Dialogue Types, Argumentation Schemes, and Mathematical Practice: Douglas Walton and Mathematics
aa r X i v : . [ m a t h . HO ] N ov Dialogue Types, Argumentation Schemes,and Mathematical Practice:Douglas Walton and Mathematics
Andrew Aberdein
School of Arts & Communication, Florida Institute of Technology, Melbourne FL [email protected]
Abstract
Douglas Walton’s multitudinous contributions to the study of argumen-tation seldom, if ever, directly engage with argumentation in mathematics.Nonetheless, several of the innovations with which he is most closely associatedlend themselves to improving our understanding of mathematical arguments. Iconcentrate on two such innovations: dialogue types (§1) and argumentationschemes (§2). I argue that both devices are much more applicable to mathe-matical reasoning than may be commonly supposed.
Several decades ago, Douglas Walton proposed a classification of dialogue types:different contexts in which argumentation may arise [67, 68]. His elegant presen-tation of the key differences between the most central types (from joint work withErik C. W. Krabbe) is summarized in Table 1. Dialogue types are distinguished bytwo main factors: the initial situation or circumstances in which the interlocutorsfind themselves and their main goal in pursuing a dialogue. Some situations admitmore goals than others: if the situation is strongly adversarial, the disputants maybe seeking a full determination of the matter at hand, requiring one to persuade theother; or they may need to decide on a course of action and negotiate a practicalconsensus; or they may have little intent beyond airing their respective positions,however quarrelsome or eristic the exchange. Whereas, if the interlocutors are ad-dressing an open problem where neither has any prior commitments, the last of thesegoals would be incoherent, but the disputants may still inquire into the problem or deliberate on how best to act. And if the interaction arises simply because one partyhas knowledge the other lacks, only the first sort of outcome makes any sense: an
Vol. \ jvolume No. \ jnumber \ jyearJournal of Applied Logics — IfCoLog Journal of Logics and their Applications berdein Main Goal Initial Situation
Conflict Open problem Unsatisfactoryspread ofinformationStable agreement/resolution
Persuasion Inquiry InformationSeeking
Practical settlement/decision (not) to act
Negotiation Deliberation
N/AReaching a (provisional)accommodation
Eristic
N/A N/A
Table 1: Walton and Krabbe’s systematic survey of dialogue types [73, p. 80] information-seeking dialogue. Thus we arrive at six principal dialogue types. How-ever, some of these types may be further subdivided or combined [72, p. 31], andthe classification is not intended to be exhaustive.Walton itemizes the goals of the interlocutors, individual or collective, and thepotential benefits that may accrue from dialogues of each of the main types inTable 2 (taken from [71, p. 605]; see also [68, p. 413], [73, p. 66]). Different patternsof argument may be appropriate in different dialogue types: what is reasonable ina negotiation would be improper in a persuasion dialogue; almost anything goes ina quarrel but well-conducted inquiries require respect for procedure, and so forth.Another important feature of the picture that Walton and Krabbe present is the dialectical shift : in the course of a dialogue its type may change [73, pp. 100 ff.].This can be a positive development—as a conversation unfolds, its participants canproductively shift their attention to different ends. But dialectical shifts can alsobe troublesome, especially if they go unnoticed by one or more of the participants,leading to the use of argumentative tactics that are now contextually inappropriate.Walton does not discuss mathematical dialogues but, in other work, his collab-orator Krabbe observes that proofs may occur in several different contexts:1. thinking up a proof to convince oneself of the truth of some theorem;2. thinking up a proof in dialogue with other people (inquiry dia-logue. . . );3. presenting a proof to one’s fellow discussants in an inquiry dialogue(persuasion dialogue embedded in inquiry dialogue. . . );4. presenting a proof to other mathematicians, e.g. by publishing it2 alton & Mathematics
Type ofDialogue InitialSituation IndividualGoals ofParticipants CollectiveGoal ofDialogue Benefits
Persuasion Difference ofopinion Persuade otherparty Resolve differ-ence of opinion UnderstandpositionsInquiry Ignorance Contributefindings Prove ordisproveconjecture ObtainknowledgeDeliberation Contemplationof futureconsequences Promotepersonal goals Act on athoughtfulbasis Formulate per-sonal prioritiesNegotiation Conflict ofinterest Maximizegains(self-interest) Settlement(withoutundueinequity) HarmonyInformation-Seeking One partylacksinformation Obtaininformation Transfer ofknowledge Help in goalactivityQuarrel(Eristic) Personalconflict Verbally hitout at andhumiliateopponent Reveal deeperconflict Vent emotionsDebate Adversarial Persuade thirdparty Air strongestarguments forboth sides SpreadinformationPedagogical Ignorance ofone party Teaching andlearning Transfer ofknowledge Reservetransfer
