By Emily R. Grosholz
Emily Grosholz bargains an unique research of demonstration in arithmetic and technological know-how, interpreting the way it works and why it's persuasive. targeting geometrical demonstration, she indicates the jobs that illustration and ambiguity play in mathematical discovery. She offers a variety of case reports in mechanics, topology, algebra, good judgment, and chemistry, from old Greece to the current day, yet focusing rather at the 17th and 20th centuries. She argues that reductive tools are potent now not simply because they scale down yet simply because they multiply and juxtapose modes of illustration. Such problem-solving is, she argues, most sensible understood by way of Leibnizian "analysis"--the look for stipulations of intelligibility. Discovery and justification are then features of 1 rational method of continuing, which produces the mathematician's formal experience.
Grosholz defends the significance of iconic, in addition to symbolic and indexical, symptoms in mathematical illustration, and argues that pragmatic, in addition to syntactic and semantic, issues are critical fore mathematical reasoning. by way of taking a detailed examine the way in which effects are awarded at the web page in mathematical (and organic, chemical, and mechanical) texts, she indicates that once or extra traditions mix within the carrier of challenge fixing, notations and diagrams are subtly altered, improved, and juxtaposed, and surrounded by way of prose in typical language and is the reason the unconventional blend. considered this fashion, the texts yield awesome examples of language and notation which are irreducibly ambiguous and efficient simply because they're ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, should be of substantial curiosity to philosophers and historians of arithmetic and technological know-how, and now have far-reaching outcomes for epistemology and philosophy of language.
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Emily Grosholz deals an unique research of demonstration in arithmetic and technology, analyzing the way it works and why it's persuasive. concentrating on geometrical demonstration, she indicates the jobs that illustration and ambiguity play in mathematical discovery. She provides quite a lot of case reviews in mechanics, topology, algebra, common sense, and chemistry, from historical Greece to the current day, yet focusing rather at the 17th and 20th centuries.
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Additional info for Representation and Productive Ambiguity in Mathematics and the Sciences
Galileo, Discorsi, 248–50. 5. 4 divided in just the same proportions. ’¹⁴ The proof of Theorem I, Proposition I in the section ‘The Motion of Projectiles’ shows that the rest of the diagram stems from the superposition ¹⁴ Galileo, Discorsi, 244. ’¹⁵ In order for the reasoning in the proof of Theorem I, Proposition I of the Fourth Day to proceed, the line abcde must mean both time and distance; it must represent time symbolically in order for the application of the results achieved in the Third Day, and it must represent distance qua displacement in order for the diagram to make sense as the icon of a trajectory, the movement of a body across a plane in space.
B. Rolf and A. George (Berkeley: University of California Press, 1967/69); Der Logische Aufbau der Welt (Hamburg: Meiner, 1928). productive ambiguity: galileo contra carnap 17 mathematics, many of its presuppositions nonetheless remain unnoticed in our discourse and thinking. I now return brieﬂy to Carnap’s classic work, in order to uncover some of these assumptions and to question them more closely. Then I will give a brief sketch of what became of Carnap and Hempel’s view of scientiﬁc and mathematical knowledge in the work of their academic children (Bas van Fraassen, Nancy Cartwright, Margaret Morrison, Ian Hacking) and grandchildren (Robin Hendry, Ursula Klein, myself), a development that will bring us back to the project of this essay in an unexpected way.
Note that in Corollary I, Galileo does not compare the interval-terms directly, but is careful to refer to them in ratios. Even if inﬁnitesimal intervals (instants and points, to use Galileo’s vocabulary) are mathematically suspect—as they surely were in the early seventeenth century—the geometry of the diagrams supports the rationality of holding that ratios between them are ‘like’ the ratios between their ﬁnite counterparts. That is, AD : DE :: AO : OP no matter what size the conﬁguration is; or, to use the other diagram, HL : LM :: 1 : 3 no matter what the size of the conﬁguration.
Representation and Productive Ambiguity in Mathematics and the Sciences by Emily R. Grosholz