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Mathematics as Sign (Writing Science) by Brian Rotman
Two features of mathematics stand its menagerie of seemingly eternal objects (numbers, spaces, patterns, functions, categories, morphisms, graphs, and so on), and the hieroglyphics of special notations, signs, symbols, and diagrams associated with them. The author challenges the widespread belief in the extra-human origins of these objects and the understanding of mathematics as either a purely mental activity about them or a formal game of manipulating symbols. Instead, he argues that mathematics is a vast and unique man-made imagination machine controlled by writing. Mathematics as Sign addresses both aspects—mental and linguistic—of this machine. The opening essay, “Toward a Semiotics of Mathematics” (long acknowledged as a seminal contribution to its field), sets out the author’s underlying model. According to this model, “doing” mathematics constitutes a kind of waking dream or thought experiment in which a proxy of the self is propelled around imagined worlds that are conjured into intersubjective being through signs. Other essays explore the status of these signs and the nature of mathematical objects, how mathematical ideograms and diagrams differ from each other and from written words, the probable fate of the real number continuum and calculus in the digital era, the manner in which Platonic and Aristotelean metaphysics are enshrined in the contemporary mathematical infinitude of endless counting, and the possibility of creating a new conception of the sequence of whole numbers based on what the author calls non-Euclidean counting. Reprising and going beyond the critique of number in Ad Infinitum, the essays in this volume offer an accessible insight into Rotman’s project, one that has been called “one of the most original and important recent contributions to the philosophy of mathematics.”
- GenresPhilosophyMathematics
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First published August 1, 2000
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Displaying 1 - 4 of 4 reviews
December 20, 2024
A specialized but fairly accessible book about mathematics and the way that it operates semiotically. Rotman’s principal aim in the book is to examine how we talk about mathematics and use math and numbers to make assertions about the world and how those mathematically-supported assertions appear to draw upon different kinds of truth conditions.
Rotman begins by tracing out the meaning of math along three different positions. One is a formalist position through which we can see mathematics as a rule-based practice of manipulating marks on paper. Another is an intuitionist position in which numbers and the mathematical syntax bringing them together are abstract and rational concepts without necessary physical signification in the world. Then there is the Platonist position in which numbers and math are idealistic forms that can be used, heuristically, to engage with the world and make truth statements about it. Because we use mathematics in sentences about the world — we name numbers as nouns and assert relationships between them as verbs it is clear that we use mathematics semiotically, to express meaning to others.
To develop his semiotic understanding of mathematics, Rotman relies on C.S. Pierce whose work on signs specified how communication utilizes language that asserts different kinds of relationships between the sign and the thing it references. Pierce discuss icons (isomorphic reference) symbols (non-necessary conventional or customary reference) and indexes (necessary, factual or physical reference). Because mathematics is a part of the world and because it requires engagement by a reader/counter/manipulator mathematics operate as all three kinds of sign.
Math and numbers can be thought of as referents existing in three arenas: Code (including mathematical symbols and nomenclature) which is spoken by a Subject who utilizes numbers and can speak their particular formalist meaning (e.g. addition, subtraction, multiplication, commutation, distribution etc.); meta-Code (related objects, diagrams, visualizations, objects, outcomes, that are implied by or epiphenomenal with the Code) spoken by the Person through stories, narratives, pictures, legitimating the number in natural language; Virtual Code (what is valid, derivable, and possible based on the math) spoken by the Agent in terms of what is possible or idealized (pp.51-52).
Although Rotman does not use these terms, he is describing the use of mathematics as speech acts, where people use words not just to assert something that is true about the world but also to declare things to be true or to create relationships with things in the world. In John Searle’s words, the speech acts assert either a word to world or world to word direction of fit. As a Subject one may assert something mathematically true and as a Person insert that mathematical expression into discourse where it means something derived from a formalistic understanding of the math but also something that is specifically or potentially true in a given contextual moment.
Rotman pushes against the notion of seeing mathematics and numbers as being things that are true in and of themselves and in doing so, seems to borrow on another aspect of speech acts, which is the “condition of satisfaction.” For mathematics to be utilized semiotically in the production of propositions it needs to become meaningful against the conditions of satisfaction that make it true. For example, if I say that I have two apples and you have two apples and I give (add) my two apples to you then you will have (equals) four apples. The conditions of satisfaction is that upon completion of the operation, you have four apples. Without this grounding in the world we are left to deal with ever more abstract, context-free numbers that eventually move beyond the realm of experience and in doing so stop being “true” or semiotically meaning in the sense that they are possible: we may be able to process the numbers as code and virtual code but only problematically as meta-code. “Numbers no longer simply are, either in actuality or in some idealized potentiality; they are materio-symbolic or technosemiotic entities that have to be made by materio-symbolic creatures” (123).
