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## viernes, 15 de junio de 2012

### John Mumma · Marco Panza Diagrams in mathematics: history and philosophy

Synthese (2012) 186:1–5

DOI 10.1007/s11229-011-9988-3

1 Introduction

Diagrams are ubiquitous in mathematics. From the most elementary class to the most

advanced seminar, in both introductory textbooks and professional journals, diagrams

are present, to introduce concepts, increase understanding, and prove results. They

thus fulfill a variety of important roles in mathematical practice. Long overlooked by

philosophers focused on foundational and ontological issues, these roles have come

to receive attention in the past two decades, a trend in line with the growing philosophical

interest in actual mathematical practice. Seminal contributions include the

historical/philosophical analysis of diagrams in Euclid’s geometry offered byManders

in his 1995 paper (2008),The date of publication of Manders’ paper is misleading.

Though it was not published until 2008, it was written in 1995 and widely circulated,

and became highly influential as a manuscript. the logical studies of diagrammatic

reasoning contained in Gerard Allwein and Barwise’s compilation (1996), and Netz’s

historical study on the place of diagrams in Greek geometry (1999).

These works exhibit a broad range of intellectual perspectives. To bring people

from these separate perspectives together, two workshops devoted to the history and

philosophy of diagrams inmathematics were jointly organized by Departments of Philosophy

and Classics at the University of Stanford, and REHSEIS (a research center

of the CNRS and the University of Paris 7). The first workshop was held at Stanford

in the fall of 2007, the second in Paris in the fall of 2008. The present special issue is

the direct result of these workshops.

Because of the central position of Euclid’s geometry, in both the history and philosophy

of mathematics, the diagrams of Euclid’s geometry have been a major topic in

the recent research on diagrams inmathematics. This is reflected in themake-up of the

issue. Part I, which contains over half of the contributions to the issue, concerns Euclidean

diagrams, addressing either their place and nature in the manuscript tradition, their

role in Euclid’s geometry, or their relevance to early modern philosophy and mathematics.

The contributions of part II examine mathematical diagrams in more modern

and/or advanced settings. In the remainder of this introduction, we briefly describe the

content of these papers.

The two articles included in section I.a are devoted to geometric diagrams in the

manuscript tradition of ancient Greek texts. As Saito points out in the beginning of

his article, the diagrams we find in modern versions of ancient Greek mathematical

texts are far from representative of those appearing in the manuscripts from preceding

centuries. These manuscripts result from various scribal traditions that exhibit various

practices in the production of diagrams. The general aim of both Saito’s article and

Gregg De Young’s is to illuminate significant features of such practices.

Saito’s concern is with how the diagrams of the manuscript tradition relate to the

geometrical content of the arguments they accompany. They often satisfy conditions

not stipulated to hold (e.g. by instantiating a parallelogram as a square) or alternatively

misrepresent conditions stipulated or proven to hold (e.g. by depicting angles

stipulated as equal as clearly unequal). There are, further, notable features with the

diagrams representing reductio ad absurdam arguments, and the diagrams representing

arguments that range over many cases. To shed light on the different manuscript

traditions in these respects, Saito presents a case study of the manuscript diagrams

that accompany two propositions in book III of Euclid’s Elements.

De Young focuses on a specific manuscript tradition. He explores the basic “architecture”

(i.e. the relation of diagrams with white space and text) of medieval manuscripts

and early printed editions of Euclidean geometry in the Arabic transmission.

Many features of Euclidean diagrams remain constant through a long transmission history,

although differences in scribal abilities exist. Yet there also seems to be a degree

of freedom to adapt diagrams to the architectural context. Adaptation to print brings

subtle changes, such as title pages, page numbers, and fully pointed text. The demands

of a long-held calligraphic aesthetic ideal and the necessity to compete against traditional

manuscripts within the educational marketplace combined to favor the use of

lithography over typography in Arabic printed geometry.

The articles included in section I.b all share the general goal of deepening our philosophical

understanding of the diagrams of Euclid’s geometry. In his paper, Marco

Panza advances an account where geometric diagrams are fundamental to Euclid’s

plane geometry in fixing what the objects of the theory are. Specifically for Panza

geometric diagrams perform two indispensable theoretical roles. First, they provide

the identity conditions for the objects referred to in Euclid’s propositions; second,

they provide the basis via their concrete properties and relations for the attribution

of certain properties and relations to these objects. After explicating this conception

in general terms, Panza demonstrates what it amounts to vis-à-vis the Elements with

a detailed and thorough analysis of the definitions and the first 12 propositions of

book I.

