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Mathematics as Sign: Writing, Imagining, Counting

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Two features of mathematics stand out: 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 mat Two features of mathematics stand out: 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."


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Two features of mathematics stand out: 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 mat Two features of mathematics stand out: 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."

36 review for Mathematics as Sign: Writing, Imagining, Counting

  1. 4 out of 5

    Adam

    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 f 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.

  2. 4 out of 5

    Rhys

    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, contin 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.

  3. 5 out of 5

    Omar Agha

    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 m 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.

  4. 5 out of 5

    Manton

  5. 5 out of 5

    Fai

  6. 4 out of 5

    Ilya

  7. 5 out of 5

    Woo Tang

  8. 4 out of 5

    Kristin Villalobos

  9. 4 out of 5

    Murat Yıldız

  10. 5 out of 5

    Tim

  11. 5 out of 5

    Alexander

  12. 4 out of 5

    Benjamin

  13. 5 out of 5

    Maciek

  14. 4 out of 5

    Christopher Daniel

  15. 4 out of 5

    Kevin K

  16. 4 out of 5

    Summer

  17. 5 out of 5

    John Gage

  18. 4 out of 5

    Jill Perry

  19. 5 out of 5

    Gerry Stahl

  20. 5 out of 5

    Davis Pan

  21. 5 out of 5

    Jake

  22. 5 out of 5

    Incipio

  23. 5 out of 5

    Derrel Blain

  24. 5 out of 5

    Amanda

  25. 5 out of 5

    Shawn

  26. 4 out of 5

    Kellyjosephc

  27. 4 out of 5

    Alexan Martin-Eichner

  28. 5 out of 5

    Derek Spencer

  29. 4 out of 5

    M

  30. 5 out of 5

    Nemo

  31. 5 out of 5

    Ellery

  32. 5 out of 5

    Greg

  33. 4 out of 5

    Nitin Rughoonauth

  34. 5 out of 5

    Uxküll

  35. 4 out of 5

    Ash

  36. 5 out of 5

    ZacharySicilian

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