Abstract
This general introduction focuses on the origin of spatial representations, the structures of perception, cognitive sciences, and the way the neurogeometry of vision fits into them.
Without a proper shape mathematics for biology, we are in the position that physics would have been in trying to develop mechanics without Euclidean geometry.
Harry Blum
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Notes
- 1.
Named after Niels Henrik Abel.
- 2.
- 3.
‘Tatsachen’ or ‘facts’ are contrasted with ‘Hypothesen’.
- 4.
This is a clear reference to Kant’s opposition between sensibility and understanding.
- 5.
For the relations between Mathematics and Physics in Poincaré, see La Valeur de la Science [16].
- 6.
See Sect. 1.1 of the Introduction.
- 7.
This problem already arose with the invention of rational mechanics. In order to mathematize the physical motions of material points, one must choose a frame of reference. But neither absolute positions, nor absolute directions , nor absolute velocities have any physical meaning, hence the advent of Galilean relativity.
- 8.
Named after Joseph-Louis Lagrange.
- 9.
See, for example, Marre [21].
- 10.
However, in Sect. 4.7.3 of Chap. 4, we shall discuss the latest methods of two-photon confocal microscopy, which can in fact distinguish individual neurons.
- 11.
There are many excellent introductory Websites to find out about cognitive science, such as The MIT Encyclopedia of the Cognitive Sciences [22].
- 12.
For an introduction to this scientific revolution, the reader is referred to the reflections of Dupuy [23].
- 13.
This is actually a common theme throughout the history of science. From Galileo and Newton , we learnt that types of motion as apparently disparate as ‘sublunar’ ballistic motions and ‘superlunar’ celestial motions could be understood using a single mathematical theory, the universal theory of gravitation. This formal similarity between empirical areas hitherto considered to be ontologically incommensurable was once culturally traumatic, but it eventually became commonplace in science. The same can be said here. For example, the fact that neural networks carrying out cognitive operations of categorization are formally analogous to spin glasses may look quite bewildering, given the gulf that separates these two ontologically incommensurable areas. But it is already a scientific commonplace for the young generation.
- 14.
These are Gabor patches, not geometric segments, as we shall explain.
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Petitot, J. (2017). Introduction. In: Elements of Neurogeometry. Lecture Notes in Morphogenesis. Springer, Cham. https://doi.org/10.1007/978-3-319-65591-8_2
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