Topoi https://doi.org/10.1007/s11245-017-9530-4

Carroll’s Infinite Regress and the Act of Diagramming John Mumma1

© Springer Science+Business Media B.V., part of Springer Nature 2017

Abstract The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry. Keywords  Mathematical proof · Inference · Diagrams · Carroll’s regress A central task of epistemology is to provide an account of justification. A person in Philadelphia may hold the true belief that it is raining in Buenos Aires, but she is not justified in the belief if it came to her from a dream the night before. The epistemological question is: what processes of belief acquisition would justify the belief for her? The topic of this paper is the analogous question in the case of deductive inference. A person may be justified in believing the premises of a deductively valid argument, and may come to believe the argument’s conclusion. But she is justified in believing the conclusion on the basis of a deductive inference only if the right kind of process led her to the conclusion from the premises. The question is: how is this process to be characterized? I examine this question with respect to the following deductive inference 𝛼 concerning the ordering of points on an oriented geometric line: α

Point a is before point b and point b is before point c. Point a is before point c.

Specifically, I characterize a process whereby the deductive inference is achieved by engaging the geometric content of its propositions. This process is non-logical—since more than just the logical form of the sentences in 𝛼 plays a role in the inference—and diagrammatic—since it is by diagramming the premises of the inference that their * John Mumma [email protected] 1



Philosophy Department, California State University San Bernardino, 5500 University Parkway, San Bernardino, CA 92407, USA

geometric content is engaged. The account is advanced as a possible starting point for the investigation of two more general questions in the philosophy of mathematics. First, a detailed analysis of how 𝛼 is performed via a diagram may provide a foothold for understanding how diagrams facilitate inference in mathematics generally. Second, the account suggests a line of inquiry into non-logical deductive inference in mathematics (be it diagrammatic or non-diagrammatic). The paper has four parts. In the first, I motivate the question of how deductive inference is to be characterized by discussing the infinite regress of Lewis Carroll’s ‘What the Tortoise Said to Achilles’ (Carroll 1895). In the second, drawing on the work of Prawitz, I present the distinction between non-logical and logical deductive inference. In the third, I give my account of a non-logical, diagrammatic process for performing 𝛼 . And finally, in the fourth part, I comment on the broader relevance of the account for investigations into diagrammatic and/or non-logical deductive inference in mathematics.

1 Carroll’s Regress and Mathematical Proof The starting point for the discussion between the Tortoise and Achilles in Carroll (1895) is an inference found in the first proposition of Euclid’s Elements: (A) Things that are equal to the same are equal to each other. (B) The two sides of this Triangle are equal to the same. (Z) The two sides of this Triangle are equal to each other.

The Tortoise, invoking the principle that all premises of a deductive inference ought to be explicit, pulls Achilles

13

Vol.:(0123456789)

J. Mumma

into an infinite regress in which the task of adding premises to the inference is never completed. He asks Achilles to consider a reader of the A–B–Z inference from Euclid. What is required of such a reader, the Tortoise asks, for her to be forced to accept the conclusion by ‘logical necessity’? Both agree that two things have to occur. First, she must accept the premises A and B as true. Second, she must accept the A–B–Z sequence as a ‘valid’ one. With Achilles’ assent, the Tortoise asserts that the latter condition amounts to the acceptance of the ‘hypothetical’ proposition