Table 2: Walton’s types of dialogue [71, p. 605]in a journal (persuasion dialogue. . . );5. presenting a proof when teaching (information-seeking and persua-sion dialogue) [44, p. 457].The primary, if not exclusive, concern of Krabbe’s account (and of my own earlierapplication of dialogue types to mathematics [2]) is with proof. This may reflect whathas been called in another context “proof chauvinism”—a tendency in philosophers3 berdein of mathematics to privilege proof over other aspects of mathematical practice [19].Nonetheless, proof is the aspect of mathematical practice where the applicability ofinformal logic is most unexpected. Hence I shall again begin with proofs.Are proofs always dialogues or can they be monologues? The conception of themathematician as isolated genius has a firm grip on the popular imagination [37].It is true that mathematicians coauthor papers less than most other scientists, andthere are some celebrated examples of solitary endeavour, such as Srinavasa Ra-manujan labouring in obscurity or Andrew Wiles’s years of solo work prior to hissurprise announcement of a proof of Fermat’s Last Theorem. Nonetheless, this im-pression is incomplete at best: Ramanujan only began to fulfil his potential aftertravelling to Cambridge to collaborate with Hardy and Littlewood [42]; Wiles dis-covered gaps in his solo work which were eventually bridged by a collaboration withRichard Taylor [53]. On the other hand, as Paul Ernest suggests, there are manyways in which mathematics is underpinned by “symbolically mediated exchangesbetween persons”—conversations or dialogues:First, the ancient origins as well as various modern systems of proofuse dialectical or dialogical reasoning, involving the persuasion of oth-ers [see also [23].] . . . Second, mathematics is a symbolic activity usingwritten inscriptions and language; it inevitably addresses a reader, realor imagined, so mathematical knowledge representations are conversa-tional. Third, many mathematical concepts [such as epsilon-delta defi-nitions of limit in analysis and hypothesis testing in statistics] have aninternal conversational structure. Fourth, the epistemological founda-tions of mathematical knowledge, including the nature and mechanismsof mathematical knowledge genesis and warranting, utilise the deploy-ment of conversation in an explicitly and constitutively dialectical way.Fifth, . . . mathematical facts stand on the basis of collective agreementand are part of institutional reality . . . built on interpersonal commu-nicative interactions, that is, through conversation [25, p. 74].Once we agree that mathematical proof is dialogical, we may ask in what dialoguetype it characteristically arises. As Krabbe indicates, the proving process, at leastat its inception, might best be thought of as an inquiry dialogue: a collaborativeexchange between mathematicians with the shared goal of settling an open question,which neither of them has prejudged. Certainly such exchanges can be found inmathematics, at least in the context of discovery of mathematical results (see, forexample, [66]). However, there is also an unavoidable element of adversariality inthe epistemology of mathematical proof: mathematicians only trust proofs that havegained wide assent from the mathematical audience [14]; the value of that assent4 alton & Mathematics lies in the assumption that the proofs have been sufficiently challenged. CatarinaDutilh Novaes has sought to capture this idea in terms of prover/sceptic dialogues[21, 22, 23]. Prover and sceptic are (idealizations of) two complementary roles inthe process that leads to the eventual acceptance of a proof by the mathematicalpublic: the prover presents a putative proof; the sceptic responds with searchingbut fair questions; through their successive exchanges the proof is improved wherenecessary and eventually comes to be generally accepted (or is exposed as unsound).Prover/sceptic dialogues are persuasion dialogues because the parties start from adifference of opinion, as required by their contrasting roles.Journal referees can play the sceptic role, at least if they are sufficiently thoroughin their scrutiny [10]. But so can collaborators, at an earlier stage in the develop-ment of a proof, or other mathematicians, at a later stage, who expand or refine thepublished proof in their own work. Imre Lakatos’s celebrated imaginative recon-struction of the development of a proof of the Descartes–Euler conjecture (linkingthe numbers of vertices, edges, and faces of polyhedra) takes the form of a dialoguebetween characters loosely representing various nineteenth-century mathematicians[47]. Lakatos identifies a range of dialectical manoeuvres whereby mathematicianseither present apparent counterexamples of various kinds to a working conjectureor respond to such apparent counterexamples. (Alison Pease and colleagues haveshown how these Lakatosian manoeuvres can be captured in terms of dialogue games[55].)But perhaps the dialogue type in which proofs are most frequently presentedis neither inquiry nor persuasion, but pedagogical information-seeking. Proofs arepresented in countless classrooms at school and university level and even researchmathematicians will attend essentially didactic presentations of novel but settled re-sults. Such exchanges are best understood as information-seeking dialogues. Hence,as Krabbe notes, the development of a proof may be seen as a sequence of dialecticalshifts, from an initial inquiry phase, to a more verification-focussed persuasion di-alogue, and eventually, if the proof survives these earlier stages, to a disseminationphase characterized by information-seeking dialogues. Of course, the progress ofmost significant proofs is seldom this smooth, so the dialectical shifts are likely tobe more numerous, as failed attempts at verification send mathematicians back tomore open-ended inquiry, or at least open up subsidiary discussions of how localizedproblems may be addressed. Michael Barany and Donald MacKenzie, in an ethno-graphic treatment of mathematical research, describe how some of these processescan work:When a suitable partial result is obtained and researchers are confident inthe theoretical soundness of their work, they transition to “writing up”.5 berdein
