# Inconsistent Mathematics

Inconsistent Mathematics

Inconsistent mathematics is the study of commonplace mathematical objects, like sets, numbers, and functions, where some contradictions are allowed. Tools from formal logic are used to make sure any contradictions are contained and that the overall theories remain coherent. Inconsistent mathematics began as a response to the set theoretic and semantic paradoxes such as Russell’s Paradox and the Liar Paradox—the response being that these are interesting facts to study rather than problems to solve—and has so far been of interest primarily to logicians and philosophers. More recently, though, the techniques of inconsistent mathematics have been extended into wider mathematical fields, such as vector spaces and topology, to study inconsistent structure for its own sake.

To be precise, a mathematical theory is a collection of sentences, the theorems, which are deduced through logical proofs. A contradiction is a sentence together with its negation, and a theory is inconsistent if it includes a contradiction. Inconsistent mathematics considers inconsistent theories. As a result, inconsistent mathematics requires careful attention to logic. In classical logic, a contradiction is always absurd: a contradiction implies everything. A theory containing every sentence is trivial. Classical logic therefore makes nonsense of inconsistency and is inappropriate for inconsistent mathematics. Classical logic predicts that the inconsistent has no structure. A paraconsistent logic guides proofs so that contradictions do not necessarily lead to triviality. With a paraconsistent logic, mathematical theories can be both inconsistent and interesting.

This article discusses inconsistent mathematics as an active research program, with some of its history, philosophy, results and open questions.

Introduction
An Example
Background
Motivations
Perspectives
Methods
Proofs
Geometry
Set Theory
Arithmetic
Analysis
Computer Science
References
1. Introduction

Inconsistent mathematics arose as an independent discipline in the twentieth century, as the result of advances in formal logic. In the nineteenth century, a great deal of extra emphasis was placed on formal rigor in proofs, because various confusions and contradictions had appeared in the analysis of real numbers. To remedy the situation required examining the inner workings of mathematical arguments in full detail. Mathematics had always been conducted through step-by-step proofs, but formal logic was intended to exert an extra degree of control over the proofs, to ensure that all and only the desired results would obtain. Various reconstructions of mathematical reasoning were advanced.

One proposal was classical logic, pioneered by Giuseppe Peano, Gottlob Frege, and Bertrand Russell. Another was paraconsistent logic, arising out of the ideas of Jan Łukasiewicz and N. A. Vasil’év around 1910, and ﬁrst realized in full by Jaśkowski in 1948. The ﬁrst to suggest paraconsistency as a ground for inconsistent mathematics was Newton da Costa in Brazil in 1958. Since then, his school has carried on a study of paraconsistent mathematics. Another school, centered in Australia and most associated with the name of Graham Priest, has been active since the 1970s. Priest and Richard Routley have forwarded the thesis that some inconsistent theories are not only interesting, but true; this is dialetheism.

Like any branch of mathematics, inconsistent mathematics is the study of abstract structures using proofs. Paraconsistent logic offers an unusually exacting proof guide that makes sure inconsistency does not get out of hand. Paraconsistency is not a magic wand or panacea. It is a methodology for hard work. Paraconsistency only helps us from getting lost, or falling into holes, when navigating through rough terrain.

a. An Example

Let T be all the sentences in the language of arithmetic that are true of N; then let T(n) similarly be all the sentences true of the numbers up to n, an inconsistent number theory. Since T(n) does not contradict T about any numbers below n, if n > 0 then T(n) is non-trivial. (It does not prove 0 = 1, for instance.) The sentences of T(n) are representable in T(n), and its language contains a truth predicate for T(n). The theory can prove itself sound. The Gödel sentence for T(n) is provable in T(n), as is its negation, so the theory is inconsistent. Yet as Meyer proved, the non-triviality of T(n) can be established in T(n) by a finite procedure.

