Argumentation in Mathematics
KeywordsArgumentation History of logic Logic Modal logic Proof theory
Argumentation refers to the process of making an argument, that is, drawing conclusions based on a chain of reasoning. Götz Krummheuer suggests that argumentation can be thought of as a social process in which the cooperating individuals “adjust their intentions and interpretations by verbally presenting the rationales for their actions” (Cobb and Bauersfeld 1995, p. 13). In mathematics, unlike any empirically based discipline, the validity of an argument in its final form is judged solely on whether it is logically consistent.
Characteristics of Argumentation
The origins of logic, a key component of mathematical argumentation, can be traced back to Aristotelian logic and his use of syllogisms, with thinkers making improvements to this method over time as they were confronted with paradoxes. Argumentation was primarily the domain of theologians and medieval and postmedieval scholastics for over 1700 years after Aristotle. Some well-known examples of theological argumentation are the Italian prelate St. Anselm of Canterbury’s (1033–1109) “ontological argument” in the Proslogion, which was later revised by Leibniz and Gödel. Today, sophisticated versions of the ontological argument are written in terms of modal logic, a branch of logic which was familiar to the medieval scholastics. Modal logic today is a useful language for proof theory, the study of what can and cannot be proved in mathematical systems of deduction. Issues of completeness of mathematical systems, the independence of axioms from other axioms, and the consistency of formal mathematical systems are all part of proof theory. One also finds the use of logical argumentation to prove the existence of God in the theological works of Descartes, Leibniz, and Pascal.
The importance of the role of formal logic in mathematical argumentation continued to increase and reached its apex with the work of David Hilbert and other formalists in the nineteenth and first half of the twentieth century. The Principia Mathematica, by Alfred North Whitehead and Bertrand Russell, was a three-volume work that attempted to put the foundations of mathematics on a solid logical basis (Whitehead and Russell 1927). However, this program came to a definitive end with the publication of Gödel’s incompleteness theorems in 1931, which subsequently opened the door for more complex views of mathematical argumentation to develop.
Given this historical preview of the development of logic and its role in mathematical argumentation, we now turn our attention to contemporary views of mathematical argumentation, and in particular its constituent elements. Efraim Fischbein claimed that intuition is an essential component of all levels of an argument, with qualitative differences in the role of intuition between novices [students] and experts [mathematicians]. For novices, it exists as a primary component of the argument. Fischbein (1980) referred to this use of intuition as anticipatory, i.e., “…while trying to solve a problem one suddenly has the feeling that one has grasped the solution even before one can offer an explicit, complete justification for that solution” (p. 10). For example, in response to why a given solution to a problem is correct, the novice may respond “just because … it has to be.” The person using this type of intuition accepts the given solution as the truth and believes nothing more needs to be said. In a more advanced argument, intuition plays the role of an “advanced organizer” and is only the beginning of an individual’s argument. In this sense, a personal belief about the truth of an idea is formed and acts as a guide for more formal analytic methods of establishing truth. For example, a student may “see” that the result of a theorem is obvious, but realize that deduction is needed to establish truth publicly. Thus intuition serves to convince oneself about the truth of an idea while serving to organize the direction of more formal methods.
In an attempt to determine how mathematicians establish the truth of a statement in mathematics, Kline (1976) found that a group of mathematicians said they began with an informal trial and error approach guided by intuition. It is this process which helped these mathematicians convince themselves of the truth of a mathematical idea. After the initial conviction, formal methods were pursued. “The logical approach to any branch of mathematics is usually a sophisticated, artificial reconstruction of discoveries that are refashioned many times and then forced into a deductive system.” (p. 451). There definitely exists a distinction between how mathematicians convince themselves and how they convince others of the truth of mathematical ideas. Another good exposition of what constitutes argumentation in mathematics is found in Imre Lakatos’ (1976) Proofs and Refutations, in the form of a thought experiment. The essence of Lakatos’ method lies in paying attention to the casting out of mathematical pathologies in the pursuit of truth. Typically one starts with a rule and clearly identifies the hypothesis. This is followed by an exploration of the possibility of its truth or falsity. The process of conjecture-proof-refutation results in the refinement of the hypothesis in the pursuit of truth in addition to the pursuit of all tangential hypotheses that arise during the course of discourse. The Lakatosian exposition of mathematical argumentation brings into focus the issue of fallibility of a proof, either due to human error or inconsistencies in an axiomatic system. However, there are self-correcting mechanisms in mathematics, i.e., proofs get fixed or made more rigorous and axiomatic systems get refined to resolve inconsistencies. For example, non-Euclidean geometries arose through work that resolved the question of whether the parallel postulate is logically independent of the other axioms of Euclidean geometry; category theory is a refinement of set theory that resolves set theoretic paradoxes; and the axioms of nonstandard analysis are a reorganization of analysis that eliminates the use of the law of the excluded middle.
However, the mathematical community has on numerous occasions placed epistemic value on results before they were logically consistent with other related results that lend credence to its logical value. For instance, many of Euler and Ramanujan’s results derived through their phenomenal intuition and self-devised methods of argumentation (and proof) were accepted as true in an epistemic sense but only proved much later by mathematicians using a more rigorous form of mathematical argumentation to meet contemporary standards of proof. If one considers Weyl’s mathematical formulation of the general theory of relativity by using the parallel displacement of vectors to derive the Riemann tensor, one observes the interplay between the intuitive and the deductive (the constructed object). The continued evolution of the notion of tensors in physics/Riemannian geometry can be viewed as a culmination or a result of the flaws discovered in Euclidean geometry. Although the sheer beauty of the general theory of relativity was tarnished by the numerous refutations that arose when it was proposed, one cannot deny the present day value of the mathematics resulting from the interplay of the intuitive and the logical. Many of Euler’s results on infinite series have been proven correct according to modern standards of rigor. Yet, they were already established as valid results in Euler’s work. This suggests that mathematical argumentation can be thought of as successive levels of formalizations as embodied in Lakatos’ thought experiment. Such a view has been expressed in the writings of prominent mathematicians in Hersh’s (2006) 18 Unconventional Essays on Mathematics.
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