WHAT IS A PROOF? A LINGUISTIC ANSWER TO AN EDUCATIONAL QUESTION

Keith Weber Rutgers University

Proof is a central concept in mathematics education, yet mathematics educators have failed to reach a consensus on how proof should be conceptualized. I advocate defining proof as a clustered concept, in the sense of Lakoff (1987). I contend that this offers a better account of mathematicians practice with respect to proof than previous accounts that attempted to define a proof as an argument possessing an essential property, such as being convincing or deductive. I also argue that it leads to useful pedagogical consequences.

Key words: Cluster model; linguistics; proof

It is widely accepted that having students successfully engage in the activity of proving is a central goal of mathematics education (NCTM, 2000; Harel & Sowder, 1998). Yet mathematics educators cannot agree on a shared definition of proof (Balecheff, 2002; Reid & Knipping, 2010). This is recognized as problematic: without a shared definition, it is difficult for mathematics educators to meaningfully build upon each anothers research and it is impossible to judge if pedagogical goals related to proof are achieved (e.g., Balacheff, 2002; Weber, 2009). Until now, most mathematics educators have sought to define proof as an argument possessing one or more desirable properties (e.g., being deductive or being convincing) without a consensus on which property or properties are the essence of proof. The main thesis of this paper is that viewing proof as a clustered model in the sense of Lakoff (1987) offers a better account of how proof is practiced by mathematicians.

Previous attempts to characterize proof Through most of the 20th century, the philosophy of mathematics was dominated by the study of logic and the foundations of mathematics. Proof was defined as a formal object: a proof was situated in a formal language with explicit rules for well-formed formulae and logical inference; a sequence of well-formed formulae was a proof if each formula was a premise or the result of applying a rule of inference to previous formulae. While this characterization has the virtue of being an objective method for identifying a proof, it had the drawback of bearing little relevance to mathematical practice. It is rare for a mathematician to produce a proof that satisfies these criteria. More recently, philosophers and mathematics educators have sought to characterize the proofs that mathematicians actually produce. In this line of research, it is typical for a researcher to propose a property (or occasionally multiple properties) that arguments must satisfy to be a proof. Inevitably, another researcher will challenge this definition by producing a counterexample that either satisfies these properties but is not a proof or is a proof but fails to satisfy these properties. For instance, Hanna (1995) argued that computer-assisted proofs (e.g., Appel and Hakens proof of the four-color theorem) challenge the notion that a proof is open to public inspection. Similarly, Larvor (2012) dismissed the notion that proofs being non-ampliative (i.e., not supplying conclusions that exceed the premises or being deductive) as he claimed that some authoritative arguments are non-ampliative but not proofs. Harel and Sowder (1998) defined a mathematical proof as an argument that convinces a mathematician, but others have argued there are convincing arguments that do not constitute proofs (e.g., Tall, 1989). (For instance, Eccheveria (1996) said the mathematical community is convinced that the unproven Goldbachs Conjecture is true based on empirical evidence).

I believe that the reason that mathematics educators have collectively failed at the problem of defining proof is due to a faulty assumption of what a solution to this problem must look like. The search is for a set of properties that will distinguish whether an argument is a proof. It is natural to attempt to define proof this way. This is, after all, how mathematical categories are defined (Alcock & Simpson, 2002). However, I claim that a search for this type of definition is doomed for failure. Aberdein (2009) coined the term, proof*, as species of alleged proof where there is no consensus that the method provides proof, or there is a broad consensus that it doesnt, but a vocal minority or an historical precedent point the other way. As examples of proof*, Aberdein included picture proofs*, probabilistic proofs*, computer-assisted proofs*, [and] textbook proofs* which are didactically useful but would not satisfy an expert practitioner. This presents a paradox for creating a definition that lists properties that distinguish proofs from non-proofs. Take picture proofs*, for instance. Such a definition would either permit some convincing picture proofs* to count as proofs or it would say no picture proofs* were proofs. If the former occurred, one could rebut the definition by citing the large number of mathematicians who said these picture proofs* were not real proofs. If the latter occurred, one could rebut the definition by citing the large number of mathematicians (or at least the vocal minority) who disagree. Similar arguments could be made for all types of proofs*. Of course, the proponent of the definition could resort to saying that the mathematicians who disagree with his or her perspective are mistaken (e.g., computer-assisted proofs* are proofs but probabilistic proofs* and picture proofs* are not and any mathematician who disagrees is in error). However, aside from making the dubious argument that a considerable number of mathematicians do not really understand their craft, this would challenge the claim that his or her definition was descriptive of mathematical practice.

