The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning.

Philosophy

Evidence, Proofs, and Derivations

EVIDENCE, PROOFS, AND DERIVATIONS ANDREW ABERDEIN∗ Abstract. The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and nonrigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning. 1. Evidence versus Proof The traditional view of evidence in mathematics is that evidence is just proof. Donald Martin poses the question, 'What then does count as mathematical evidence? There is, of course, an obvious answer to the question of how one can come to know the truth of a mathematical proposition: namely, proof. Indeed, this may seem the only way to establish mathematical truth' (Martin, 1998, 216). However, as Martin acknowledges, there are good reasons for not accepting this view. Crucially, it does not comport with actual mathematical practice. Here, for example, is a short, melancholy narrative of an unsuccessful proof attempt, unusual only in being so self-contained. The mathematician Vašek Chvátal proposed as a conjecture a generalization of the well-known Sylvester-Gallai theorem. Unable to supply a proof, he comments that 'we present meagre evidence in support of this rash conjecture' (Chvátal, 2004, 175). However, the last lines of his published paper read: 'Received July 14, 2002, and in revised form April 14, 2003. Online publication December 31, 2003. Note added in proof. In September 2003 Xiaomin Chen proved Conjecture 3.2' (Chvátal, 2004, 195). Conjecture 3.2 is Chvátal's 'rash conjecture'; where Chvátal had evidence, Chen had proof. If proof were the only evidence ∗School of Arts & Communication, Florida Institute of Technology, Melbourne FL. Date: April 4, 2019. Key words and phrases. argument; argumentation schemes; derivation; evidence; proof. Forthcoming in ZDM mathematics education 51(4), 2019. 1 2 ANDREW ABERDEIN there is in mathematics, then either Chvátal would have had nothing to say, or Chen nothing to add. Here is a more protracted exhibition of the use of evidence beyond proof in mathematical practice. One of the last publications of the celebrated mathematician (and Nobel laureate) John Nash was a collection of essays on open problems, edited in collaboration with Michael Rassias. The editors remark of their choice of problems that 'Some were chosen for their undoubtable importance and applicability, others because they constitute intriguing curiosities which remain unexplained mysteries on the basis of current knowledge and techniques, and some for more emotional reasons' (Nash and Rassias, 2016, vi). Their selection includes many of the best known open problems in mathematics, each discussed by a leading expert. Not every essay explicitly references the evidence for these conjectures, but many do. Here is a sample (boldface emphasis mine throughout; internal citations omitted): P versus NP: 'Of course, this gives even more dramatic evidence that GraphIso is not NP-complete: if it was, then all NP problems would be solvable in npolylog n time as well' (Aaronson, 2016, 20). Montgomery's Pair Correlation Conjecture: 'The agreement with the first million zeros is poor, but the agreement near zero number 1012 is close, near perfect near zero number 1016, and even better near zero number 1020. These results provide massive evidence for Montgomery's conjecture' (Barrett et al., 2016, 155). The Birch-Swinnerton-Dyer Conjecture: 'Today, the numerical evidence in support of both the weak and full Birch-Swinnerton-Dyer conjecture is overwhelming, and probably more extensive than for any other conjecture in the history of mathematics' (Coates, 2016, 213). The Erdős-Szekeres Problem: 'The special cases A = I and A = {i1, in} correspond to minimal evidence for existence of an n-cup and an n-cap' (Morris and Soltan, 2016, 363). The Hadwiger-Nelson Problem: 'Ronald L. Graham . . . cites a theorem of Paul O'Donnell (see [34, 49]) showing the existence of 4-chromatic unitdistance graphs of arbitrarily large girth (Theorem 28 below) as "perhaps, the evidence that χ is at least 5."' (Soifer, 2016, 441). Erdős's Unit Distance Conjecture: 'There is no strong evidence supporting the assumption that such an example cannot exist' (Szemerédi, 2016, 468). Goldbach's Conjecture: This is the strongest evidence we have that for even n, r2(n) ∼ nS2(n)' (Vaughan, 2016, 486). The Hodge Conjecture: 'The following theorem proved in [7] is the best known evidence for the Hodge conjecture' (Voisin, 2016, 538). None of these examples of evidence refers to proof; indeed, in many of them the evidence is for a conjecture notable for being unproven. Nonetheless, we may see that mathematical evidence can be dramatic, massive, overwhelming, strong-or, conversely, minimal. But even overwhelming evidence falls short of proof. Notice also that evidence can diverge from proof on more than one dimension. Some of these cases fall short of proof since they are proofs of some weaker conjecture that provides bounds on the headline conjecture, or is otherwise suggestive of its truth (e.g. Hadwiger-Nelson, Hodge); others differ in kind, rather than degree, by offering numerical, that is empirical, evidence EVIDENCE, PROOFS, AND DERIVATIONS 3 (e.g. Montgomery, Birch-Swinnerton-Dyer). Nor are considerations of evidence short of proof confined to the contemplation of stubbornly unsolved problems. On the contrary, mathematicians must often consider such evidence when making decisions about their own careers and those of their students. Crucially, as James Franklin notes, every Ph.D. supervisor must ensure that the problems their students tackle are open, but likely to yield to a few years work (Franklin, 1987, 2). Similar considerations of evidence bear on grant awards, tenure decisions, and many other administrative aspects of mathematical practice. 2. Proof versus Derivation The 'standard view' of the relationship between informal, or everyday proof and derivation, its formal counterpart, is that 'informal proofs are just sloppy, incomplete versions of formal proofs' (Pawlowski and Urbaniak, 2018). On this account, the only proofs worthy of the name would either already be derivations or be rewritable as derivations in a more or less trivial manner. Nonetheless, it has lately become 'a common observation that the proofs that mathematicians write on blackboards and publish in journals are not like the derivations that appear as objects in proof theory' (Larvor, 2016, 401). However, it can be a challenge to pin down exactly where the divergence arises. Before going any further, I should clarify what I am not saying. Nothing in this paper is intended to contradict what is sometimes termed the Formalizability Thesis: that every proof can (in principle, at least) be formalized as a derivation.1 Crucially, the Formalizability Thesis is an existence claim: it states that for every proof there is a derivation. It makes no claim as to whether the derivation is known, or accessible, or surveyable. It does not require that the internal structure of a proof bear any resemblance to that of the corresponding derivation.2 It also makes no claim as to the metaphysical status of the relationship between the proof and the derivation. For present purposes it suffices to observe that accepting that there is a derivation for every proof is consistent with proofs and derivations being profoundly different entities and specifically does not entail that knowing a proof has any connection to knowing a derivation. The proof/derivation distinction has been drawn in multiple, conceptually distinct ways. I shall survey some of the most important-semantic/syntactic; normative/theoretical; act/object-before addressing how they are related. One influential account of the proof/derivation distinction situates it in the different manner of expression of proofs and derivations. Thus Yehuda Rav, in a highly influential paper, declares that he will 'understand by proof a conceptual proof of customary mathematical discourse, having an irreducible semantic content, and distinguish it from derivation, which is a syntactic object of some formal system' (Rav, 1999, 11). Likewise, Jody Azzouni draws the distinction between 'formal derivations, which occur in artificial languages, and mathematical proof, which occurs in natural languages' (Azzouni, 2013, 247). Analogously, Keith Weber and Lara Alcock 'define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way' and 1This is sometimes referred to as Hilbert's Thesis, although that name is more properly reserved for the narrower claim that every proof can be formalized as a derivation in first-order logic (Kahle, 2019). 