(C) If A and B are true, Z must be true. Thus the reader (who the Tortoise eventually identifies with himself) must accept three propositions—A, B and C—in order to be forced to accept Z by logical necessity. But now, the Tortoise observes, the reader’s position is similar to the initial one. Along with accepting the truth of the propositions A, B and C, the reader must also accept the A–B–C–Z sequence as valid, which means accepting the truth of a fourth proposition D (D) If A and B and C are true, Z must be true. But then this means that the reader must accept a fifth proposition E, and following that a sixth proposition F, and so on. If the relation of logical validity is conceived as a relation between propositions wholly separate from human thought and agency, then Carroll’s regress does not raise any issues with respect to it. The key idea in the Tortoise’s generation of the regress is the state of being forced to accept a proposition by other propositions. The relation whereby this state is realized has propositions and a person as relata: propositions A1 , A2 … An force person  to accept proposition Z. It is common, especially in classroom settings where the concept of validity is being introduced, to talk of the premises of a valid argument as forcing acceptance of the conclusion. With his piece Carroll is exploring whether there is anything substantial behind this talk. It may seem at first that he has demonstrated with a reductio that there is not. Achilles and the Tortoise begin their dialogue under the assumption that it’s possible for propositions to force the acceptance of others, and from this assumption are led to the absurdity of an infinite regress. Thus the assumption is false. This reading however is too hasty. The state of being forced to accept a proposition Z by other propositions A1 , A2 , … , An seems to have, independently of Carroll’s regress, genuine epistemological significance. There is a genuine epistemological distinction to draw between a person who is forced to accept Z by his acceptance of A1 , … , An and one who merely accepts A1 , …, An and Z without recognizing any connection between the propositions. This comes out clearly in considering mathematical knowledge, which rests on mathematicians’ capacity to produce and check deductive proofs. Altering the situation Carroll presents us

13

with, consider two readers of the proof of the first proposition in the Elements. The first accepts the premises of the proof and follows Euclid’s deductive inferences step by step and thereby comes to accept that the proof’s conclusion must be true. The second also accepts the proof’s premises, but rather than take the time to read the argument accepts the conclusion on the authority of the Elements as a mathematical text. There is a definite sense in which the first reader is forced to the accept the proof’s conclusion, where the second is not. Moreover, the distinction between the two is not incidental to the methods by which mathematical knowledge is amassed. A mathematical proof would not, it seems, count as a proof unless it succeeds at forcing a mathematically trained reader to accept its conclusion in this sense. Altering the example again, suppose the proof being read is one submitted to a mathematical journal for publication, the two readers are referees reviewing it, and the review is not blind. In engaging with the proof so that it forces her to accept its conclusion, the first reader is fulfilling her professional duties. On the other hand, in accepting the conclusion on the authority of the proof’s prestigious author, the second reader is not. Given, then, that propositions A1 , A2 ... An force person P to accept proposition Z

is a genuine, epistemologically significant relation, Carroll’s piece then amounts to a negative result on how to characterize it. The Tortoise and Achilles get mired in a regress because of a flawed conception of what it is to be forced to accept the conclusion of a valid argument. Specifically, the difference between the first and second readers cannot consist merely in the first believing the proposition that the proof is valid. If it did, the question would then remain how we can distinguish the first reader from a third who also believes that the proof is valid, but is not forced to accept the conclusion. It is not hard to imagine such a reader. One who accepts the conclusion of a mathematical proof on the authority of its author would most likely also be willing to accept the proposition that the proof is valid. The general moral is that characterizing beliefs justified by deductive inference cannot, on pain of infinite regress, be done by just providing a list of propositions believed. As Stroud observes in his insightful discussion (Stroud 1979) of Carroll’s piece: ...even if the person is said to believe or accept some statement R linking things he already believes with his conclusion we still must attribute to him something else in addition if we are to represent his belief in that conclusion as based on those other beliefs. That additional factor cannot be identified as simply some further proposition he accepts or acknowledges. There must always exist some ‘non-propositional’ fac-

Carroll’s Infinite Regress and the Act of Diagramming

tor if any of his beliefs are based on others. (Stroud 1979, pp. 188–189) Stroud later elaborates on this non-propositional factor, and asserts it to consist in a person’s understanding of believed propositions, identified as a ‘disposition, competence, or practical capacity’ (Stroud 1979, p. 194). Stroud’s comments forge a link between understanding and deductive inference. Understanding the premises of a deductive inference amounts—in part—to the disposition/practical capacity/competence to perform the deduction. Following Stroud, we can frame the question Carroll’s piece raises as follows: given that the premises of a deductive inference are accepted, what processes of understanding force the acceptance of the conclusion?

