SYMPOSIUM
Anti-Realism and Infinitary Proofs
Diego Tajer
Universidad de Buenos Aires - CONICET
diegotajer@gmail.com
Abstract
In the discussion about Yablo's Paradox, a debated topic is the status of infinitary proofs. It is usually considered that, although a realist could (with some effort) accept them, an anti-realist could not do it at all. In this paper I will argue that there are plausible reasons for an anti-realist to accept infinitary proofs and rules of inference.
KEY WORDS: Anti-realism; Realism; Infinitary logic.
Resumen
En la discusión sobre la Paradoja de Yablo, un tópico debatido es el estatus de las pruebas infinitarias. Se suele considerar que, aunque un realista podría (con cierto esfuerzo) aceptarlas, un anti-realista no podría hacerlo en absoluto. En este artículo, argumento que hay razones plausibles para que un anti-realista acepte pruebas y reglas de inferencia infinitarias.
PALABRAS CLAVE: Anti-realismo; Realismo; Lógica infinitaria.
1. Introduction
One of the deepest topics of discussion around Yablo's Paradox is
the debate about the possible length of arithmetical proofs. The very
concept of proof is usually associated with the idea of an always finite set
of sentences, where there are permitted transformations between them.
Nevertheless, the finiteness of proofs is not conceptually unavoidable:
infinitary systems, for example, make use of infinitary proofs. In fact, the
non-circular proof of Yablo's paradox that Cook designs (which was also
announced by Priest, Sorensen and others) consists in an infinite number
of steps, and uses sentences of infinite length and infinitary rules. It's easy
to see that a proof like that could raise some objections.
Graham Priest, for example, says that nobody can actually perform
an infinitary rule, and that when we talk 'as if' we performed one, it's
actually finite information what is justifying us. Sorensen (1998) and Cook
(forthcoming), on the other side, defend a somewhat realist position, which
says that even if no human being could perform an inference like that, that doesn't mean that the proof doesn't exist. The proof would be, then,
one of the many things that exist but we can't know (or maybe some superhuman
entity can). Bringsjord and van Heuveln (2003) make an original
point: they claim that human beings are not Turing machines but
supermachines, that is, machines that can perform infinite tasks in a
finite amount of time; if that is true, they say, there are no reasons to
reject infinitary proofs.
Iin this article I will focus on this discussion, but I will take a
different account. Following Sorensen, who observed that any realist can
be justified to reject infinitary proofs (claim with which i don't necessarily
agree1), I will ask which position can an anti-realist take with respect to
this issue. Prima facie, it's unconceivable that an anti-realist would accept
infinitary rules or sentences. But, I think that if we focus on the internal
discussions of anti-realism (which authors like Tennant, Wright and
Dummett developed), the rejection of this kind of rules is much less
obvious than it appears.
2. Manifestation
One of the central elements of anti-realism (or at least, of a
Dummettian anti-realism) is the manifestation requirement2. It says that
a subject knows the meaning of a sentence only if she can recognize a proof
of it, and evaluate whether it is correct or not (that is, when she can manifest
her knowledge of the verification-conditions of the sentence). Nevertheless,
this requirement seems too restrictive, and was discussed inside the antirealist
tradition. In particular, there are some statements whose proofs are
much longer than anything a human being (or the whole number of human
beings) could read in their entire life (Ketland 2005). In those cases, it's
insensitive to ask people (or even experts!) to recognize a proof when they
see it; because, plausibly, they won't be able to see the whole proof in order
to recognize it in its parts and check it step by step. So, an expert
competence could be insufficient for comprehension of a statement.
The traditional way of proceeding in these cases is idealization: it's
supposed that the speaker, if she had a great amount of time and a
memory that exceeds by far human capacities, could recognize if the proof
is correct or not3.
Nevertheless, the idealization seems to go too far away from the
actual competence: what is going to be evaluated is if the speaker has
potentially a level of comprehension and capacities that may not reach
in a lifetime. Much worse, because of the difficulty for evaluating this kind
of idealized potential capacities, that makes difficult to fulfill of an
important manifestationist desideratum: that we should be able to
verify whether a speaker understands a statement.
Because of those reasons, I think that it's better to think of
manifestation in a different way. A better response to this problem, which
doesn't appeal to idealization, was proposed by Tennant (1997) by the
name of 'factorization'. He says that an individual understands a
mathematical statement only if he can recognize the relevant aspects of
the proof of it, and is able to verify, locally, the correctness of any of its
parts, so that he understands if he is dealing with a valid proof or not.
With this solution, it seems less clear why we should reject
infinite statements whose proofs are infinitary. for, (e.g.) regarding Cook's
proof of Yablo's Paradox, it seems that a competent logician, even if she
can't evaluate that proof step by step, is able to differentiate the relevant
aspects of it, and recognize it as a valid proof. That is, i think, much closer
to our intuitions. if we are faithful to the traditional intuitionistic
account, we should say that we actually don't understand infinite
sentences like S1 ('S2 is false and S3 is false and...'), because in order to
recognize a proof of it, we should pay attention to infinite applications of
the universal instantiation rule or the conjunction introduction rule. But
it seems reasonable to hold that we do understand a sentence like that,
partly because in a manifestationist sense, we can recognize a proof of it,
not checking every step in particular, but understanding in a general way
the relevant aspects and checking if they are right or not.
