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Research in History and Philosophy of Mathematics The CSHPM 2017 Annual Meeting in Toronto Ontario Maria Zack
Research in History and Philosophy of Mathematics: The CSHPM 2018 Volume (Proceedings of the Canadian Society for History and Philosophy of ... et de philosophie des mathématiques) 1st Edition Maria Zack (Editor)
From Logic to Practice Italian Studies in the Philosophy of Mathematics Boston Studies in the Philosophy and History of Science 308 2015th Edition Gabriele Lolli
Annals of the Canadian Society for History and Philosophy of Mathematics
Société canadienne d’histoire et de philosophie des mathématiques
Maria Zack
David Waszek
Editors
Research in History and Philosophy of Mathematics
The CSHPM 2022 Volume
Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et
de philosophie des mathématiques
Series Editor
Maria Zack, Department of Mathematical Information & Computer Sciences, Point Loma Nazarene University, San Diego, CA, USA
The books in the series contain selected papers written by members of the Canadian Society for History and Philosophy of Mathematics. Founded in 1974, this society promotes research and teaching in the history and philosophy of mathematics, as well as in the connection between the two. Volumes in this series cover a broad range of topics from a variety of time periods and cultures. They will be accessible to anyone who has had exposure to mathematics at the university level and will appeal to scholars of the history and/or philosophy of mathematics, graduate and undergraduate students undertaking research projects, and anyone with a general interest in mathematics.
Maria Zack • David Waszek
Editors
Research in History and Philosophy of Mathematics
The CSHPM 2022 Volume
Editors Maria Zack
Mathematical, Information & Computer
Sciences
Point Loma Nazarene University
San Diego, CA, USA
David Waszek École Normale Supérieure-PSL Paris, France
ISSN 2662-8503
ISSN 2662-8511 (electronic)
Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques
ISBN 978-3-031-46192-7 ISBN 978-3-031-46193-4 (eBook) https://doi.org/10.1007/978- 3- 031- 46193- 4
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Preface
Since 1988, the Canadian Society for History and Philosophy of Mathematics (Société canadienne d’histoire et de philosophie des mathématiques) has been publishing a yearly collection of papers. One virtue of these volumes is that they offer a window into the thematic and methodological breadth of contemporary scholarship in the field. This is particularly manifest in this year’s installment, consisting of eight chapters which approach the history, philosophy, and teaching of mathematics from a great variety of angles.
***
The volume opens with two chapters whose concerns are primarily foundational or philosophical.
Nicolas Fillion reflects on the way logic is currently being taught to philosophy students. He argues that the reliance in philosophy courses on substantial mathematical tools, in particular elements of set theory, to rigorously ground logic makes the standard curriculum dangerously circular since axiomatic set theory itself cannot be grasped without some understanding of logic. Instead, Fillion proposes to start from a rejuvenated version of the Aristotelian idea of an argument schema and use this idea to develop a limited set of concepts from logic that is sufficient for the introduction of the axioms of set theory. Once set theory is in place, mathematical logic can then be taught as usual.
The question that underlies Mario Bacelar Valente’s chapter is the following: From the perspective of mathematical practice (rather than from that of logical foundations), what are mathematical proofs good for? His answer is that one of the important roles of proofs is to enable their readers to ascertain the correctness of the statement proved. To explain how they achieve this, he offers a cognitive model of the process of working through a proof. According to Bacelar Valente, mathematical proofs offer guidance for a reasoning process made of steps that are sufficiently small for each of them to elicit a strong metacognitive feeling of correctness. He tests his models on several proofs from ancient Greek mathematics, including the quadrature of a lune by Hippocrates of Chios.
One of the central goals of recent historiography of mathematics has been that of reconstructing the social, institutional, and cultural setting of past mathematics. This is quite difficult to do if looking solely at mathematical texts. The next three chapters showcase the diversity of sources and methods that can be enlisted to help place mathematical work in its appropriate context.
Megan Briers takes a quantitative approach, using data on mentions in encyclopedia articles to study collaboration in nineteenth-century British mathematics. Her chapter is in the spirit of previous efforts to reconstruct the collective dynamics of mathematics using, for instance, citation or co-authorship networks. Such work, however, usually relies on the onerous construction of tailor-made databases and on limited kinds of formal co-authorship or citation relationships. Instead, Briers shows how interesting patterns can be extracted by creatively exploiting information that is readily available online (in this instance, on Wikipedia and Wikidata, Wikipedia’s data back-end). Integrating young scholars into the profession is an important goal of the Society, and it is always a point of pride for us to be able, as in this case, to publish an author’s first paper.
Cynthia Huffman turns to imagery to study how the scientific contributions of a woman, Émilie Du Châtelet, were depicted and perceived in eighteenthcentury Europe. Her richly illustrated paper focuses on the frontispieces designed by publishers for books by Du Châtelet or connected to her, especially some books by Voltaire to which Du Châtelet is believed to have made substantial contributions. Huffman shows how these images served to credit Du Châtelet visually, even when she did not share formal authorship for the work, but in a way that marked out Du Châtelet’s contributions as distinctively feminine, combining traditional emblems of scientific investigations with the figure of the Muse.
Shawn McMurran and James Tattersall are working on a long-term project that studies how women slowly gained access to scientific education in Britain in the nineteenth and twentieth centuries. Their chapter in this volume fills a further blank in our knowledge of this protracted process: it focuses on women’s access to peer discussion societies, in particular the Cambridge Women’s Research Club (1919–1979). By exploiting institutional archives, including minutes of meetings, McMurran and Tattersall are able to reconstruct the club’s activities and get a sense of its role in socializing women into a community of researchers, at a time when full participation in existing male-dominated spaces would have been difficult. ***
The last three chapters in this volume underscore how valuable it can be to study the development and circulation of mathematics at very different scales, both temporally and geographically.
Emmylou Haffner puts, so to speak, individual texts under a magnifying glass. Through a variety of examples, ranging from Leibniz to Elie Cartan and Dedekind, she shows that much insight can be gained into the research and writing process of mathematicians by looking at their drafts and other preparatory notes, many of which have been preserved in archives, yet are rarely looked at by historians. Among other things, she argues, drafts allow us to see mathematicians taking notes and
reworking other sources; performing exploratory computations and playing with notation or diagrams as a research strategy; and re-elaborating their concepts while preparing their work for publication.
Roger Godard and Mark Lewis take as their focus two individual textbooks on vector analysis from the early twentieth century, an American one from 1901 by Edwin Wilson (based on lectures by Josiah Gibbs), and a German one from 1919 by Carl Runge. Comparing and contrasting them allows Godard and Lewis to highlight how differently a similar field could be covered, at roughly the same period, in distinct national and institutional contexts.
Finally, Jeffrey Oaks takes the long view in his wide-ranging chapter “The Legacy of al-Khwarazmı in History of Algebra.” What he offers is a kind of global history of one particular algebra treatise, the famous Kitab al-jabr wa-l-muqabala, tracing its reception through its numerous re-elaborations and translations over the course of about eight centuries—from later algebra books written in Arabic to van Roomen’s commentary at the end of the sixteenth century, through Latin translations, Italian trattati d’abaco, and German cossist treatises. ***
The 2022 annual meeting of the Society, at which a number of the chapters in this volume were presented, once more had to take place online because of the lingering effects of the coronavirus pandemic. In such conditions, the Annals have been all the more important in preserving the intellectual exchanges that are essential to any scientific society. We are grateful to our authors and reviewers; their hard work is what makes it possible for the Annals to continue to thrive.