2 Non‑logical Deductive Inference One thing that can said at the outset is that an answer is not to be found in the notion of valid inference given by the standard, Tarskian definition of logical consequence. To see this, consider a nontrivial mathematical theorem formalizable in a first-order theory. Dirichlet’s prime number theorem, for instance, can be derived within a (very weak) firstorder theory of arithmetic.1 It is thus a logical consequence, in Tarski’s sense, of some basic arithmetical facts—i.e. the arithmetical facts serving as axioms of the first order theory of arithmetic. Yet there is no sense in which it could be said that there is one step proof of the theorem from the axioms. In other words, a person who accepts the basic arithmetical facts but does not accept the theorem could not be faulted for failing to understand the basic arithmetical facts. In fact, as Prawitz (2013) observes, the unsuitability of the Tarskian conception of validity in meeting the challenge posed by Carroll’s regress on its own is apparent once it is recognized that the conception is fully captured by the proposition: if the premises of an inference are true in a model  , then the conclusion is true in model  . In asserting that an inference is a valid by the Tarskian conception, we are not doing anything more than what the Achilles and the Tortoise do when they add the premise C to the inference A–B–Z. It would be possible, of course, to stay within the standard conception of logical consequence, and consider formal systems of proof that are sound and complete relative to Tarski’s model-theoretic semantics. A more radical approach, pursued by Prawitz, is to dispense with Tarski’s analysis of logical consequence altogether, and develop a theory of deductive inference that is epistemic from the ground up. (See, in addition to Prawitz 2005, 2009, 2012

1

  See (Avigad 2003).

and 2013). Prawitz’s starting point is a general notion of deductive inference, conceived as an act in which the truth of the premises is seen to guarantee the truth of the conclusion. Prawitz’s perspective on deductive inference thus aligns with Stroud’s. A central task of Prawitz’s project is to characterize the inferrer’s active role in ‘seeing’ the necessity of the inference’s conclusion. And he seeks to do this by appealing to the inferrer’s understanding of what the premises, or components thereof, mean. His notion of deductive inference is general in that not all such inferences are logical. Among deductive inferences, the logical ones are those which are “invariant under variations of non-logical notions (p. 190 of Prawitz 2012)”. In other words, they are those deductive inferences that depend only on the meaning of their constituent logical notions.2 There are thus for Prawitz deductive inferences that rely on more than just the meaning of their constituent logical notions. The one example he develops in detail is inference by mathematical induction. He specifies the form of such an inference in Prawitz (2012) with the scheme:

IND

[A(x)] .. .. A(0) A(suc(x)) A(a)

where x and a are arbitrary natural numbers, suc is the successor function and A is a formula in the language of arithmetic with one free variable. Prawitz describes a process whereby a reasoner comes to see that the conclusion A(a) must hold given what is above the line in the inference scheme. It is thus, according to Prawitz’s conception of validity, a deductively valid argument. It is not however a logically valid one. Its validity depends essentially on the meaning attributed to x, a and suc. If the domain of discourse is no longer understood to be the natural numbers or suc is understood to be a different function on the natural numbers, the process cannot be carried out. My aim in the next section is to establish that inference 𝛼 is in the same category as IND in Prawitz’s taxonomy by specifying a process whereby a reasoner comes to see that its conclusion must hold given that its premise holds. Before moving on to this, it is worth taking a moment to examine the relation of 𝛼 to the following inference 𝛼 ′ :

2

  Prawitz analyzes the meaning of these logical notions in the terms of the notion of a proposition’s ground. The result is an account where the basic logical inferences—i.e. those logical inferences where the necessity of the conclusion is immediately seen—correspond directly to the the introduction and elimination rules of natural deduction. For the details, see (Prawitz 2012).