Only then do most of the formalisms associated with official mathemat-ics emerge, often with frustrating difficulty. Every researcher interviewedhad stories about conclusions that either had come apart in the attemptto formalize them or had been found in error even after the paper hadbeen drafted, submitted, or accepted. Most saw writing-up as a pro-cess of verification as much as of presentation, even though they viewedthe mathematical effort of writing-up as predominantly “technical”, andthus implicitly not an obstacle to the result’s ultimate correctness orinsightfulness [15, pp. 111 f.].Although inquiry, persuasion, and information-seeking dialogues are perhaps thedialogue types most hospitable to proof, they do not exhaust the range of dialoguetypes in which mathematical argumentation may occur. By analogy with the deviceused to indicate problematic sporting records, I have elsewhere used “proof*” to re-fer to “species of alleged ‘proof’, where there is either no consensus that the methodprovides proof, or there is broad consensus that it doesn’t, but a vocal minority oran historical precedent which points the other way” [3, p. 2]. Amongst the proofs*I included “proofs* predating modern standards of rigour, picture proofs*, proba-bilistic proofs*, computer-assisted proofs*, textbook proofs* which are didacticallyuseful but would not satisfy an expert practitioner, and proofs* from neighbouringdisciplines with different standards”. Each of these cases can be seen two ways: ei-ther as a (perhaps very) disputed form of mathematical proof or as an undisputedform of mathematical reasoning that ought to be characterized as something otherthan proof. Hence, if our goal is to repudiate proof chauvinism and characterizemathematical reasoning in general, then we must pay attention to proofs*.Table 3, adapted from [2, p. 148], summarizes the principal mathematical di-alogue types discussed so far: inquiry, persuasion, and pedagogical information-seeking. It also lists three dialogue types in which proof* is likely to be more at homethan proof: deliberation, negotiation, and a non-pedagogical form of information-seeking. Deliberation and negotiation abandon the goal of stable resolution thatwe would normally expect of proof whereas oracular information-seeking pursuesthat goal in an unconventional manner. In one of his foundational papers on com-putability, Alan Turing briefly considers the case of a machine “supplied with someunspecified means of solving number-theoretic problems; a kind of oracle as it were”[64, p. 172]. Subsequent authors expanded this remark into a theory of relative com-putability [60]. There is nothing necessarily supernatural about an oracle machine:a laptop with access to an online database would meet the broad definition (if weignore Turing’s statement that the oracle “cannot be a machine” [64, p. 173]). How-ever, an oracle is by definition a “black box”: its inner workings are inscrutable to6 alton & Mathematics
DialogueType InitialSituation Main Goal Goal ofProponent Goal ofRespondent
Inquiry Open-mindedness Prove ordisproveconjecture Contribute tomain goal ObtainknowledgePersuasion Difference ofopinion Resolve differ-ence of opinionwith rigour Persuaderespondent PersuadeproponentPedagogicalInformation-Seeking Respondentlacksinformation Transfer ofknowledge Disseminateknowledge ofresults andmethods ObtainknowledgeOracularInformation-Seeking Proponentlacksinformation Transfer ofknowledge Obtaininformation InscrutableDeliberation Open-mindedness Reach aprovisionalconclusion Contribute tomain goal ObtainwarrantedbeliefNegotiation Difference ofopinion Exchangeresources for aprovisionalconclusion Contribute tomain goal Maximizevalue ofexchange
Table 3: Some mathematical dialogue typesthe local machine; in principle, they could be inscrutable to any analysis. Scepticsof the proof status of unsurveyably large computer-assisted proofs, such as ThomasTymoczko, have suggested that the appeal such proofs make to a computer shouldbe seen in similar terms [65]. Analogously, Yehuda Rav proposes as a thought exper-iment a computer that could answer any mathematical question with certainty butwithout proof. For Rav, such a machine would be “a death blow to mathematics, forwe would cease having ideas and candidates for conjectures” [57, p. 6]. Tymoczkoand Rav are both concerned about fallacious appeal to authority in mathematicalproof, an issue I will return to below.The combinatorialist Edward Swart was also concerned with “lengthy proofs(whether achieved by hand or on a computer)”. He coined the term “agnograms”7 berdein to refer to the resulting “theoremlike statements” since we are, at least for the im-mediate future, required to be agnostic about their truth value, as they “have beenneither adequately formalized nor adequately surveyed and are suggestive ratherthan definitive”, due to the limitations of our available resources [62, p. 705]. Estab-lishing an agnogram is thus more of a practical settlement than a stable resolution,suggesting that the dialogue in which it results may better be seen as delibera-tion or even negotiation, rather than inquiry or persuasion. Likewise, in a widelydiscussed polemical proposal, Arthur Jaffe and Frank Quinn sought a clear demar-cation between “speculative and intuitive work” in “theoretical mathematics” and a“proof-oriented phase” of “rigorous mathematics” [41, p. 2]. Of course, speculativeand intuitive work is characteristic of the earlier, inquiry phase of a proof dialogue.However, Jaffe and Quinn anticipate an outlet for responsibly labelled speculation;since this is provisional in character, the process by which it is derived might be seenas deliberation. Something similar might also be said about conjectures, particularlythe wide-ranging, fruitful conjectures that comprise the framework of mathemati-cal research programmes, sometimes called “architectural conjectures” [49, p. 