San Diego, CA, USAMaria Zack Paris, FranceDavid Waszek
Editorial Board
The editors wish to thank the following people who served on the editorial board for this volume:
Amy Ackerberg Hastings Convergence, Mathematical Association of America
Eisso Atzema University of Maine, Orono
Christopher Baltus State University of New York, College of Oswego
Michael Barany University of Edinburgh, United Kingdom
Daniel Curtin Northern Kentucky University
Thomas Drucker University of Wisconsin, Whitewater
Craig Fraser University of Toronto
Cynthia Huffman
Pittsburg State University
Dirk Schlimm
McGill University
James Tattersall
Providence College
Glen Van Brummelen
Trinity Western University
David Waszek
École Normale Supérieure, Paris
Maria Zack
Point Loma Nazarene University
Contributors
Mario Bacelar Valente Pablo de Olavide University, Seville, Spain
Megan Briers Department of Mathematics and Statistics, University of St Andrews, St Andrews, UK
Nicolas Fillion Department of Philosophy, Simon Fraser University, Burnaby, BC, Canada
Roger Godard Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ON, Canada
Emmylou Haffner Institut des textes et manuscrits modernes, ÉNS/CNRS, Paris, France
Cynthia J. Huffman Pittsburg State University, Pittsburg, KS, USA
Mark Lewis Saint Lawrence College, Kingston, ON, Canada
Shawnee L. McMurran Department of Mathematics, California State University, San Bernardino, San Bernardino, CA, USA
Jeffrey A. Oaks University of Indianapolis, Indianapolis, IN, USA
James J. Tattersall Department of Mathematics and Computer Science, Providence College, Providence, RI, USA
Logical Methodology and the Structure of Logic Syllabi
Nicolas Fillion
Abstract This paper starts from a dissatisfaction with the way logic is currently taught in philosophy departments. In such a context, we would want logic to be developed from first principles, in such a way that it can then be applied to any other domain of study. But standard “rigorous” logic courses make it depend on a substantial amount of set theory, which itself seems to require notions from logic— confusing students, if not striking them as circular. Instead, this paper proposes to start from a rejuvenated version of the Aristotelian notion of an argument schema, and to use this idea to develop a limited collection of concepts from logic that is sufficient for the introduction of the axioms of set theory. Once set theory is in place, mathematical logic can be taught as usual.
Most colleges and universities offer at least one course in elementary formal logic. Such courses are often offered by philosophy departments, but they are also quite commonly offered by other departments such as mathematics, computer science, and linguistics. This fact reflects the plurality of reasons for which logic is included in educational curricula. Since a course syllabus ought to be responsive to the reason(s) for which the course is included in the curriculum, it is clear that distinct logic syllabi are suited to logic courses having different functions in educational curricula. This paper’s objective is to discuss some important respects in which the structure of elementary logic syllabi ought to be improved. The emphasis will not be on the selection of subjects that should be included in a well-conceived elementary logic course, which nowadays standardly includes curricular items such as the concepts of argument, argument form, validity, deduction, interpretation, etc., systems such as syllogistic logic, propositional logic, and predicate logic, their elementary metatheory, and symbolization of simple natural language expressions.
N. Fillion ()
Department of Philosophy, Simon Fraser University, Burnaby, BC, Canada e-mail: nfillion@sfu.ca
Instead, the paper’s objective is to make critical remarks about the organization of those curricular items or, in other words, about the structure of logic syllabi.1 For the critique to be well-grounded, it is first necessary to make clear what sort of logic course is targeted: logic courses devoted to students learning elementary logical theory and methodology, as opposed to logic courses that presuppose their knowledge. As I will argue, the pedagogical plan deployed in such courses fails to live up to the promises we make in the name of logic in that it presents a circular view of the discipline. I will also sketch an alternative plan that does live up to the promises of logic.
1 The Constitutive Promise of Logic
It may prima facie seem that all elementary logic courses are introduced in educational curricula to make students learn elementary logical theory and methodology. This, however, is not the case. Indeed, modern educational curricula typically contain multiple courses that cursorily introduce students to some elements of logic as a means rather than as an end. As such, the objective is for students to acquire the knowledge of some collection of logical facts relevant to the study of something other than logic, without there being a need for a comprehensive well-grounded perspective on what logic is and on how it works.
For example, in Altshuler et al. (2019), a textbook used at my institution for a third-year course on semantics offered by the linguistics department, a 42-page chapter introduces students to “symbolic logic.” Although the chapter does a fine job of introducing students to the rudiments of syntax and semantics for predicate logic (but not for the whole of first-order logic), it hardly teaches students even a minimally comprehensive view of logic. Indeed, the concepts of argument, validity, and deduction are not even mentioned,2 although logic is usually considered to be a discipline that primarily seeks to study the validity of arguments, where the validity of an argument is in most cases shown by means of a deduction. However, the logic syllabus it contains can hardly be criticized on that basis, for the purpose of the courseistosimplyintroducestudentstoafewofthemanyconcepts oflogicaltheory and methodology, i.e., those relevant to a linguistics course on semantics, rather than to present students a compelling and comprehensive picture of logic as a discipline.
As a second example, junior mathematics students are often offered a course (or part of a course) that contains elements of logic that—so it seems to be believed by
1 According to the Cambridge Advanced Learner’s Dictionary & Thesaurus, the definition of ‘syllabus’ is as follows: “(a plan showing) the subjects or books to be studied in a particular course, especially a course that leads to an exam.” For the purpose of this paper, I will use the word ‘syllabus’ to refer specifically to the list of subjects included in a course plan, and the phrase ‘structure of the syllabus’ to refer specifically to the organization of the subjects listed as part of the syllabus.
2 For another textbook commonly used in linguistics for the same purpose, and to which this observation applies, see Kratzer and Heim (1998).
curricula and syllabi designers—should suffice for students to become proficient at understanding and producing proofs of mathematical statements. At my institution, this is part of a first-year discrete mathematics course offered in mathematics & computer science. Such courses usually give fairly loose definitions of key logical terms such as ‘argument,’ ‘validity,’ ‘deduction,’ and ‘proof,’ and introduce students to truth-tables and the interpretation of simple quantified sentences. On that basis, a collection of logical facts that ought to be known is given, e.g., De Morgan laws, associative laws, distributive laws, double negation, contraposition, quantifier negation, etc. Logical deductions are presented as a process of going from sentence to sentence, making transformations based on equivalence rules learned as basic facts. For one of the best examples of this, see Velleman (2019). Logic is once again not presented as a discipline with a distinctively structured theory and methodology, but only as a means for having an ability to produce mathematical proofs.
This should suffice to show that there are courses teaching introductory logic that are not per se devoted to teaching logical theory and methodology; they cursorily introduce students to some aspects of logic with the aim of applying it in specific predetermined ways. As such it is sufficient to introduce students to a small number of logical facts and rules, and the time pressure associated with the needs to move on to other topics may justify not engaging with logic in a more comprehensive manner. For the objective is not to understand logic and how it relates to other intellectual disciplines, but only to become competent users of a few elements of logic. This is the full (and clearly limited) extent of the promises made in the name of logic in such courses. As such, although logicians will likely find their coverage of logic superficial and unsatisfactory, those courses are not ill-conceived. But the fact that such courses deliver on their promises merely reflects the fact that so little of the breath, depth, and richness of logic as a discipline is touched on. In other words, they deliver on their promises, but merely because they promise so little.
It is appropriate, at this point, to contrast such situations with one involving the more ambitious promises traditionally made in the name of logic. Let us start with the father of logic himself. Aristotle wrote six logical treatises that are collectively referred to as the Organon, following the edition of his work by Andronicus of Rhodes around 40BC; in what is not uncontroversially thought to be their chronological order, they are: Categories and Topics, Sophistical Refutations, On Interpretation, Prior Analytics, and Posterior Analytics (Boche ´ nski, 1961; Kneale and Kneale, 1962). We can appreciate the ambitious role promised for logic by considering what Aristotle says in the very first sentence of Topics about the method he seeks to develop:
The purpose of the treatise is to discover a method from which we will be able to reason about any problem set before us from accepted opinions and, when maintaining an argument, avoid saying anything self-contradictory. (100a163 )
The need for such a method became quite pressing in fourth century BC Athens— where Aristotle studied and taught—since, to an unprecedented degree, large bodies of argumentative discourses were found in written form. Such argumentative discourses intended to systematically establish some theses on the basis of other theses that were thought to be acceptable. In Aristotle’s time, this included flourishing bodies of mathematical knowledge (especially geometry), elaborate discussions of questions of metaphysics, cosmogony, and science, as well as debates of political and legal natures.