Most striking with respect to Hilbert’s program, there is a way, in principle, to figure out for any arithmetic sentence Φ whether or not Φ holds, just by checking all the numbers up to n. This means that T(n) is decidable, and that there must be axioms guaranteed to deliver every truth about the collapsed model. This means that an inconsistent arithmetic is coherent and complete.

6. Analysis

Newton and Leibniz independently developed the calculus in the 17th century. They presented ingenious solutions to outstanding problems (rates of change, areas under curves) using infinitesimally small quantities. Consider a curve and a tangent to the curve. Where the tangent line and the curve intersect can be though of as a point. If the curve is the trajectory of some object in motion, this point is an instant of change. But a bit of thought shows that it must be a little more than a point—otherwise, as a measure a rate of change, there would be no change at all, any more than a photograph is in motion. There must be some smudge. On the other hand, the instant must be less than any finite quantity, because there are inﬁnitely many such instants. An infinitesimal would respect both these concerns, and with these provided, a circle could be construed as infinitely many infinitesimal tangent segments.

Infinitesimals were essential, not only for building up the conceptual steps to inventing calculus, but in getting the right answers. Yet it was pointed out, most famously by Bishop George Berkeley, that infinitesimals were poorly understood and were being used inconsistently in equations. Calculus in its original form was outright inconsistent. Here is an example. Suppose we are differentiating the polynomial f(x) =ax2+bx+c. Using the original definition of a derivative,

In the example, ε is an infinitesimal. It marks a small but non-trivial neighborhood around x, and can be divided by, so it is not zero. Nevertheless, by the end ε has simply disappeared. This example suggests that paraconsistent logic is more than a useful technical device. The example shows that Leibniz was reasoning with contradictory information, and yet did not infer everything. On the contrary, he got the right answer. Nor is this an isolated incident. Mathematicians seem able to sort through “noise” and derive interesting truths, even out of contradictory data sets. To capture this, Brown and Priest (2004) have developed a method they call “chunk and permeate” to model reasoning in the early calculus. The idea is to take all the information, including say ε = 0 and ε ≠ 0, and break it into smaller chunks. Each chunk is consistent, without conflicting information, and one can reason using classical logic inside of a chunk. Then a permeation relation is defined which controls the information flow between chunks. As long as the permeation relation is carefully defined, conclusions reached in one chunk can ﬂow to another chunk and enter into reasoning chains there. Brown and Priest propose this as a model, or rational reconstruction, of what Newton and Leibniz were doing.

Another, more direct tack for inconsistent mathematics is to work with infinitesimal numbers themselves. There are classical theories of infinitesimals due to Abraham Robinson (the hyperreals), and J. H. Conway (the surreals). Mortensen has worked with differential equations using hyperreals. Another approach is from category theory. Tiny line segments (“linelets”) of length ϵ are considered, such that ϵ2 = 0 but it is not the case that ϵ = 0. In this theory, it is also not the case that ϵ ≠ 0, so the logical law of excluded middle fails. The category theory approach is the most like inconsistent mathematics, then, since it involves a change in the logic. However, the most obvious way to use linelets with paraconsistent logics, to say that both ϵ = 0 and ϵ ≠ 0 are true, means we are dividing by 0 and so is probably too coarse to work.

In general the concept of continuity is rich for inconsistent developments. Moments of change, the ﬂow of time, and the very boundaries that separate objects have all been considered from the standpoint of inconsistent mathematics.

7. Computer Science

The questions posed by David Hilbert can be stated in very modern language:

Is there a computer program to decide, for any arithmetic statement, whether or not the statement can be proven? Is there a program to decide, for any arithmetic statement, whether or not the statement is true? We have already seen that Gödel’s theorems devastated Hilbert’s program, answering these questions in the negative. However, we also saw that inconsistent arithmetic overcomes Gödel’s results and can give a positive answer to these questions. It is natural to extend these ideas into computer science.