Rotman is a mathematician by training and is a professor emeritus in the department of Comparative Studies of the Ohio State University. This affiliation seems to explain the various sources that he draws upon for this book, including ancient metaphysics, semiotics, American pragmatism, media theory, and human-computer interaction studies. It’s a heady mix, mostly on point, but with some academic tangents and asides that take a long, winding route to get to a point that Rotman already used the shortcut for earlier in the chapter.
Rotman begins by tracing out the meaning of math along three different positions. One is a formalist position through which we can see mathematics as a rule-based practice of manipulating marks on paper. Another is an intuitionist position in which numbers and the mathematical syntax bringing them together are abstract and rational concepts without necessary physical signification in the world. Then there is the Platonist position in which numbers and math are idealistic forms that can be used, heuristically, to engage with the world and make truth statements about it. Because we use mathematics in sentences about the world — we name numbers as nouns and assert relationships between them as verbs it is clear that we use mathematics semiotically, to express meaning to others.
To develop his semiotic understanding of mathematics, Rotman relies on C.S. Pierce whose work on signs specified how communication utilizes language that asserts different kinds of relationships between the sign and the thing it references. Pierce discuss icons (isomorphic reference) symbols (non-necessary conventional or customary reference) and indexes (necessary, factual or physical reference). Because mathematics is a part of the world and because it requires engagement by a reader/counter/manipulator mathematics operate as all three kinds of sign.
Math and numbers can be thought of as referents existing in three arenas: Code (including mathematical symbols and nomenclature) which is spoken by a Subject who utilizes numbers and can speak their particular formalist meaning (e.g. addition, subtraction, multiplication, commutation, distribution etc.); meta-Code (related objects, diagrams, visualizations, objects, outcomes, that are implied by or epiphenomenal with the Code) spoken by the Person through stories, narratives, pictures, legitimating the number in natural language; Virtual Code (what is valid, derivable, and possible based on the math) spoken by the Agent in terms of what is possible or idealized (pp.51-52).
Although Rotman does not use these terms, he is describing the use of mathematics as speech acts, where people use words not just to assert something that is true about the world but also to declare things to be true or to create relationships with things in the world. In John Searle’s words, the speech acts assert either a word to world or world to word direction of fit. As a Subject one may assert something mathematically true and as a Person insert that mathematical expression into discourse where it means something derived from a formalistic understanding of the math but also something that is specifically or potentially true in a given contextual moment.
Rotman pushes against the notion of seeing mathematics and numbers as being things that are true in and of themselves and in doing so, seems to borrow on another aspect of speech acts, which is the “condition of satisfaction.” For mathematics to be utilized semiotically in the production of propositions it needs to become meaningful against the conditions of satisfaction that make it true. For example, if I say that I have two apples and you have two apples and I give (add) my two apples to you then you will have (equals) four apples. The conditions of satisfaction is that upon completion of the operation, you have four apples. Without this grounding in the world we are left to deal with ever more abstract, context-free numbers that eventually move beyond the realm of experience and in doing so stop being “true” or semiotically meaning in the sense that they are possible: we may be able to process the numbers as code and virtual code but only problematically as meta-code. “Numbers no longer simply are, either in actuality or in some idealized potentiality; they are materio-symbolic or technosemiotic entities that have to be made by materio-symbolic creatures” (123).
Rotman is a mathematician by training and is a professor emeritus in the department of Comparative Studies of the Ohio State University. This affiliation seems to explain the various sources that he draws upon for this book, including ancient metaphysics, semiotics, American pragmatism, media theory, and human-computer interaction studies. It’s a heady mix, mostly on point, but with some academic tangents and asides that take a long, winding route to get to a point that Rotman already used the shortcut for earlier in the chapter.
March 20, 2019
It occurred to me that one of the many philosophical debates semiotics could bring some clarity to was the nature of mathematical objects, and I was excited to find a whole book dedicated to the question. But while it's provocative and occasionally interesting, I can't say this was a completely satisfying take on the question. Part of the problem is perhaps that Rotman uses an unholy blend of Saussurean and Peircean semiotics, which keeps him from developing his ideas as clearly as he needs to for this difficult subject. I was on board with some of the broad goals he laid out (anti-Platonism, constructing math from basic perceptual activities like Bloor did in Social Imagery) but I can't say I found most of his arguments very convincing. The whole bit about the Agent doing infinite operations in the mind of the mathematician seems pretty dumb and not necessarily helpful, and I just don't know if his anti-infinity crusade really follows from semiotics/is right/would lead to anything productive. Otherwise the discussion of information theory and physics and quanta all seems competent and above board, nothing outside the scope of productive philosophical discourse, but I'm not sure it really went anywhere.
August 17, 2014
What an interesting book!
It took a while, but once it got a finger-hold in my skull, it ripped my mind wide open.
