Abstract
No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity.
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Notes
The P \(=\) NP problem deals with the complexity of an algorithm that solves a particular problem. If the problem is characterized by a parameter n, then a P-problem has a solution such that the time required for its calculation is a polynomial in n. For a NP-problem, the solution is non-polynomial, but, when a solution is suggested, it takes polynomial time to check whether it is a solution or not. The importance of this problem is shown by the fact that it is on the list of problems of the Clay Mathematics Institute, promising a million dollars for an answer to the problem.
Although I am not sure at all what a poll would reveal in this case.
In addition, this type of reconstruction is not limited to contradictory theories. It is also applied to mathematical theories that are supposed to be consistent. The best known example surely must be the reconstruction in full formal detail of Euclid’s Elements. One of the finest, earliest examples is (Mueller 1981), but see also, more recently, (Avigad et al. 2009).
Due to the wealth of literature, I mention here only (Priest and Tanaka 2009) as an excellent starting point.
For an excellent introduction and starting point, see http://logica.ugent.be/adlog/al.html. The originator of this program is Diderik Batens.
Such as the fact that in the paper itself, Priest and Brown use lambda-abstraction to talk about functions, so strictly the formulation here is not correct, but, as far as the examples goes, it is, I believe, not essential.
E.g., in the Theodicy, he writes in § 70: “...every number is finite and specifiable, every line is also finite and specifiable. Infinite [magnitudes] and the infinitely small only signify magnitudes which one can take as big or as small as one wishes, in order to show that the error is smaller than the one that has been specified.”
Do note however that this is not the only possibility. I refer to Van Bendegem (2002) for a brief discussion of (a) Synthetic Differential Geometry, where infinitesimals have the property that \(e\ne 0\), yet \(e^{2} =\) 0, (b) Analysis with Big 0, developed by Donald Knuth, and (c) Non-non-standard Analysis, developed by J.M. Henle, where real numbers are replaced by series of real numbers and infinitesimals are those series that tend to zero.
Two additional remarks. First, it is trivial to remark that several geometrical interpretations exist of the complex numbers at the present moment, e.g., in terms of rotations, but do note that the interpretation proposed here is different from the known versions and it is a plausible one given the mathematical knowledge of Cardano’s days. Secondly, it also perfectly possible that, instead of complex numbers, something entirely different could have been proposed. The interesting attempt in Martinez (2006) starts from the simple observation to restore symmetry in our basic calculation rules, especially concerning the use of signs. If plus times plus gives plus, should not minus times minus produce minus? If so, what kind of mathematics results? Among other things, a mathematics where \(\sqrt{-1}=-1\).
This is a thought to be explored further. In mathematics itself, the burden of proof is quite simple: he or she who claims to have a proof of a particular mathematical statement, is obliged to produce the proof. It is not up to the reader to “prove” that he or she does not have a proof. (However in 17th century French mathematics, it was not uncommon to write a letter to someone and to challenge him to find a proof whereof you claimed in your letter that you had proven it, which was not necessarily the case, in the hope that the return letter would contain a proof.) As soon as we talk about mathematics, then the whole matter of the burden of proof becomes quite important. Is it, e.g., the burden of the platonist to show that his or her view is correct or is it the burden of the opponent or of both?
In fact, it is possible to generate such sentences. A fine example is a variation on the Gödel-sentence, that states: “For this statement, there is no proof shorter than \(k\) lines”, for some constant \(k\). If there is a proof, it will have more than \(k\) lines. In this way, one can “force” the length of a proof. See Parikh (1971) for a detailed presentation.
He also calls the numbers \(x\) “engineering” numbers. This in itself invites a quite interesting thought to ponder. Once we leave the save realm of “pure” mathematics, where we can uphold the belief that all is well and everything is consistent, and enter into the arena of applied mathematics, where all is muddled and approximations reign, what kind of mathematics are we thinking of? Surely an inconsistent version seems the most appropriate as it seems to be a reliable reflection of their practices.
In terms of the collapsing lemma, one splits up a classical model into a finite number of parts, namely [0], [1], [2], ... [\(n\), \(n+\)1, \(n+\)2, ...]. All numbers up to \(n-\)1 behave as their classical counterparts, but upwards from \(n\), statements become true-false, as both \(n=n\) \(+\) 1 and \(n\ne n+1\) are true (and false).
References
Avigad, J., Dean, E., & Mumma, J. (2009). A formal system for Euclid’s elements. Review of Symbolic Logic, 2(4), 700–768.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford: Oxford University Press.
Beall, J. C. (1999). From full blooded platonism to really full blooded platonism. Philosophia mathematica, 7(3), 322–325.
Broome, T. H. (2010). Metaphysics of engineering. In I. Van De Poel & D. E. Goldberg (Eds.), Philosophy and engineering: An emerging agenda (pp. 295–304). New York: Springer.
Cardano, G. (1968). The great art or the rules of algebra. Cambridge(Mass): The MIT Press, (translated and edited by T. R. Witmer).
