive, it would include itself in itself: 1(1, 2, 3).  Yet this is a contradiction: it would mean the set of all non-reflexive sets includes a reflexive set.  If 1 is non-reflexive –  1(2, 3, 4) – then a single non-reflexive set could not be included in the set of all non-reflexive sets.  Another contradiction.  There can be no complete set of non-reflexive sets: therefore there can be no complete set of all sets.  In other words, there is no one infinity that includes all the rest.

What does this bizarre detour into math gain us?  It shows us that there is a thinkable difference between the given possibilities of the dice, that is, the 1:6 chance of falling on a three, and the infinite possibilities posed by logic.  In order to make a probabilistic calculation for the changing of laws, we need to presuppose a single infinity in order to give us the X in 1:X.  Yet there is no single or largest infinity. The hypothesis of the stability of laws being like millions of dice being rolled over and over again for billions of years requires a largest infinity in order to make sense, and no such largest infinity exists.

In other words, assigning probabilities to the changing of laws (as opposed to events controlled by those laws, like the rolling of a single, real die) is impossible.  The stability of laws is not a matter of probability, and therefore not a statistical marvel.

One more attempt to explain this.  Take the well-known gambler’s fallacy: when we roll dice repeatedly, but our desired number three does not come up, it is very easy to assume that the chance of finally getting a three increases with each throw.  But of course it does not; no matter how many times we miss the three, there is still only a 1:6 of getting it.

Meillassoux is claiming that believing in a frequent change of natural laws is a cousin to this fallacy.  In order to believe that contingent laws would change frequently, we have to step outside the possibilities given by a law (the possibilities that form the X of 1:X) and assume that there is a single larger set of possibilities that have an equal probability of taking place – yet there is no such single larger set, and so these potential outcomes (more precisely, these virtualities), while equally logical, are not equally probable.  More precisely, they are not probable at all – they are contingent, and entirely outside the gambler’s calculations.

It is is a logical fallacy to believe that the up-until-now unseen three has an ever greater probability of appearing – and yet the three could still appear; it is the expectation of the three that relied on a false assumption. Similarly, the expectation of frequent changes in laws relies on false assumption, despite it being possible.  We cannot step outside experience and assume that there is an infinite set of outcomes that are equally probable – because they are not probable at all.  Probability has no purchase on laws themselves, because laws are contingent.

This is the refutation of the frequentialist implication: it is false to say that contingent laws would change frequently.  We do not need to live in fear that the coffee we are drinking will suddenly change into poison, just as it is false to believe that this time, the three must appear.  A world of contingent yet stable laws is entirely thinkable, contrary to Kant’s assumptions.  We do not need to retreat into metaphysics, and we do not need to ignore the problem of stability, like a helpless parent in a supermarket ignoring their screaming child.

So back to our hypothetically faster-than-light neutrino.  If such a thing did occur, then it occurred for no reason, because no probability governed this event.  A faster-than-light neutrino would not be the chance actualization of a potentiality governed by law, but rather the emergence of a virtuality out of the non-totalizable infinities of contingency.

In our final discussion of Meillassoux, we will have a look at the ethical implications of the principle of factiality.  If anything can happen for no reason, what does this teach us about the ethical and political elements of the world?

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