# Grosholz 2007 Oxford Univ Press

Grosholz Emily R (2007) Representation and productive ambiguity in mathematics and the sciences. Oxford Univ Press 312 pp. |

Grosholz Emily R (2007) Oxford Univ Press

*Abstract*: Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian 'analysis' - the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience. Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable for mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are subtly altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

* β’ Bioblast editor:* Gnaiger E

- Larvor Brendan P (2012) Book review. Philosophia Mathematica(III) 20:245β74. - Research Gate book review

## Selected quotes

- ..primacy and canonicity of certain geometrical objects, like the Euclidean line, right triangle and circle.

- I defend the importance of iconic, as well as symbolic and indexical, signs in mathematical representation.

- no representation is ever perfectly expressive, for if it were it would not be a representation but the thing itself.

- Galileo's proofs are canonical ..

- The philosophical use of the polar terms 'icon' and 'symbol' is due to C. S. Peirce, who distinguished the former as similar to their objects, and the latter as linked to their objects only by convention.

- Mathematics often requires the combination of different modes of representation in the same argument: equations, diagrams, matrices, tables, proportions, schemata, natural language.

- .. distance that is re-conceptualized to mean displacement, since we are instructed to suppose that a moving particle is traversing it.

- In Greek mathematics, ratios cannot hold between lines and numbers, between finite and infinitesimal magnitudes, or between curved lines and straight lines. The Euclidean tradition treats ratios as relations, different from the things related, and proportions as assertions of similitude (not equality) between ratios. There is, however, a second, medieval tradition of handling ratios and proportions that originates with Theon, a commentator of Ptolemy's
*Almagest*, .. It associates each ratio with a 'denomination,' that is, a number which gives its size, and in general treats the terms occurring in ratios as well as the ratios themselves uniformly as numbers. Thus ratios are just quotients and the distinction between ratio and term is abolished insofar as they are all numbers.

- In Greek mathematics, ratios cannot hold between lines and numbers, between finite and infinitesimal magnitudes, or between curved lines and straight lines. The Euclidean tradition treats ratios as relations, different from the things related, and proportions as assertions of similitude (not equality) between ratios. There is, however, a second, medieval tradition of handling ratios and proportions that originates with Theon, a commentator of Ptolemy's

- Galileo's treatment of the proportions and diagrams later on becomes carefully ambiguous; and therein lies the innovation.

- .. a vertical line CD .. representing space traversed (again, not just distance but displacement) ..

- The unit intervals are intended to be counted as well as measured.

- Read as finite, the triangles are the iconic representations of geometrical figures; read as infinitesimal, the triangles are the symbolic representation of a dynamical process, free fall. .. natural language explains the ambiguous configuration.

- The symbolic language of logistics is allegedly an ideal mode of representation that makes all content explicit; it stands in isomorphic relation to the objects it describes.

- .. the adequacy of a scientific theory is characterized in terms of a relation of isomorphism between theory and model. .. It is often assumed that the best way to think of representation in mathematics is in terms of isomorphism between structures, ..

- Whereas isomorphism is reflexive and symmetric, representation is irreflexive and asymmetric due to its intentionality. β
*Comment: therefore,***isomorphic representation**may be considered as the unifying concept.

- Whereas isomorphism is reflexive and symmetric, representation is irreflexive and asymmetric due to its intentionality. β

- When distinct representations are juxtaposed and superimposed, the result is often a productive ambiguity that expresses and generates new knowledge. β
*Comment: Carefully ambiguous representations provide isomorphic connections.*

- When distinct representations are juxtaposed and superimposed, the result is often a productive ambiguity that expresses and generates new knowledge. β

- Some mathematical representations are iconic, that is, they picture and resemble what they represent; some are symbolic and represent by convention, without much resemblance; and some are indexical, representing for the sake of organization and ordered display. .. But all icons have a symbolic dimension, as all symbols have an iconic dimension; and all representations to a certain extent organize, order, and display.

- .. the letters in Berzelian formulas are signs that convey a plurality of information simultaneously. Depending on context, .. The ambiguity of the notation allowed chemists to move back and forth between the macroscopic and microscopic worlds as needed.

- U. Klein 'The fact that a letter is a visible, discrete, and indivisible thing (unlike a written name) constitutes a minimal isomorphy with the postulated object it stands for, namely, the indivisible unit or portion of chemical elements.

- Thus Klein sees here a strong interplay between paper tools and theory construction, .. between laboratory apparatus and the modes of data analysis they foster, and theory construction.

## Cited by

- Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5
^{th}ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002

*Labels:*

Ambiguity crisis, Number, BEC 2020.2, X-mass Carol