2This reflects what has been called Tait's Maxim: 'The notion of formal proof was invented to study the existence of proofs, not methods of proof' (Baldwin, 2013, 114). 4 ANDREW ABERDEIN 'define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws' (Weber and Alcock, 2004, 209). While all of these points are correct, we may feel that they do not strike to the heart of the distinction. By contrast, Gila Hanna draws the distinction in a manner that brings out the social aspects of the contrast: (1) Formal proof: proof as a theoretical concept in formal logic (or metalogic), which may be thought of as the ideal which actual mathematical practice only approximates. (2) Acceptable proof: proof as a normative concept that defines what is acceptable to qualified mathematicians (Hanna, 1990, 6). The mathematician John Baldwin, asserts that he takes Hanna's 'formal/acceptable to be the same distinction' (Baldwin, 2016, 71) as that he draws between 'two degrees of formalization', respectively into a formal syntax and rules of inference or within (sufficiently mathematical) natural language (Baldwin, 2013, 89). So, for Baldwin at least, Hanna's distinction is coextensive with that between proofs and derivations. However, Hanna stresses the essentially social role of acceptability in proof which has no counterpart in derivation. Authors who emphasize the argumentational nature of proof, to which we will return below, specifically draw attention to this factor. Thus Todd CadwalladerOlsker states that the 'formal view of proof is contrasted with the view of proofs as arguments intended to convince a reader' (CadwalladerOlsker, 2011, 33) and Trevor Bench-Capon stresses that 'argumentation is . . . an activity which has to be actively engaged with, whereas a proof is an object to be understood and admired' (Bench-Capon, 2012). The latter observation points towards an aspect of the proof/derivation contrast that Hanna's distinction overlooks: derivations are objects; proofs are acts. In support of an account of proofs as acts, Joseph Goguen argues that the only sort of proofs 'that can actually happen in the real world are proof events, or provings, which are actual experiences, each occurring at a particular time and place, and involving particular people, who have particular skills as members of an appropriate mathematical community' (Goguen, 2001; see also Stefaneas and Vandoulakis, 2012, 489 ff.). Goguen is not alone in drawing the distinction this way. Oswaldo Chateaubriand also distinguishes between 'provings and idealized proofs' (Chateaubriand, 2003, 41), while tracing the temporal conception back to Brouwer. Goguen also observes that provings often have internal temporal structure-their components must be executed in the right order-and are thus ultimately proof processes: The efficacy of some proof events depends on the components of a proof object being seen to be given in a certain temporal order, e.g., Euclidean geometric proofs, and commutative diagrams in algebra; in some cases, the order may not be easily infered from just the diagram. Therefore we must generalize from static proof objects to proof processes, such as diagrams being drawn, movies being shown, and Java applets being executed (Goguen, 2001). Lest this conception of proof seem to have drifted too far from what is conventionally labelled as proof, it is important to reflect on what is meant by an act. Göran Sundholm draws a useful distinction between an act, the subjective process EVIDENCE, PROOFS, AND DERIVATIONS 5 that comprises the act, and the trace that the act leaves behind (Sundholm, 2012, 948). In the case of proof, we may further distinguish the concrete traces (marks on the blackboard, empty coffee cups, and so on) from the informational trace, a written proof (see Fig. 1). The latter signifies the act and serves as a blueprint or recipe whereby the act may be repeated. Just as recipes aren't very nourishing unless you carry them out, likewise the written proof only works as a proof if it is actually carried out. Although the contrast is more stark in the case of proofs, we may draw a similar contrast between derivations and derivation traces: the derivation is a mathematical object; its trace is a written counterpart to that object (a formal proof, or a sequence of code, or the like). Act of Proof Theorem Process of getting to know (exercised act) (trace) set of coffee cups, chalk marks, scrap paper (concrete trace) written proof in mathematical text (signified act) Figure 1. Göran Sundholm's act/process/trace distinction (after Sundholm, 2012, 948) In summary, we shall say that proofs make irreducible use of the semantics of natural language; involve a normative appeal to acceptance by a mathematical audience; and are events that involve participants and extend over time. Conversely, derivations are mathematical objects that represent formal relationships between propositions in an artificial language. Of course, this definition of proof is stipulative. If we treat the three contrasts discussed above-semantic/syntactic; normative/theoretical; act/object-as separate axes, then my definitions of proof and derivation occupy diagonally opposite vertices: {0, 0, 0} and {1, 1, 1}, as it were. That leaves six other, unlabelled vertices. I shall not attempt a complete taxonomy here, but some conceptions of proof rival to that defended here may be seen to correspond to some of these other vertices. 3. Argument versus Proof We saw in §1 that mathematicians make use of a concept of evidence distinct from proof and we saw in §2 that the practice of mathematical proof cannot be reduced to derivation. But nor can we discard derivations: they are mathematical objects in good standing (whatever that means-I make no ontological claim), and acknowledged as such by the mathematicians who study them. Furthermore, at least in the eyes of many mathematicians, it is the existence of a derivation that is the ultimate guarantor of the truth of a theorem.3 So, if we are to respect actual mathematical practice, we must accommodate all three concepts: evidence, proof, 3For example, for Saunders Mac Lane, 'the test for the correctness of a proposed proof is by formal criteria and not by reference to the subject matter at issue' (Mac Lane, 1986, 378) and Thomas Hales characterizes formal proof as providing 'a thorough verification of my own research that goes beyond what the traditional peer review process has been able to provide' (Hales, 2008, 1378). 6 ANDREW ABERDEIN and derivation. Is there a common framework into which they may be subsumed? One candidate is argument. I have already alluded to an argumentation-based account of mathematical reasoning. In its barest form, this states that a mathematical argument is 'a more liberalized version of the notion of mathematical proof' (Van Bendegem, 2005, 157). That is, proofs are a special case of mathematical argument. If these arguments are understood as similar in kind to the arguments of non-mathematical discourse, it follows that an account of mathematical reasoning may be given using the tools of argumentation theory. Elsewhere I have referred to proofs∗, where the asterisk indicates that the proof does not command universal assent (Aberdein, 2009, 2). These include unsurveyably long proofs∗, diagrammatic proofs∗, proofs∗ that depend on contested axioms, computer-assisted proof∗, experimental proof∗, proof∗ by probabilistic methods, and so on. While it is controversial in each case whether proofs∗ should lose the asterisk (for relevant discussion, see Fallis, 1997; Van Bendegem, 2005; Paseau, 2015), each of these is uncontroversially a mathematical argument. Some proofs∗ also uncontroversially correspond to derivations. For others this is less obvious; their admission as proofs would pose a challenge to the Formalizability Thesis discussed in §2. The Formalizability Thesis, that 'every proof can be