Aconsequence of Panza’s account is that Euclid’s geometry is in a sense a constructive

theory. It does not concern a pre-existing domain of objects, but instead objects

brought into existence by the constructions of geometers. The issue of John Mumma’s

paper is the extent to which such an interpretation of Euclid can be given in formal

terms. At the center of his discussion is a proof system he developed where Euclid’s

diagrams are formalized as part of the system’s syntax. Mumma argues that his formalization

is superior to others in accounting for the spatial character of the theory’s

constructions. It is not immediate, however, that the resulting picture of the theory

qualifies as a constructive one. Mumma closes the article by pointing to the philosophical

work that is needed for it to be understood as such.

A prima facie feature of the diagram based reasoning of Euclid’s geometry is that

perception is involved in a distinctive and essential way. But how is it so involved? The

purpose of Annalisa Coliva’s paper is to develop the philosophical concepts necessary

to address the question. The key concept for Coliva, as the title of her paper indicates,

is that of seeing-as. A segment joining opposite vertices of a square in a diagram can,

for instance, be seen as a diagonal of the square, or the side of a triangle composing

the square. Coliva articulates seeing-as as philosophically sharp notion, and argues

that the ability is required to engage in diagram based geometric reasoning. She also

offers some initial thoughts on how her analysis bears on the status of geometrical

knowledge with respect to the a priori versus a posteriori and analytic versus synthetic

distinctions.

In her paper, Sun-Joo Shin considers Euclid’s diagrammatic arguments in order to

examine commonly held views on the strengths and weaknesses of diagrams in comparison

with symbolic/linguistic representations—namely the views that the latter are

superior for giving proofs and the former are superior for brainstorming and discovery.

Shin relates the first view to Locke and Berkeley’s opposing accounts of the generality

of Euclid’s proofs, and argues that Berkeley’s account offers a way for understanding

diagrams as a legitimate means for general proofs. She proposes that a major reason

behind the second view, aptly illustrated by Euclid’s proofs, is that diagrammatic representations

of individual mathematical objects are effective in generating the right

associations and connections with what is already known.

Section I.c contains papers that explore how the diagrammatic method of Euclid’s

geometry influenced and/or was interpreted by thinkers in the early modern period.

Though mathematics was pushing far beyond classical Greek geometry in this period,

Euclid’s Elements nevertheless maintained its position as a foundational text and thus

formed part of the intellectual context inwhich philosophical or methodological reflections

on mathematics were carried out.

Such reflections in Hume’s Treatise form the subject of Graciela De Pierris’s paper.

In part II of Book I of the Treatise Hume asserts that perceivable yet indivisible minima

compose geometric continua, but do so in a confused manner. Consequently, for

Hume, geometry does not possess the exactness of arithmetic, which concerns clearly

distinct discrete quantities. De Pierris argues that these doctrines follow in a principled

way from an epistemological model in which phenomenologically given sensory

images are fundamental. She also explains how Euclid’s geometry for Hume, despite

its relative inexactitude, qualifies as a demonstrative science in being based on simple,

easily surveyable diagrams.

In her paper, Katherine Dunlop explores the role of diagrams in Newton’s Principia

by considering Newton’s claim that the reasoning of the book is distinctively

geometrical. She notes that Newton conceives geometry as the science of measurement,

and argues that his characterization applies to the propositions of Section II,

Book I. As other scholars have noted, these constitute a foundation in the sense that

they show certain spatial quantities to be related as measures to nonspatial quantities,

in particular, force and time. Dunlop takes Newton’s point to be that the measures

he supplies can be manipulated in the same way as elements of diagrams in classical

geometry. In the case she examines in detail, Newton’s proof of Kepler’s area law, the

principal measure of time is just the diagrammed trace of an inertial motion. Dunlop

contends that Newton aims to share in geometry’s certainty by resting determinations

of equality and relative size (between nonspatial magnitudes) on relationships that are

open to sensory inspection.

In the Critique of Pure Reason Kant often illustrates his claims with geometric

proofs from Euclid’s Elements. One thus might look to recent work on the diagrammatic

character of these proofs to gain a better understanding of Kant’s philosophy of

geometry. In his article, Michael Friedman argues that such a project would be misdirected.