It is not the place to discuss why so much of it was present in Aristotle’s time, but it should be clear that a systematic methodology to scrutinize discourses involving many interrelated arguments would then feel desirable. Indeed, if one is to develop a method to adjudicate between different answers to legal, political, mathematical, or metaphysical questions, the method in question should not presuppose one of the answers to what is being questioned. Otherwise, using the method to adjudicate such questions would only be begging the question. Hence, such a method should remain neutral even with respect to some of the most fundamental metaphysical questions. For example, one of the hotly debated questions in Aristotle’s time was whether the Heraclitean view that nothing ever remains—everything being constantly changing—or the Parmenidean view that nothing ever changes—everything remaining constantly and necessarily the same—is correct. The early dialectical account of logic developed by Aristotle in the Topics was designed so as to not even make commitments with respect to such questions, so that the method could be used to weigh in with the intent of non-circularly determining the correct answer, this being done throughout without running into contradictions. Let us refer to this as the constitutive promise of logic. Indeed, we find Aristotle vigorously trying to fulfill this promise in the six treatises belonging to the Organon.
2 The Promise of Logic in the Liberal Arts Tradition
Until the middle of the nineteenth century, logic as a discipline was traditionally understood to be a development of the ideas first advanced by Aristotle. It must be said, however, that distinct traditions of Aristotelian logic were developed, partly as a result of the complicated history of the availability of the works of Aristotle in the centuries that followed his death (e.g. Lord, 1986; Spade, 2007). In some sectors of the historical literature, canonical Aristotelian logic was primarily devoted to the study and elaboration of themes such as Aristotle’s analysis of the predicative relation ‘to be’ in Categories and his account of the predicaments in Categories and Topics. It ought to be said that this endeavour has at times confused logic and metaphysics, in a way that did not live up to the constitutive promise of logic. In some other sectors of the historical literature, canonical Aristotelian logic was largely devoted to the complex theory of axiomatic knowledge that
Aristotle developed in the Posterior Analytics which, famously, was to a large extent implemented by Euclid in The Elements 4
There is another part of Aristotelian logic that has been universally accepted to be an essential component of the canon of Aristotelian logic, namely, the theory of categorical syllogisms presented in Prior Analytics I, 1–2 and 4–7. In the Medieval period, this is the logical theory that became encoded using the letters A, E, I, O, and the various mood names such as Barbara, Celarent, Darii, etc. (e.g., see Spade, 2007). Except for a few wrinkles concerning conversion per accidens and subalternation, this remains a universally accepted core element of logical theory.
The theory of syllogisms, and sometimes the other logical doctrines developed in the Organon, occupied a central place not only in the research-oriented scholarship, but also in the tradition of liberal arts education. This is because logic is both a science and an art. When we say that logic is an art, the word ‘art’ is not used to refer to the fine arts. Rather, it is used in a broader, classical sense which includes things such as fine art, industry, craftsmanship, skills, and even some things we now consider to be sciences. This sense has its roots in the ancient Greek word τέχνη— the etymological root of our current word ‘technique.’ Accordingly, the study of logic as an art is the study of the correct usage of a set of techniques governing correct reasoning. Bonevac (1990) puts the point as follows:
[...] logic is both an art and a science. Logic is concerned with constructing a theory of correct reasoning, making it a science. Indeed, modern symbolic developments have led to sophisticated mathematical theories of reasoning. But logic is also concerned with applying theory to practice, making it an art. Most people study logic to improve their ability to reason: to argue, to analyze, and to think critically about issues that concern them.5 [...]
But it’s important to remember that without theories of reasoning, there would be nothing to apply. The art of logic depends crucially on the science of logic. (p. vii–viii)
Just as was the case in Aristotle’s Organon, the objective remained that of developing an increasingly comprehensive method that lives up to the constitutive promise of logic, including a comprehensive treatment of both logical theory and of its applications.
Traditionally, the liberal arts together composed an entire educational curriculum. The education obtained from the liberal arts has been meant to be a general-purpose (as opposed to a technical or professional) education. In other words, the point of liberal arts is to equip people with the tools to articulate a general perspective that applies to everything—a point that directly echoes the promise made by Aristotle at the beginning of the Topics
It is thus not surprising that logic has occupied a central place in the traditional liberal arts curriculum, in such a way that it was not taught with particular disciplinary objectives in mind (in contrast to the linguistics and mathematics &
4 Such points of focus may appear somewhat strange, from a modern point of view, since these topics are not contemporarily considered to be parts of logic per se, but it is perhaps not as strange as it might prima facie seem; in any case, it is not the purpose of this paper to decide this question.
5 As one would expect, there seems to be empirical support for the idea that studying logic improves one’s skills in logic. See, e.g., Attridge et al. (2016).
Trivium
expression/organization of the intellect
developed by studying
grammar, logic, rhetoric
Fig. 1 The seven traditional liberal arts
Liberal Arts
Quadrivium
Intellect per se / Judgment
developed by studying
arithmetic, music, geometry, astronomy
computer science courses mentioned above). To better grasp the intended structure of the liberal arts curriculum, consider as an example the list and division of disciplines presented in Fig. 1, which have taken their form in the Middle Ages, as the teachings of Thierry de Chartres (c. 1150)—who anticipated the universities of the Renaissance—testify: “Philosophy has two main instruments, namely intellect and its expression. Intellect is illumined by the quadrivium (arithmetic, music, geometry, and astronomy). Its expression is the concern of the trivium (grammar, logic and rhetoric).” The study of the liberal arts is thus meant to develop the intellect in general, not merely as it applies to particular topics. As we see, the purpose is to have a method to argue about any topic whatsoever, without producing circular arguments or running into contradictions. This being said, my objective in what follows is not to examine syllabi proposals in this tradition, but rather to make critical remarks about the current approaches to teaching logic. The main purpose of the section was merely to make more visible the enduring importance of what I called the constitutive promise of logic. Moreover, in Sect. 4, I’ll suggest that we revive some aspects of the logical methodology found in the liberal arts tradition.
3 Premature Attempts at Rigour in Logic
Nowadays we do not teach logic as was traditionally done, owing to our reliance on mathematics from the start. It is certainly true that the scope of logic has expanded, as well as the rigour with which logical theory and methodology are developed. Although our typical courses sometimes touch upon topics from Aristotelian logic, they normally focus on sentential and predicate logic. Of course, the scope and rigour of such courses vary wildly. Some institutions have an introductory class that presents the material in a rather loose way, with an almost exclusive focus on the art component of logic. Sometimes such a course would be followed by a course designed toward the study of the science of logic. Such courses would typically be more mathematically involved, and would present what we “officially” (or, at least,
very commonly) take to be the rigourous treatment of elementary logic.6 It is with respect to this official take on how one ought to rigorously teach logic that I would like to raise the question: to what extent does our presentation of logical theory and methodology deliver on the constitutive promise of logic?
To be clear, I do not think that there is anything inconsistent or otherwise factually incorrect in what is presented. The purpose of asking the question is pedagogical: does it constitute a well-ordered curricular sequence, where assumptions and results are organized linearly and cumulatively, and especially in a way that avoids circularities? Does the presentation of elementary material presuppose nontrivial knowledge of things whose knowledge is itself grounded in the elementary material we seek to rigorously teach students?
The standard approach to rigorously teaching logic (especially first-order logic) relies on a few key metatheoretical strategies:
• Before logic is officially introduced, a preamble (or an appendix) “reviews” some of the notions of set theory needed. Among other things, it includes basic set-theoretic operations (∅, ∪, ∩, ℘ , etc.) and their properties, the definition of functions, the principle of mathematical induction, and sometimes something is said about uncountable sets.