formalized as a derivation', is one clause of what has been called Leibniz's Thesis; the other clause states that 'every acceptable argument of (informal) mathematics is a proof' (Berk, 1982, 17). This implies that mathematical arguments that are not proofs are unacceptable. But what does it mean for a mathematical argument to be acceptable? Interpreted widely, as 'acceptable in some mathematical practice', then the thesis is demonstrably false; interpreted narrowly, as 'acceptable as proof', then the thesis seems almost tautological. Nonetheless, as we have acknowledged above, being acceptable to a mathematical audience is a necessary requirement for a proof. Below I shall sketch what that might mean. Arguments resemble proofs, at least as defined above, much more closely than they do derivations. Both arguments and proofs are acts (or processes comprised of acts) whereas derivations are objects. Yet, as we saw in the last section, proofs and derivations both leave traces: written blueprints or recipes. Likewise, we can consider argument traces. The passing resemblance between a proof trace and a derivation trace explains how two such dissimilar things as proofs and derivations ever came to be conflated. A derivation is a directed graph. Its nodes are (formal counterparts of) truths of mathematics and its edges represent logical deductions. A determination of whether a derivation is sound requires a choice of axioms and of logical system. Relative to that choice, the derivation is sound if all its source nodes (those which do not depend on other nodes) are axioms and all its edges are valid. Proof traces (and argument traces in general) are also directed graphs. The nodes of a proof trace are mathematical truths, expressed in a suitably augmented natural language, and the edges are arguments of some kind. So, the same theorem will be linked to (at least) two structures: a proof trace and a derivation, each of which is comprised of directed graphs. How do these two structures relate to each other? In principle, any node in either structure could be a node in the other, at least assuming the Formalizability Thesis holds. In practice, there is rather less overlap. Derivations are substantially more verbose than proofs: they contain many more intermediate statements and they take everything back to axioms, as proofs characteristically do not. What's more, they may follow an EVIDENCE, PROOFS, AND DERIVATIONS 7 entirely different path from any recognisable proof. (Recall Tait's Maxim, cited above.) And, for that reason, proof traces may contain nodes that do not figure in any corresponding derivation. But at least the nodes of the two structures are held to the same standard: in each case, they must be true. The edges, on the other hand, must be judged differently. As deductions in a system of formal logic, the edges of a derivation can be assessed by the canons of that system: do they instantiate an admissible rule of inference? Of course, the edges of the proof structure may also be logical deductions, in which case they can be held to that standard too. But, with some rarefied exceptions, mathematical proofs are seldom purely logical proofs. So the edges of the proof structure must be judged instead by standards appropriate to arguments of the relevant kind; that is, by the standards of (some suitably localized form of) argumentation theory. Multiple methodologies for the analysis and evaluation of informal arguments have been proposed, but here I shall focus on the method of argumentation schemes. This is an ancient idea in origin, deriving from the topoi or loci of classical rhetoric, but it has been reinvigorated in recent years. An argumentation scheme is a template that captures a stereotypical pattern of reasoning. Different schemes are fine-tuned to capture the idiosyncrasies of different types of argument-and can be brought to bear to determine when they have been used cogently. This evaluative function is largely the role of the critical questions, which most schemes contain. If in some instantiation of a scheme these questions cannot be adequately addressed, then the argument fails. Deductive inference rules can be understood as argumentation schemes, but the method comes into its own when applied to non-deductive reasoning. Douglas Walton and colleagues characterize many such schemes as special cases of the very general scheme that they call Defeasible Modus Ponens (Walton et al., 2008, 366). In Scheme 1, I have adapted their presentation of this scheme to bring out the resemblance between argumentation schemes and Toulmin layouts (Toulmin, 1958, 101):4 Argumentation Scheme 1. Defeasible Modus Ponens Data: P. Warrant: As a rule, if P, then Q. Therefore, . . . Qualifier: presumably, . . . Conclusion: . . . 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? In principle, any argument that can be represented using Toulmin layouts could be represented using Scheme 1. However, the great strength of the argumentation scheme methodology lies in its diversity: there are many more schemes to choose from. One of the most extensive surveys of general purpose schemes distinguishes more than 90 different types (Walton et al., 2008, 308 ff.). Some of these off-the-shelf schemes are directly applicable to mathematics, but yet more can be produced to order. 4For a more protracted discussion of how these two models of reasoning are related, see (Pease and Aberdein, 2011, 28 ff.). For an alternative account, see (Konstantinidou and Macagno, 2013, 1070). 8 ANDREW ABERDEIN Elsewhere I draw a distinction between three classes of scheme that are of immediate relevance to proof (Aberdein, 2013b, 366 f.): • 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 are formal counterparts of its data and claim.) • B-schemes are exclusively mathematical arguments: high-level algorithms or macros. Their instantiations correspond to substructures of derivations rather than individual derivations (and they may appeal to additional formally verified propositions). • C-schemes are even looser in their relationship to derivations, since the link between their data and claim need not be deductive. Specific instantiations may still correspond to derivations, but there will be no guarantee that this is so and no procedure that will always yield the required structure even when it exists. Thus, where the qualifier of Aand B-schemes will always indicate deductive certainty, the qualifiers of C-schemes may exhibit more diversity. Indeed, different instantiations of the same scheme may have different qualifiers. We are now in a position to analyze what it means for a proof to be acceptable. For a given mathematical audience, that is a community of mathematicians who share the same standards, a proof will count as acceptable if the schemes it instantiates are ones which that audience judges consistent with mathematical rigour. In other words, the audience must be convinced that each step of the proof instantiates a scheme in such a way that it earns the qualifier 'rigorously' (or stronger). Exactly which schemes are deemed consistent with rigour will vary by audience. Different areas of mathematics can have somewhat different standards and even within the same area there are different audiences: the audience for a research article is not identical to that for an undergraduate lecture. Nonetheless, we may generalize. For most audiences, A-schemes are invariably admissible. Likewise, most audiences will admit many B-schemes. However, some B-schemes can be highly complex. As such they may only be admissible to a narrow audience: professionals in a particular subspecialty, say. Such schemes may need to be broken into simpler steps (probably also instantiating simpler B-schemes) for consumption by a wider audience of research mathematicians, let alone for a student audience. Indeed, in some pedagogic contexts, it may be expedient to substitute a less (than) rigorous C-scheme for an intricate Bscheme, thereby 'handwaving' through an aspect of the proof unsuitable for a given audience. C-schemes themselves pose a further challenge. Since not all their instantiations are deductive, it may be tempting to assume that they have no place in rigorous argument, at least as most mathematical audiences