198].Even more speculative is the suggestion of Doron Zeilberger that in the not so distantfuture “semi-rigorous mathematics” may essentially assign price tickets to proofs,indicating the quantity of computational resources needed for certainty, thereby sit-uating mathematical proof within a negotiation dialogue [78]. This proposal hasnot generally been well received [12]. Nonetheless, in a weaker form it reflects atruism: even the purest of mathematicians cannot ignore issues of funding, even ifthe link to their work is not as intimate as Zeilberger suggests. Lastly, even eristicdialogues have had some role to play in mathematical reasoning, as witnessed bysuch celebrated quarrels as that between the early modern mathematicians GirolamoCardano and Niccolò Tartaglia [59]. The salient detail is not the asperity of theirexchange, which ultimately turned on an accusation of theft of intellectual prop-erty, but the adversarial strategy mathematicians of that era adopted to convincethe mathematical public of their successes. Rival mathematicians would keep theirmethods (in this case of solving cubic equations) secret but challenge each otherto public contests, each solving problems set by the other until the winner posed aproblem the loser could not solve.
An argumentation scheme is a stereotypical pattern of reasoning. In recent decades,the study and classification of argumentation schemes has been the most influentialaspect of Douglas Walton’s work [70, 75]. Although antecedents of the argumenta-8 alton & Mathematics tion scheme can be traced back millennia to the tradition of loci or topoi, Walton’swork set it on a new foundation of rigour and clarity. Building on that foundation,Hans Hansen has proposed the following definition of argumentation scheme: “(i) apattern of argument, (ii) made of a sequence of sentential forms with variables, ofwhich (iii) at least one of the sentential forms contains a use of a schematic constantor a use of a schematic quantifier, and (iv) the last sentential form is introduced bya conclusion indicator like ‘so’ or ‘therefore’ ” [32, p. 349]. Schemes also generallyinclude ‘critical questions’, which itemize possible lines of response. The criticalquestions are key to the evaluation of defeasible schemes: whether the argumentshould be judged to have succeeded or whether it has been defeated will turn onwhether the questions can receive a satisfactory answer.Walton argues that in principle all defeasible argumentation schemes could beunderstood as special cases of a defeasible version of modus ponens [75, p. 366]:
Argumentation Scheme 1 (Defeasible Modus Ponens)
Data: P . Warrant:
As a rule, if P , then Q .Therefore, . . . Qualifier: presumably, . . .
Claim: . . . Q . Critical Questions:1. Backing:
What reason is there to accept that, as a rule, if P , then Q ?2. Rebuttal:
Is the present case an exception to the rule that if P , then Q ?I have reconstructed Walton’s scheme for defeasible modus ponens so as to bringout its resemblance to another very general model of defeasible reasoning, the Toul-min layout [7, p. 829]. (On the relationship of schemes to layouts, see also [54,pp. 22 ff.]; for a contrasting view, see [38].) This is not an accidental choice: Toul-min layouts have lately found widespread employment in the analysis of mathemat-ical argumentation (for recent surveys, see [43, 46]). Although Walton emphasizeddefeasible schemes, deductive rules of inference can also be seen as argumentationschemes: the schemes framework is “illatively neutral” [32, p. 355]. This means thatargumentation schemes can provide a unified treatment of a wide range of argumentsemployed in mathematics. Indeed, a number of authors have applied argumentationschemes to mathematical reasoning [1, 4, 5, 6, 7, 20, 50, 51, 54].The illative neutrality of the schemes framework licences scepticism about the“standard view” [13] of mathematical argumentation as purely comprised of deriva-tions, that is arguments in which every step instantiates a deductive inference rule.9 berdein To that end, I have elsewhere proposed a threefold distinction between A-, B-, andC-schemes:•
A-schemes correspond directly to derivation rules. (Equivalently,we could think in terms of a single A-scheme, the ‘pointing scheme’which picks out a derivation whose premisses and conclusion areformal counterparts of its data and claim.)•
B-schemes are exclusively mathematical arguments: high-level al-gorithms or macros. Their instantiations correspond to substruc-tures of derivations rather than individual derivations (and theymay appeal to additional formally verified propositions).•
C-schemes are even looser in their relationship to derivations, sincethe link between their data and claim need not be deductive. Spe-cific instantiations may still correspond to derivations, but therewill be no guarantee that this is so and no procedure that will al-ways yield the required structure even when it exists. Thus, wherethe qualifier of A- and B-schemes will always indicate deductivecertainty, the qualifiers of C-schemes may exhibit more diversity.Indeed, different instantiations of the same scheme may have differ-ent qualifiers ([7, p. 829]; cf. [6, pp. 366 f.]).So the widespread “standard” view of mathematical proof, that it is identical toderivation, could be expressed as denying C-schemes a place in proofs. I have arguedagainst that view [6, p. 375], but even if it were to be conceded, it would still leaveroom for C-schemes in other forms of mathematical reasoning.What sort of schemes might C-schemes be? Some of them may be unique tomathematics, but we should expect others to resemble schemes that have beenfound useful in addressing non-mathematical reasoning. Walton and his collabo-rators have made a number of attempts to classify