• A language is recursively defined for a given primitive vocabulary; this enables us to assert that metavariables have a well-defined range.
• A semantics is provided based on an interpretation function that satisfies various recursive conditions for interpreting expressions containing primitive logical vocabulary.
• Validity is defined by quantifying over all possible interpretation functions: an argument ∑/C is valid if and only if there is no interpretation function under which v(S) = T for all S ∈ ∑ and v(C) = F
• Some set of rules of inference is proposed, which will be shown to be sound by induction over the recursively defined language. (The matter of completeness will not matter in what follows.)
Note that non-trivial elements of set theory are presupposed for such metatheoretical strategies to work. Firstly, the notion of function is presupposed. Given the standard definition of what a function is, it is at least necessary to understand sentences of predicate logic with three quantifiers and identity. This seems like an awkward thing to presuppose if the intent is to teach someone predicate logic. Secondly, the strategies presuppose the abstract and non-trivial notion that there is a determinate set of all functions from a set to another set; more specifically, that there is a determinate set of all interpretation functions satisfying given recursive conditions. Until the nineteenth century, I do not think such a notion was part of the arsenal of either logicians or mathematicians, let alone the knowledge that such a notion
6 Although I prefer to not mention examples of recent textbooks that adopt the approach I criticize in the paper, here are example of older but still widely used textbooks exemplifying the problematic approach in question: Mendelson (1964), Enderton (1972), and Bell and Machover (1977).
is logically unproblematic. But following a standard treatment of set theory (e.g. Halmos, 1960), one can show the existence and properties of this set of functions based on the following argumentation. Firstly, the axiom of union and the axiom of separation enable one to show that for any two sets A and B , the Cartesian product A × B exists. By applying the axiom of separation, functions can then be explicitly introduced as subsets of A × B such that every element of A is paired with at most one element of B . In addition to this construction, the power set axiom and the axiom of separation are applied to show that there exists a unique set that contains all the functions with domain A and codomain B (i.e., all the subsets of A × B associating with every element of A at most one element of B ), which is usually denoted B A . The definition of validity mentioned above quantifies over all interpretation functions and thus presupposes the existence of a set such as B A . Accordingly, teaching validity on the basis of such a definition presupposes an understanding of this sort of set-theoretic construction. But again, presupposing that a student who does not yet know predicate logic can have a hold of those axioms, of their correctness, and of how they ought to be used seems odd at best.
Thirdly, it is presupposed that the method of recursive definitions actually works, i.e., that there exists a set of objects that it defines, and that this set is unique. In his book on set theory, Suppes (1972) lays down two principles that definitions must satisfy: the criterion of eliminability and the criterion of non-creativity. The point of the second criterion is merely that “a definition should not function as a creative axiom” (p. 16). Later in the book comes the time of rigorously defining the notion of addition of natural numbers:
To illustrate the problems which face us, we may for the moment concentrate on defining addition. Given simply Peano’s axioms P1–P5 formulated in predicate logic with identity and without set theory, it may be shown (J. Robinson [1949]) that a proper, explicit definition of addition cannot be stated within this framework. The customary procedure is to adopt two further axioms:
P6. If x is a natural number, then x + 0 = x P7. If x and y are natural numbers then x + y ' = (x + y)'
After a brief discussion of the procedure in question, Suppes continues:
A pair of postulates like P6 and P7 is said to provide an inductive or recursive “definition” of addition. From the standpoint of the theory of definition [...], such “definitions” are not proper ones at all. [...] Recursive definitions, however, are close to being proper ones, and what we want to show is that given the additional apparatus of set theory, we can replace any recursive definition by an explicit, proper definition. (p. 137)
Thus, from the point of view of set theory, we cannot merely assume the correctness of recursive definitions. This is something to be proved; and the proof requires means that go beyond first-order logic. Once again, presupposing the correctness of mathematical induction and recursive definitions is an odd choice if the purpose is to rigorously teach first-order logic.
Some students who somehow succeed at pursuing their study of this style of rigorously presented logic may go ahead in their education and take a course on the foundations of mathematics. Perhaps they will take a fairly standard course in set theory such as Halmos’ or Suppes’. The course would include the following:
• A first-order symbolic language containing only one non-logical symbol, i.e., the binary predicate ‘∈,’ of which set theory is a theory.
• The logically non-trivial axioms of extensionality, separation, pairing, union, and power will then enable us to precisely describe the notion of an arbitrary function between two sets, and of the set of all such functions.
• Eventually, the axiom of infinity will be introduced and enable the definition of the set of natural numbers ω à la von Neumann. On that basis, it will be possible to prove the correctness of mathematical induction for countably infinite sets and to prove the recursion theorem, which legitimizes recursive definitions.
A student completing this curricular sequence will be in a position to make the following summary: We grounded our logic in set theory, recursive definitions, and induction; then we have become in a position to rigorously study set theory, which enabled us to show that set theory, recursive definitions, and induction are indeed justified.
This understandably often leaves students in a deep state of confusion, and they occasionally feel cheated. In the face of these observations, what should we reply to a student, if we are asked: are you not peddling vicious circles? How can you teach logic in this way, knowing full well that it fails so patently to live up to the constitutive promise of logic, i.e., that promise of developing a method to reason about any topic without running into circles or contradictions? Perhaps we should just admit that they are not entirely wrong. But if so, what would be a better way to proceed?
4 Revisiting the Traditional Methodology of Formal Logic
As we have seen, there is an important circularity in how contemporary logic is commonly taught—one that can reasonably trouble students in the process of learninglogic—inthatwhatoughttobeshownbymeansofourlogicalmethodology is instead assumed by the methodology itself. This difficulty is a symptom of a widespread tendency to present logic in a way that is too mathematically involved too quickly. That appeals to distinctively mathematical substantive facts ought to be postponed is required by the promise that logic be a method to discuss all topics, including mathematics, without incongruities such as contradictions and circularities. It is also required by sound pedagogy.
However, we should not exaggerate the gravity of the situation. Indeed, the existence of a circularity in the common structure of logic syllabi does not imply that there is a circularity in the foundations of logic, as there might very well be another way of characterizing the dependency relations between the various
elements of logic. As such, attempts at correcting our pedagogical oversights call for a different—and potentially better—point of view on the foundations of logic.
To begin, we must emphasize that, following modern terminology, it is not the same to say what alogic is and to say what logic itself is. For example, in their recent article on classical logic, Shapiro and Kouri Kissel (2022) do not introduce readers to logic itself (as it was classically thought of in the tradition), but rather to a system of logic known as ‘classical logic’ (as opposed to, say, three-valued or intuitionistic logic). Following this usage, the usual way of defining alogic (short for: a logical system) is to define a quadruple
〈L, I ,V,D 〉
where L is a set of well-formed formulas corresponding to a given operational type (tousethe terminology of Johnstone, 1987), I isan interpretation function assigning to each relevant expression an element of a set-theoretical structure rich enough to interpret a language of the given operational type, V is a validity relation (which is a subset of ℘(L) × L) that may be defined intensionally or extensionally, and D is a deduction system containing a given set of rules of inference and sometimes axiom schemata. This is in line with what was described in Sect. 3
Consider the quadruple corresponding to classical first-order logic. The purpose of introducing classical logic in this way is not to teach logic, for knowledge of logic is patently presupposed by the definition.7 Nevertheless, this is an immensely important mode of characterization of classical logic, for it enables the serious study of logical metatheory. Among other things, logical metatheory enables researchers to carefully re-examine their current and potential logical practices, so as to discover limitations of certain methods or new possibilities of application. From this point of view, the basic elements of logic are presupposed to be already known, but this is not a problem, for the knowledge of elementary logical principles is not in question. This being said, given the pedagogical concern of this paper, the simple way out is to not characterize logic itself in this way, i.e., to not require that L be a (recursively defined) set, that I be a (recursively defined) function, that V be a subset of ℘(L) × L, and that the metavariables occurring in the description of D be ranging over a predefined set of expressions contributing to the definition of L (e.g., terms, n-ary predicates, sentences, etc.). Indeed, none of those requirements were traditionally imposed on the methodology of formal logic.