understand it. Even were this so, they would still be welcomed by some mathematical audiences, notably those that see no need for the asterisks on some of the more outré species of proof∗. However, in practice, even quite conservative mathematical audiences can find some C-schemes admissible. Some of the general purpose C-schemes that may be accepted by some mathematical audiences are listed in Table 1 This is by no means an exhaustive list; nor is the classification of schemes it presumes unarguable. In the article from which Table 1 is taken, I discuss the application to informal mathematical reasoning of ten EVIDENCE, PROOFS, AND DERIVATIONS 9 Table 1. Summary of reasoning types and schemes (Aberdein, 2013a, 247) (1) Reasoning (a) Retroduction (i) Argument from Gradualism (ii) Argument from Positive Consequences (iii) Argument from Evidence to a Hypothesis (2) Source Based (a) Citation (i) Appeal to Expert Opinion (ii) Argument from Danger (b) Intuition (i) Argument from Position to Know (c) Meta-Argument (i) Ethotic Argument (d) Closure (i) Argument from Ignorance (3) Rule Based (a) Generalization (i) Argument from Example (b) Definition (i) Argument from Definition to Verbal Classification schemes drawn from (Walton et al., 2008). I only have space to address a couple of them in comparable detail here, but I will briefly survey the rest. I classified three schemes as retroduction, that is reasoning backwards from a conclusion, a common strategy in mathematical argument since antiquity. Argument from Gradualism comprises multi-step argumentation. This can be fallacious when the steps are individually dubious or collectively improbable, yielding a slippery slope fallacy. Conversely, if all the steps are deductively valid, it would correspond to a derivation. Some instances of this scheme, however, are neither fallacious nor deductively valid: their admissibility will turn on the standards of rigour of the audience. Argument from Positive Consequences is well described by its name. It finds a place in mathematics in the informal justification of axioms and hypotheses.5 I will discuss Argument from Evidence to a Hypothesis at greater length in the next section. Several of the schemes in Table 1 are source-based: that is, they involve reasoning to a mathematical conclusion from a source external to the reasoning process. The most familiar of these reasoning patterns may be the use of citation, which is ubiquitous in most mathematical research. Appeal to Expert Opinion is an obvious fit for such reasoning, and also one of the most widely studied schemes in the argumentation scheme literature. Argument from Danger, which I also included under citation, is of much narrower application. It captures arguments 5The mathematical uses of Argument from Positive Consequences are also discussed, together with some other schemes not in Table 1, in work by Nikolaos Metaxas and colleagues (Metaxas, 2015, 84; Metaxas et al., 2016, 387). 10 ANDREW ABERDEIN to the effect that one should not act in a way that would be a danger (to oneself or others). I argued that this is employed in what is sometimes (facetiously) called 'proof by intimidation': dismissing objections as trivial, for example. Argument from Position to Know describes a pattern of argument where an informant is taken to have a privileged source of information. The obvious non-mathematical example would be an eye witness; in a mathematical context, this scheme might be employed to characterize appeals to intuition, which some authors analogize to perception (for example, Chudnoff, 2013). Ethotic Arguments are appeals to the character (or ethos) of some individual. It has been shown empirically that such factors can also influence how mathematical results are received (Inglis and Mejıa-Ramos, 2009). Argument from Ignorance is generally treated as a fallacy, since not knowing that something is false is not usually a legitimate reason for treating it as true. Yet it can be a reliable inference in special circumstances, some of which can be found in mathematical reasoning, such as reporting the outcome of an exhaustive search. The last group of schemes in Table 1 are those involving the justification or application of rules. Argument from Definition to Verbal Classification could potentially correspond to a step in a derivation, but it can also be used more casually, and is vulnerable to outright misuse, for example when a definition is subtly misstated. For a more considered example, consider the following: Argumentation Scheme 2. Argument from Example Premise: In this particular case, the individual a has property F and also property G. Conclusion: Therefore, generally, if x has property F, then it also has property G. Critical Questions (1) Is the proposition claimed in the premise in fact true? (2) Does the example cited support the generalization it is supposed to be an instance of? (3) Is the example typical of the kinds of cases the generalization covers? (4) How strong is the generalization? (5) Do special circumstances of the example impair its generalizability? (Walton et al., 2008, 314) Many instantiations of Argument from Example will not be acceptable at least to modern audiences. Enumerative induction, for instance, can be characterized in terms of this scheme. But there are other cases where Argument from Example meets modern standards of rigour. In order to do so, adequate answers to all the critical questions will need to be provided. In particular, question (4) will require an exceptional answer: one justifying the revision of the qualifier from 'generally' to 'necessarily'.6 4. Argument versus Evidence We have explored the relationship between proofs and derivations and between each of these and arguments. We have not yet addressed the relationship between 6For a more extensive discussion of mathematical uses of Scheme 2, see (Aberdein, 2013a, 244). EVIDENCE, PROOFS, AND DERIVATIONS 11 evidence and argument, but alert readers may have spotted the following scheme in Table 1: Argumentation Scheme 3. Argument from Evidence to a Hypothesis Major Premise: If A (a hypothesis) is true, then B (a proposition reporting an event) will be observed to be true. Minor Premise: B has been observed to be true, in a given instance. Conclusion: Therefore, [presumably,] A is true. Critical Questions (1) Is it the case that if A is true, then B is true? (2) Has B been observed to be true? (3) Could there be some reason why B is true, other than its being because of A being true? (Walton et al., 2008, 331 f.) This is the pattern of reasoning called abduction by Charles Peirce (for a direct comparison, see Pease and Aberdein, 2011, 22; for an alternative approach to abduction in terms of argumentation schemes, see Metaxas et al., 2016, 386). To observe how this scheme might work in practice, consider the following example of evidence for Goldbach's Conjecture (GC), as reconstructed by the philosopher Alan Baker: Another line of auxiliary argument might be based on the various partial results relating to GC. In 1931, Schnirelmann proved that every even number can be written as the sum of not more than 300,000 primes(!). This upper bound on the number of primes required has since lowered to 6 (Ramaré 1995). In addition, Chen (1978) proved that all sufficiently large even numbers are the sum of a prime and the product of two primes. Such results do not seem to make the truth of GC any more likely. But perhaps they provide evidence that GC is provable (Baker, 2007, 71). Here the hypothesis A is the proposition that GC is provable and the 'event' (or events) that B reports on are a series of proofs of weaker, but related, conjectures. As Baker acknowledges, this is at best suggestive. Critical question (3) is completely open and the argument would only sustain a very weak qualifier. On the other hand, some of the examples of argument from evidence cited in §1 are much more convincing. For instance, the argument for Montgomery's Pair Correlation Conjecture can easily be reconstructed as an instance of Scheme 3, with A a statement of the conjecture and B the observation that 'the agreement near zero number 1012 is close, near perfect near zero number 1016, and even better near zero number 1020' (Barrett et al., 