One might object that when we meet an application of an infinitary
rule, no factorization can be done4. in fact, no human being can see an infinite numbers of premises, In order to evaluate if the rule was
correctly applied; there seems to be no finite part to separate and
evaluate step by step. That seems different from what happens with
normal, finitary rules, where a normal subject can go back and see the
finite (and maybe few) steps that served as premises. But actually,
factorization does not mean exclusively separating the parts of the proof
and checking the correctness of the rules applied. It means also to see the
relevant structure of the proof and evaluate it. Surely, it's impossible to
check all the infinite steps needed for an application of the omega-rule;
but yet it's possible to check if there is something like a uniform method
which can be applied for each number at its time (and check, for any of
the infinite instantiations of that method, if the subproof is valid), as
normally do infinitary logicians. That is empirically the same way in
which a regular mathematician evaluates the validity of an unfeasible
finite proof; so, if in that case we could argue that the subject recognizes
it as valid, the same could be said in the infinitary case.
Obviously, it remains a substantial difference between infinitary
proofs and unfeasible finite proofs: the former is infinite, and the latter
is finite. So, the rejection of infinitism from an anti-realist standpoint
should be, not in the manifestation principles, but in the infinitism itself.
3. Finitism
Anti-realism is deeply committed to finitism. Since its birth, with
Brouwer, the claim that we can't make up infinite mental constructions
was related to the claim that time and reality are finite (Brouwer 1928).
With Dummett, the concept of an infinite proof would go, conceptually,
against the verificationism that characterizes his position. Every proof
should be decidable in a finite time; that is, for every proof, in a finite
amount of time it should be possible to determine whether it is correct.5
Nevertheless, I think that anti-realism can, not easily but
reasonably, dismiss this old commitment. The main reason is that the
finiteness of proofs, or in anti-realist terms, the 'in principle decidability' of them, obeys to an idealization that is, as C. Wright says, bizarre: proofs
that no one could ever perform, because incredibly great memory,
concentration, time and inferential capacity are needed, are decidable 'in
principle', and it's supposed that they are legitimate because the needed extension of cognitive capacities is only finite (in a not so clear way6). The
most natural anti-realist reaction to that idealization is the one of the
strict finitist: they consider that there's a limit for human feasibility,
established by biological and physical laws, and that no proof exceeding
that limit could be legitimate.7
Strict finitism has serious difficulties to ground arithmetics; in spite
of that, and as Wright (1982) observes against Dummett (1975), it can be
reasonably defended and characterized. Which is to say that it cannot be ruled out as incoherent. The strategy of a traditional finitist to overcome
the strict finitist objections has to be grounded, then, in the same
idealization as the one of the last section: even if it's true that no person
(or machine) could perform proofs of more than n steps (let's suppose,
10100), she can do it 'in principle', in a (incredibly large) finite time, with
infinite memory, etc. That draws a line between what is feasible in a finite
time with 'finite capacities' and what is not.
But, if we forget all the physical and cognitive boundaries for the'decidable in principle' requirement, and we claim that every finitely
conceivable proof is acceptable, there cease to be strong reasons to stay
there and don't go further. For on the one side, if the world was physically
different and, e.g. the time were infinitely divisible, an ideal human being
could perform supertasks (i.e. infinite tasks in a finite time)8. On the other
hand, even if the very notion of performing infinite tasks in a finite time
was an inconsistent concept, there aren't strong reasons to be always
committed to finiteness. If we are talking from an ideal point of view,
because we have stopped to worry about what a human being can
actually do, we could broaden our notion of 'decidable in principle',
including not only what can be shown to be false or true in a finite amount
of time, but also what can be verified or falsified by intuitively acceptable
rules and axioms in an infinite time. After all, the anti-realist who tries
to answer to the strict finitist has to relax substantively the commitment
to actual human capacities; so the attachment to what is finitely decidable is not as well motivated as it was, and it's reasonable to look
for less restrictive principles. This is obviously different from the
traditional anti-realism, and could be seen as an argument against it9.
But I'm not saying that traditional anti-realists should embrace infinitary
proofs; they are free to narrow their concept of decidability in the typical
way, and I think that there are good motivations to do that (the whole
history of anti-realism is in their side!). My point is that those motivations
are not conclusive and can be perfectly challenged; so, apart from the
typical finitist anti-realism, there is also room for an infinitist anti-realism
that is motivated by the same philosophical considerations (semantical
manifestation principle, rejection of recognition-transcendent truths).
Something good about this anti-realist defense of infinitism is that
we don't have to ask any more if we can 'actually' apply the omega-rule.