Lakoffs clustered concepts Lakoff (1987) noted that according to classical theory, categories are uniform in the following respect: they are defined by a collection of properties that the category members share (p. 17). As noted above, this assumption underlies the ways in which most philosophers and mathematics educators seek to define proof. However, Lakoffs thesis is that most real-world categories cannot be characterized this way. In particular, he argued that some categories might be better thought of as cluster models, which he defined as occurring when a number of cognitive models combine to form a complex cluster that is psychologically more basic than the models taken individually (p. 74). As an illustrative example, Lakoff considered the category of mother. To Lakoff, there are several types of mothers, including the birth mother, the genetic mother, the nurturance mother (i.e., the adult female caretaker of the child), and the marital mother (i.e., the wife of the father). In the prototypical case, these concepts will converge-- that is, the birth mother will also be the genetic mother, the nurturance mother, and so on. And indeed, when one hears that the woman is the mother of a child, the default assumption is that she assumes all of these roles. Lakoff raised two other points that will be relevant to this paper. First, there is a natural desire to pick out the real definition of mother, or the true essence of mother. However, this does not seem possible. Different dictionaries list different conceptions of mother as their primary definition. Further, sentences such as, I was adopted so I dont know who my real mother is and I am uncaring so I doubt I could be a real mother to my child both are intrinsically meaningful yet define real mother in different ways. Second, in cases where there is divergence in the clustered concept of mother (e.g., a genetic but not adoptive mother), compound words exist to qualify the use of mother. Calling one a birth mother

typically indicates that she in not the nurturance mother; calling one an adoptive mother or a stepmother indicates that she is not the birth mother.

Proof as a clustered concept The main thesis in this paper is that proof can be productively characterized as a cluster model. Exactly what cognitive models compose this cluster can be the subject of interesting debate. What I present is a first attempt at categorizing proofs:

A proof is an argument that is deductive and non-ampliative argument: Each statement in a proof should be a premise or a necessary logical consequence of previous statements.

A proof is an argument that would convince a contemporary mathematician who knew the subject (adapted from Davis & Hersh, 1981)

A proof is an argument in a natural language and symbolic representation system where there are socially sanctioned rules of inference. Its noteworthy that some researchers view a proof as an argument that can be translated into a formal proof (e.g., Azzouni, 2004; Mac Lane, 1986). If so, the distance between the language of formal proofs and the proofs that are actually produced should not be too great.

A proof is an argument that convinces a particular community at a particular time (adapted from Balacheff, 1987). This definition emphasizes the social role of proof and situates proof as dependent on time and culture.

A proof is an argument that is a blueprint that knowledgeable mathematicians can use, in principle, to write a complete proof with no logical gaps.

My claim is that an argument that satisfies all these properties would be judged by (nearly) all knowledgeable mathematicians as constituting a proof and an argument that satisfied none of these conditions would be rejected by (nearly) all mathematicians as a non-proof. Arguments that satisfied some, but not all of these conditions, such as Aberdeins (2009) proofs*, would be regarded as controversial. Further, it would be desirable, if possible, to improve these arguments so they satisfied the properties that they lacked. The usual stratagem of rejecting a conception of proof-- citing a counterexample where the conception did not reflect reality-- would not apply here. In this characterization, there is no guarantee that every proof satisfies every property. However, this conception of proof does make some concrete predictions about proof that can be tested. First, arguments satisfying all the criteria above would be considered more proof-like or representative of proof than those that would not. A proof in knot theory that relied on diagrams or kinesthetic motion (see Larvor, 2012 or Rav, 1999) might be accepted as a proof, but the mathematical community at large would consider these proofs unusual in this respect. Second, if a mathematician was told a proof of a particular theorem existed, his or her default assumption would be that the argument satisfied each of the constraints, even though he or she would be aware that this might not necessarily be so. Third, there should be compound words that qualify proof-like arguments that satisfy some, but not all, of these criteria. Indeed, Aberdein (2009) provided a list including picture proofs* (convincing deductive arguments in a non-symbolic/natural language representation system) and probabilistic proofs* (arguments that are arguably convincing but not deductive). Fourth, if mathematicians were asked what the true essence of a proof was, they should not all list one of the criteria above. Rather, their responses should be heterogeneous (i.e., the essence of proof varies by mathematician) or many should say the question is unanswerable. That Aberdein (2009) alleged that his proofs* have both their adherents and detractors is evidence toward this point.

Benefits of this conception of proof

Previous attempts to define proof Previous attempts to define proof in mathematics education have usually latched on to one or two of the properties above and treated it as the sole criterion on which to judge proofs. Harel and Sowder (1998) defined mathematical proof in terms of arguments that convince mathematicians, Hoyles and Kucheman (2002) treated proofs as arguments that are deductive, Weber and Alcock (2009) defined proof as arguments within a representation system, and Balacheff (1987) defined a proof as time- and community-dependent. Stylianides (2007) desired arguments to be both deductive and in an age-appropriate representation system to constitute a proof. Each definition essentially says the true essence of proof lies in one or two of the attributes described above, treating the other listed criteria as either tangential or a consequence of a proofs essence. For instance, Hanna (1991) referred to the formality of proof, or placing the argument in a representation system, as merely a hygiene factor and Harel and Sowder (1998) believed the fact that mathematical proofs are deductive is a necessary consequence of these proofs convincing mathematicians. As noted earlier, such characterizations cannot account for mathematicians practice with respect to proofs* and thus seem not to categorize mathematical practice.