Many textbook authors realize that, pedagogically speaking, the mathematically defined notion of validity is a non-starter for students not already knowing elemen-
7 I am confident that the textbook authors mentioned in footnote 6 would agree with this assessment. Their textbooks are clearly written with the assumption that students will be ready for this sort of discussion of logic. In contrast, this paper asks how a pedagogical sequence to get students in this state of readiness can be articulated, and emphasizes that for this to be the case validity cannot be introduced in terms of quantification over the set of all recursively defined interpretation functions.
tary logic. The predominant alternative is to define validity as follows:
An argument is logically valid if and only if it is not possible for all the premises to be true and the conclusion false. (Bergmann et al., 2009)
A large number of authors (e.g., Restall, 2004, Klenk, 2007, and Baronett, 2012) provide the same account. But this definition of validity is patently incorrect, unless one is inclined to reject the possibility of there being non-tautological necessary truths. Acknowledging the existence of such truths immediately leads one to see the problem with this account. Let us consider the example introduced by Kripke (1972), namely, that water is necessarily H2 O. Provided that this claim is factually the case, it is not possible for “I am currently drinking water” to be true and for “I am currently drinking H2 O” to be false, and thus the argument “I am currently drinking water, therefore I am drinking H2 O” would be deemed valid. Yet, it is not valid unless the necessary fact in question is explicitly stated as a premise. A similar situation would obtain by considering things from an Aristotelian point of view. If we consider humans, there would be a definition of the predicate ‘human’ stating its essence. Suppose that, as a matter of fact, the essence of human is rational animal. Then the argument ‘Socrates is a human, therefore Socrates is a rational animal’ would be deemed valid. Yet, it is not valid unless the statement of essence in question is explicitly added as a premise. Grounding elementary logical theory and methodology in this account of validity would be disastrous.
The natural course correction is to specify which narrower sense of necessity is required for validity to obtain: the necessity connecting the premises and the conclusion is linguistic necessity. With this modification, we would say that an argument is valid if and only if it is not possible in virtue of the meaning of the premises and of the conclusion that the premises be true and the conclusion false. Although this account is on the right track, the obvious problem is that the way in whichitmightbeoperationalizedisnotclearatall,fortherearemanydisagreements about the meaning of particular terms and phrases, let alone disagreements about meaning itself. Aristotle’s ground-breaking insight was that this difficulty can, to a large extent, be circumvented by only focusing on patterns of occurrence of a few syncategorematic words whose meaning can be safely assumed, or at least explicitly stipulated (as he does in Prior Analytics 1.2). This is the insight that gave rise to formal logic.
From the traditional point of view, the notions of form of a sentence and of argument form play the most central role in logical methodology. In this sense, ‘formal’ is not to be equated with ‘symbolic.’ Rather, the formal aspect of logic is to be understood in terms of schemas. For instance, whereas Medieval accounts of syllogisms heavily rely on the notion of mood (i.e., modus), Aristotle restricts himself to discussing “the arguments in the figures,” where the Greek term for figure is schema. 8 It seems that the same emphasis is found in the Stoics’ logical
8 E.g., “ἐν τούτῳ
σχήματι” (in this figure) in Prior Analytics I.4. For a more extensive discussion, see Smith (2020).
writing, both for sentential and for predicate logic (see, e.g., Bobzien and Shogry, 2020). Thus, perhaps the point of view on the foundations of logic needed for a pedagogically sound approach to logic lies in a reappraisal of the traditional role of ‘forms’ in logical methodology. It could be what is needed if, at any rate, it does a good-enough job of giving a self-contained account of validity, although perhaps limited in ways that need to be highlighted.
Here is an account of the notion of schema that it is in line with the traditional one, adapted from Corcoran and Hamid (2022):
A schema (plural: schemata, or schemas) consists of two things:
1. a linguistic expression composed of significant words and/or symbols and also of placeholders (such as blanks, ellipses, letters, syntactic variables).
2. a side condition specifying how the placeholders are to be filled to obtain instances (as well as the language, whether natural or symbolic, to which the instances of the schema are to belong).
What we obtain by substituting for the placeholders in a schema according to the side condition is an instance of the schema.
An equivalent term for ‘syntactic variable’ in the above is ‘metavariable.’ It is important that metavariables not be confused with what is commonly called ‘individual variables’ in modern predicate logic (i.e., x , y , z,...). Individual variables do not refer to anything in the domain of interpretation, but rather are said to range over the domain; with suitable rules, they are combined with quantifiers (∀, ∃)to express propositions about the domain of interpretation. In contrast, metavariables do not range over the domain; they are merely indications specifying how one ought to substitute linguistic expressions uniformly in order to obtain meaningful expressions. Importantly, this does not require a fully determinate set of objects from which substituents may be picked. Thus, for the notion of schema to play its key role in logical methodology, it is not required that we assume that linguistic objects suitable for substitution for a metavariable together constitute a determinate (perhaps recursively defined) set. If the conditions sufficient for substitution for each metavariable are satisfied, then the result is called an instance of the schema (or form).
Following this traditional methodology, an argument is said to be valid provided that it has a valid argument form. More precisely: it is valid provided that there is a valid argument form of which it is an instance. According to this definition, all that is required to acquire the knowledge that an argument is valid is that we find one valid argument form of which it is an instance. Arguments may be instances of multiple argument forms, some valid and some invalid, but this is irrelevant: as soon as a valid form of which the argument is an instance is found, we can conclude that the argument is valid. However, what if our search for such a valid form fails? It may be because there simply is no valid form of which this argument is an instance, or it may be that there is such a valid form but that we have been so far unfortunate. One limitation of the traditional approach to formal logic is that, owing to the fact that there is no fully determinate set of logical forms, it may not always be clear which of the two is the case. Of course, this will not be a
problem if we relativize our judgments to limited collections of logical forms (e.g., syllogistic, propositional, quantificational, alethic modal, etc.), for there will then be a small number of argument forms of which the argument might be an instance to consider. But this is what one usually does in contemporary logic as well (by imposing limitations on the logical vocabulary under consideration), and thus the traditional approach is not inferior to the modern one in this respect.
This sketch of the traditional formal account of the validity of arguments presupposes a definition of what it is for an argument form to be valid. Following this traditional methodology, an argument form is said to be valid provided that there is no instance of that form with true premises and a false conclusion. According to this definition, all that is required to acquire the knowledge that an argument form is invalid is that we find one argument of that form with true premises and a false conclusion. Thus, as soon as an argument of this form with true premises and a false conclusion is found, we can conclude that the argument form is invalid. What, however, if our search for such an instance fails? It may be because there simply is no such instance (in which case the argument form is valid), or it may be that there is such an instance but that we have been so far unfortunate (in which case the argument form is invalid). One limitation of the traditional approach to formal logic is that, owing to the fact that there is no fully determinate set of arguments to consider, it may not always be clear which of the two is the case. Of course, there nevertheless remains an important resource: for simple argument forms, we can analyze the side-conditions of the metavariables occurring in the form in order to determine whether an instance with true premises and a false conclusion is possible or not. This is fairly straightforward in the case of the simpler rules of inference of syllogistic and propositional logic. For more complex forms, the method of deductions can be relied upon to establish validity, but showing invalidity cannot be done in this way.