13

J. Mumma

α

(A) For any three points, if the first is before the second and the second is before the third, then the first is before the third. (B) Point a is before point b and point b is before point c. (Z) Point a is before point c.

This inference is the result of modifying 𝛼 into a logically valid inference by explicitly stating the transitivity of the ‘x is before y’ relation on an oriented geometric line. It fits naturally into the A–B–Z form of the initial inference in Carroll’s dialogue. In both Carroll’s initial inference and 𝛼 ′ , the first premise A is a universal conditional statement, the second premise B is the antecedent of A′s conditional instantiated by a particular group of objects, and the conclusion Z is the consequent of A′s conditional instantiated by the same group of objects. Moreover, in both Carroll’s inference and 𝛼 , the second premise B seems sufficient to infer the conclusion Z from a non-logical standpoint—i.e. a standpoint that takes into account the particular binary relation composing B and Z. In both, the universal conditional A is needed as a premise to make the evident but logically invalid inference from B to Z logically valid. This would not seem to be an accidental feature of Carroll’s example. His regress is of philosophical interest, in part, because it is not obvious how the methodological imperative requiring that B be supplemented with A is different from the Tortoise’s demand that a premise be added at each stage of the regress. In claiming that 𝛼 is (in Prawitz’s terminology) a nonlogical valid inference, I am claiming that with respect to 𝛼 there is a sense in which there is no difference. That is, there is a sense in which requiring the addition of A to 𝛼 is akin to requiring in the vein of the Tortoise that a premise

(C) If A and B are true, Z must be true be added to 𝛼 ′.3 Just as the premises of 𝛼 ′ provide enough information for one to be justified in carrying out a valid and 3  One may wonder if an analogous claim holds also of A in Carroll’s A–B–Z inference. (One may wonder this even even if identity is understood to be a logical notion. The notion of equality in the inference is not the notion of identity. Rather, it concerns whether two possibly distinct lengths have the same magnitude.) I believe a case can be made that an inference from B to Z is valid and non-logical, but this would require an in-depth analysis of Euclid’s notion of equal magnitude, which is beyond the scope of this paper.   On the other hand, if we go by the treatment of the equality of magnitude in the Elements, the inference from B to Z is not on its own justified. In contrast to order relations like ‘x is before y’, the equality of magnitude is explicitly declared to be transitive in an axiom given in the common notions of book I (the axiom is in fact premise A in Carroll’s syllogism). This fits with Manders’ analysis of Euclid’s proof method in Manders (2008). ‘x is before y’ is a co-exact relation and thus the method permits one to use diagrams to reason about it. On the other hand, equality of magnitude is an exact relation, and the method demands that all reasoning involving it be made explicit in propositions. I will discuss Manders’ analysis in more detail in the conclusion.

13

logical inference, the single premise of 𝛼 provides enough information for one to be justified in carrying out a valid and non-logical inference.4 This is not to say that premise A of inference 𝛼 ′ provides no information with respect to inference 𝛼 . It identifies an abstract logical property of the relation ‘x is before y’ and thus brings to light how 𝛼 belongs to a class of inferences concerning relations with this property. The point is that this information is not necessary in order for the inference to be justified. One can recognize that the relation ‘x is before y’ is transitive, and then perform the valid and logical inference 𝛼 ′ . But one can also invoke one’s understanding of ‘x is before y’ and use the transitivity of the relation to perform the valid and non-logical inference 𝛼 . I now turn to the specifics of the latter task.5

3 Diagramming 𝛼 There is temptation to characterize a diagram depicting the premises of 𝛼 , such as

c b a

4

  Prawitz in fact identifies an inference akin to 𝛼 as valid and nonlogical in Prawitz (2013). It is: Adam is longer than Beatrice, Beatrice is longer than Carlo. Therefore, Adam is longer than Carlo.

