Abstract
It is commonly accepted that “actual” (not astronomically too large or sub-microscopically too small) geometrical properties and relations of physical objects follow the axioms and theorems of the three-dimensional Euclidean geometry. But what about perceptions (and related judgments) that concern these geometrical relations and properties of physical objects?
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cf. also the considerations given in chapter 3, section 4, this book.
Moreover, we do not have to confuse this two-dimensional continuum of “visibles” with the images of perspective geometry of the two-dimensional Euclidean geometry of the flat Euclidean plane (for instance on the paper) in three space.
On the role of spatial and geometrical models in perception and in science cf. also Myrstad, 2000.
And of the so-called naturalized phenomenology, that is phenomenology supported by scientific explanations, neurophysiological, mathematical, physical, etc.
On the role of adumbrations in the genesis of ideal space and on their abductive and non-monotonic character cf. below section 2.3.
The problem of the “origins of geometry”, examined in this chapter from the point of view of a strong Husserlian philosophical theory, has been already examined in chapter 1, in that case by analyzing some “factual” issues belonging to the area of cognitive psychology, cognitive anthropology, and history.
Moreover, Husserl thinks that space is endowed with a double function: it is able to constitute a phenomenal extension at the level of sensible data and also it furnishes an intentional moment. Petitot says: “Space possesses, therefore, a noetic face (format of passive synthesis) and a noematic one (pure intuition in Kant’s sense)” (1999, p. 336).
Cf. also Husserl, 1931 [1913], § 40, p. 129.
On some results of neuroscience that corroborate and improve several phenomenological intuitions cf. Pachoud, 1999, pp. 211–216, Barbaras, 1999, and Petit, 1999.
The ego itself is only constituted thanks to the capabilities of movement and action.
Husserl considered it impossible to give a scientific account (for instance mathematical, neurobiological, or in terms of dynamical systems) of operations and events at the level of the descriptive eidetics and of morphological essences in perception. The so-called naturalized phenomenology tries to fill this gap (Petitot, Varela, Pachoud, and Roy, 1999). For instance Petitot demonstrates that there exist “a geometrical descriptive eidetics able to assume for perception the constitutive tasks of transcendental phenomenology and to mathematize the correlations between the kinetic noetic synthesis and the noematic morphological Abschattungen” (1999, p. 371); the mathematical apparatus he exploits is partially derived from Thom’s theory of catastrophes (Thom, 1975).
This point of view is confirmed by recent research on stereopsis (cf. Ninio, 1989).
Of course we do not have to confuse the prescientific world with the more elementary prepredicative world of appearances and primordial and immediately given experiences. The prescientific world is already characterized by predications, values, empirical manipulations and techniques of measurement.
“According to Kant, geometry is not imaginary because it is grounded on the universal form of pure sensibility, on the ideality of sensible space. But according to Husserl, on the contrary, geometrical ideality is not imaginary because it is uprooted from all sensible ground in general. […] Husserl remains then nearer to Descartes than to Kant. It is true for the latter, as has been sufficiently emphasized, that the concept of sensibility is no longer derived from a ‘sensualist’ definition. We could not say this is always the case for Descartes or Husserl” (Derrida, 1978, pp. 124–125, footnote 140).
Cf. below section 2.4.5.
Cf. chapter 1, section 3, this book.
A very simple example is given by the fact it is used to improve methods of measurement increasing the approximation of geometrical tools to the geometrical ideals.
It is the so-called CRUM (Computational-Representational Understanding of Mind) illustrated and criticized by Thagard (1996).
Cf. also the notion of “conceptual” space we introduced in chapter 1.
In Magnani (2001), chapter 6, I have illustrated the importance of contradictions, novelty,
and unexpected findings in various kinds of abductions in reasoning.
Cf. the previous sections of this chapter and chapter 6.
This approach in computer science, involving the use of diagram manipulations as forms of acceptable methods of reasoning, was opened by Gelernter’s Geometry Machine (1959), but the diagrams played a very secondary role. Other contributions are given by Furnas, 1992, and by Anderson and McCartney, 1995 and 1996 (transformations of pixel arrays). Computations are not done algebraically, but by making use of edge-following and color-spreading methods implemented as pixel-array representations — with their eight neighborhood relations (north, northeast, east, etc.) and operations.
A method already suggested by Johnson-Laird to deal with generalization using the so-called “mental models” (1983).
The program is able to “discover demonstrations”, that is to find sequences of manipulations that achieve a particular end.
This is not the case of the two-dimensional depictions of a three dimensional object (like in the clear cases illustrated by Escher’s drawings) (Lindsay, 1998, p. 266).
Byrne, 1847. See the web site http://sunsite.ubcxa/DigitalMathArchive/Euclid/byrne.html. The home page provides links to other web sites where it is possible to find Java editions of Euclid and other “visual” information. Particularly interesting is the web site devoted to illustrating a list of many proofs of Pythagoras’ theorem in Java, where the user can “construct” geometrical demonstrations by clicking at the figures presented, moving points and features of geometrical diagrams (http://sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagoras.html).
Cf. the Banchoff s web page http://www.geom.umn.edu/~banchoff/, where it is possible to manipulate diagrams that correspond to mathematical problems. The appealing Joyce’s Java edition of Euclid where it is possible to “construct by clicking”, is given at the web site http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
The role of spatial abilities in the embodied and mediated cognition related to manipulative abduction is illustrated in section 3 above; details on imagination and spatial reasoning in situated cognition and robotics (cognition as “imagined interaction”) are given in Stein (1995).
The Journal Computational Intelligence devoted a special issue to the so-called “imagery debate” (9, 1993). Moreover, the AAAI Society organized the “Spatial and Temporal Reasoning” Workshop in 1994, and a spring Symposium on “Cognitive and Computational Models of Spatial Representation”, in 1996, not to mention the many Journals dealing with visual imagery, diagrammatic reasoning, and spatial ability in the area of cognitive psychology.
I have considered in Magnani, 2001, chapter 5 (section 1), the whole problem of imagery representations and what I call “visual” abduction, a kind of model-based abduction.
Cf. Magnani, 2001, chapter 6.
Chapter 4 is mainly devoted to analyze Proclus’s philosophy of geometry.
But it is well-known that verbal/symbolic proofs and argumentations can be misleading too.
In some cases (when for instance there are not deductive proofs of mathematical conjectures) there is only inductive evidence (enumerative induction), in the sense of a suitable degree of confirmation provided by individual diagrams.
That it is the milestone of the axiomatization of geometry and of the elimination of diagrams, considered a flaw in a formal system (cf. chapter 3, this book).
The case of Euclidean geometry is considered by Miller, 1999.
However, cf. the previous chapter (section 4), where it is also observed that deductive reasoning can be seen as consisting of the employment of logical rules in a heuristic manner, by means of abductive steps. Finally, cf. the interesting and fruitful applications of heterogeneous logic to the historical cases of Venn diagrams (Shin, 1994, 1996) and Peirce graphs (Hammer, 1996), shown as formal systems. An interesting research, not excluding logical aspects, that aims to delineate the representation of spatial primitives and “entities” exploiting a mixture of cognitive, ontological and topological problems is the theory of parts and wholes (Casati and Varzi, 1997). The formal representation of qualitative spatial relations about regions is related to the AI tradition of qualitative reasoning (Cohn, et al, 1997).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Magnani, L. (2001). Geometry and Cognition. 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_7
Download citation
DOI: https://doi.org/10.1007/978-94-010-9622-5_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-0241-0
Online ISBN: 978-94-010-9622-5
eBook Packages: Springer Book Archive