“The term `geometry'...refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics.”

— William Thurston

Foreword to Teichmüller Theory

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William Thurston 4

mathematician 1946–2012

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“Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.”

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“[O]ur purpose is to give a presentation of geometry... in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems... beyond this, it is possible... to depict the geometric outline of the methods of investigation and proof, without... entering into the details... In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes... and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists.”

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“The idea that different aspects of visual perception might be handled in separate areas of the brain was predicted by Freud at the end of the nineteenth century. …a cortical defect that affected [the] ability to combine aspects of vision into a meaningful pattern. …defects, which Freud called agnosias (loss of knowledge)…”

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“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

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Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit. http://books.google.com/books?id=QF0ON71WuxEC&q=%22Insofern+sich+die+S%C3%A4tze+der+Mathematik+auf+die%22&pg=PA3#v=onepage

Geometrie and Erfahrung (1921) pp. 3-4 link.springer.com http://link.springer.com/chapter/10.1007%2F978-3-642-49903-6_1#page-1 as cited by Karl Popper, The Two Fundamental Problems of the Theory of Knowledge (2014) Tr. Andreas Pickel, Ed. Troels Eggers Hansen.
Ref: en.wikiquote.org - Albert Einstein / Quotes / 1920s

http://books.google.com/books?id=QF0ON71WuxEC&q=%22beziehen+sind+sie+nicht+sicher+und+insofern+sie+sicher+sind+beziehen+sie+sich+nicht+auf+die+Wirklichkeit%22&pg=PA4#v=onepage
1920s, Sidelights on Relativity (1922)

“The really important facts were that spatial relationships had ceased to matter very much and that my mind was perceiving the world in terms of other than spatial categories. At ordinary times the eye concerns itself with such problems as where? — how far? — how situated in relation to what? In the mescaline experience the implied questions to which the eye responds are of another order. Place and distance cease to be of much interest. The mind does its perceiving in terms of intensity of existence, profundity of significance, relationships within a pattern.”

Aldous Huxley book The Doors of Perception

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“Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities.”

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“The term "cognition" refers to all processes by which the sensory input is transformed, reduced, elaborated, stored, recovered, and used. It is concerned with these processes even when they operate in the absence of relevant stimulation, as in images and hallucinations. Such terms as sensation. perception. imagery. retention. recall. problem-solving. and thinking. among others. refer to hypothetical stages or aspects of cognition.”

Ulric Neisser (1928–2012) American psychologist

Source: Cognitive Psychology, 1967, p. 4

“The term `geometry'...refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics.”

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William Thurston 4

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