Hilbert’s program demands certain algorithms—a step-by-step procedure that can be carried out without insight or creativity. A Turing machine runs programs, some of which halt after a ﬁnite number of steps, and some of which keep running forever. Is there a program E that can tell us in advance whether a given program will halt or not? If there is, then consider the program E*, which exists if E does by deﬁning it as follows. When considering some program x, E* halts if and only if x keeps running when given input x. Then

E* halts for E*
if and only if
E* does not halt for E*,

which implies a contradiction. Turing concluded that there is no E*, and so there is no E—that there cannot be a general decision procedure.

Any program that can decide in advance the behavior of all other programs will be inconsistent.

A paraconsistent system can occasionally produce contradictions as an output, while its procedure remains completely deterministic. (It is not that the machine occasionally does and does not produce an output.) There is, in principle, no reason a decision program cannot exist. Richard Sylvan identifies as a central idea of paraconsistent computability theory the development of machines “to compute diagonal functions that are classically regarded as uncomputable.” He discusses a number of rich possibilities for a non-classical approach to algorithms, including a fixed-point result on the set of all algorithmic functions, and a prototype for dialetheic machines.

Important results have been obtained by the paraconsistent school in Brazil—da Costa and Doria in 1994, and Agudelo and Carnielli in 2006. Like quantum computation, though, at present the theory of paraconsistent machines outstrips the hardware. Machines that can compute more than Turing machines await advances in physics.

Priest’s In Contradiction (2006) is the best place to start. The second edition contains material on set theory, continuity, and inconsistent arithmetic (summarizing material previously published in papers). A critique of inconsistent arithmetic is in Shapiro (2002). Franz Berto’s book, How to Sell a Contradiction (2007), is harder to find, but also an excellent and perhaps more gentle introduction.

Some of da Costa’s paraconsistent mathematics is summarized in the interesting collection Frontiers of Paraconsistency (2000)—the proceedings of a world congress on paraconsistency edited by Batens et al. More details are in Jacquette’s Philosophy of Logic (2007) handbook; Beall’s paper in that volume covers issues about truth and inconsistency.

Those wanting more advanced mathematical topics should consult Mortensen’s Inconsistent Mathematics (1995). For impossible geometry, his recent pair of papers with Leishman are a promising advance. His school’s website is well worth a visit. Brady’s Universal Logic (2006) is the most worked-out paraconsistent set theory to date, but not for the faint of heart.

If you can find it, read Routley’s seminal paper, “Ultralogic as Universal?”, reprinted as an appendix to his magnum opus, Exploring Meinong’s Jungle (1980). Before too much confusion arises, note that Richard Routley and Richard Sylvan, whose posthumous work is collected by Hyde and Priest in Sociative Logics and their Applications (2000), in a selfless feat of inconsistency, are the same person.

For the how-to of paraconsistent logics, consult both the entry on relevance and paraconsistency in Gabbay & Günthner’s Handbook of Philosophical Logic volume 6 (2002), or Priest’s textbook An Introduction to Non-Classical Logic (2008). For paraconsistent logic and its philosophy more generally see Routley, Priest and Norman’s 1989 edited collection. The collection The Law of Non-Contradiction (Priest et al. 2004) discusses the philosophy of paraconsistency, as does Priest’s Doubt Truth be a Liar (2006).

For the broader philosophical issues associated with inconsistent mathematics, especially in applications (for example, consequences for realism and antirealism debates), see Mortensen (2009a) and Colyvan (2009).