5

 A similar point concerning C′s relation to 𝛼 ′ can be made. As observed above, it is natural to interpret C as a model-theoretic statement—i.e. if A and B are true in a model  , then Z is true in model  . It thus provides some information about inference 𝛼 ′ . This information, however, is not necessary to justify the inference.  The idea here is formulated from a more general perspective in Michael Detlefsen’s interpretation of Poincaré’s philosophical writings on mathematics in Detlefsen (1992). According to Detlefsen, non-logical mathematical inferences such as 𝛼 are central to Poincaré’s epistemology of mathematics. “In the case of Poincaréan proof, the key feature is the grasping or intuiting the mathematical architecture between p [the premise of a mathematical inference] and c [the conclusion of a mathematical inference] (p. 368).” In making a non-logical mathematical inference logical by adding a premise that expresses a universal feature of the given mathematical subject—as is done in the move from 𝛼 to 𝛼 ′—one “abstracts away from this grasp of architecture itself and instead focuses on its net classical effect...[a logical proof] separates or detaches the net classical effect of a warrant or justification from that warrant or justification itself. It then replaces the actual grasp or comprehension of the underlying justification with reflection on the semantic or epistemic status thus abstracted from it (p. 369).”

Carroll’s Infinite Regress and the Act of Diagramming

as some kind of intuitive evidence that compels a reasoner to infer the conclusion of 𝛼 , analogously to the way visual inspection compels acceptance of statements of fact. After being told that there is no milk in the refrigerator, a person who opens the refrigerator door and sees only a carton of orange juice is compelled to believe the statement. Accordingly, in consulting the diagram next to the premises of 𝛼 , the reasoner in some sense sees that the conclusion holds, and cannot believe its negation. The difficulty with fleshing this general characterization out, of course, is specifying the sense in which the conclusion is seen to hold. If completely analogous to visual perception, the diagram is an object separate and independent from the reasoner in the same way that the contents of the refrigerator are separate and independent from the person opening the refrigerator door. And so nothing seems to be gained with respect to the issue Carroll’s regress raises. There is still a gap between reasoner and conclusion that the Tortoise can exploit. He could assert that the reasoner needs a second diagram to see that the first diagram applies to the premise, and then a third diagram to see that the second diagram applies to the first, and so on. Conceived as static piece of evidence, a diagram for 𝛼 does not differ in status from the propositions that the Tortoise demands Achilles to add to 𝛼.6 This observation illustrates the general position advanced by Stroud and Prawitz. Inference must be characterized as an act. And so, simply adjoining the diagram above—call it 𝛽 —to the propositions of 𝛼 would be of no help in understanding how the acceptance of 𝛼 ′s premises force acceptance of its conclusion. If 𝛽 is to be of any help in blocking Carroll’s regress with respect to 𝛼 , it must be explained how the inference is accomplished through 𝛽  , as opposed to the diagram merely serving as evidence for the inference. Accordingly, the starting point for understanding the epistemic relation of 𝛽 to 𝛼 should be the procedure of diagramming 𝛼 with 𝛽 . Understood epistemically, 𝛼 records a transition from an initial state—where the reasoner accepts the premise of 𝛼 —to a final state—where she accepts the premise of 𝛼 and its conclusion. The desired account of the diagramming procedure, then, would link the two states. It ought to characterize the procedure as having two stages: one where the reasoner moves from her initial state to 𝛽  , and another where she moves from 𝛽 to the final state. The first stage demands of the reasoner an ability to produce a diagram from the premise of 𝛼 . She must (given a pencil and paper) be able: to draw a line and identify an orientation on it; to mark two points on the line and label them a and b so that a is before b according to the orientation; and to mark a third point after b according to the orientation