He argues specifically that Kant’s notion of a schema for geometrical concepts

and his distinction between pure and empirical intuition precludes a diagrammatic

interpretation of his theory of geometrical intuition. Such an interpretation would fail,

for Friedman, to explainwhy Kant takes the constructions of Euclid’s geometry to provide

an a priori framework for empirical space. Friedman relates these points to Kant’s

theory of space, and the role of geometry and spatial intuition in the transcendental

deduction of the categories.

Dominique Tournès’s paper, the first of section II.a, serves nicely as bridge between

the papers of part I and those of part II. In the article Tournès provides a survey of

the changing role of diagrams in the history of differential equations from the seventeenth

century—when there were still vestiges of the diagrammatic approach of

Euclid’s geometry—up until the time of Poincaré—when the influence of Euclid on

mathematics was entirely absent. The article is a useful resource for those interested in

understanding diagrams inmathematical subjects that are not ostensibly or exclusively

geometrical in content.

The other contributions of section II.a examine diagrams in the thought and work

of two giant figures in the history and philosophy of modern mathematics.

Danniele Macbeth’s proposes a new way of reading Frege’s logical notation, as

a sort of diagrammatic language within which to exhibit the contents of concepts.

Taking Frege’ s proof of theorem 133 in Part III of his 1879 treatise Begriffsschrift as

an illustrative example, Macbeth aims to show just how a proof in Frege’ s language

(so read) can extend our knowledge despite being strictly deductive.

In his paper, Ivahn Smadja seeks to clarify how Hilbert understood the relative

positions of diagrams and axioms in mathematics. An easy, and overly simplistic,

characterization of Hilbert on the issue would be just to attribute to him the view the

view Shin examines in her contribution—i.e. diagrams belong to the realm of heuris-

tics and discovery while axiomatics belong to the separate realm of justification. That

it would be overly simplistic is revealed by Smadja’s study of the connections between

the diagrams found in Minkowski’s work in number theory and Hilbert’s work in the

foundations of geometry. Smadja specifically describes the links the former have to

Hilbert’s axiomatic investigations of the notion of a straight line, and Hilbert’s concern

with the conceptual compatibility of geometry and arithmetic.

The papers included in section II.c, the last group of papers, concern contemporary

mathematics and linguistics. At the heart of Solomon Feferman’s discussion are

infinite diagrams for the proofs of central theorems in set theory, model theory and

homological algebra. The diagrams are infinite in the sense that they depict infinitely

iterated constructions. Feferman contends that they are more or less essential for

understanding and accepting the proofs, and considers whether this serves as evidence

against the so-called formalizability thesis: the idea that allmathematical proofs admit

of formalization. He argues for the conclusion that it does not in the light of recent

discussions of the formalizability thesis.

Sketch theory, a mathematical theory that treats the diagrams of category theory

as mathematical objects, is the topic of Brice Halimi’s paper. Halimi presents a technically

sophisticated discussion of the theory in order to bring out its relevance to a

variety of philosophical and foundational issues. These are: the capacity of a mathematical

diagram (in category theory and elsewhere) to represent not just a static object

but also the dynamics of a proof; the role and nature of diagrams within category

theory; and the alternative ways category theory and set theory provide a semantics

for mathematics.

A general point of many of the papers discussed so far is that diagrams can have

important theoretical functions within mathematics. Arancha San Gines’s paper provides

evidence that the same is true in linguistics, in the most direct way possible.

In it she presents a diagrammatic account of the role played by anaphoric pronouns

in natural discourse. The account, specifically, is a diagrammatic system of representation

for sentences containing such pronouns. After motivating and describing the

system, she shows how it accounts for a variety of linguistic phenomena that have so

far resisted uniform treatment.

John Mumma & Marco Panza

Stanford, Paris, May 2011

References

Allwein, G., & Barwise, J. (Eds.). (1996). Logical reasoning with diagrams. New York: Oxford University

Press.

Manders, K. (2008). The Euclidean diagram (1995). In P. Mancosu (Ed.), The philosophy of mathematical

practice (pp. 80–133). New York: Oxford University Press.

Netz, R. (1999). The shaping of deduction in Greek mathematics. A study in cognitive history. Cambridge,

USA: Cambridge U. P.

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