Colyvan, M. (2009). Applying Inconsistent Mathematics. In Bueno, O. & Linnebo, Ø. (Eds.), New waves in philosophy of mathematics. (pp. 160–172). Basingstoke: Palgrave MacMillan.
Costa, Da, & Newton, C. A. (2004). Paraconsistent logics and paraconsistency: Technical and philosophical developments. CLE e-prints (Section Logic), 4(3), 105.
Gasarch, W. I. (2002). The P=?NP Poll. Electronic version at http://www.cs.umd.edu/gasarch/ papers/poll.pdf, Accessed 2 February 2010
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Maddy, P. (1988a). Believing the axioms I. Journal of Symbolic Logic, 53(2), 481–511.
Maddy, P. (1988b). Believing the axioms II. Journal of Symbolic Logic, 53(3), 736–764.
Mares, E. (1997). Paraconsistent probability theory and paraconsistent Bayesianism. Logique et Analyse, 42(160), 375–384.
Martinez, A. A. (2006). Negative math. How mathematical rules can be positively bent. Princeton: Princeton University Press.
Meyer, R. K., & Mortensen, C. (1984). Inconsistent models for relevant arithmetics. The Journal of Symbolic Logic, 49, 917–929.
Mortensen, C. (1995). Inconsistent mathematics. Dordrecht: Kluwer.
Mueller, I. (1981). Philosophy of mathematics and deductive structure in Euclid’s elements. Cambridge, Mass: MIT. (New edition: 2006, New York: Dover Publications).
Parikh, R. (1971). Existence and feasibility in arithmetic. The Journal of Symbolic Logic, 36(3), 494–508.
Priest, G. (1982). To be and not to be: dialectical tense logic. Studia Logica, 41(2—-3), 157–176.
Priest, G. (1987). In contradiction. Dordrecht: Nijhoff.
Priest, G. (1991). Minimally inconsistent LP. Studia Logica, 2, 321–331.
Priest, G. (1997). Inconsistent models for arithmetic: I finite models. The Journal of Philosophical Logic, 26, 223–235.
Priest, G. (2000). Inconsistent models for arithmetic: II the general case. The Journal of Symbolic Logic, 65, 1519–1529.
Priest, G., & Brown, B. (2004). Chunk and permeate, a paraconsistent inference strategy. Part I: The infinitesimal calculus. Journal of Philosophical Logic, 33, 242–263.
Priest, G., Tanaka, K. (2009). Paraconsistent logic. The Stanford Encyclopedia of Philosophy (Winter 2009 ed.), E. N. Zalta (Ed.), http://plato.stanford.edu/archives/win2009/entries/logic-paraconsistent/
Priest, G., Berto, F. (2010). Dialetheism. The Stanford Encyclopedia of Philosophy (Summer 2010 Edition), E. N. Zalta (ed.), http://plato.stanford.edu/archives/sum2010/entries/dialetheism/
Robinson, A. (1979). The metaphysics of the calculus. In W. A. J. Luxemburg, & S. Körner (eds.), Selected papers of Abraham Robinson: nonstandard analysis and philosophy (vol. 2, pp. 537–549). New Haven: Yale University Press
Routley, R., & Meyer, R. (1976). Dialectical logic, classical logic and the consistency of the world. Studies in Soviet Thought, 16, 1–25.
Van Bendegem, J. P. (1994). Strict finitism as a viable alternative in the foundations of mathematics. Logique et Analyse, 37, 145, (date of publication: 1996, pp. 23–40)
Van Bendegem, J. P. (2002). Inconsistencies in the history of mathematics: The case of infinitesimals. In: Joke Meheus (ed.) Inconsistency in science, (pp. 43–57). Dordrecht: Kluwer Academic Publishers (Origins: Studies in the Sources of Scientific Creativity, vol. 2).
Van Bendegem, J. P. (2008). What-if’ stories in mathematics: An alternative route to complex numbers. In Danblon, E., Kissine, M., Martin, F., & Vogeleer, S. (Eds.), Linguista Sum: Mélanges offerts à Marc Dominicy à l’occasion de son soixantième anniversaire. (pp. 391–402). Paris: Harmattan.
Van Bendegem, J. P. (2010). Finitism in geometry. The Stanford Encyclopedia of Philosophy, E. N. Zalta (ed.), The Metaphysics Research Lab at the Center for the Study of Language and Information. Stanford, CA: Stanford University. http://plato.stanford.edu/entries/geometry-finitism/
Weber, Z. (2010). Transfinite numbers in paraconsistent set theory. The Review of Symbolic Logic, 3, 71–92.
Acknowledgments
I wish to thank the members of my research center where a first reduced version of this paper was presented, especially Patrick Allo and Francesca Poggiolesi for helpful comments. The same holds for the referee’s suggestions that further improved and enriched this contribution.
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van Bendegem, J.P. Inconsistency in mathematics and the mathematics of inconsistency. Synthese 191, 3063–3078 (2014). https://doi.org/10.1007/s11229-014-0474-6
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DOI: https://doi.org/10.1007/s11229-014-0474-6