On Philosophy

December 5, 2007

What If Arithmetic Was Inconsistent?

Filed under: Logic — Peter @ 12:00 am

Considering the consequences of the inconsistency of arithmetic may seem to be an enterprise of questionable value. After all, don’t we know that arithmetic is consistent? Well it turns out that we don’t. And not only are we unable to demonstrate the consistency of arithmetic but it has been proven that such a demonstration is essentially impossible. Nor has anyone proven that arithmetic isn’t inconsistent, because obviously being able to do that would amount to a consistency proof. But the possibility of proving or demonstrating the inconsistency of arithmetic remains an open possibility. This asymmetry makes the consistency of arithmetic seem at least questionable when we think about it; if it had been proven that nothing could be conclusively said about the consistency or inconsistency of arithmetic in one breath then perhaps our unease could be set to rest, but as things we are guaranteed that either the possibility that arithmetic is inconsistent will always be open or that it will be shown definitively to be inconsistent. In such a situation isn’t it at least reasonable to entertain that possibility?

Of course it might be objected that the consistency of arithmetic can be proven in other systems, and that such proofs are reason to believe in the consistency of arithmetic, even though a proof of that consistency is impossible within arithmetic itself. But to me this simply seems to beg the question, at best. It is equally well known that the problem with the consistency of arithmetic arises for any system of equal expressive power, which includes all those in which a consistency proof for arithmetic could be given. Suppose then that Z proves the consistency of arithmetic. It is then an open question whether Z is consistent. And if Y proves Z consistent than Y’s consistency is an open question, and so on. If it was the case that Z is in fact inconsistent then obviously Z could prove anything, including arithmetic’s inconsistency as well as its consistency. Additionally, if any of these systems are actually inconsistent then the inconsistencies will likely arise in certain applications of induction, of the kind that are needed to construct these very consistency proofs. And so the original proof of the consistency of arithmetic from Z might very well exhibit the “error” in Z that makes it inconsistent, meaning that, again, whether we take the proof from Z to mean anything depends to a large extent in our confidence in Z.

Another natural objection to make to this idea is that we can simply observe that arithmetic works, that we don’t get inconsistent results when we add or subtract, and that in the uses we put arithmetic to it doesn’t seem to have failed us yet. But this is an odd kind of argument to give for the idea that arithmetic is consistent. It is not necessarily the case that every inconsistent system must demonstrate its inconsistency immediately. Consider, for example, someone who held both that the arithmetical axioms were true and that their calculator accurately reflected arithmetical operations. Taken in conjunction these beliefs form as system which we know is inconsistent; calculators can only work with numbers up to a certain size and precision, and thus if we take the overflow values literally as the result of certain operations we can produce arithmetical identities that express contradictions. And yet this person may never come to realize such contradictions exist, assuming they use their calculator only on real problems and don’t immediately try to break it by giving it numbers that are too large as I did when I was a child. Indeed all the practical applications of arithmetic that are supposed to justify its consistency could easily have been carried through by our fictitious person and their calculator, assuming it could handle certain large numbers, a system that we know to be inconsistent. The moral is that inconsistency does not necessarily cause a formal system to “explode”, it may lie hidden because, being finite beings, all of our investigations are confined to some finite area that the inconsistency may lie hidden outside. For similar reasons it can’t be argued that the existence of the universe somehow proves that arithmetic, or any other mathematical system, must be consistent. It is true that if we can exhibit a model of some system then it must be consistent. The universe, however, is finite, or at least we only know of a finite universe. Thus the universe can, at best, model a finite set of mathematical truths. But such finite mathematical systems are not the problem, all the problematic cases, arithmetic included, are problematic specifically because an infinite number of unique formulas might be proven. And thus the existence of the universe demonstrates that, at best, only some finite subset of those systems is consistent. At least that is what I would say were I to consider that possible defense seriously, but it seems to have some serious flaws all on its own regardless of the size of the universe. Mathematical truths are used to model the universe, but the universe is not compelled to model mathematical truths. The universe simply exists, as a collection of independent facts, and thus talking about its consistency is, to an extent, meaningless.