This methodology does not at once solve all the logical questions one may have. Indeed, there are many aspects of the syntax and semantics of the expressions under consideration that remain underdetermined. As such, many metatheoretical questions cannot at once be solved. Nevertheless, this methodology has at its disposal the resources required to acquire knowledge of a large swath of logical facts. It is indeed sufficient to construct a syllabus sequence that reaches the treatment of dyadic predicates in multiply quantified sentences, which are precisely the elements of logic required to introduce the axioms of set theory. Once axioms of set theory have been understood and the property of set-theoretic operators demonstrated, it then becomes pedagogically possible to use this very set-theoretical apparatus to re-examine the logical facts that are already known, with an eye toward metatheoretical concerns. This approach thus makes it possible for instructors to live up to the constitutive promise of logic—i.e., to provide a method to study all problems without circularity or contradiction. However, as we see, this curricular sequence resembles more the shape of a spiral than that of a straight line. It will perhaps occasionally lead to confusion and debate, but this is the sort of honest depiction of logical investigations that our students deserve.
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Ancient Greek Mathematical Proofs and Metareasoning
Mario Bacelar Valente
Abstract We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feelingknowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness.
1 Introduction
There is no unique definition of mathematical proof. Besides establishing the truth of a mathematical statement, a mathematical proof has many functions or roles. For example, convincing (that that is the case), explaining (how that is the case), and others (Dutilh Novaes 2020). Depending on the roles we might want to stress, we might adopt somewhat different definitions (see, e.g., Tall et al. 2021; Krantz 2011; Beck, Geoghegan 2010). However, it is still the case that if a mathematical argumentation is not correct, strictly speaking, it is not a proof. So, the defining role of a mathematical proof is that it establishes, or enables us to establish, the truth of a certain mathematical claim.
M.
B. Valente () Pablo de Olavide University, Seville, Spain
Depending on the context in which we address a mathematical proof, different definitions might be valuable. For example, we might consider the following “definition”:
A proof is a piece of discourse that puts forward a chain of arguments for public scrutiny, whose core function is to establish the truth of a mathematical claim.
Here, we will adopt a somewhat different working definition. The point is that a proof does not really establish the truth of a mathematical statement; the proof is an enabler for us to ascertain its truth. The reasoning is in us, not in the proof. As Rav emphasized:
Mathematical texts abound in terms such as “it follows from that” [ ] a mathematical proof in general only says that it follows, not why [it follows] [ ] why the consequent follows from the antecedents has to be figured out by the reader of a proof. (Rav 2007)
So, the proof consists of a sort of scaffolding of antecedents and consequents. And it is us who must carry out the reasoning connecting them and come to the realization that a consequent (a conclusion) does follow from an antecedent (a premise).
A definition that, in our view, stresses this aspect of a mathematical proof is as follows:
A proof is a piece of discourse that puts forward a chain of arguments for public scrutiny, whose core function is to enable the audience to ascertain the correctness of the chain of arguments.
Here, we want to address the question of how a proof functions, for us, as an enabler to ascertain the correctness of its argument. As we will see, this question relates to the issue of how we ascertain this correctness. In the present work, we will answer both of these related questions.
We will consider the most ancient proofs for which there are records. The reason is simple. If the approach developed here is to be of general application to informal mathematical proofs, it must work for the case of extant early proofs. If this is the case, then in the future, we might address how this approach works in the case of mathematical practices from other periods (which we will not do here).
The present work is structured as follows. In Sect. 2, we will look in some detail into a mathematical proof by Hippocrates of Chios. In Sect. 3, we will present the framework adopted in this work. We will consider a schematic model of intentional reasoning, in which metareasoning gives rise to a feeling of correctness associated with each reasoning process. We will address Hippocrates’ proof as a form of (guided) intentional reasoning. At this point, we will see how a proof functions, for us, as an enabler to ascertain the correctness of its argument, and how we ascertain its correctness. In Sect. 4, we will address, using this framework, two proofs from Euclid’s Elements and see how they can also be conceived as a form of (guided) intentional reasoning.
2 Hippocrates of Chios’ Proof of the Quadrature of a Lune
The earliest extant mathematical proofs are those of Hippocrates of Chios about the quadrature of lunes, which were made by Hippocrates around 450–430 BCE. We do not have Hippocrates’ text, but only an account of it in a text by Simplicius from the sixth century CE (Høyrup 2019a). Simplicius’ text reports on what Alexander of Aphrodisias wrote around 200 CE, and on what Eudemus wrote in the late fourth century BCE (or possibly a later version of this text) (Høyrup 2019a). Here, we consider the part of Simplicius’ text related to the older Eudemian text. We rely on the reconstruction by Becker of the Eudemian text and on Netz’s translation of it into English (Netz 2004).
It is said that Hippocrates taught geometry and wrote the first collection of elements of geometry. This collection, not yet in the axiomatic format of Euclid, “is likely to have been connected to Hippocrates’s teaching” (Høyrup 2019b). Most likely, it consisted of a loose collection of known results and techniques –which were taken for granted – and newer developments obtained using these (Høyrup 2019b). Among the newer developments, we might expect that there was Hippocrates’ quadrature of lunes.
Here, we will consider the proof of the first quadrature as given in the Eudemian account of Hippocrates’ text (the numbering is not part of the ancient text; it is included by Netz to ease reference). It is as follows:
(2) So he made his starting point by assuming, as the first among the things useful to the quadratures, that both the similar segments of the circles, and their bases in square, have the same ratio to each other [ ] [(4)] he first proved by what method a quadrature was possible, of a lunule having a semicircle as its outer circumference. (5) He did this after he circumscribed a semicircle about a right-angled isosceles triangle and, about the base, <he drew> a segment of a circle, similar to those taken away by the joined <lines>. (6) And, the segment about the base being equal to both <segments> about the other <sides>, and adding as common the part of the triangle which is above the segment about the base, the lunule shall be equal to the triangle. (Netz 2004)
Part 2 of the text mentions geometric knowledge that is useful for arriving at the intended result: both the similar segments of the circles, and their bases in square, have the same ratio to each other. This “similar segments principle” is the only background knowledge made explicit in the text. Not mentioned are the related Pythagorean rule or the simple arithmetic of areas (additivity and subtractivity of areas). According to Høyrup, this principle, like the Pythagorean rule, and the simple arithmetic of areas, “were known since well above a millennium [BCE] in Near Eastern practical and scribal geometry” (Høyrup 2019a).
Part 4 of the text specifies what quadrature is being considered, namely that of a lune having a semicircle as its outer circumference. Afterward, the text, in part 5, gives indications of how to draw that figure–alunehavingasemicircleasitsouter circumference. We circumscribe a semicircle about a previously drawn right-angled isosceles triangle. This gives rise to two segments of circle, each having a side of the triangle as its base (see Fig. 1 left). Then we draw about the base of the triangle
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Frank was in the box, while Bart Hodge adjusted his mitt behind the plate.
The batting-order of the two teams is here given: MERRIES. REDS.
Merriwell placed his foot upon the pitcher’s plate and prepared to deliver the ball. Every man was ready.
Frank was cautious about using speed at first, and he tried Jones on a slow drop.
Crack! The bat met the ball, and Jones lifted a pretty single just over the infield, prancing down to first like a long-geared race-horse, while the crowd gave a shout of satisfaction.
“The very first one!” laughed Morley. “Why, I knew it was a snap!”
“Mr. Umpire,” said Frank quietly, “if that gentleman is going to make remarks, kindly ask him to leave the players’ bench.”
“That’s right, Morley,” said the umpire, “you will have to keep still while you are on the bench.”
This caused the crowd to howl derisively, and it seemed that the Merries had very few friends present.
Davis was ready to strike, and Frank gave him a wide out drop. He let it pass, and Jones took the opportunity to hustle for second in an
attempt to steal.
Hodge took the ball, did not swing, but seemed to pull his hand just back to his ear, and then threw to second. It was a quick, easy throw, and it did not seem that Bart put enough force into it to send the ball down.
“Slide!” yelled the coacher.
Jones had been running like a deer, for he was the best base-stealer on the team, as well as the surest hitter. Forward he flung himself, sliding gracefully along the ground with his hands outstretched.
The ball came into Rattleton’s hands about two feet from the ground, and Harry had it on the runner when Jones’ hands were yet a foot from the bag.
“Man is out!” announced the umpire.