Never have numbers been so interesting: "Numbers no longer simply are, either in actuality or in some idealized potentiality: they are materio-symbolic or technosemiotic entities that have to be made by materio-symbolic creatures. They and their arithmetic are always part of the larger and open-ended human initiative of constant becoming – an enterprise never free from choice, contingency, the limits of our (always material) resources, and the arbitrariness of history."
The final chapter on Deleuze & Guattari really made the work resonate.
It took a while, but once it got a finger-hold in my skull, it ripped my mind wide open.
Never have numbers been so interesting: "Numbers no longer simply are, either in actuality or in some idealized potentiality: they are materio-symbolic or technosemiotic entities that have to be made by materio-symbolic creatures. They and their arithmetic are always part of the larger and open-ended human initiative of constant becoming – an enterprise never free from choice, contingency, the limits of our (always material) resources, and the arbitrariness of history."
The final chapter on Deleuze & Guattari really made the work resonate.
June 29, 2020
The first two essays give a focused and insightful discussion of how Peirce's semiotics can be applied to analyze the practice of mathematics. (Both familiarity with Peirce and familiarity with university-level mathematics are required to follow the argument.) What are mathematicians doing when they invent and prove theorems? Rotman investigates this question by decomposing the mathematician into three agencies that use different codes and observe different rules: a Person, who understands the motivation, diagrams, and stories behind a theorem; a Subject, who writes only "rigorous" statements (syntacto-semantically correct, formally verifiable); and an idealized Agent who conjures well-defined objects in a virtual, imagined space.
The Person observes the Subject-Agent relation, and the proof succeeds if the Person judges that the Agent can stand in for the subject (that the formal semantics of the proof is coherent and unambiguously denoted by the Subject's proof-code). The Person uses a vernacular Meta-Code (the motivation), the Subject uses a rigorous Code (the proof), and the Agent uses a formal Virtual Code within the imagined realm of mathematical objects.
The rest of the book moves quickly between diverse topics: alphabets, computers, virtual reality, biology, and Deluze and Guattari. These sections lack the focus of the first two chapters, but contain a few interesting bits. Overall, the first half of the book felt more worthwhile to me.
Certain subjects require more elaboration:
(a) The idea of doing math as performing an idealized experiment to determine whether a "future" Subject will arrive at the same conclusion relies on a notion of "future" that seems to smuggle Platonism back into the picture.
(b) The "structural sense" of contextualized mathematical symbols is not fully described or appreciated. For example, the numerals 1 and 0 takes on a different meaning in the integers than they do in cyclic groups of finite order, and 1 and 0 take on yet a different meaning in the context of ring theory (multiplicative and additive identity symbols), where they stand as structural slots to be filled by more concrete/particular objects.
(c) Proofs contain a limited set of acceptable discourse relations (relations between clauses, sentences, and sections in a proof) that allow proofs to be legible as coherent ordered sequences of instructions to the Agent. However, these discourse relations are never analyzed---such a study would be interesting from the perspective of Rotman's project, and its absence is felt.
(d) The distinction between the three codes of the Person, Subject, and Agent is not always clear.
I look forward to reading Rotman's later work.
The Person observes the Subject-Agent relation, and the proof succeeds if the Person judges that the Agent can stand in for the subject (that the formal semantics of the proof is coherent and unambiguously denoted by the Subject's proof-code). The Person uses a vernacular Meta-Code (the motivation), the Subject uses a rigorous Code (the proof), and the Agent uses a formal Virtual Code within the imagined realm of mathematical objects.
The rest of the book moves quickly between diverse topics: alphabets, computers, virtual reality, biology, and Deluze and Guattari. These sections lack the focus of the first two chapters, but contain a few interesting bits. Overall, the first half of the book felt more worthwhile to me.
Certain subjects require more elaboration:
(a) The idea of doing math as performing an idealized experiment to determine whether a "future" Subject will arrive at the same conclusion relies on a notion of "future" that seems to smuggle Platonism back into the picture.
(b) The "structural sense" of contextualized mathematical symbols is not fully described or appreciated. For example, the numerals 1 and 0 takes on a different meaning in the integers than they do in cyclic groups of finite order, and 1 and 0 take on yet a different meaning in the context of ring theory (multiplicative and additive identity symbols), where they stand as structural slots to be filled by more concrete/particular objects.
(c) Proofs contain a limited set of acceptable discourse relations (relations between clauses, sentences, and sections in a proof) that allow proofs to be legible as coherent ordered sequences of instructions to the Agent. However, these discourse relations are never analyzed---such a study would be interesting from the perspective of Rotman's project, and its absence is felt.
(d) The distinction between the three codes of the Person, Subject, and Agent is not always clear.
I look forward to reading Rotman's later work.
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