b. References
Arruda, A. I. & Batens, D. (1982). “Russell’s set versus the universal set in paraconsistent set theory.” Logique et Analyse, 25, pp. 121-133.
Batens, D., Mortensen, C. , Priest, G., & van Bendegem, J-P., eds. (2000). Frontiers of Paraconsistent Logic. Kluwer Academic Publishers.
Berto, Francesco (2007). How to Sell a Contradiction. Studies in Logic v. 6. College Publications.
Brady, Ross (2006). Universal Logic. CSLI Publications.
Brown, Bryson & Priest, G. (2004). “Chunk and permeate i: the inﬁnitesimal calculus.” Journal of Philosophical Logic, 33, pp. 379–88.
Colyvan, Mark (2008). “The ontological commitments of inconsistent theories.” Philosophical Studies, 141(1):115 – 23, October.
Colyvan, Mark (2009). “Applying Inconsistent Mathematics,” in O. Bueno and Ø. Linnebo (eds.), New Waves in Philosophy of Mathematics, Palgrave MacMillan, pp. 160-72
da Costa, Newton C. A. (1974). “On the theory of inconsistent formal systems.” Notre Dame Journal of Formal Logic, 15, pp. 497– 510.
da Costa, Newton C. A. (2000). Paraconsistent mathematics. In Batens et al. 2000, pp. 165–180.
da Costa, Newton C.A., Krause, D´ecio & Bueno, Ot´avio (2007). “Paraconsistent logics and paraconsistency.” In Jacquette 2007, pp. 791 – 912.
Gabbay, Dov M. & Günthner, F. eds. (2002). Handbook of Philosophical Logic, 2nd Edition, volume 6, Kluwer.
Hinnion,Roland & Libert, Thierry (2008). “Topological models for extensional partial set theory.” Notre Dame Journal of Formal Logic, 49(1).
Hyde, Dominic & Priest, G., eds. (2000). Sociative Logics and their Applications: Essays by the Late Richard Sylvan. Ashgate.
Jacquette, Dale, ed. (2007). Philosophy of Logic. Elsevier: North Holland.
Libert, Thierry (2004). “Models for paraconsistent set theory.” Journal of Applied Logic, 3.
Mortensen, Chris (1995). Inconsistent Mathematics. Kluwer Academic Publishers.
Mortensen, Chris (2009a). “Inconsistent mathematics: Some philosophical implications.” In A.D. Irvine, ed., Handbook of the Philosophy of Science Volume 9: Philosophy of Mathematics. North Holland/Elsevier.
Mortensen, Chris (2009b). “Linear algebra representation of necker cubes II: The routley functor and necker chains.” Australasian Journal of Logic, 7.
Mortensen, Chris & Leishman, Steve (2009). “Linear algebra representation of necker cubes I: The crazy crate.” Australasian Journal of Logic, 7.
Priest, Graham, Beall, J.C. & Armour-Garb, B., eds. (2004). The Law of Non-Contradiction. Oxford: Clarendon Press.
Priest, Graham (1994). “Is arithmetic consistent?” Mind, 103.
Priest, Graham (2000). “Inconsistent models of arithmetic, II: The general case.” Journal of Symbolic Logic, 65, pp. 1519–29.
Priest, Graham (2002). “Paraconsistent logic.” In Gabbay and Günthner, eds. 2002, pp. 287–394.
Priest, Graham (2006a). Doubt Truth Be A Liar. Oxford University Press.
Priest, Graham (2006b). In Contradiction: A Study of the Transconsistent. Oxford University Press. second edition.
Priest, Graham (2008). An Introduction to Non-Classical Logic. Cambridge University Press, second edition.
Priest, Graham, Routley, R. & Norman, J. eds. (1989). Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag.
Routley, Richard (1977). “Ultralogic as universal?” Relevance Logic Newsletter, 2, pp. 51–89. Reprinted in Routley 1980.
Routley, Richard (1980). “Exploring Meinong’s Jungle and Beyond.” Philosophy Department, RSSS, Australian National University, 1980. Interim Edition, Departmental Monograph number 3.
Routley, Richard & Meyer, R. K. (1976). “Dialectical logic, classical logic and the consistency of the world.” Studies in Soviet Thought, 16, pp. 1–25.
Shapiro, Stewart (2002). “Incompleteness and inconsistency.” Mind, 111, pp. 817 – 832.
Author Information

Zach Weber
Email: [email protected]
University of Sydney, University of Melbourne
Australia

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