A hush fell on the crowd, and then somebody started the clapping, which was rather generous.
“Say, that catcher can throw!” cried a man on the bleachers. “Bet you don’t steal many bags on him to-day.”
The first man was out, and the Reds had been taught a lesson they would not fail to profit by. They had found that Hodge was a beautiful thrower, so that it was dangerous to try to steal.
“Hard luck, old man,” said several of the players, as Jones came in. “But he got you, all right.”
“And I thought I had a good start, too,” said Jones. “I’d bet my shirt I had that bag.”
One ball had been called on Frank. He tried a high one next time, and another ball was called.
Then Davis fouled, which caused the umpire to call a strike on him.
“Put another in the same place,” invited the batter.
Frank seemed to accommodate him, and Davis cracked it out, driving it past Carson, who did not touch it.
Another base-hit had been made off Merry
“That’s two of the five!” exclaimed Elrich, in satisfaction. “When three more are made I’ll have won one thousand dollars, anyhow.”
Croaker was a heavy hitter. Merry suspected it, and he tried his arts to pull the fellow, but three balls were called.
Davis had not attempted to steal, for he remembered the fate of the man ahead of him, and Merry held him close to the bag.
It seemed, however, that Frank was certain to give the next batter a base on balls. He was forced to put the ball over, which he did.
Mahoney, the captain of the team, had advised the batter to “play the game,” which prevented him from striking, although he afterward declared that the ball came sailing over the plate “as big as a house.”
A strike was called. Frank calmly put another in the same place, and it was another strike.
Croaker gripped his bat. The coachers warned Davis to run, as the batter would be out on the third strike, anyhow, if the first base was occupied.
So, as soon as Merriwell drew back his arm, Davis started hard for second. The ball was a swift high one, but Croaker met it and drove it out for a single that landed Davis on third.
“Here is where we score a hundred!” cried the coachers. “Oh, say! is this the wonder we have been hearing about?”
Hodge called Merry in, and said to him, in a low tone:
“Speed up!”
“But——” said Frank.
“No buts,” said Bart.
“Your hand.”
“I’ll hold them.”
“All right.”
Then they returned to their places.
“Down on the first one,” was the advice of the coacher near first. “With Davis on third, he’ll never throw to second.”
Frank sent in a swift in shoot, having compelled Croaker to keep close to first. Croaker, however, confident that Bart would not throw to second, scudded for the bag.
Hodge seemed to throw to Merriwell, and Frank put up his hands, as if to catch the ball, which had been thrown high.
Seeing Davis had not started from third, Frank did not bring his hands together, but let the ball pass between them over his head. The ball struck the ground about ten feet from second and bounded straight into Harry’s waiting hands.
The runner slid, but Harry touched him out, and then sent the ball whistling home, for, having seen the ball go over Merry, Davis had started to score.
Davis had been fooled into clinging close to the base too long. The trick had worked well, for Hodge had thrown the ball so that Merry could catch it in case Davis started, but with sufficient force to take it to second on a long bound, if Merry saw fit to let it go. Had Davis started, Frank would have caught the ball and cut him off.
Now, although Davis ran as if his life depended on the issue, he could not get home in time, and Bart was waiting for him with the ball.
“Out second and home!” cried the umpire.
The spectators gasped, for they had been treated to a clever piece of work that showed them the Merries knew a thing or two about baseball.
Three hits had been made by the first three men at bat, yet the side had been retired without a run, through the clever work of Hodge, Merriwell, and Rattleton.
The Reds were disgusted over the result, but Black Elrich said: “They can’t keep that up, and Merriwell is fruit for the Reds. Every man can hit him. Two more hits mean a cool thousand for me, and
there are eight innings to make them in.”
“They’re going to get twenty off him,” said Dan Mahoney. “My brother Pete is the worst hitter in the bunch, but he can lace that fellow all over the lot.”
On the bleachers Old Joe Crowfoot was grimly smoking his pipe, but by his side sat an excited boy, whose face was flushed and whose eyes shone.
“They didn’t get a run, did they, Joe?” asked the boy eagerly.
“Ugh!” grunted the Indian. “Don’t know. White man’s game. Injun don’t know him.”
“But they did hit the ball,” said Dick, in disappointment. “I didn’t think Frank would let them do that.”
“He throw um ball pretty quick,” said Joe.
“He’s afraid to do his best, I’m sure,” said Dick. “He’s afraid Hodge can’t catch it.”
“Hodge he heap big catch,” asserted Crowfoot. “Not afraid of stick when it swing. Him good.”
“We got out of a bad hole that time, fellows,” said Frank, as the team gathered at the bench. “If we keep on playing ball like that we’ll win this game.”
“Those fellows will know better than to chance such takes—take such chances,” said Rattleton.
“How is your hand, Bart?” asked Merry.
“All right,” said Hodge.
Ready had chosen a bat.
“I’m going to drive the first one over Old Baldy,” he said, with a motion toward the distant mountains. But he walked up to the plate and proceeded to strike out on the first three balls pitched.
“Speed!” he said, as he came back to the bench. “Whew! That fellow’s got it! They didn’t look larger than peas as they came over.”
Carson went out and fouled twice, getting strikes called on him. Then he drove a short one to the pitcher and was thrown out.
“See if you can’t start the ball rolling, Bruce,” urged Merry
Browning, however, did not seem much more than half-awake, and he, too, fell before the speed and sharp curves of Park, making the third man.
Favor took his place at the plate, and Merry faced him in the box. Frank gave the fellow a high one to start with, but Favor was confident and hit it safely past Ready.
“Four hits!” counted Elrich exultantly. “One more gives me a thousand.”
Before the ball could be fielded in Favor had reached second and was safe.
“Everybody hits him!” shouted a voice from the bleachers. “Is this the great Frank Merriwell?”
Tears of rage came into Dick Merriwell’s eyes, and his hands were tightly clenched.
“Why doesn’t he use the double-shoot?” panted the boy. “He hasn’t tried it once.”
Frank was as calm as ever. Gresham, a stout, solid-looking chap, grinned tauntingly as he took his place to strike. Frank tried to pull him, but two balls were called. Then Merry put one over the corner, and Gresham batted it down to Ready.
Jack should have handled the ball, but he did not get it up in time to cut Gresham off at first. Seeing he was too late, he took no chances of a wild throw, and did not throw at all.
“Oh, wow! wow!” roared the crowd. “All to pieces! How easy! how easy!”
Hodge was looking black as a thundercloud. The game was not pleasing him at all. Was it possible Frank has lost some of his skill?
Arata, a stocky young Indian, advanced to the plate. He showed his teeth to Merry, who gave him a pretty one on the outside corner.
Arata smashed it hard, driving it on a line over Frank’s head.
Like a flash Merriwell shot into the air and pulled down the ball with one hand. Like a flash he whirled round and threw to Rattleton.
As the bat met the ball, both Favor and Gresham had started to run. They did not realize Merry had caught the ball until Frank threw to second.
Rattleton took the throw, touched the bag and drove the ball whistling to first.
Gresham had stopped and was trying to scramble back to first, but the ball got there ahead of him, being smothered in Browning’s mitt.
“Batter out!” announced the umpire. “Out second and first!” It was a triple play!
Dick Merriwell flung his hat into the air, giving a shrill yell of joy. The yell was taken up by the crowd, for this was the sort of ball-playing to delight the cranks.
The Merries were fast winning friends.
The shout of applause having subsided, somebody cried:
“Why, you fellows don’t need a pitcher! You can play the game with any kind of a man in the box!”
Mahoney, the captain of the Reds, was sore, but he told his men that it would not happen again in a thousand years.
Gamp was the first hitter of the Merries, and the long youth from New Hampshire drove the ball out to Gresham, who made a very pretty catch.
Hodge hit savagely, but his temper was not right to connect with Park’s curves, and he fanned.
Then came Swiftwing. Again the collection of boys whooped like a lot of Indians from the bleachers. The Indian put up an infield fly, and
was out.