such general purpose argumenta-tion schemes. Table 4 is based on a recent classification he developed with FabrizioMacagno. Walton and Macagno employ a series of binary distinctions: first betweensource-dependent arguments and source-independent arguments; then subdividingthe latter into practical reasoning and epistemic reasoning; which is in turn dividedinto discovery arguments and arguments applying rules to cases. Each of the fourresulting headings are then further subdivided into various thematic groups of in-dividual schemes. However, Walton and Macagno concede that this classification isincomplete, notably omitting some linguistic arguments [74, p. 24].I have annotated Table 4 with citations to works in which mathematical versionsof each scheme are discussed. As may be seen, mathematical arguments have been10 a l t o n & M a t h e m a t i c s Discovery arguments Applying rules to cases Practical reasoning Source-dependent arguments1. Arguments establishing rules• Argument from a randomsample to a population• Argument from best ex-planation2. Arguments finding entities• Argument from sign [20,50, 51]• Argument from ignorance[5] 1. Arguments based on cases• Argument from an estab-lished rule• Argument from verbalclassification [4, 50, 51]• Argument from cause toeffect2. Defeasible rule-based arguments• Argument from example[5, 7, 50, 51]• Argument from analogy[6, 51]• Argument from precedent[54]3. Chained arguments connectingrules and cases• Argument from gradual-ism [5]• Precedent slippery slopeargument• Sorites slippery slope ar-gument 1. Instrumental argument frompractical reasoning• Argument from action tomotive2. Argument from values• Argument from fairness3. Value-based argument frompractical reasoning(a) Argument from positiveor negative consequences[5, 50, 51]• Argument fromwaste• Argument fromthreat• Argument fromsunk costs 1. Arguments from position to know(a) Argument from expertopinion [1, 5, 50](b) Argument from position toknow [5]• Argument from witnesstestimony2. Ad hominem arguments(a) Direct ad hominem(b) Circumstantial ad hominem• Argument frominconsistentcommitment• Arguments attackingpersonal credibilityi. Arguments fromallegation of biasii. Poisoning thewell by alleginggroup bias3. Arguments from popular acceptance• Argument from popularopinion [4]• Argument from popularpractice [4]
Table 4: Walton & Macagno’s partial classification of schemes (adapted from [74, p. 22]), with applications tomathematical argumentation indicated. berdein identified under each of Walton and Macagno’s four main headings. In addition,mathematical applications have been found for several schemes that are missingfrom Table 4 but which are found in the more exhaustive (but less structured) listin [75]. These include linguistic arguments, such as arguments from arbitrarinessor vagueness of a verbal classification [54] and argument from definition to verbalclassification [5], but also source-dependent arguments, such as ethotic argument[5], practical reasoning arguments, such as argument from positive consequences [5],discovery arguments, such as abductive argument [54] and argument from evidenceto a hypothesis [5, 7, 54], and arguments applying rules to cases, such as argumentfrom an exceptional case [54]. Conversely, not all of the individual schemes in Table 4have yet been found useful in discussing mathematics. While some of these omissionsmay merely be oversights, others are to be expected. For example, causal reasoning,whether argument from cause to effect or the various kinds of slippery slope, isunlikely to be of direct application to mathematics, since mathematical objects aregenerally understood to be causally inert. In the remainder of this section, I willdiscuss the mathematical applications of a sample of schemes, chosen in part toremedy some of the omissions in Table 4. Walton and Macagno subdivide epistemic reasoning into two subcategories, discov-ery arguments and arguments applying rules to cases. Arguments of both kinds canbe readily found in mathematical reasoning. In particular, many discovery argu-ments are broadly abductive in character and abduction has been proposed as anaccount of mathematical reasoning in a wide range of situations, including class-room discussion [29, 52]; concept formation in mathematical practice [35]; and theselection and defence of axioms [36]. There are several abductive schemes in Wal-ton’s catalogue, including multiple subtypes of abductive argument [75, p. 329] andargument from evidence to (verification of) a hypothesis, which I have discussedelsewhere [5, 7, 54]. Another such scheme is argument from sign:
Argumentation Scheme 2 (Argument from Sign [75, p. 329])
Specific Premise: A (a finding) is true in this situation. General Premise: B is generally indicated as true when its sign, A , is true. Conclusion: B is true in this situation.Critical Questions:1. What is the strength of the correlation of the sign with the event signified?12 alton & Mathematics
