85020 - Methods of Scientific Knowledge (1)

Learning outcomes

Students are guided to focus on and analyze topics and issues emerging from a methodological reflection on logic-mathematical knowledge and scientific investigation.

Course contents

In his Trattato di pittura, Leonardo explains that painting is mental, because – like music and geometry – it considers not only the proportions of continuous quantities and the arithmetic of discontinuos ones, but also all continuous quantities together with the qualities of proportions between shadows, light, and distances from its perspective. For Hilbert (1930) instead, the empirical nature of geometry is a secure result of science. “The Einsteinian theory of gravitation makes it manifest: geometry is nothing more than a branch of physics; geometric truths are in no principled way whatsoever different from physical truths." What is then geometry, a "mental" construction like painting or an "empirical" science like physics? How far are these two conceptions from each other? Is there a "perspectival view" shared by Leonardo and Hilbert?

This course aims to reconstruct the emergence of the concept of "representation space", and to analyse how its transformations impact on our understanding of geometry. 

The discussion will cover the following topics:

• The Euclidean Vision
• Painting as a Science
• Linear Perspective
• Invariants
• Visual Geometry
• Transcendental Aesthetics
• Non Euclidean Geometry
• The Erlangen Program
• Foundations of the Relativity Theory
• Spacetime Physics

 

Readings/Bibliography

• Alberti L. B. [1436], On Painting, ed. by M. Kemp, Penguin, London 
• Leonardo da Vinci, A Treatise on Painting, ed. by J. F. Rigaud, London 1877 [https://archive.org/stream/treatiseonpainti001974mbp#page/n5/mode/2up]
• Kant I. [1781], "Transcendental Aesthetic," Critique of Pure Reason, Cambridge UP 1998 [http://strangebeautiful.com/other-texts/kant-first-critique-cambridge.pdf]

Kemp M. (1990), The Science of Art, Yale UP, New Haven (chaps I-II)

Stillwell J. (2018), Da Pitagora a Turing, ETS, Pisa (capp. 3-5, 7)

• Cassirer E. (1921), Einstein's Theory of Relativity, Open Court, Chicago 1923
• Cassirer E. (1929), The Philosophy of Symbolic Forms, Vol. III.2, Yale UP
• Hilbert, D. (1918), “Axiomatic Thought”, From Kant to Hilbert, vol. II, ed. by W. Ewald, Oxford UP 1996
• Hilbert, D. (1930), “Logic and the Knowledge of Nature” (Ewald 1996)
• Taylor E. F., Wheeler J. A. (1992), Space-Time Physics, W. H. Freeman and Company, New York

Approfondimenti

Jammer M. (1954), Concepts of Space. The History of Theories of Space in Physics, Harvard Univ. Press, Cambridge MA

Stillwell J. (2005), The Four Pillars of Geometry, Springer, New York

van Fraassen B. C. (2008), Scientific Representation: Paradoxes of Perspective, Oxford UP

Weyl, H. (1949), Philosophy of Mathematics and Natural Science, Princeton UP

Teaching methods

Lectures

Students are advised to attend classes regularly.

Assessment methods

Oral examination

 

Marks:

30 cum laude - excellent as to knowledge, philosophical lexicon and critical expression.

30 – Excellent: knowledge is complete, well argued and correctly expressed, with some slight faults.

27-29 – Good: thorough and satisfactory knowledge; essentially correct expression.

24-26 - Fairly good: knowledge broadly acquired, and not always correctly expressed.

21-23 – Sufficient: superficial and partial knowledge; exposure and articulation are incomplete and often not sufficiently appropriate

18-21 - Almost sufficient: superficial and decontextualized knowledge. The exposure of the contents shows important gaps.

Exam failed - Basic skills and knowledge are not sufficiently acquired. Students are requested to show up at a subsequent exam session.

Office hours

See the website of Rossella Lupacchini