If we relax the requirements of manifestation and decidability in
principle, it results that our use of rules like those in some occasions (for
example, where we know that they preserve truth) doesn't alter any
fundamental principle of anti-realism. Because, as in other mathematical
proofs, in this one we can see its relevant aspects, analyze every one of
its finite parts (although not the entire proof), and determine if they are
correct or not. This also can be useful to answer to the JC. Beall's claim
that we 'can't see and infinite sequence of sentences' (cf. 2001): we can
say that, actually, we can see it in the same way that we can see
schematically finite proofs that we can't write with all the paper in the
world.
A last objection that any anti-realist could raise is that, even if we
concede that we can evaluate infinite proofs and that they can exist, we
can't actually construct infinite objects. To this objection we can answer
the same as before. It's true that we can't construct a proof of omega steps
as we construct a proof of three or four steps (that is, writing it in a sheet
of paper), but any version of finitism that wants to avoid strict finitism
has to admit proofs that can't be physically constructed. and so, if we can
actually recognize proofs in a purely general and schematic way (by
factorization), we can also construct them in the same way: a schema of
proof (or the proof for the first steps, followed by 'and so on...') may
perfectly be enough. This idea of construction seems to be closer to the
practice of infinitary mathematicians. in this way, some proofs will be in
strict sense larger than anything we could see o construct physically, but
they will be as well constructed, recognizable and evaluable as every other.
1 In particular, there use to be a strong rejection between realists of infinitary proofs. That is mainly because even if they consider that this kind of inferences can preserve truth (Tarski's 1936 case is paradigmatic), it's supposed to be something 'human' or'feasible in principle' in the notion of proof. This article is, in part, an answer to the realists who hold that position.
2 I am consciously ignoring a vital point of this account: the rejection of bivalence. But i follow Tennant (1997) in drawing a difference between anti-realism as the rejection of recognition-transcendent truths and anti-realism as the rejection also of bivalence.
3 Idealization often includes some increase in the mathematical capacities of the subject, specially in the practice and the familiarity with the theorems and axioms that are used in the proof. Only with that idealization we can explain that a lot of mathematicians actually understand what fermat's theorem says, but can't evaluate if Wiles' proof is correct or not.
4 This objection was raised by Ignacio Ojea.
5 Hilbertian formalism has a less visible commitment to finitism; in some papers, like (Hilbert 1931), Hilbert suggest the inclusion of rules whose finiteness is matter of discussion (see Ignjatovic 1992).
6 In Wright's words (1982, p. 223) : 'it is not obviously sensible to speak of a finite extension of at least some of the particular capacities determining the scope of our overall practical powers. (...) The only purchase, it seems, which we have on the idea of a finite increase in such powers is via the finiteness of the tasks that would then be within our compass; now, though, on pain of circularity, an independent explanation is required of the finiteness of tasks -independent , that is, of any appeal to the finiteness of the capacities of creatures who could actually perform'.
7 Positions as those were plausibly defended by Wittgenstein (1964) and van Dantzig (1956), although it's obviously not the predominant anti-realist standpoint.
8 In fact, Moore (1990) claims that this is the main challenge for intuitionism.
9 This was suggested by Lucas Rosenblatt.
References
1. Beall, JC. (2001), 'Is Yablo's paradox non circular?', Analysis, 61, pp. 176-187.
2. Bringsjord, S. and van Heuveln, B. (2003), 'The 'mental eye' defense of an infinitized version of Yablo's paradox', Analysis, 63, pp. 61-70.
3. Brouwer, L. (1928), 'Mathematics, science and language', in Ewald, W. (1996), From Kant to Hilbert, vol. 2, Oxford University Press, pp. 1170-1785.
4. Cook, R. (forthcoming), Yablo's Paradox: An Essay on Circularity, Oxford, Oxford University Press.
5. Dummett, M. (1975), 'Wang's Paradox', Synthese, 30, pp. 301-324.
6. Hilbert, D. (1931), 'The grounding of elementary number theory', in Ewald, W. (1996), From Kant to Hilbert, vol. 2, Oxford University Press, pp. 1148-1157.
7. Ignjatovic, A. (1992), 'Hilbert's program and the omega-rule', Journal of Symbolic Logic, 59 (1), pp. 322-343.
8. Ketland, J. (2005), 'Some more curious inferences', Analysis, 65, pp 18-24.
9. Moore, A. (1990), 'A problem for intuitionism: The apparent possibility of performing infinitely many tasks in a finite time', Proceedings of the Aristotelian Society, 90, pp. 17-34.
10. Sorensen, R. (1998), 'Yablo's paradox and kindred infinite liars', Mind, 107, pp. 137-155.
11. Tarski, A. (1936), "On the Concept of following logically", Stroińska, M. and Hitchcock, D. (trads.), History and Philosophy of Logic, 2002, 23, pp. 155-196.
12. Tennant, N. (1997), The taming of the true, Oxford, Oxford University Press.
13. van Dantzig, D. (1956), 'Is 101010 a finite number?', Dialectica, 10, pp. 273-277.
14. Wittgenstein, L. (1964), Remarks on the Foundations of Mathematics, von Wright, G. H., Rhees, R. and Anscombe, G. E. M.(eds), Anscombe, G. E. M. (trad.), Oxford, Blackwell.
15. Wright, C. (1982), 'Strict finitism', Synthese, 51 (2), pp. 203-282.