Abstract
Thanks to developments in the last few decades in mathematical logic and computer science, it has now become possible to formalize non-trivial mathematical proofs in essentially complete detail. we discuss the philosophical problems and prospects for such formalization enterprises. We show how some perennial philosophical topics and problems in epistemology, philosophy of science, and philosophy of mathematics can be seen in the practice of formalizing mathematical proofs.
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We should be clear that no computer can act as a “complete” arbiter of questions like these in the sense that it could correctly answer arbitrary questions of the form: “Is the sentence Ø a logical consequence of the set Γ of assumptions?” For some notions of logical consequence, such as classical propositional logic, the logical consequence relation is, of course, decidable. The standard notion of first-order logical consequence, though, is, however, undecidable. Some theories expressed in the language of first-order logic are decidable, but most theories of foundational interest, such as Peano Arithmetic or Zermelo-Fraenkel set theory, are themselves undecidable.
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According to this definition, the number 1 is a prime number, which is not correct. We may repair the definition by replacing the constraint that p be positive (p > 0) with the condition that p be greater than 1 (p > 1).
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This is in fact how it is defined in the Mizar Mathematical Library, the collection of mathematical knowledge that has been formalized in the Mizar system. See http://mizar.org/version/current/html/sin_cos.html.
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ZFC is Zermelo-Fraenkel set theory with the axiom of choice.
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The actual implementation of the notion of obviousness in Mizar diverges somewhat from the definition that we are about to take up. Nonetheless, the notion we are about to define is the main feature of the actual, implemented notion of obviousness in Mizar.
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One might object to Davis’s proposal at this point because arbitrary classical propositional reasoning is known to be an NP-complete problem. In other words, all tautologies are deemed obvious by Davis’s notion. We do not take up the problem of whether this fact conflicts with our ordinary notion of “obvious”.
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I thank Artur Korniłowicz for this example, which comes from the Mizar Mathematical Library. See http://mizar.org/version/current/html/xboole_1.html.
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Acknowledgement:
Both authors were supported by the ESF research project Dialogical Foundations of Semantics within the ESF Eurocores program LogICCC (fundedby the Portuguese Science Foundation, FCT LogICCC/0001/2007). The second author was also supported by the FCT-project Hilbert’s Legacy in the Philosophy of Mathematics, PTDC/FIL-FCI/109991/2009.
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Alama, J., Kahle, R. (2013). Computing with Mathematical Arguments. In: Andersen, H., Dieks, D., Gonzalez, W., Uebel, T., Wheeler, G. (eds) New Challenges to Philosophy of Science. The Philosophy of Science in a European Perspective, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5845-2_2
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