and label it c. If a diagram is given beside the text listing the premise and conclusion of 𝛼 , she is spared the task of actually producing a diagram. An ability similar to the ability to construct the diagram is however still in play in her verification that the given diagram depicts the premises. She must be able: to recognize an orientation on the given line; to verify that a is before b according to the orientation; and to verify that b is before c. Having either constructed 𝛽 or verified its agreement with 𝛼 ′s premise, the reasoner enters into the second stage, and considers 𝛽 with respect to the issue addressed by the conclusion: the position of a to c on the line. 𝛽  ′s depiction of the configuration is unambiguous on the issue. The point marked a on it is before the point marked c on it. The reasoner’s perception of this however is not enough to force her, in the Tortoise’s sense, to the conclusion that a is before c. She is not in the position of a person who has performed a numerical calculation on paper, and reads off the result. Such a person can be said to be forced to accept the result seen at the end of the calculation. For this to happen, however, she must have already accepted prior to the calculation that the algorithm underlying the calculation yields correct results. This is lacking with the reasoner diagramming 𝛼 with 𝛽  . In fact, the reasoner has reason to believe that with respect to geometric facts of linear order the procedure of simply reading off a conclusion from a diagram leads to error. Consider the incorrect inference γ

Point a is before point b and point a is before point c. Point b is before point c.

A diagram 𝛿 that depicts the premise of 𝛾 is:

c b a

One is indeed forced to the proposition that the point marked b on 𝛿 is before the point marked c on 𝛿 . But this does not (or at least should not) force one to accept the incorrect conclusion of 𝛾 . Contrasting 𝛼 and 𝛾 is instructive, as it brings out that the premise-to-diagram stage is not cleanly separated from diagram-to-conclusion stage. The diagram has probative value as it is being constructed from the premise.7 The difference

7 6   I assume in what follows that a similar difficulty does not arise with respect to the visual verification of statements of fact.

  For ease of exposition, I describe only the case where the diagram is constructed from the premises in the first stage. The same point applies mutatis mutandis to the case where the diagram is verified to satisfy the premise.

13

J. Mumma

between the way the conclusions of 𝛼 and 𝛾 relate to their respective premises is modal. In the context of 𝛼 ′s premise, a is necessarily before c in 𝛽  , while in the context of 𝛾  ′s premise b is possibly but not necessarily before c in 𝛿 . The necessity of 𝛼 cannot, of course, be seen in the completed diagram 𝛽  . But it can be seen before c is marked on 𝛽 . What is seen, specifically, are ranges of possible positions of c to both a and b. 𝛽 serves not only to depict each point of the configuration, but also the partition each point induces on the line—i.e. the region of points after the point, and the region of points before it. And crucially, it depicts the partitions induced by a and b together, in a single representation. This allows the reasoner to compare the regions, and thus ascertain the modal relationship between being positioned after a and being positioned after b. The region of points after b in the diagram is, literally, contained in the region of points after a. The reasoner thus sees that any possible position after b is also a possible position after a. She recognizes that in placing a point c after b to represent the second conjunct of the premise she is forced to place c after a. In contrast, she is not so forced in diagramming the second conjunct of the premise of 𝛾  . In 𝛿 , the region of points after a is not contained in the region after b, and the reasoner thus sees that c can fall before or after b. This procedure of diagramming provides, I propose, the ‘non-propositional factor’ that according to Stroud must exist in any full account of deductive inference. To perform 𝛼 , the reasoner must have in Stroud’s terms ‘the practical capacity’ to carry out the procedure. The key act is the combination of the separate geometric conditions in the premise into a single diagram.8 An obstinate, Tortoise-like reasoner could depict the premise of 𝛼 with two distinct diagrams

c a

b

b

and then claim that there is nothing preventing her to have c before a in a third diagram

a c

8   A similar procedure, termed integration of premises, is posited as fundamental in the psychological theory of reasoning presented in Knauff’s (2013). Knauff’s theory, in fact, is built around an account of inferences that exploit the transitivity of a linear order like 𝛼.

