Abstract
In this paper I will argue that Tait’s axiomatic conception of mathematics implies that it is in principle impossible to be justified in believing a mathematical statement without being justified in believing that statement to be provable. I will then show that there are possible courses of experience which would justify acceptance of a mathematical statement without justifying belief that this statement is provable.
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Notes
(Tait 2001) p. 11.
I’d like to thank Warren Goldfarb, Peter Koellner, Ned Hall and Peter Gerdes for help fine-tuning my Tait exegesis and much lively debate about the larger philosophical questions at issue in this paper.
(Tait 2001) p. 4.
He writes, “if we should discover a contradiction in Peano Arithmetic, say, that would … undermine the sense of existence assertions concerning numbers (and so the sense of their negations, as well)” (Tait 2001) p. 3.
It will plausibly also include more than this. For although all propositions which can be derived by applying first order logic to the Zermelo–Fraenkel axioms would be accepted by mainstream mathematicians, there are further statements, such as the arithmetical sentence Con (ZF), which cannot be proved from these axioms yet are taken as genuine items of mathematical knowledge and perfectly acceptable starting points for proofs.
“But, of course, completeness fails and must fail. Nor is the essential incompleteness due simply to Gödels incompleteness theorem” (Tait 2001) p. 3.
i.e. if one cannot derive both a statement and its negation using the premises and inference rules permitted by that practice.
For instance, the existence of a model demonstrates the coherence of a theory but, when working in strong logics, it may not be the case that all coherent collections of axioms have a set model e.g. if one can uniquely describe the structure of the sets then this description will not apply to any structure inside the universe of sets.
e.g., the statement that every set has a powerset intuitively requires something that isn’t true in a countable model of ZFC.
In the case of the plenitudinous platonist I am speaking loosely with regard to the claim that all coherent theories are acceptable topics for investigation. For appeal to quantifier restriction, broad limits on abstraction or something more is needed to deal with the fact that not all internally coherent descriptions of mathematical objects are compatible with one another.
For instance, they believe the numbers are as small as possible while satisfying certain basic principles of arithmetic. However, no consistent collection of first-order axioms can fully express this idea. Any first-order theory that describes the natural numbers will also be satisfied by some non-standard model including infinite ‘numbers.’ Thus, it would seem that our real axioms for the numbers go beyond what is first-order expressible, in ruling out these spurious infinitary ‘numbers.
(Tait 2001) p. 13.
(Tait 2001) p. 13.
That is, on the fact that these descriptions uniquely determine a mathematical structure.
see (Tait 2001) pp. 4, 8–9.
(Tait 2001) p. 11.
(Tait 2001) p. 13.
Thus, for example, the Goldbach conjecture states that every even number greater than 2 is writable as the sum of two primes. This qualifies as a \(\Uppi_1^0\) sentence because it requires that ∀n n = 2 or n is odd or ∃x ≤ n ∃y ≤ n and x is prime and y is prime and x + y = n, where the property of being prime is itself expressible using only bound quantifiers.
The incompleteness theorem applies to any collection of mathematical statements, such as those which could be derived using a particular mathematical practice, which is syntactically consistent, algorithmically enumerable, and sufficiently powerful to capture certain basic facts of number theory. It tells us that, any such collection will fail to include both some \(\Uppi_1^0\) sentence and its negation.
The standard definition of an MH machine in the literature requires that the machine only go through stages corresponding to natural numbers, i.e. it computer accepts the \(\Uppi_1^0\) sentence iff no actual integer provides a counterexample. However, one thing that’s at issue here is whether we can think thoughts which distinguish a unique structure ω from various ‘nonstandard models’. Therefore, I adopt a less restrictive notion which merely requires an MH machine to include a computer which checks 0 and then (provided no counterexample has been found) checks the successor of any stage it checks.
The reader may wonder why I don’t take the much simpler route of simply arguing that our continued failure to find a contradiction while working with certain proof systems (e.g., ZFC set theory) gives us reason to accept the sentence expressing the arithmetical consistency sentences for these practices.
