On the force of "in principle"

"But even if -- imagining a man quite exempt from all influences, examining only his momentary action in the present unevoked by any cause -- we were to admit so infinitely small a remainder of inevitability as equalled zero, we should even then not have arrived at the conception of complete freedom in man, for a being uninfluenced by the external world, standing outside time, and independent of cause, is no longer a man."

Count Leo tolstoy
Second epilogue to War and Peace

It strikes some people as perverse that some, such as Edward Nelson, seriously doubt the consistency of Peano arithmetic (and even weaker theories such as EFA). Such doubts, and Nelson not only doubts the consistency of EFA but is serious about finding a contradiction, are indeed perverse from any ordinary point of view, but they are not unprincipled, that is, they are not examples of arbitrary skepticism. It is an instructive philosophical exercise to make sense of these doubts, not to address or refute them, but rather to articulate the basic preconceptions inherent in our thinking, to see what rests on what. To that end, I'll say a few words about the force of "in principle" in such turns of phrases as "in principle, we can find a counter-example to Goldbach's conjecture if there is one", "if ZFC is inconsistent we can in principle find this out" and so on.

We are in principle capable of wonderful feats. Perusing texts in logic and the philosophy of mathematics we learn for example that we are in principle capable of verifying the correctness of arbitrarily complex formal proofs, of finding a counter-example to Goldbach's conjecture if there is one, of finding a contradiction in ZFC should it be inconsistent, of multiplying arbitrarily large natural numbers, of normalising arbitrarily complex intuitionistic proofs to extract witnesses for existential statements, and so on. It is not a particularly original observation that even though we are in principle able to do all these things, we are, in reality, quite incapable of doing so. It is nevertheless an instructive exercise to examine this apparent tension, and ask just what is the force of ``in principle'', and how it is that we are, so it seems, in principle capable of doing all sorts of things it is utterly fantastic to consider to lie within our actual abilities.

Let's call abilities such as ours of factoring arbitrarily large numbers ("in principle") counterfactual abilities. Such abilities are counterfactual in being counter to actual facts about our abilities. In contrast, I have the real ability to factor any number less than 1000, in that, should I so choose, I could in fact, with some patience or with the aid of a computer or a calculator, go through all numbers less than 1000 and check whether any of them divides the given number. In case of arbitrarily large naturals, my counterfactual ability is obviously not grounded in any such facts about my real abilities. So, what is it grounded in? The answer is rather trivial: we are in principle capable of factoring arbitrarily large naturals in the sense that we are in possession of an algorithm for doing so, an algorithm of which we can mathematically prove that it returns the factors of any given number, and a model of mechanical computability on which it is obvious that in any step of computation we are not required to do anything that is beyond our actual capabilities. That is, even though we will never find ourselves in the position of having carried out 2^65536 long divisions, what is required of us in the hypothetical situation is something we can actually do, such as writing down a single digit, striking out a symbol and so on. (In contrast, with such notions as "a mathematical principle in principle acceptable to us" or "unassailably true arithmetical statement" we have no idea what the relevant counterfactual situations are, or what sort of idealisations regarding our abilities are in question.)

There is a strong intuition that the truth or falsity of Goldbach's conjecture is determined as a matter of mathematical fact, that either there is a natural with these or those properties or not (and, for the intuitionistically minded reader, that either there is a canonical proof of this or that or there is not -- Dummett is one of the few thinkers who have seriously rejected an implicit sort of realism about intuitionistic proofs, leading in the good philosophical tradition to many very obscure musings he blithely ignores when going on about other matters). One source of this intuition is our image of ourselves going through the naturals, carrying out computations and so on. It seems almost unintelligible to suppose there is no fact to unearth, since we can picture ourselves having found out that an even number greater than two is not the sum of two primes, by calculation. (I don't recall who it was, but it has been observed that talk about "metaphors" and "pictures" in philosophy is usually metaphorical...) But, since we are not dealing with our actual abilities, all of our reasoning about arbitrarily complex calculations and such like is necessarily theoretical, grounded on some conception of what it would be like if we were not limited the way we are. On closer inspection, it turns out that in fact this conception does not turn on our nature, that is, it is not based on any physiological, psychological, neurological analysis. Rather, it is based on a mathematical image. In other words, the claim that if Goldbach's conjecture is false we can in principle find a counter-example is not really about us at all, but is merely a restatement of the fact that being a counter-example to Goldbach's conjecture is a decidable property.

Let us return to radical finitism, or ultra-finitism. The rejection of the "in principle" is nicely captured by Gandy in Limitations of Mathematical Knowledge

In discussions about the foundations of mathematics it's usual to suppose that the inscriptions which are used to communicate mathematics (e.g. numerals, formulae, proofs) form a potentially innite totality. Despite the fact that this suppostion runs directly counter to everyday experience, no effort is made to justify it. It is of course admitted that in practice concrete inscriptions are limited in length; but an author who has made this assertion is likely to claim in the very next sentence that in principle arbitrarily long inscriptions may be written and the operations and tests of elementary syntax can in principle effectively performed on them. This seems to me as objectionable as to say that in principle pigs can fly while admitting that in practice they have no wings.

The rejection of the "in principle" is not non-sensical or unprincipled. As noted, our abilities "in principle" to do this or that in fact depend on mathematical theorems, on mathematical pictures. We can't address the ultra-finitist doubts by these "in principles" without running foul of circularity. Of course, this is not to say there is something objectionable or dubious in claims about our counterfactual abilities of this sort -- as noted, it is obvious in what sense we have the ability to factor arbitrarily large naturals etc. The point is rather to make the ultra-finitist position understandable, and to note that our reasoning about counterfactual abilities presupposes a substantial amount of mathematics. It therefore cannot serve as a justification of our mathematical thinking, even though it is undoubtedly very useful in explaining our thinking, and making it palatable. We should be thankful for such people as Wittgenstein and Nelson for providing us with actual examples of real anti-realism, of really taking the notion that understanding of mathematics should be grounded in our actual practice seriously. They serve as a healthy corrective to such thinkers as Dummett, who happily go about "our" abilities and "our" reasoning while obviously describing some unknown species of immortal and indefatigable demi-gods.

Incidentally, Peter Smith now has a link to this blog in his. Peter's blog is recommended to everyone interested in logic and philosophy.