2. Are there other events that would more reliably account for the sign?Argument from sign has been discussed since antiquity, particularly in the con-text of medical reasoning [9]. The explicit application of Scheme 2 to mathematicsis due to Ian Dove, who uses it to analyse a surprising but widely discussed classof proofs*: those employing molecular computation [20]. This consists in encodinga mathematical problem into strands of DNA which are then subject to standardlaboratory assays that determine the solution of the problem with high likelihood[8]. Hence the outcome of the assay is a sign of the mathematical problem having aspecific solution, and the mathematician infers the latter from the former in accor-dance with Scheme 2. This, and other less esoteric probabilistic methods, such as theMiller–Rabin primality test, are generally viewed by mathematicians as heuristicallyuseful but falling short of the standards of rigour required for proof. Nonetheless,the intellectual defensibility of this perspective has also been the subject of a debatein philosophy of mathematics [24, 26, 28]. Much of this debate could be understoodas offering competing answers to the critical questions for Scheme 2. Dove also sug-gests that what I have referred to above as oracular information seeking could beanalysed as employing the same scheme [20, p. 144].Argument from sign also illustrates the importance of the illative neutralityof argumentation schemes: not all its instances need be defeasible. We can finddeductive instances of Scheme 2. For example, much of twentieth and twenty-firstcentury mathematics employs increasingly complex mathematical infrastructure ortools: that is, mathematical theories designed to help us investigate other areasof mathematics. Mathematical tools, such as Galois theory or K -theory, establishrigorous relationships between outwardly unrelated classes of mathematical objects.As Jean-Pierre Marquis observes, the function of such tools is to “reveal important properties of the objects studied, and only these properties” [48, p. 264]. In otherwords, a result in one of the two related areas may be taken as a sign that one ofthe presumably less tractable objects in the other area has a particular property.However, since the relationship between the areas can be rigorously established, thesign is not merely generally indicative, but infallibly so.Applying rules to cases is also a very widespread practice in mathematics. Severalof the schemes that fall under this heading, such as argument from verbal classi-fication, argument from example, and argument from analogy, have mathematicalapplications that I have discussed elsewhere [4, 5, 6]. Walton and Macagno also in-clude chained arguments, which comprise a substantial proportion of mathematicalreasoning [5, p. 235]. But here I shall focus on a different scheme: Argumentation Scheme 3 (Argument from an Established Rule [75, p. 343])13 berdein
Major Premise:
If carrying out types of actions including A is the established rulefor x , then (unless the case is an exception), x must carry out A . Minor Premise:
Carrying out types of actions including A is the established rule for a . Conclusion:
Therefore, a must carry out A .Critical Questions:1. Does the rule require carrying out types of actions that include A as an in-stance?2. Are there other established rules that might conflict with or override this one?3. Is this case an exceptional one, that is, could there be extenuating circum-stances or an excuse for noncompliance?Scheme 3 is framed in terms of actions to be carried out. That might initiallyappear to be an obstacle to its application to mathematics. However, as WilfridHodges has observed, informal mathematical arguments include not only the “objectsentences”, in which some mathematical content is explicitly given, and “statedor implied justifications for putting the object sentences in the places where theyappear” but also “instructions to do certain things which are needed for the proof”[39, p. 6]. The last of these, carrying out actions, has perhaps received least attentionfrom logicians, but it is ubiquitous and important. Many proofs instruct us to“ ‘Suppose C ’, ‘Draw the following picture, and consider the circles D and E ’, ‘Define F as follows’ ” and so forth [39, p. 6]. Language of this sort is phrased conventionallyas an instruction to the reader, but it is also a description of the actions undertakenby the deviser of the proof. But how did the proof’s author know which actions tocarry out? At least in some cases, by application of Scheme 3.Carrying out a rule has also been the focus of a significant debate in the philos-ophy of mathematics, inspired by the work of Ludwig Wittgenstein. Wittgensteinconsiders the case of a pupil who learns to follow a rule whereby he writes downa series of natural numbers each greater than its predecessor by 2. However, afterhe gets to 1000 he increments the numbers by 4, instead of 2, but takes himselfstill to be following the same rule [76, §185]. What ought we to make of such be-haviour? It has been suggested that Wittgenstein’s intent was to suggest a generalscepticism about rule-following [45]. If that were to be the case, Critical Question2 in Scheme 3 would always receive an affirmative answer: there would always beanother rule which might override any rule we may consider. Less radically, wecould read Wittgenstein as counselling against a platonist interpretation of rules asexisting independently of the practices they govern [77, p. 91]. Rather we should14 alton & Mathematics understand rules as implicit within our practice but nonetheless as carrying norma-tive force. The ontological status of rules is the subject of a difficult and importantdebate. Fortunately, Scheme 3, and related rule-establishing and applying schemes,are neutral as to the outcome of that debate. Practical reasoning is an inevitable component of resource-sensitive mathematicaldeliberation dialogues whether limited by time, money, or processor capacity. Ifnumerical approximation methods are easy and cheap and an exact answer wouldbe expensive and slow, we may settle for the former. More broadly, a dialogue canshift to addressing this sort of question within reasoning about a problem whenevera choice of methods arises. So practical reasoning is not just a project managementphase to be completed before the real work begins, but potentially a recurrent phe-nomenon throughout the research process. For example, James Franklin points outone context in which practical reasoning occurs in the career of almost every re-search mathematician: choice of Ph.D. topic. A Ph.D. thesis is expected to addressan open question which must also be “tractable, that is, probably solvable, or atleast partially solvable, by three years’ work at the Ph.D. level” [30, p. 2]. Determin-ing whether a problem is tractable is not something which can be established withcertainty. But it is critical to the success of the Ph.D. Elsewhere I have addressedsome special cases of practical reasoning, including argument from positive conse-quences: if important results would follow from a conjecture, that at least providesgood reason to investigate it more thoroughly than similar, but less consequentialconjectures [5, pp. 235 f.]. However, I have not directly discussed the most generalpractical reasoning scheme in Walton’s taxonomy (see also [69, p. 131]):
