Abstract
We have to analyze in this chapter what is the role of Euclidean geometry in the Critique of Pure Reason. Oversimplifying things for the sake of clarity, we can, first of all describe the dynamic nature that exists between the three faculties of knowledge — sensibility, imagination, understanding.
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References
For the complex differences on the treatment of the faculty of imagination in the two Kantian deductions cf. for example Cassirer, 1907, vol. II, and 1918; Heidegger, 1929; Palumbo, 1985; Scaravelli 1973; Wolff, 1943.
From this passage some authors have drawn their interpretation of imagination as a sort of unconscious understanding.
We overlook the problem of time.
On the construction of a concept as regarding the constitution of a single object (and therefore on the “singular” character of intuition), see the considerations of chapter 3, section 2, this book.
Examples from the area of computational medical knowledge-based systems (able to simulate diagnostic reasoning), machine discovery (able to simulate scientific discovery), visual and analogical reasoning, etc. are given for instance in Magnani, 1988, 1999a and b, in my recent book Magnani, 2001, and in Magnani, Nersessian, and Thagard, 1999.
Like the slave boy of Plato’s Meno dialogue, that is in front of an analogous inferential problem. The story of Socrates that teaches geometry to the slave is illustrated in chapter 6, section 1.
It is well-known that by axiomatizing Euclidean geometry in a rigorous modern way theorems can be deduced through logical reasoning without resorting to real or imagined figures, beginning with a relatively small number of primitive propositions and using inference rules. The problems implicated by this possibility, which seem to attenuate the importance of “constructions”, are treated in chapter 3, sections 1 and 2. On recent logical models of “diagrammatic” geometrical reasoning see chapter 7, section 6.2.
Cf. on these topics and the differences between the First and Second deduction, Palumbo, 1985, pp. 35–50, and Broad, 1988, pp. 180–224.
Notwithstanding its tendency to a kind of psychologism of the innate dispositions of human beings.
And I would add, by “creative abduction”, cf. above.
On the role of frame-like descriptions of concepts in scientific conceptual change cf. Magnani, 1999a and b.
At least this is the interpretation imposed by his followers that sustain the central role of the linguistic “dimension”, of course the problem in Wittgenstein is much more subtle and complex, cf., for example, Bouveresse, 1987.
On the importance of the transcendental tradition in emphasizing the specific role of mathematics in scientific knowledge see below chapter 3, section 6.
It is Euclid’s Proposition XXXII, Book I, cf. Figure 1.
“Constructing by clicking” is the recent Java version of this ancient problem of geometrical construction. Cf. the web site http://sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagoras.html, devoted to illustrating many proofs of Pythagoras’ theorem, where the user can “construct” geometrical demonstrations by clicking on the figures presented, moving points and features of geometrical diagrams. (Cf. also footnotes 26 and 27, chapter 7, this book).
I would say, not just a mere definition, that does not “pass beyond” the concept. 17 This is a precise rule.
Cf. chapter 3, sections 1 and 2, this book.
Cf. also on this topic Palumbo, 1985, pp. 62–68, Parsons, 1969, and Winterburne, 1988.
This passage was corrected by Vaihinger and it is accepted by the majority of interpreters.
The original passage is: “The roundness which is thought in the former can be intuited in the latter”. Isaac (1968) considers the original version to be more acceptable since, in the geometrical concept, the roundness is intuited, because constructed in intuition. On the contrary, he states that in the plate the roundness is thought, because this object cannot be constructed. Both interpretations seem plausible.
In chapter 6 we will see that this kind of reasoning can be usefully illustrated by means of the concept of “selective abduction”.
Cf. also chapter 3, section 5, this book.
The history of non-Euclidean geometries and related philosophical and epistemological problems, as a background to better understand many of the argumentation’s of this book, are provided in Magnani, 1978. The following is a list of texts related to the history of non-Euclidean geometry and philosophy of geometry and space. Fundamental books, treatises, and bibliographies: Barbarin, 1902; Bonola, 1899–1900–1902 (bibliography), 1906, 1925; Enriques, 1918 and 1923–1927; Fano, 1898 and 1935; Halsted, 1878–1879 (bibliography); Sommerville, 1911 (bibliography) and 1914; Stäckel and Engel, 1895–1913. More recent books and articles: Boi, 1991; Capec, 1–976 (including many historical texts from the origins of geometry up to now); Coxeter, 1945; Gray, 1979; Greenberg, 1980; Guéridon and Dieudonné, 1978; Kline, 1972 (chapters 34, 37–39); Kulczycki, 1961; Lanczos, 1970; Magnani, 1975, 1977a (on Lobachevsky); Moise, 1963; Rosenfeld, 1988; Toth, 1972 and 1991 (on the cultural and scientific controversies related to the non-Euclidean geometries); Van Fraassen, 1985 (on the related problem of the philosophy of space); Weyl, 1967 (chapters 3–4).
Cf. also the considerations given in the subsection 5.3.1 of chapter 3, this book.
On this old-fashioned aspects of Kantian philosophy see chapter 3, section 6.
On the interesting question of the so-called “mathematical schematism”, and the related Petitot’s interpretation of Kantian thought about space and mathematics, see chapter 3, section 7, this book.
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Magnani, L. (2001). Geometry: the Model of Knowledge. In: Philosophy and Geometry. The Western Ontario Series in Philosophy of Science, vol 66. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9622-5_2
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