However, there is a significant line of worry in the literature about whether merely using a system without encountering a contradiction can give us reason to believe that that system is consistent, or whether the reason it gives us can be sufficient to let us qualify as having knowledge. Following Frege (1980), some philosophers have argued the numbers differ from one another so radically that “in the absence of proof, we should not expect numbers (in general) to share any interesting properties.
And hence that dealings with any number of finite cases where some number has failed to code a proof of 0=1 in a given proof system S can never provide us with any justification at all for the belief that some (untried) number fails to code a proof of contradiction in S. Less radically, it is sometimes argued that dealings with particular cases always provide us with a biased sample - with knowledge of what holds for small numbers and short proofs, and that such knowledge provides no basis for justified generalization to the claim that all numbers have a certain property or that no larger proof is possible.(Alan Baker. Non-deductive methods in mathematics. Stanford Encyclopedia of Philosophy (2013)
I avoid this obstacle by providing an apparent counterexample to Tait’s thesis which does not depend on the claim that a history of safe use of a mathematical theory can provide justification sufficient for knowledge of the claim that that theory is consistent.
At least near earth for the kind of low-tech uses that can ignore relativistic effects.
i.e. if one can add a \(\Uppi_1^0\) sentence to any proof practice which (like ours) contains the Peano Axioms and first order logic, then this sentence expresses a truth [finalcheck!].
(Tait 2001) p. 4.
Admittedly, adopting this line of response raises serious problems of its own. For example, what sense are we to make of Tait’s own talk of consistency when saying that, e.g., inconsistency debars an axiom system from giving meaning to our mathematical claims? If claims about consistency are only determined to have a particular truth value by being derived in some axiom system, what axiom system is relevant to Tait’s claim? If the relevant axiom system is the total collection of mathematical claims we are inclined to accept, there’s a prima facie problem. This system (presumably) cannot prove its own consistency (Gödel 1931). If there is not a finite demonstrable inconsistency in our axioms, then the question of whether the total collection of axioms that we are inclined to accept determine a consistent and hence true mathematical system, or an inconsistent (and hence meaningless) one will turn out to have the same status as independent \(\Uppi_1^0\) sentences. This seems like an odd consequence. It also seems odd that it in stipulating facts about arithmetic we could thereby determine facts about what alternative choices of axioms would have been meaningful.
References
Alan B. Non-deductive methods in mathematics. In E. N. Zalta (Ed.) Stanford encyclopedia of philosophy (Fall 2013 Edn.). <http://plato.stanford.edu/archives/win2012/entries/davidson/>.
Button, T. (2009). Sad computers and two versions of the church–turing thesis. The British Journal for the Philosophy of Science, 60(4), 765–792.
Earman, J., & Norton, J. (1996). Infinite pains: The trouble with supertasks (Vol. 11, p. 271). Cambridge, Mass.: Blackwell Publishers.
Earman, J., & Norton, J. D. (1993). Forever is a day: Supertasks in Pitowsky and Malament–Hogarth spacetimes. Philosophy of Science, 60(1), 22–42.
Etesi, G., Németi, I. (2002). Non-turing computations via Malament–Hogarth space-times. International Journal of Theoretical Physics, 41(2), 341–370.
Frege, G. (1980). The foundations of arithmetic: A logico-mathematical enquiry into the concept of number. Evanston, IL: Northwestern University Press.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik, 38(1), 173–198.
Hogarth, M. (2004). Deciding arithmetic using SAD computers. The British Journal for the philosophy of Science, 55(4), 681–691.
Kaye, R. (1991). Models of Peano arithmetic, volume 15 of Oxford logic guides. New York: Oxford University Press.
Rosser, B. (1936). Extensions of some theorems of Gödel and Church. The Journal of Symbolic Logic, 1(3), 87–91.
Tait, W. W. (2001). Beyond the axioms: the question of objectivity in mathematics. Philosophia Mathematica, 9(1), 21–36.
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Berry, S. Malament–Hogarth Machines and Tait’s Axiomatic Conception of Mathematics. Erkenn 79, 893–907 (2014). https://doi.org/10.1007/s10670-013-9571-z
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DOI: https://doi.org/10.1007/s10670-013-9571-z