Argumentation Scheme 4 (Practical Inference [75, p. 323])
Major Premise:
I have a goal G . Minor Premise:
Carrying out this action A is a means to realise G . Conclusion:
Therefore, I ought (practically speaking) to carry out this action A .Critical Questions:1. What other goals that I have that might conflict with G should be considered?2. What alternative actions to my bringing about A that would also bring about G should be considered?3. Among bringing about A and these alternative actions, which is arguably themost efficient? 15 berdein
4. What grounds are there for arguing it is practically possible for me to bringabout A ?5. What consequences of my bringing about A should also be taken into account?Scheme 4 could be used to analyse much of the embedded negotiations aboutresource allocation discussed above. However, it can also play a more direct rolein mathematical reasoning: Yacin Hamami and Rebecca Morris have proposed anaccount of plans and planning in proving in terms of intentions and practical rea-soning, building on Michael Bratman’s work in the philosophy of action [16]. Theprocess of finding a proof, at least if the proof is of any complexity, may involve theconstruction and execution of a carefully devised proof plan, which Hamami andMorris define as “an ordered tree whose nodes are proving intentions, whose rootis the proving intention corresponding to the theorem at hand, and where each setof ordered children consists of a subplan obtained from the parent node through aninstance of practical reasoning” [31]. The plan is not the proof, any more than themap is the journey. But, they suggest, the plan is essential not only to successfullyfinding the proof, but also to subsequently understanding the proof. Mathematicians have an ambivalent attitude to authority: there is “a schism in themathematical community . . . [between those] who think that one should never usea result without having understood its entire proof . . . [and those who] don’t sharethat view” (anonymous mathematician, interviewed in [11]). Unlike the empiricalsciences, where replication of experiments can require substantial resources, it is inprinciple always possible for mathematicians to work through every step of everyproof they use. But, for many mathematicians, a division of labour is unavoidableand even welcome. Hence there is a place in mathematics for one of the most dis-cussed argumentation schemes, that for argument from expert opinion [1, 5, 50].Notably, there are several disputes over the role of testimony in mathematical rea-soning that may be understood in terms of the critical questions of this scheme.“Folk theorems” are one such troublesome case. These are results which are widelyused and accepted despite lacking a clear source in the literature. In a pioneer-ing study drawing attention to their prevalence, the theoretical computer scientistDavid Harel suggests that “popularity, anonymous authorship, and age . . . seem tobe necessary and sufficient for a theorem to be folklore, [although] the ways in whichthey appear and can be established are by no means clear-cut” [33, p. 379 f.]. DonFallis raises the concern that the citation of folk theorems may represent “univer-sally untraversed gaps” in mathematical reasoning since “everyone is convinced that16 alton & Mathematics these theorems are provable, but no one has bothered to work through all the detailsof a proof” [27, p. 62]. And, of course, everyone could be wrong. Colin Rittbergand colleagues raise a different problem: the ambiguous status of folk theorems,neither rigorously proved nor strictly open problems, presents a hazard to youngresearchers [58]. Actually proving a folk theorem can be an unrewarding project,since referees may reject such work as unoriginal—despite being unable to cite anyprior proof. A possible resolution to this and related problems lies in the work ofKenny Easwaran, who has posited a property of “transferability” that may distin-guish the proofs which safely support argument from expert opinion from those thatdo not. Transferable proofs are those that “rely only on premises that the compe-tent reader can be assumed to antecedently believe, and only make inferences thatthe competent reader would be expected to accept on her own consideration” [24,p. 354]. Hence folk theorems lack transferable proofs, unless there is a proof simpleenough for any competent mathematician to reconstruct. Many other proofs* wouldbe untransferable too, including unsurveyably long proofs, probabilistic proofs, andproofs that rely on empirical procedures. This suggests a revision, or precisification,of the critical questions of the expert opinion scheme.Argument from expert opinion is not the only source-dependent argument rel-evant to mathematics. Walton draws an important distinction between argumentfrom expert opinion and argument from position to know. The distinction is familiarfrom legal practice, as that between expert and fact witnesses. I have suggested else-where that argument from position to know provides a model for appeals to intuitionin mathematics [5, p. 240 f.]. In this I follow philosophers, such as Elijah Chudnoff,for whom intuition is analogous to perception [17, 18]. Reports on (reliable) per-ceptions support cogent instances of argument from position to know, so if intuitionmay be treated analogously, then the same scheme should apply. Arguments frompopular opinion and popular practice also have important applications to mathe-matics [4, p. 283 ff.]. However, here I will focus on yet another source-dependentargument:
