Abstract
It is characteristic of some quite important discourses in mathematics, having to do with computability and constructivity, to appeal to what appear to be possibilities of human thought and action, where, however, these possibilities go beyond what is “practically” possible. In particular, they do not rest on actual human abilities. Such possibilities are what I have in mind in asking what we can “in principle” do. Such possibilities are also invoked about nonhuman agents, in particular computers. I am primarily concerned, however, with real agents, human beings like you and me and computers like the ones we have on our desks, or at most supercomputers. But talk about the “in principle” capabilities of real agents is roughly equivalent to talk about the capabilities of ideal agents.
In philosophy it is can in particular that we seem so often to uncover, just when we had thought some problem settled, grinning residually up at us like the frog at the bottom of the beer mug.
J.L. Austin, “Ifs and cans”1
Earlier versions of this paper were presented as the Skolem Lecture, University of Oslo, and to philosophy colloquia at Stanford and UCLA. I have received valuable comments from a number of hearers; I am painfully aware of not having done justice to all of them. Special thanks are owed to Joseph Almog, Tyler Burge, John Carriero, Solomon Feferman, Dagfinn Føllesdal, David Kaplan, Per Martin-Löf, and Grigori Mints. The earlier versions of the paper were written while I was a Fellow of the Center for Advanced Study in the Behavioral Sciences, with the support of the Andrew W. Mellon Foundation. I wish to thank both institutions.
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References
Philosophical Papers (Oxford: Clarendon Press, 1961), p.231.
In the course of explaining the notion of necessity that concerns him, Alvin Plantinga remarks of “leaping tall buildings in a single bound or ... travelling faster than a speeding bullet”, “These things are impossible for us; but not in the broadly logical sense” (The Nature of Necessity (Oxford: Clarendon Press, 1974), p.2). In talking of “jumping”, I am assuming that the jumper relies on his own power and does not use such aids as balloons. It’s not clear that all talk of jumping satisfies this constraint; consider (where the problem is the downward journey rather than reaching a height) parachute jumping.
The time demands of computing 257729 should not have daunted the Elves of The Lord of the Rings, although it is hard to imagine one of them devoting himself to such an undramatic task. The space demands might have posed a problem.
I ignore possible shortcuts, but the fact that the problem is NP-complete shows that there are serious limits of what they can accomplish.
S.C. Kleene, Introduction to Metamathematics (New York: Van Nostrand, 1952), p.136, emphases mine.
S.C. Kleene, Introduction to Metamathematics (New York: Van Nostrand, 1952), , p.47, emphases again mine. In formalized intuitionistic mathematics, it is possible to prove AV A for quantifier-free A, and then derive it for any A containing only bounded quantifiers. The pro makes essential use of primitive recursion, but not in any very strong form. It suggests that it may be possible to look at the grounds for the truth of this restricted law of excluded middle in a different way.
For simplicity I will assume that the procedure is deterministic and can thus yield at most one number as output. Functions of more than one argument can be accommodated by conventions about the representation of n-tuples of numbers.It can of course happen that the procedure does not yield a number value for any number argument; that is f is undefined for all numbers. Moreover, for a given argument we may not be able to determine whether or not the procedure terminates. That, for a given canonical form, it is not effectively decidable whether a procedure terminates is one of the basic results of the theory of computability.
In fact, sometimes it is only this direction that is called Church’s Thesis, for example by Kleene, op. cit., p.300. The equivalence fits better with Church’s original formulation. He states that he is proposing a “definition of effective calculability” as either A-definability or general recursiveness (“An unsolvable problem of elementary number theory”, American Journal of Mathematics 58 (1936), 345–363, p.90 of the reprint in Martin Davis (ed.), The Undecidable (Hewlett, N.Y.: Raven Press, 1965); cf. also p.100 and “An unsolvable problem of elementary number theory: Preliminary report” (abstract), Bulletin of the American Mathematical Society 41 (1935), 332–333). On the other hand Church’s most developed argument, in §7 of the paper, is for the claim that all effectively calculable functions are recursive.
In the Turing machine case, the machine is in a “terminal state”; in the case of the formalism of recursive functions, we have reached an equation of the form f(n) = m, where f is the principal function letter.
To be sure, this formulation fits best with an embedding of the notion of general recursiveness in classical mathematics. A constructivist would demand a constructive proof that the computation terminates in general. Cf. Kleene, op. cit., p.319.
Van Dantzig writes, “Modern physics implies an upper limit, by far surpassed by 101010, for numbers which actually can be constructed in this way.” (“Is 101010 a finite number?” Dialectica 9 (1955–56), 273 277, p.273). This point was later developed further by Robin Gandy. Although van Dan zig’s point of view is akin to what is now called strict finitism, he does not propose to ba sh exponentiation from mathematics; rather he argues for a kind of relativity of the notion of natural number and the irreducibility of formal expressions involving functors introduced by primitive recursion to canonical 0 — S numerals, or even to expressions introduced earlier in a chain of primitive recursions. These ideas are developed in more recent strict finitist writing.
Such a view is usually attributed to Brouwer, not unreasonably in spite of the strongly naturalistic tendencies in his philosophy. Cf. the remarks below on Heyting.
John Carriero asked the reasonable question whether some notion of human capacity in principle already underlies the conception of mathematical construction in Euclid. Certainly lines, for example, are thought of in traditional geometry as indefinitely extendible, and we naturally think of the constructions postulated by Euclid as indefinitely iterable. Euclid’s language in stating his postulates seems to me quite reserved; for example postulate 2 is stated “To produce a finite straight line continuously in a straight line”. (T.L. Heath, The Thirteen Books of Euclid’s Elements, 2d ed. (Cambridge University Press, 1926), vol. 1, p.154.) Although verbs of action are used, nothing is said about what this implies. The question is not addressed in Heath’s commentary on Euclid or a couple of others I have consulted.