2016, 155). Of course, as stressed in §1, the authors of this argument are not claiming that this is a proof. Hence this instance of Scheme 3 is not one that they would treat as acceptable. Are there any instances of Scheme 3 which a mainstream mathematical audience would treat as acceptable? One possible affirmative answer brings us back to derivations. In his discussion of the relationship between proofs and derivations, Richard Epstein makes the following bold assertion: 'A proof in a fully formal system of logic that a claim follows from some axioms is not a proof in mathematics. It is evidence that can be used in a mathematical proof' (Epstein, 2013, 274). In the terminology of this paper, Epstein may be read as saying that a derivation, or more properly, a derivation trace, should not be mistaken for a proof (or proof trace), but 12 ANDREW ABERDEIN can be employed in a proof as evidence. On the account of proof defended above, such employment would require use of an argumentation scheme, presumably along the lines of Scheme 3. For proofs where the corresponding derivation is comparatively short and straightforward, this may seem to be an unnecessarily scrupulous point. Yes, in principle, the derivation is a mathematical object, but its trace will be similar enough to a proof trace for it to be treated as such with comparatively little additional effort, at least for audiences familiar with such things. In these sorts of cases, the more characteristic response to a derivation trace would be to 'reverse engineer' it into a proof in this manner. But, for longer or more technical derivations, this process is less practical. In such cases, Epstein's analysis seems correct. If there is a sufficiently strong reason for accepting the derivation trace as reliable, then this would be an acceptable instance of Scheme 3. Unsurveyable computer-assisted proof∗ represents a similar application of Scheme 3. Proofs such as Kenneth Appel and Wolfgang Haken's proof of the Four Colour Conjecture or Thomas Hales's proof of Kepler's Conjecture depend essentially on unsurveyably vast computer calculations. Within the human readable component of the proof, these calculations play an evidential role. Thus, if these proofs are acceptable to the mathematical community, then the steps introducing the computer calculations can be seen as acceptable instances of Scheme 3. Of course, while the consensus seems now to be in their favour, I have flagged such results as proof∗ because their acceptability remains a topic of debate. However, if the existence of a derivation is what ultimately underwrites the acceptance of a theorem, then a mechanically verified derivation trace may be an (even more) acceptable instance of Scheme 3. Successful formalization projects, such as Georges Gonthier's work on the Four Colour Theorem or Hales's Flyspeck Project, have provided just such corroboration (Gonthier, 2008; Hales et al., 2017).7 5. Conclusions for Education One of the morals of this paper is that it is important to carefully distinguish proof from derivation. This observation certainly has profound implications for the teaching of mathematics. However, it is scarcely novel: I have already cited a paper from almost thirty years ago which addresses the educational implications of the distinction (Hanna, 1990). The role of evidence in relation to proof and derivation has received rather less attention. I have suggested that it can be successfully accommodated within the framework of argumentation schemes. Specifically, appeals to evidence can be understood as C-schemes: argumentation schemes drawn from natural language reasoning that generally fall short of rigorous proof, but can sometimes be used rigorously. In this manner, mathematical arguments that rely on evidence to provide less than rigorous support for their conclusions can be understood as belonging to the same genus as mathematical proofs, but not the same species. Thereby the importance of rigour in proof is maintained, but without misleading the student into imagining that proof is somehow entirely alien from ordinary reasoning. 7It may be objected that this results in a regress, since the software checking the derivation trace must itself be checked. However, it is what Hales has called 'a rather manageable regress' (Hales, 2008, 1376). The kernel of such proof checking software is very carefully designed to be small enough and clear enough to be amenable to thorough human checking. EVIDENCE, PROOFS, AND DERIVATIONS 13 The framework of argumentation schemes, however, is itself a potentially valuable instrument for the mathematics educator. Toulmin layouts, which are also drawn from the toolbox of argumentation theory, have lately been widely applied to mathematical reasoning by educational theorists (see, for example, Knipping and Reid, 2013). At least in their current form, argumentation schemes are a much more recent invention, and they have yet to receive much attention from mathematics educators: I am only aware of a few studies from one group of researchers specifically applying argumentation schemes to mathematics education (Metaxas et al., 2009; Metaxas, 2015; Metaxas et al., 2016; Koleza et al., 2017). In addition, Aikaterini Konstantinidou and Fabrizio Macagno have discussed the application of argumentation schemes to science education (Konstantinidou and Macagno, 2013; Macagno and Konstantinidou, 2013). They argue that argumentation schemes can have a particular value in 'discovering the implicit beliefs affecting a student's learning process' (Macagno and Konstantinidou, 2013, 235). This is an important task in mathematics education too, and their study would seem to naturally generalize to mathematics education. Perhaps the most exciting opportunity that argumentation schemes represent for mathematics education is that they are already widely implemented within what is sometimes called the 'argument web' (Reed et al., 2017). In recent years, a growing number of software tools of increasing sophistication have been designed for the analysis and evaluation of argument. These tools represent a potentially invaluable resource for the study of mathematical argument, but their application to mathematics has only just begun (Pease et al., 2017; Corneli et al., 2019). Again, I am unaware of any application of these resources in mathematics education: there is a growing body of work applying digital tools to mathematics education (for example, Modeste, 2016; Durand-Guerrier et al., 2019), but not the tools specific to argumentation. They represent an as yet untapped resource, of considerable scale and importance. A final, more speculative benefit may accrue from an educational approach which emphasizes the continuity of mathematical argument and argument in other areas. Some authors have urged that mathematics educators promulgate the value of an education in mathematics as a source of the intellectual virtues and skills necessary for successful navigation of contemporary society (recent examples include Su, 2017; Cheng, 2018). When mathematics is taught in a fashion that emphasizes its differences from everyday reasoning this can be a tough case to make. But argumentation schemes represent a plausible bridge between mathematical and other conversations; and mathematics may thereby provide an invaluable testbed for the acquisition and mastery of argumentation techniques of much wider application. Acknowledgements I presented an earlier version of this paper at the interdisciplinary symposium on Mathematical Evidence and Argument held at the University of Bremen in 2017. I am grateful to the participants for their comments and particularly indebted to Christine Knipping and Eva Müller-Hill for their invitation and their hospitality in Bremen. I am also grateful to three anonymous referees for insightful and thorough comments. 14 ANDREW ABERDEIN References Aaronson, Scott. 2016. P ?= NP. In Open Problems in Mathematics, eds. John Forbes Nash, Jr. and Michael Th. Rassias, 1-122. Cham: Springer. Aberdein, Andrew. 2009. Mathematics and argumentation. Foundations of Science 14(1-2): 1-8. Aberdein, Andrew. 2013a. 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There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book.