More appropriate pedagogical suggestions At a broad level, the components of the clustered model of proof are correlated with one another. For instance, in general, as an argument becomes more deductive, it tends to become more convincing, easier to translate into a formal proof, and more likely to be sanctioned by ones peers. Hence, encouraging students to make their arguments more deductive would usually make their arguments more proof-like in other respects as well. However, this is not the case if we take some of these criteria to extremes. For a first example, suppose we strive to present students with arguments that are as convincing as possible in geometry. In many cases, an exploration on a dynamic geometry package would be extremely convincing, both for mathematicians and for students (de Villiers, 2004). For a student, such explorations would probably be more convincing than a complicated deductive argument because the student may worry that he or she has overlooked an error in the argument. If we view the mode of reasoning (deductive vs. perceptual) and the representation system in which an argument is couched as irrelevant, it is difficult to argue why demonstrations on dynamic geometry software packages are not proofs. A similar claim relates to how formal an argument is. Increasing the formality of an argument usually makes the argument more deductive and more acceptable to the mathematical community. However, it is generally accepted that there is a point where an argument is formal enough and making it more rigorous would be detrimental. An argument with large logical gaps might be judged as not convincing and not accepted by mathematicians. However, filling in all the gaps would make the proof impossibly long and unwieldy. The result would be a proof that masks its main ideas. As understanding these ideas is important for determining the validity of the proof, increasing the rigor of the proof would lessen its persuasive power. If we want students and teachers to present proofs that satisfy all or most of the criteria above, it would be best not to focus on a single criterion. Not only would the other criteria be ignored, a singular focus on one criterion might actually lessen the possibilities of the other criteria being achieved.

References

Aberdein, A. (2009). Mathematics and argumentation. Foundations of Science, 14, 1-8. Alcock, L. & Simpson, A. (2002). Definitions: Dealing with categories mathematically. For

the Learning of Mathematics, 22(2), 28-34. Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia

Mathematica, 12, 81-106. Balacheff, N. (1987). Processus de preuves et situations de validation. Educational Studies in

Mathematics, 18(2), 147-176. Balacheff, N. (2002). The researcher epistemology: A deadlock from educational research on

proof. In F.L. Lin (ed.) 2002 International Conference on Mathematics Education- Understanding proving and proving to understand. NCS and NSUT: Taipei.

Davis, P. J. and Hersh, R. (1981). The mathematical experience. New York: Viking Penguin Inc.

de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24. de Villiers, M. (2004). The role and function of quasi-empirical methods in mathematics.

Canadian Journal of Science, Mathematics, and Technology Education, 4, 397-418. Echeverria, J. (1996). Empirical methods in mathematics. A case study: Goldbachs

Conjecture. In G. Munevar (ed) Spanish studies in the philosophy of science,. (pp. 19-56). Dordrecht: Kluwer

Hanna, G. (1991). Mathematical proof. In D. Tall (ed.) Advanced Mathematical Thinking. Dordrecht: Kluwer.

Hanna, G. (1995). Challenges to the Importance of Proof. For the Learning of Mathematics, 15(3), 42-50.

Harel, G. & Sowder, L. (1998). Students proof schemes. Research in Collegiate Mathematics Education, 3, 234-282.

Hoyles, C. & Kuchemann, D. (2002). Students understanding of mathematical implication. Educational Studies in Mathematics, 51, 193-223.

Lakoff, G. (1987). Women, fire, and dangerous things. Cambridge University Press: Cambridge.

Larvor, B. (2012). How to think about informal proofs. Synthese, 187, 715-730. Mac Lane, S. (1986). Mathematics: Form and function. Springer: Berlin. National Council of Teachers of Mathematics (2000). Principles and Standards for School

Mathematics. Reston, VA: NCTM. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7, 5-41. Reid, D. & Knipping, C. (2010). Proof in mathematics education. Sense Publishers:

Rotterdam. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in

Mathematics Education, 38, 289-321. Tall, D. (1989). The nature of mathematical proof. Mathematics Teaching, 127, 28-32. Weber, K. (2009). Review of Boeros Theorems in schools. Mathematical Thinking and

Learning, 11, 289-294. Weber, K. & Alcock, L. J. (2009). Proof in advanced mathematics classes: Semantic and

syntactic reasoning in the representation system of proof. In D.A. Stylianou, M.L. Blanton & E. Knuth (Eds.), Teaching and Learning Proof Across the Grades: A K-16 Perspective (pp. 323-338). New York, NY: Routledge.