Dummett’s
Intuitionism and Bifurcated Supertasks
How bifurcated supertasks would affect Intuitionists (the issue which links to this essay, from my 2007) would depend upon how their Intuitionism was justified. One major philosophical defence of Intuitionism having been given by Dummett (via his general argument against semantic realism), I shall here consider how bifurcated supertasks might affect that sort of Finitism (see Fletcher 2007 for other sorts of Constructivism).
Mathematical Platonists believe that arithmetical propositions describe (truly or falsely) an objective mathematical realm, so that every arithmetical proposition is either true or false independently of a mathematician’s capacity to acquire knowledge of it, e.g. via proofs. Platonists are semantic realists, in Dummett’s terminology, as opposed to the Intuitionists, who are mathematical anti-realists. But because Dummett defends Intuitionism via a general argument against semantic realism, and since a similar argument might defend Strict Finitism (which outlaws the infinitude of the natural number sequence, and which is therefore of doubtful coherence), he must allow certain concessions towards semantic realism. And since mathematical Platonists might reasonably demand similar concessions, hence Dummett’s argument is vulnerable to Earman and Norton’s (1996) bifurcated supertasks.
Consider Goldbach’s Conjecture (GC) that each even composite number equals the sum of two primes, and which is neither refuted nor proven. GC’s meaning is that 4 is the sum of two primes (2 + 2) and 6 is the sum of two primes (3 + 3) and 8 is the sum of two primes (5 + 3) and so forth. So GC is true if all of those conjuncts are true, and false if at least one of them is false. Platonists think that GC is either true or false, and hence that GC might (for all we currently know) be true for reasons that cannot be expressed by any finite sentence in our language, and it is to such a possibility that Dummett objects:
Since the theory of meaning underlying classical mathematics, as conceived by the platonist, requires that the understanding of a sentence consists in a knowledge of the conditions for it to be true (or for it to be true in a particular model), that is, in an awareness of what has to be the case for it to be true, we must possess an understanding of quantification over an infinite domain which does not relate to our own restricted means of recognizing as true sentences formed by such quantification, but does yield a conception of truth for such sentences as something which they, determinately, either do or do not possess. The nub of the intuitionistic critique of classical mathematics is the contention that we do not, and could not, have any such conception of mathematical truth; that we suppose ourselves to have it only by an illusion based upon a false analogy. (Dummett 1977: 373)
The infinite is indeed unlike the finite, in many ways, but Platonists believe that transfinites (small infinities) are also akin to natural numbers in many ways. Now, Dummett (1993) believes that the concept of a natural number is an indefinitely extensible concept (a concept such that, given any legitimate collection of objects falling under it, there would exist other objects that fall under it), but his (1993) reason for that belief is based upon an analogy—between the transfinite set of all the finite cardinals and the proper class of all the cardinals—that Platonists would not accept. And Dummett needs to argue from premises that are Platonistic, since he wants to show that Platonism is incoherent.
Dummett’s semantic argument against Platonism is that truth-conditional accounts of meaning (e.g. the above, for GC) conflict with the assumption that (language being nothing if not public) “a grasp of the meaning of an expression must be exhaustively manifested by the use of that expression” (Dummett 1977: 376). Such usages are necessarily constrained by our limitations, and so Dummett does not see how we could display our knowledge of the truth-conditions of propositions like GC. E.g., even if a community of mathematicians existed forever and examined each of the conjuncts of GC at some time, at no time could it have surveyed them all. Consequently, Dummett concludes, we cannot have a coherent conception of how the Platonistic meaning of GC could be manifested.
Is the demand for exhaustive manifestation too strict—might not our linguistic practices be guiding us towards the meanings of our expressions, rather than constituting them? Dummett considers that:
[Perhaps] we first learn the meanings of the quantifiers […] for finite and surveyable domains [… and then] form the conception of the condition for the truth of a statement involving unbounded quantification by analogy with this, by imagining a being who, unlike ourselves, could in a finite time check the truth-values of denumerably many instances of such a statement. (Dummett 1977: 379)
But he thinks that such analogies are unacceptable:
The language that we use, when we are engaged in mathematics as in other activities, is our language, and its meaning must be connected with our own capacities: it cannot be derived from the hypothetical conception of capacities which we do not have, and the attempt to explain it by such means only illustrates the illusions implicit in our misunderstanding of our own language. (Dummett 1977: 380)
Platonists misunderstand their language, in Dummett’s opinion, because they “treat certain of our sentences as if their use resembled that of other sentences in certain respects in which it in fact does not” (Dummett 1976: 62).
But language is all about communication and so, were communicants to associate the same meanings with the same words, that would surely be good enough—it could not really matter how such correlations arose. E.g. in the case of realism about the physical world, Craig (1982: 557) noted how “one firmly expects one’s companions, who share one’s environment, also to share a high number of one’s beliefs about it.” Similarly in the case of mathematics, for us to be able to display our mathematical knowledge to each other it would be sufficient that we held similar beliefs about mathematics (whatever their origin). And so because Platonists do share similar beliefs about infinity there is at least one flaw in Dummett’s argument against Platonism.