Philosophy

Introduction

Andrew Aberdein & Matthew Inglis Advances in Experimental Philosophy of Logic and Mathematics Introduction There has been very little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to logic or the philosophy of mathematics is impractical or unavailing. That this would be a mistake is exhibited by at least three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a 'practice turn' in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. Before turning to the specific contribution that we hope this book will make, we will offer a snapshot of each trend and address how they intersect with some of the standard criticisms of experimental philosophy. Firstly, although experimental philosophy is often thought of as a twenty-first-century phenomenon primarily focussed on questions in ethics and epistemology, it has some important anticipations in earlier projects in the philosophy of logic. The most significant is the work of Arne Naess and the Oslo Group (Naess, 1938, 1959, 1982; Tönnessen, 1951). For instance, Ingemund Gullvåg argued that to understand the meaning of a word such as 'truth', it was 'hardly sufficient that a single person registers his own reactions to this or that sentence, or makes pronouncements based on intuitions' (Gullvåg, 2 1955, 343). Instead, the Oslo group argued, systematic empirical investigations were required. The connections between the 'empirical semantics' developed by the Oslo group and experimental philosophy have now begun to be made explicit by historians of philosophy and further developed by a new generation of researchers (Murphy, 2014; Barnard and Ulatowski, 2016; Chapman, 2018). This productive connection between the empirical methods of two different generations is continued in Barnard and Ulatowski's chapter in the present volume, discussed in greater detail below. Secondly, in recent decades there has been a 'practice turn' in the philosophy of mathematics, focussing on how mathematical research is actually conducted, rather than on the search for foundations for mathematics (Van Kerkhove and Van Bendegem, 2007; Mancosu, 2008). This has naturally led to an interest in empirical data about mathematical practice, a programme dubbed 'Empirical Philosophy of Mathematics' by some of its practitioners (Buldt et al., 2008; Löwe et al., 2010; Pantsar, 2015). There are several distinct axes along which the connections between the philosophy of mathematical practice and empirical work have been drawn. A significant body of work applies cognitive science research on mathematical reasoning to philosophical questions (Pease et al., 2013). This includes work on the status of mathematical knowledge (Cappelletti and Giardino, 2007; Pantsar, 2014); on the symbol systems of mathematics (De Cruz and De Smedt, 2013; Dutilh Novaes, 2013; Marghetis and Núñez, 2013); and on the role of diagrams and visualization in mathematics (Giaquinto, 2007; Hamami and Mumma, 2013). Moreover, modern mathematicians increasingly employ online tools for collaboration. This produces a considerable amount of 3 potential data for researchers interested in mathematical practice, giving rise to another strategy for the investigation of that practice (Martin and Pease, 2013; Martin, 2015; Pease et al., 2017). Although logical practice may have received less attention than its mathematical counterpart, some researchers in the philosophy of logic have pursued a practice turn of their own, modelled on that in the philosophy of mathematics (Dutilh Novaes, 2012). As with its sister programme in philosophy of mathematics, advocates of the philosophy of logical practice stress that too much attention has been paid to foundational issues at the expense of philosophical questions that arise elsewhere, such as in the application of logic to artificial intelligence, game theory, linguistics, and other disciplines. For example, the burgeoning research programme of 'argumentation mining', which applies corpus-based techniques to extract and analyse arguments across large bodies of text, may be seen as a logical counterpart to the use of big data techniques in analysis of mathematical practice (Moens, 2018). Thirdly, there is a growing awareness of how much research in adjacent disciplines has anticipated the research questions of experimental philosophy of logic and mathematics. Philosophers of logic have an extensive body of research on the psychology of reasoning to draw upon (Johnson-Laird, 2006). Lately, some work in the intersection of philosophy of logic and psychology of reasoning has made the relationship to experimental philosophy explicit (Pfeifer, 2012; Pfeifer and Douven, 2014; Ripley, 2016). There is also a substantial research tradition in mathematics education that addresses questions of immediate relevance to the philosophy of mathematical practice (Heinze, 2010; Weber et al., 2014; Weber and Mejía4 Ramos, 2015; Alcock et al., 2016). Understanding mathematical practice is important for education researchers for at least two reasons. First, understanding the behaviour of expert mathematicians helps to decide what the purpose of a mathematics curriculum should be. If a particular activity is highly valued in expert mathematical practice then this perhaps provides a reason for mathematics students to be exposed to some appropriate version of it (see, for example, Ball and Bass, 2000; Harel and Sowder, 2007; Lampert, 1990; Weber et al., 2014). Second, studying the in-the-moment strategies adopted by expert practitioners (in any domain) might provide suggestions for how to develop interventions that assist learners to develop expertise. An example of this approach can be found in the work of Alcock, Hodds, Roy and Inglis (2015). They studied the reading behaviour of research mathematicians, and used these insights to develop training materials that encouraged undergraduates to adopt similar strategies. These training materials significantly increased the amount students learned from reading a mathematical text. In addition, there is now an emerging tradition of interdisciplinary work, applying quantitative techniques to address traditionally philosophical questions, such as mathematical aesthetics (Inglis and Aberdein, 2015, 2016). Some of this work has been presented as an enquiry into 'mathematical cultures' (Löwe, 2016; Larvor, 2016). Likewise, Reuben Hersh, one of the forerunners of the practice turn, has lately called for 'a unified, distinct scholarly activity of mathematics studies: the study of mathematical activity and behavior' (Hersh, 2017, 335). We regard the present volume as, in part, a contribution to the integrative work required for this project. 5 The advent of experimental philosophy has not been without controversy and has provoked a salutary debate on the proper methods of philosophical enquiry. One of the most prominent critiques is the 'expertise defence' of traditional philosophical practice (Nado, 2014; Mizrahi, 2015). This maintains that surveys of non-philosophers have limited bearing on the arguments of philosophers since, as experts, philosophers can be expected to be immune from the errors and biases exhibited by non-experts. This debate has given rise to a substantial literature. However, the experimental philosophies of mathematics and logic seem to have ready responses to the expertise defence. Many studies of mathematical practice focus on professional mathematicians, placing the expertise of the participants essentially beyond dispute. Nonetheless, this is not universally true; for instance, some philosophers (for example, De Cruz, 2016) have used results from the numerical cognition literature to draw conclusions about the ontology of natural numbers. Participants in numerical cognition studies include non-mathematical adults, children and even non-human animals. An important difference between mainstream experimental philosophy and work focused on mathematics is that studies in the latter tradition typically ask their participants-be they mathematicians, children or animals-about mathematics, not about philosophy. (This is just as well, for in Hersh's famous formulation, 'the typical working mathematician is a Platonist