Argumentation Scheme 5 (Ethotic Argument [75, p. 336])
Major Premise: If x is a person of good (bad) moral character, then what x saysshould be accepted as more plausible (rejected as less plausible). Minor Premise: a is a person of good (bad) moral character. Conclusion:
Therefore, what a says should be accepted as more plausible (rejectedas less plausible). Critical Questions:1. Is a a person of good (bad) moral character?17 berdein
2. Is character relevant in the dialogue?3. Is the weight of presumption claimed strongly enough warranted by the evi-dence given?Superficially, this scheme may appear to be a poor fit for mathematical argument,but there is empirical research that suggests the applicability of something much likeit. Matthew Inglis and Juan Pablo Mejía-Ramos gave an informal mathematicalargument for the presence of one million sevens in the decimal expansion of π tosamples of undergraduates and research mathematicians and asked them to rate howpersuasive they found it [40]. Participants in both groups for whom the argumentwas correctly attributed to the prominent mathematician Tim Gowers ranked it asmore persuasive than those for whom it was presented anonymously, significantlyso for the researchers (who were presumably more likely to have heard of Gowers).Of course, it is not Gowers’s (doubtless exemplary) moral conduct which leads usto trust his arguments, but rather his demonstrably high standards as a workingmathematician. As one of the research subjects in this study comments, “We aretold the argument is made by a reputable mathematician, so we implicitly assumethat he would tell us if he knew of any evidence or convincing arguments to thecontrary” [40, p. 42]. This suggests that we should localize Scheme 5 to mathematics,replacing instances of “moral” with “mathematical”: Argumentation Scheme 6 (Ethotic Mathematical Argument)
Major Premise: If x is a person of good (bad) mathematical character, then what x says should be accepted as more plausible (rejected as less plausible). Minor Premise: a is a person of good (bad) mathematical character. Conclusion:
Therefore, what a says should be accepted as more plausible (rejectedas less plausible). Critical Questions:1. Is a a person of good (bad) mathematical character?2. Is mathematical character relevant in the dialogue?3. Is the weight of presumption claimed strongly enough warranted by the evi-dence given?A (presumably implicit) invocation of Scheme 6 would explain Inglis and Mejía-Ramos’s finding, but it raises other questions, most centrally: what is mathematicalcharacter? This is a question which recent work applying virtue epistemology tomathematical practice has sought to answer [63]. Mathematicians have also em-ployed virtue talk to describe their activities. For example, George Pólya assertsthat the following “moral qualities” are required of a mathematician:18 alton & Mathematics • First, we should be ready to revise any one of our beliefs.• Second, we should change a belief when there is a compelling reasonto change it.• Third, we should not change a belief wantonly, without some goodreason [56, vol. 1, p. 8].He goes on to expand on these points in explicitly virtue-theoretic terms, telling usthat intellectual courage is required for the first, intellectual honesty for the second,and wise restraint for the last. The first of these is a widely discussed intellectualvirtue and the second is, at least in this context, closely related to the even morewidely discussed intellectual humility. Wise restraint is perhaps more familiar asprudence or practical wisdom. More recent mathematicians who have discussedcharacter virtues relevant to their profession include Michael Harris [34] and Fran-cis Su [61]. Of course, these mathematicians would not necessarily endorse everyapplication of Scheme 6. Indeed, as we saw above, some mathematicians consider itto be a virtue to never take mathematical results on trust and insist on convincingthemselves of the proof of any result they cite. Nonetheless, many other mathemati-cians, especially when reasoning speculatively rather than writing up proofs, relyon a division of labour which makes essential use of the informal arguments of theirpeers, and may be expected to take those arguments more seriously when they havemore reason to trust their authors. Recent work in the philosophy of mathematical practice has drawn attention tomathematical reasoning in contexts other than proof and challenged the traditionalconception that mathematical proof is essentially reducible to formal derivation.This leaves a conspicuous lacuna in our understanding of how mathematics works.Formal logic is an excellent tool for the analysis of formal derivations, but it is lesswell adapted to the analysis of informal reasoning. However, the tools developedby informal logicians such as Douglas Walton are a rich source for remedying thisdeficit. In particular, as we have seen, dialogue types help to contextualize thedifferent levels of rigour that mathematical argument can exhibit and argumentationschemes provide a valuable taxonomy of the steps that comprise such arguments.
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