13

The appropriate response would be to ask her to reflect on the modality of depicting the two conjuncts of the premise together on one and the same line. If she still maintains earnestly that she is not forced to the conclusion from the premises, then one would be justified in concluding that she lacked the ability to engage in such reflection, and moreover that her grasp of the concept of an oriented geometric line was deficient. If such a concept exists at all, competence with it requires, in part, an understanding of how the spatial form of a line constrains the stipulation of order relations between points on it. Without this competence—i.e. without the competence to see that c cannot be stipulated to be before a once the premise of 𝛼 is diagrammed in a single diagram like 𝛽  —a person could not be said to possess the concept.

4 Concluding Remarks If the given account of 𝛼 is correct, then an inference of elementary geometry can be performed by diagramming it. This of course runs counter to the received modern view on the place of diagrams in geometry. Diagrams have been inextricably a part of elementary geometry since its ancient origins. Yet with the advent of modern axiomatic treatments of the subject at the end of the 19th century (e.g. Pasch 1882; Hilbert 1971) they came to be separated from the subject’s proofs. Their role, accordingly, is to illustrate proofs about geometric objects, not to justify anything within them. Recent work, beginning with Manders’ seminal (Manders 2008), has called this widespread view into question. Manders’ analysis shows that Euclid’s use of diagrams in the Elements is not open-ended and ad-hoc, but systematic and controlled. His diagrams serve to record and ground inferences regarding a specific kind of geometric information. In Manders’ terminology, this information concerns the co-exact features of geometric configurations—roughly their qualitative spatial properties and relations. In subsequent work, the proof systems Eu and E were developed to confirm this analysis in formal terms (see Mumma 2010 and Avigad et al. 2009). These proof systems provide with their rules a precise account of what can be inferred from the diagrams of elementary geometry. They do not however provide (nor did they set out to provide) any account of how the inferences are carried out. They do not, in the terms of Carroll’s dialogue, provide an account of how geometric diagrams force one to accept certain geometric conclusions. The analysis of 𝛼 given here can, I believe, be generalized to provide such an account. Consider for example the following inference 𝜖 , codified as a intersection rule 3 in E. (The corresponding rule in Eu is P15, in which L is a ray, not a line.)

Carroll’s Infinite Regress and the Act of Diagramming

γ

A diagram 𝜃 that depicts the premise of 𝜂 and the issue addressed by the conclusion is:

Point a is inside circle C. Point a is on line L. Line L intersects circle C.

X

A diagram 𝜁 that depicts the premises of 𝜖 is:

h1

h2

a

A

f B

C g

L C A relationship analogous to that spelled out between 𝛼 and 𝛽 above can be described between 𝜖 and 𝜁  . The moment to focus on in the production of 𝜁  , accordingly, occurs after the depiction of the first premise and before the depiction of the second. Against the background of a geometric situation where the first premise holds, the reasoner considers, simultaneously, two ranges of geometric possibilities with respect to line L: its incidence with a and its intersection with C. As a result, she in a sense sees that if she were to place L outside the second range—i.e. if she drew L so that it did not intersect C—she would also place it outside the first range—i.e. L would not be incident with a. Diagramming 𝜁 in this way reveals the modality of the geometric situation: the premises necessitate the intersection of L and C. The key feature of both 𝛽 and 𝜁 is the representation of the geometric issue addressed by the conclusion of 𝛼 and 𝜖 together with the premises of 𝛼 and 𝜖 . It is not simply the position of c in 𝛽 or the position of L in 𝜁 that is decisive. With 𝛽  , it is the relationship displayed between the realization of 𝛼 ′s premises and the possibility of c′s position above or below a. With 𝛾  , it is the relationship displayed between the realization of 𝜖 ′s premises and the possibility of L′s intersection or non-intersection of C. The same analysis can be applied to the other diagrammatic inferences codified as rules in Eu and E.9 The analysis, moreover, suggests an approach for developing a framework for diagrammatic inference in mathematics generally. Consider the following inference from category theory:

η

e

e : A → B equalizes f, g : B → C. e : A → B is monic.