on weekdays and a formalist on Sundays' (Hersh, 1979, 32). Such insouciance would not bode well for the resolution of philosophical dilemmas.) In this respect experimental philosophy of mathematics is similar 6 to psychological work on reasoning relevant to debates in the philosophy of logic. Here too participants are typically drawn from a more general population. (Although there clearly is such a thing as logical expertise; for a start, people can be trained to be better at logical reasoning (Attridge et al., 2016).) Just as mathematicians/children/fish are asked about mathematics not philosophy, participants in reasoning studies are asked object-level questions about everyday reasoning, not specialised questions about logical hypotheses that might predict or explain such reasoning. On this basis David Ripley has argued that these studies are better placed to answer the expertise objection than studies relevant to debates in ethics or epistemology (Ripley, 2016). Nonetheless, there is a substantial body of psychological research that reveals a divergence between best practice in reasoning (at least, as defined by logicians) and how lay people actually reason. There is also a substantial body of work critiquing these results. Broadly speaking, they lend themselves to four possible responses: 1. Lay people are to blame: they routinely make damaging errors in their inferential practices; 2. Psychologists are to blame: they fail to understand the relationship between formal and informal reasoning, and thereby design experiments which show only that good reasoners can be hoodwinked by artificial examples; 3. Logicians are to blame: they persist in defending systems of formal inference which do not describe the legitimate inferential practices of ordinary folk; 7 4. No one is to blame: logicians might well be right that formal logic is the best way to reason, but in many (perhaps most) real world circumstances it takes too much cognitive effort to do so. Many such studies, especially in early psychology of reasoning work, are presented as supporting the first response. However, they can often be reinterpreted in support of one of the others. In particular, much research of this sort is implicitly deductivist (and often classicist): it presumes that the best account of human inference will always be deductive logic (and often that classical logic is the best or only viable system of deductive logic). Hence such work is undermined by the successful modelling of informal, non-deductive patterns of inference in argumentation theory (Zenker, 2018) or non-classical logics (Aberdein and Read, 2009). The moral may be that, as with many sciences, theoretical and empirical approaches should be mutually reinforcing: logicians need the empirical research conducted by psychologists of reasoning to corroborate their claim of faithfulness to actual reasoning; psychology of reasoning needs to be informed by current research in logic if it is to stay relevant. Recent research suggests that even preverbal children can exhibit behaviour consistent with logical reasoning (Cesana-Arlotti et al., 2018). Children as young as twelve months were presented with stimuli either complying with or violating simple inferential rules, such as disjunctive syllogism, p∨q,¬p⊢q. That they looked longer at violating cases than they did at stimuli consistent with those rules, just as adults do, suggests that they found those cases incongruous. This provides an echo of a far older debate. 8 The ancient logician Chrysippus argued that dogs employ disjunctive syllogism, since a scent hound, tracking a quarry to a crossroads and eliminating all but one of the exits, will (or so Chrysippus claims) immediately take the last exit without further checks of the trail. The story has been retold many times, with at least four different morals: 1. dogs use logic, so they are as clever as humans; 2. dogs use logic, so using logic is nothing special; 3. dogs reason well enough without logic; 4. dogs reason better for not having logic (for details, see Aberdein, 2008). The third option may be closest to Chrysippus's own; it may also be the best take on the empirical research. That is, such studies do not attribute conscious, reflective awareness of any system of logic to dogs (or infants). Rather, they demonstrate that logic succeeds in tracking the pre-theoretical reasoning not just of the logically educated, but of pretty much anyone capable of rational thought. It is sometimes argued that logic, as an a priori discipline, is immune from revisionary pressures that apply to natural science. If this is so, then there may be little room for empirical research in logic. On the other hand, there is a tradition, associated with W. V. O. Quine in particular, of treating logic as continuous with the natural sciences (Bryant, 2017). In recent years, this debate has been characterized in terms of 'anti-exceptionalism' about logic (Hjortland, 2017; Read, 2018). However, we do not need to resolve the debate in order to observe that it is less damaging to our concerns than it may 9 first appear. Even if one concedes that the truths of logic are analytic and necessary-that is, true in virtue of their meaning and such that they could not have been different-our knowledge of these truths is still fallible. So we may expect the methods whereby we come to know these truths to have much in common with the methods whereby we learn truths in the natural sciences, even though the truths of those disciplines are neither analytic nor necessary. This collection is intended to consolidate and develop the three trends identified above: the reappraisal of the Oslo Group; the practice turn in the philosophies of logic and mathematics; and the reintegration into these philosophies of empirical work from adjacent disciplines. The ten chapters are divided equally between the philosophies of mathematics and logic. Their authors include some of the leading figures in each of the areas of research discussed above. Several chapters are methodological analyses of the applicability of empirical techniques to these areas of philosophy, but many (also) include actual empirical results. They demonstrate a wide variety of different empirical methods, including experiments, surveys, and datamining. Benedikt Löwe and Bart Van Kerkhove's chapter, 'Methodological triangulation in empirical philosophy of mathematics', is written by two of the leading figures in the philosophy of mathematical practice. They survey the uses that have been found for a variety of different empirical methods in philosophy, emphasising that the experimental method in the strict sense is only one of them. They argue for methodological triangulation in empirical 10 philosophy, that is, the employment of a battery of different empirical methods to compensate for the biases and limitations implicit in any one of them. Their paper provides a helpful introduction to the potential that empirical methods offer for the philosopher of mathematics (or logic). In particular, they rehearse a sartorial analogy that Löwe has proposed elsewhere for the different levels of integration between philosophy and empirical methods (Löwe, 2016, 36). He distinguishes 'ready-to-wear', the philosophical exploitation of existing, independently conducted empirical research, from 'bespoke', which involves more direct collaboration, such as philosophers designing projects to be conducted by empirical researchers, and 'do-it-yourself' (homespun?), in which the philosopher conducts all aspects of the research. (We might add that such cross-disciplinary work can cut both ways: empirical researchers can develop the interest and expertise necessary to address philosophical questions. Indeed, some philosophical questions, including many posed by the philosophies of mathematical and logical practice, are already within the remit of nearby empirical disciplines.) It is important to stress, as Löwe and Van Kerkhove do, that this is not a hierarchy of quality. If you are lucky enough to find off-the-peg clothes that are a good fit, they may be much better value than bespoke. And making your own clothes is unlikely to have a good outcome