Furthermore, common beliefs are relied upon at the most basic level of language acquisition, e.g. when we learn about kinds of things. Even before we learn to count, we have to learn the natural kind terms for the things to be counted, which we can only do if we are able to regard the indicated instances of such terms as similar—as of the same kind—even though they are different. And such analogical reasoning is clearly based upon common beliefs, e.g. we learn about crows rather than observable crows (and rabbits rather than rabbit slices, etc.).
A theory of meaning in the style of our
realist ascribes to a competent speaker (amongst other things) dispositions to
respond to the obtaining of truth-conditions—circumstances of sorts which need
not be detectable—when they are detectable. (McDowell 1978: 138)
Not all crows are observed crows, of course (although all the crows that we will have observed will have been observed crows), and our knowledge of that distinction—and of the distinction between crows and observable crows—may be manifested simply by mentioning it because, as McDowell (1981) notes, our natural capacities include our linguistic capacities (which is why we can manifest our understanding via thought-experiments). Basically, language itself, by being a public affair, makes us distinguish between the crows that we have personally observed and the crows that are in principle observable.
And without any analogies we would not even get the concept of 1, so the legitimacy of analogy is clearly a matter of degree. Intuitionists must allow us to imagine ourselves with considerable capacities that we do not actually have, in order that we may acquire the concept of an arbitrarily large natural number. And so, since we are allowed the concept of the infinitude of all the natural numbers, we may consider infinite spacetimes and endless sequences of tasks.
Earman and Norton (1996: 247-50) argued that bifurcated supertasks might be possible within Malament-Hogarth spacetimes—they are called ‘bifurcated’ because an agent in one region (of such relativistic spacetimes) can have infinite time to carry out an endless sequence of tasks (a supertask), while an observer in another region can observe the entirety of that supertask in a finite time. Earman and Norton’s (1996: 250-6) agent was an infinity machine, which spent forever surveying all the natural numbers, testing a number-theoretic conjecture (e.g. GC) against each number in turn, so that after some finite time the observer (who might be a normal human) could know its truth or falsity.
We can consider, instead of an infinity machine, endless generations of a mathematical community, to which the observers also belong. Those agents send a signal to the observers iff they find a counter-example to GC, whence the observers could conclude, after some finite time (were they confident of the agents’ abilities, and of the reliability of the signalling mechanisms), either that GC was true (no signal received) or that GC was false (a signal received). That is what the observers could conclude—whether they do or not is up to them, but we can certainly think of them talking amongst themselves about those options.
Dummett’s semantic argument fails because an infinite conjunction like GC can be understood by analogy with a physical conjunction. From the point of view of the observers, the agents are simply occupying parts of spacetime, and so the assumption that GC is either true or false (independently of the possibility of proving or refuting it) is as coherent as the assumption that, within endless spacetimes, infinitely many stars could exist (which is hardly an assumption at all since it follows from how each star is possible and how there is enough spacetime for them all), as follows.
Firstly, merely human powers were required of the agents and the observers (not the superhuman powers that Dummett thought necessary for an explication of Platonistic quantification), and those two linguistic communities—agents and observers—would not have to diverge very much (since all of the agents are able to contact the observers). Secondly, only scientifically respectable spacetimes were being conjectured. An Intuitionist might question the coherence of the spacetime physics involved, but it appears to make sense as physics and if Intuitionistic mathematics could not yield all of current physics, then that would surely be a problem for Intuitionism, rather than for Platonism.
Basically, the analogy was therefore similar to those made by Intuitionists when they consider surveying finite sequences of natural numbers that are so huge that all the protons in the known Universe would have decayed long before they could finish. We can imagine spacetimes in which all the protons are renewed (e.g. via white holes) without becoming incoherent, and we can imagine, no less coherently, Malament-Hogarth spacetimes. Those are just two scientifically respectable situations in which normal human beings might, conceivably, find themselves, and about which they could (and we can) talk.
Craig, E. (1982)
‘Meaning, Use and Privacy’, Mind 91, 541-64.
Dummett, M. A. E. (1976)
‘What is a Theory of Meaning? (II)’, in his (1993) The Seas of Language,
Oxford: Clarendon Press, 34-93.
Dummett, M. A. E. (1977) Elements
of Intuitionism, Oxford: Clarendon Press.
Dummett, M. A. E. (1993)
‘What is Mathematics About?’ in his The Seas of Language, Oxford:
Clarendon Press, 429-45.
Earman, J. and J. D. Norton (1996) ‘Infinite Pains: The
Trouble with Supertasks’, in A. Morton and S. P. Stich (eds.) Benacerraf
and His Critics, Oxford: Blackwell, 231-61.
Fletcher, P. (2007)
‘Infinity’ in D. Jacquette (ed.) Philosophy of Logic, Amsterdam:
North-Holland, 523-85.
McDowell, J.
(1978) ‘On “The Reality of the Past”’, in C. Hookway and P. Pettit (eds.)
Action and Interpretation: studies in the philosophy of the social sciences,
Cambridge University Press, 127-44.
McDowell, J. (1981) ‘Anti-Realism and the Epistemology of
Understanding’, in H. Parret and J. Bouveresse (eds.) Meaning and
Understanding, New York: de Gruyter, 225-48.