9  For these rules in Eu and E, see Section  5.2 of ‘The technical notions of Eu’ at www.john.mumma.org/Writings.html and Section 3.4 of Avigad et al. (2009), respectively.

The issue of whether the arrow e is monic comes down to whether the arrows h1 and h2 are identical. The diagram displays these arrows along with the premise that e is an equalizer for f and g. It thus provides a means for evaluating how the premise bears on the issue of the conclusion. (Specifically, the diagram links the universal mapping property of equalizers to the the question of the distinctness of h1 and h2 .) Whether a framework along these lines could be fleshed out for the diagrams of category theory, and for mathematics generally, may be a fruitful question to explore. Another question that may also be fruitful to explore is whether an approach that recognizes non-logical deductive inferences can be extended beyond elementary geometry to other areas of mathematics. More specifically, can a subject matter specific, mathematical kind of modality be useful in providing a full account of how mathematical proofs prove? A successful proof of a mathematical claim shows that counter-examples to the claim are impossible. Yet the impossibility at issue is not always logical, at least on the face of things. The range of possibilities being considered at an inference step within a proof are often defined by the mathematical concepts that the proof concerns. Consider for example the inference

x2 is even. 4 divides x2 . √ that forms part of the standard proof that 2 is irrational. No counter-example to the inference is possible in the sense that there is no number x which makes the premise true and the conclusion false. The possibility at issue with the inference is thus arithmetical possibility, not logical possibility. This is not to deny that the inference can be given an illuminating logical analysis within an axiomatization of arithmetic. The point, rather, is analogous to that made with respect to 𝛼 above: the inference can be legitimately performed at the level of arithmetical possibility, whereby the reasoner’s understanding of arithmetical concepts is essential. This raises many questions of course. What does understanding of arithmetical concepts consist in? How does

13

J. Mumma

such understanding provide the basis for the judgment that it is impossible for an even square number not to be divisible by 4? In exploring them, and related ones in other areas of mathematics, we may find ourselves in a better position to explain to the Tortoise how the premises of a mathematical proof force acceptance of its conclusion.

Compliance with Ethical Standards  Conflict of interest  There were no sources of funding for this paper, and so no conflicts of interest exist.

References Avigad J (2003) Number theory and elementary arithmetic. Philosophia Mathematica 11:257–284 Avigad J, Dean E, Mumma J (2009) A formal system for Euclid’s elements. Rev Symb Log 2:700–768 Carroll L (1895) What the tortoise said to achilles. Mind 104:691–693

13

Detlefsen M (1992) Poincaré against the logicians. Synthese 90:349–378 Hilbert D (1971) Foundations of geometry. Open Court Publishing Company, La Salle Knauff M (2013) Space to reason: a spatial theory of human thought. MIT Press, Cambridge Manders K (2008) The Euclidean diagram. In: Mancosu (ed) Philosophy of mathematical practice. Clarendon Press, Oxford, pp 112–183 Mumma J (2010) Proofs, pictures, and Euclid. Synthese 175:255–287 Pasch M (1882) Vorlesungen über neure geometrie. B.G. Teubner, Leipzig Prawitz D (2005) Logical consequence from a constructivist point of view. In: Shapiro S (ed) Oxford handbook of philosophy of math and logic. Oxford University Press, Oxford, pp 671–695 Prawitz D (2009) Inference and knowedge. In: Pělis M (ed) Logica yearbook 2008. College Publications, London, pp 175–192 Prawitz D (2012) The epistemic significance of valid inference. Synthese 187:887–898 Prawitz D (2013) Validity of inference. In: Frauchiger M (ed) Reference, rationality, and phenomenlogy: themes from F ϕllesdol. Ontos Verlag, Heusenstamm, pp 179–204 Stroud B (1979) Inference, belief, and understanding. Mind 88:179–196