unless you acquire significant expertise. Likewise, when existing empirical studies address the right questions, ready-to-wear studies can be highly effective. The remaining chapters in this collection report on studies of all three varieties. Helen De Cruz's work in the philosophy of mathematics has long made use of empirical results (De Cruz, 2006, 2016). Her chapter, 'Animal 11 cognition, species invariantism and mathematical realism', is a notable piece of ready-to-wear empirical philosophy. She uses a variety of results from numerical cognition (especially neurological and animal work) to tackle a recently influential argument in the philosophy of mathematics. The 'evolutionary debunking' argument against moral realism suggests that our moral beliefs cannot be objectively true if they are the result of a highly contingent evolutionary process. If we had evolved from animals with very different social behaviour (and there are many such species) then we would have a quite different set of moral intuitions, so why imagine that those intuitions track the truth? Mathematical realism, the view that our mathematical beliefs are objectively true, has obvious similarities to moral realism. So might there not be an analogous evolutionary debunking argument against mathematical realism too? However, De Cruz demonstrates that there is significant evidence that the mathematical behaviour of animals is substantially convergent, which suggests that the analogy fails; if anything the empirical data provide support for mathematical realism. The next chapter, 'The beauty (?) of mathematical proofs', also makes extensive use of existing empirical research. We have already noted Catarina Dutilh Novaes's research on logical practice; besides the history and philosophy of logic she also works on social epistemology and the philosophy of mathematics, as in this chapter. She coordinates a number of disparate literatures to propose a novel approach to the aesthetics of mathematical proof grounded in empirical work on affective responses to unexpectedness. The key idea is that in many situations mathematical 12 judgments bring together epistemic and aesthetic components, and that we should not be surprised by this. The next two chapters are drawn from the bespoke tradition: they report on original studies that were conducted by the authors to address (at least) philosophical questions. Both chapters look at aspects of visual reasoning in mathematics. This has been a controversial subject: an influential view maintains that visuals should play no role in mathematical proof, but a growing body of work suggests that this is an unrealistic, indeed harmful, idealization (Larvor, 2013, 2018). Josephine Relaford-Doyle and Rafael Núñez are both cognitive scientists-the latter is a co-author of a landmark in the application of cognitive science to mathematics (Lakoff and Núñez, 2000). Their chapter, 'Can a picture prove a theorem? Using empirical methods to investigate visual proofs by induction', reports an empirical study that investigates how undergraduate students with and without formal mathematical training use images to justify mathematical claims. They focus on visual induction proofs, and find results that challenge James Brown's (non-empirical) claim that such proofs are immediately understandable for people without mathematical training (Brown, 1997). Keith Weber and Juan Pablo Mejía-Ramos are mathematics educators. In their chapter, 'An empirical study on the admissibility of graphical inferences in mathematical proofs', they investigate the admissibility of graphical inferences in proofs in real analysis. They conclude that the type of graphical inference is important to consider when addressing their question. In particular, Weber and Mejía-Ramos find support for the importance of distinguishing between metrical and non-metrical graphical inferences 13 (Larvor, 2018). A metrical graphical inference is one that depends for its success on the measurements of angles, lengths, and so on, being precisely correct, whereas a non-metrical graphical inference does not; that is, the latter sort of inference is unaffected by local deformations in the diagrams at issue. Where the first five chapters focus primarily on the philosophy of mathematics, the remaining five concentrate on the philosophy of logic. In their chapter, 'Does anyone really think that <⌜p⌝ is true if and only if p>?', the philosophers Robert Barnard and Joseph Ulatowski link together Arne Naess's early empirical work, their own recent replications of some of these results, and the contemporary debate on deflationary accounts of truth. As well as noting this chapter's contribution to philosophical theory, given the ongoing replication crisis in psychology (Chambers, 2017), it is worth explicitly remarking upon and celebrating Barnard and Ulatowski's successful replication of Naess's early findings. Igor Douven's research lies at the intersection of several fields, including formal epistemology and cognitive science. His chapter, 'New foundations for fuzzy set theory', seeks to rehabilitate fuzzy set theory as an account of vagueness by grounding it in recent empirical work on conceptual spaces. Conceptual spaces were developed by the cognitive scientist Peter Gärdenfors as a geometrical framework for the qualitative comparison of concepts along multiple dimensions (Gärdenfors, 2000). Douven argues that seeing fuzzy membership as the distance of a point from a prototypical point in such a space is a productive approach to fuzzy set theory and, moreover, 14 that there is empirical support for adopting this view from work conducted by cognitive psychologists. The philosopher Moti Mizrahi's chapter, 'What isn't obvious about "obvious": A big data approach to philosophy of logic', uses a corpus linguistics approach to investigate the obviousness or otherwise of logic. Mizrahi reasons that if logic really was obvious, then the frequency with which logicians use the word 'obvious' should correlate with deductive indicator words such as 'necessary' and 'certainly' but not with inductive indicator words such as 'probably' or 'likely'. By analysing a large corpus of text drawn from research papers published in logic, philosophy, mathematics and biology journals, Mizrahi empirically tests these predictions. While he finds some support for the predictions, he also finds some results that require further explanation. David Over and Nicole Cruz are both psychologists of reasoning. Their chapter, 'Philosophy and the psychology of conditional reasoning', is a wideranging discussion of recent empirical work on conditional statements and its relationship to philosophy. The authors cite an extensive array of empirical studies to argue in favour of a Bayesian account of conditionals and against a mental model account. They conclude that much psychological research on conditionals has paid too little attention to philosophical and logical work. Remedying that oversight has already led to improved empirical studies, and promises to go further. In their chapter, 'Folk judgments about conditional excluded middle', the philosophers Michael J. Shaffer and James Beebe employ empirical studies to motivate a novel analysis of so-called Bizet/Verdi conditionals: 15 • If Bizet and Verdi had been compatriots, Bizet would have been Italian. • If Bizet and Verdi had been compatriots, Verdi would have been French. Across three experiments Shaffer and Beebe find evidence for Alchourrón et al.'s (1985) 'belief revision' theory of counterfactuals, in line with the tradition of the Ramsey Test. Interestingly, they reject the alternative accounts from Lewis (1973) and Stalnaker (1981) by coordinating analyses from both quantitative and qualitative data. The experimental philosophy of logic and mathematics has been quietly thriving for some time. We hope that this collection will form an indispensable resource for future research in the field. References Aberdein, A. (2008). Logic for dogs. In Hales, S. D., ed., What Philosophy Can Tell You About Your Dog, pp. 167-181. Open Court, Chicago, IL. Aberdein, A. and Read, S. (2009). 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