“Give us the double-shoot at your best speed, Merriwell,” said Hodge, in a low tone. “Just show these chumps you can pitch a little.”
“All right,” nodded Frank; “if you can handle it with that hand, you shall have it.”
“Don’t worry about me,” said Bart.
“Now,” said Dan Mahoney, “you’ll see my brother get a hit.”
“I hope so,” said Elrich, “for it wins the first thousand for me.”
Mahoney came to the plate. He had seen others hitting Frank, and he felt fully confident he could do so. Merry gave him a swift doubleshoot to start with, and he fanned, gasped, rubbed his eyes and looked amazed.
“Do that again,” he invited.
Merry did, and again he fanned.
The third one was a slow drop that dragged him, and he did not hit it. Frank had struck out his first man.
Park was not much of a hitter, and Merry found him easy, striking him out quite as easy as he had Mahoney.
“Those are the weak men!” cried somebody. “Now, let’s see you do it to Jones.”
Being thus invited, Frank sent in his prettiest double-shoot, and Jones missed the first one.
“Hello!” muttered Jones, as he gripped the bat. “That was a queer one. If I didn’t know better, I should say——”
He did not mutter what he should say, for Frank was ready and another came buzzing past, only the curves were reversed.
Again Jones bit at it and failed to connect.
“Two strikes!” called the umpire.
“Oh, he’s doing it now!” breathed Dick Merriwell, in delight.
Every ball hurt Bart’s hand, but he held them all and showed no sign of pain.
Jones was mad and surprised, which made him easy for the third double-shoot, and he, like the two before him, struck out.
Not one of the three men had even fouled the ball.
“Well, well!” roared a spectator. “It seems that you’ve got a pitcher there, after all!”
“Thanks, most astute sir,” chirped Ready, doffing his cap and bowing.
“He hasn’t begun to pitch yet. He’s just getting warmed up.”
CHAPTER XXX. ONE
TO NOTHING.
It was the beginning of the ninth inning, and neither side had scored. Never before had there been such an exciting game in the city of Denver. The crowd was throbbing, and Merriwell’s team had won a host of friends by its clever work. Since the second inning, however, Frank had given his men no chance to show what they could do, for he had struck out man after man, just as fast as they came up. Never in all his life had he been in better form, and his work was something to amaze his most intimate friends.
Bart Hodge, with his arm paining him from the tip of his fingers to the shoulder, looked very well satisfied.
Dick Merriwell was wild with delight and admiration. He heard the crowd wondering at the work of Frank and cheering at it, and it warmed his heart toward the brother he had once thought he hated.
“Oh, Joe!” he panted, “did you ever see anything like it?”
“Ugh! No see before,” answered Crowfoot, still smoking.
“Isn’t it fine?”
“Heap big noise. Ev’rybody yell lot; nobody get killed yet.”
Three times had Merriwell’s men reached third, but, by sharp work, the home team had kept them from scoring. Now, however, Morley was desperate, and he went among the men, urging them to win the game.
“You must win it!” he said. “Elrich loses five thousand and five hundred dollars if you don’t. He won’t back the team another day. We’ll have to disband.”
“We’d win if we could hit that devil in the box,” said Mahoney bitterly. “He’s the worst man we ever went up against, and we all know it
now You’ll never hear me tell anybody after this that there is no such thing as a double-shoot. Why, that fellow can throw regular corkscrew curves!”
Morley swore.
“You’re quitting!” he growled.
“Did you ever know me to quit?” asked Mahoney angrily.
“No, but——”
“Then don’t talk! They have not scored, and we may be able to make this a draw game, if we can’t get in a run.”
Black Elrich was worried, although his face looked perfectly calm, with the strained expression of the gambler who is unchangeable before victory or defeat. At his side, Dan Mahoney was seething.
“Hang it!” he grated. “If it had only been that catcher’s right hand! The woman made a terrible blunder!”
“No one would have thought him able to catch, anyhow,” said Elrich.
“The big mitt protects his hand.”
“Still, it must hurt him every time the ball strikes, for Merriwell has been using all kinds of speed.”
Morley came up to the place where he knew Elrich was sitting.
“What do you think?” he asked, in a low tone. “The boys can’t hit Merriwell, and it’s too late to try to buy Harris, the umpire, now. Can’t you start a riot and break up the game?”
“If you start it, it is worth a hundred dollars to you,” said Elrich, “even though that will throw all bets off, and I’ll make nothing. What say?”
“I can’t!” muttered Morley. “If I did so, Harris would give the game to the other side, and you’d lose just the same. If the spectators start it, it will be all right.”
“The spectators won’t,” said the gambler. “More than three-fourths of them are Merriwell men now.”
“Then,” said Morley, “I am afraid for the result.”
Well might he be afraid. In the last inning Frank was just as effective as ever, and the batters fell before him in a way that was perfectly heart-breaking to the admirers of the home team. Denver was unable to score in the ninth.
“We must shut them out again, boys,” said Mahoney, as his men took the field.
But Merriwell’s team went after that game in their half of the ninth. Carker was the first man up. He had not been hitting, and Park considered him easy. That was when Park made a mistake, for Greg set his teeth and laced out the first ball in a most terrific manner
It was a clean two-bagger. But Carker tried to make it three, encouraged by Ready on the coaching-line. Ready believed in taking desperate chances to score, and he waved for Greg to come on.
The crowd was standing again, shouting wildly as Carker tore across second and started on a mad sprint to third.
The center-fielder got the ball and threw it to Mahoney at second. Mahoney whirled and shot it to third.
“Slide!” shrieked Ready
Greg heard the command and obeyed, but Croaker took the ball and touched him easily.
“Runner out!” decided the umpire clearly.
Then there was another roar from the bleachers.
Jack Ready fiercely doubled his fist and thumped himself behind the ear.
“All my fault!” he moaned. “I did it!”
Carker looked sorrowful.
“My last game of baseball,” he said sadly. “I do not care to play the game any more. It is a deception and a humiliation. No more! No more!”
Merriwell was the next batter. Park knew Merry was a good hitter, and he was cautious. Frank did his best to work the pitcher for a
base on balls, but, with two strikes called on him, he was finally forced to hit.
He did so sharply, sending the ball shooting along the ground between third and short.
Frank crossed first and turned to the left, knowing it was best to have all the start he could if there was any show of making second.
“Go on!” roared Browning, who had reached the coaching-line at first, Ready, having come in from near third.
Then Frank ran at his best speed. He knew it would be close, and he flung himself forward for a slide at second, which enabled him to reach the base safely a moment ahead of the ball. By fast running, he had made a two-bagger out of an ordinary single.
Everybody knew now that Merriwell’s team was out for the game in that inning if there was any possible way to capture it. Such work turned the fans into howling maniacs.
For once in his life, Jack Ready looked grave when he took his place to strike. He realized the responsibility on him, and it had driven the smile from his ruddy face.
Park was pitching at his best, and he did not let up a bit. Ready made two fouls, after which he put up a high infield fly, which dropped and remained in the hands of Croaker. Two men were out, and the admirers of the home team began to breathe easier.
Merriwell was taking all the start he could get from second when Carson got ready to hit.
Park seemed to feel absolutely sure of retiring the side without further trouble, and he did get two strikes on Berlin. Then something happened, for the cattleman’s son did a thing to delight the heart of his father. He made a beautiful safe hit to right field and won the game.
Merriwell was running when the ball and bat met. He knew it was not a high fly, and instinct told him the fielder could not catch it. As he came toward third, Hodge was on the coaching-line, madly motioning for him to go in.
Frank obeyed. The fielder threw from right to cut him off at the plate, but, by another splendid slide, he scored.
The game was over
In the newspaper accounts of the game the following day Merriwell’s team was highly praised, and the reporters took pains to mention that it was the hit of Berlin Carson, a Colorado lad, that brought in the winning run.
THE END.
“BEST OF ALL BOYS’ BOOKS”
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