David Hilbert, “Mathematische Probleme”, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1900, 253–297, Problem 10, my translation.
David Hilbert and Wilhelm Ackermann, Grundzüge der theoretischen Logik (Berlin: Springer, 1928), p.73, translation from Robin Gandy, “The confluence of ideas in 1936,” in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford University Press, 1988), pp.55–111, at p.62.
“An informal exposition of proofs of Gödel’s theorem and Church’s theorem,” The Journal of Symbolic Logic 4 (1939), 53–60, p.225 of the reprint in The Undecidable.
Gandy, op. cit., pp.66–7, quoted from J. von Neumann, “Zur Hilbertschen Beweistheorie,” Mathematische Zeitschrift 26 (1927), 1–46; I have modified Gandy’s translation.
A. Heyting, “The intuitionist foundations of mathematics,” in Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings (2d ed., Cambridge University Press, 1983), p.54. (Translation from Erkenntnis 2 (1931).)
E.g. ibid., p.53.
“The philosophical basis of intuitionistic logic” (1973), in Truth and Other Enigmas (London: Duckworth, 1978), p.224, emphases mine.
Cf. the opening remarks in “Wang’s paradox” (1975), in Truth and Other Enigmas, p.248.
Crispin Wright, “Strict finitism,” Synthese 51 (1982), 203–282, cited according to the reprint in Realism, Meaning, and Truth, 2d ed. (Oxford: Blackwell, 1993);
Alexander George, “The conveyability of intuitionism: An essay in mathematical cognition,” Journal of Philosophical Logic 17 (1988), 133–156.
“Strict finitism,” p.113. P should generalize readily to other activities than verifying or falsifying statements; in the text I understand it in this generalized way.
In spite of these reservations, Wright curiously appeals to P in another essay in the same volume where in principle capabilities are at issue; see Realism, Meaning, and Truth, 2d ed., p.326.
We are talking about jumping over the Eiffel Tower; the fact that a bird might be able to fly over it is presumably irrelevant. Cf. note 2.
I should make clear that I do not claim to be refuting this view. For present purposes it should suffice to observe that it would not be acceptable to either Dummett or Wright.
For example “Mathematical intuition,” Proceedings of the Aristotelian Society N.S. 80 (1979–80), 145–168, pp.158–159.
“On computable numbers, with an application to the Entscheidungsproblem,” Proceedings of the London Mathematical Society (2) 42 (1936–7), 230–265, cited according to the reprint in The Undecidable.
Ibid., p.135.
This formulation, as well as (2.1) and (2.2), are from Wilfried Sieg, “Mechanical procedures and mathematical experience,” in Alexander George (ed.), Mathematics and Mind (Oxford University Pres, 1994), p.93. I am much indebted to this article as well as the one by Gandy cited in note 15.
In crediting Turing with a theorem at this point, Gandy anticipated Sieg, but Sieg’s statement, given here, is more precise.
Gandy and Sieg rightly emphasize that Turing in §9 was not analyzing computation by a machine, but “mechanical” computation by a human. It is another task, undertaken by Gandy in another paper, to analyze the notion of computation by a mechanical device.
Such an inquiry might be relevant to the justification of the assumptions; cf. Gödel’s questioning of (1.2) (Collected Works, volume II, Publications 1938–1974 , Solomon Feferman et al, eds. (Oxford University Press, 1990), p.306), and Sieg’s (in my view correct) defense of Turing, op. cit., p.98.
Cf. Christos H. Papadimitriou, Computational Complexity (Reading, Mass.: Addison-Wesley, 1993), pp.6–7.The remarks in the text do not, I think, contradict Gödel’s claim that the notion of computability has an absolute character, in the context of classical mathematics. (See in particular “Remarks before the Princeton Bicentennial Conference on Problems in Mathematics,” in Collected Works, II, 150–153.) Turing computability was meant to answer to certain mathematical demands. Once one admits primitive recursive functions as computable (or, indeed, much less) and applies the principle “Finiteness is sufficient”, then one is forced to the conclusion that all general recursive functions are computable. And the latter principle was certainly very well motivated from a mathematical point of view. More constrained notions question either “Finiteness is sufficient” or some accepted classical assumption (such as that exponentiation is well-defined). The motivation is either philosophical or for some application, such as to implementation of algorithms.Mints, to whom I am indebted for questioning me on this point, remarks that although there is a unique notion of computable function, there is not a unique notion of efficient computation or computability.
Cf. Gödel’s remark that the undecidability results “do not establish any bounds for the powers of human reason, but rather for the potentialities of pure formalism in mathematics.” (Postscriptum (1964) to “On undecidable propositions of formal mathematical systems,” Collected Works, volume I, Publications 1929–1936, Solomon Feferman et al, eds., p.370.)
“Infinity and Kant’s conception of the ‘possibility of experience’,” first published 1964, in Mathematics in Philosophy: Selected Essays (Ithaca: Cornell University Press, 1983).
I do not deal at all with one possible objection to this proposed use of Turing: it is a matter of doubt and controversy whether the effective procedures called for by the intuitionistic understanding of the mathematical notion of function must be mechanical and thus whether Church’s thesis is an acceptable principle for intuitionistic mathematics. This question is clearly not relevant to the issues surrounding strict finitism.
See Petr Hájek and Pavel Pudlák, Metamathematics of First-Order Arithmetic (Springer-Verlag, 1993), ch. V, especially theorem 4.16, p.332.
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Parsons, C. (1997). What Can We Do “In Principle”?. In: Dalla Chiara, M.L., Doets, K., Mundici, D., van Benthem, J. (eds) Logic and Scientific Methods. Synthese Library, vol 259. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0487-8_17
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