Of course by suggesting that arithmetic might be inconsistent I don’t mean to suggest that somehow it might come apart with sufficiently large but still finite numbers, as my analogy with the calculator might have suggested. Certainly it may, since I haven’t personally witnessed all possible calculations. But the most suggestive possibility seems to be related to the Gödel sentence, a key part of the proof showing that consistency proofs for arithmetic are impossible, which states that there is no proof of that very sentence. Often we intuitively understand this sentence as true, and thus as expressing the consistency of arithmetic while at the same time being without provable. But of course, since it is not provable, it is also consistent to take its negation, which says that there is a proof of the affirmative version, as true. I don’t know what such a system looks like, since the negation of the Gödel sentence seems to contradict itself, but there must be some such system in which there is no real contradiction (otherwise we would have a proof by contradiction of the inconsistency of normal arithmetic). However, in trying to devise a consistent system including its negation we might stumble upon the idea of a class of infinite numbers expressing infinite length proofs that behave in odd ways in order to avoid the apparent contradiction. But the simplest way to take these infinite proofs as being actually producible and thus genuinely giving rise to contradictions. This gives us an interesting way for arithmetic to be inconsistent, because the inconsistencies can only be revealed by infinite length proofs. And obviously we will never actually come across an infinite length proof, which is perfectly consistent with our experience of arithmetic.

Suppose then that arithmetic, or mathematics in general, was inconsistent. What ought we to do in such a situation? One reaction might be to restrict ourselves to domains in which we know no contradictions will arise. Of course that might mean simply less powerful mathematical systems, but it could also mean restricting ourselves in the way we use these mathematical systems such that their inherent inconsistency won’t bubble to the surface. If contradictions only arise as a result of infinite length proofs, as has been suggested above, then obviously all we need to do is commit ourselves to never actually constructing one of those, which isn’t too hard of a task. But, more generally, we might find a way to establish for any given starting point that any contradictions that exist must lie a certain number of deductive steps away from it. And then we could simply restrict ourselves to staying within that bound when we construct our deductions, which should give us all the power we need, for all practical purposes. We also might realize that, again for all practical purposes, whether mathematics is inconsistent doesn’t really matter. All we need is mathematical models that we can put into correspondence with the world and use in certain limited ways to make calculations about it. An inconsistency that actually arises, from this point of view, is just one more way that the model can be defective, prompting a replacement. Although the model might give rise to inconsistent conclusions were we to deduce things from it that is simply not how we generally use mathematical models, we calculate with them, not deduce from them. Furthermore, there is a certain tradition in physics of gluing together mathematical models known to be inconsistent in ways that the inconsistencies can be simply swept under the rug in most situations, and physics doesn’t seem too much the worse for wear for it, although it isn’t a perfect solution. Because of such considerations you might also think that if mathematics is inconsistent that mathematicians should stop being mathematicians. But I don’t think that they necessarily should take that step, if they can reconcile themselves to the inconsistency of mathematics. Some people just like doing mathematics, and if they like mathematics they shouldn’t let deficiencies in it stop them, just as I don’t let deficiencies within philosophy stop me from doing it.

So suppose someone were to assert that all sufficiently powerful mathematical systems (all of those which can’t be proved consistent) were inconsistent. Indeed I am tempted to make such an assertion given some of the thoughts I have put forward here. Could we refute it? Indeed we could, although obviously we couldn’t demonstrate the consistency of any particular system. What we could demonstrate is a certain kind of relationship between two or more systems, such that only a certain number of them, at most, can be inconsistent. In the case of two systems this would boil down to showing that one of them is consistent if and only if the other is inconsistent, and vice versa. This would demonstrate that not all sufficiently powerful mathematical systems were inconsistent. Indeed it would even validate the both of them, since we know that by being of sufficient power they can both model the other we know that the inconsistency in the inconsistent one of them cannot be demonstrated, because if it could be then a proof could be formulated in the consistent system of its own consistency using the demonstrated inconsistency in the other system and the particular relationship between the two systems mentioned earlier to produce a proof of its own consistency. And since we know that is impossible it must be the case that the inconsistency in two such systems related in this way can never be demonstrated, making even the inconsistent one “safe” for all practical purposes. (This very fact may suggest that no two systems can be related in this way, although it isn’t conclusive evidence.)


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