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... we find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies - such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later.
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The Sleepwalkers (1959)
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The First Nonlinear System of Differential and Integral Calculus (1979) by Michael Grossman

 

Arthur Koestler

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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.

 

Hans Reichenbach

 

The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight—as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to thro over the axiom of parallels and to work out a non-Euclidean geometry.

 

John Maynard Keynes

 

...the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors.

 

Hans Reichenbach

 

The main objection to the theory of pure visualization is our thesis that the non-Euclidean axioms can be visualized just as rigorously if we adjust the concept of congruence. This thesis is based on the discovery that the normative function of visualization is not of visual but of logical origin and that the intuitive acceptance of certain axioms is based on conditions from which they follow logically, and which have previously been smuggled into the images. The axiom that the straight line is the shortest distance is highly intuitive only because we have adapted the concept of straightness to the system of Eucidean concepts. It is therefore necessary merely to change these conditions to gain a correspondingly intuitive and clear insight into different sets of axioms; this recognition strikes at the root of the intuitive priority of Euclidean geometry. Our solution of the problem is a denial of pure visualization, inasmuch as it denies to visualization a special extralogical compulsion and points out the purely logical and nonintuitive origin of the normative function. Since it asserts, however, the possibility of a visual representation of all geometries, it could be understood as an extension of pure visualization to all geometries. In that case the predicate "pure" is but an empty addition, since it denotes only the difference between experienced and imagined pictures, and we shall therefore discard the term "pure visualization." Instead we shall speak of the normative function of the thinking process, which can guide the pictorial elements of thinking into any logically permissible structure.

 

Hans Reichenbach

 

Ideas are dangerous, but the man to whom they are least dangerous is the man of ideas. He is acquainted with ideas, and moves among them like a lion-tamer. Ideas are dangerous, but the man to whom they are most dangerous is the man of no ideas. The man of no ideas will find the first idea fly to his head like wine to the head of a teetotaller. It is a common error, I think, among the Radical idealists of my own party and period to suggest that financiers and business men are a danger to the empire because they are so sordid or so materialistic. The truth is that financiers and business men are a danger to the empire because they can be sentimental about any sentiment, and idealistic about any ideal, any ideal that they find lying about, just as a boy who has not known much of women is apt too easily to take a woman for the woman, so these practical men, unaccustomed to causes, are always inclined to think that if a thing is proved to be an ideal it is proved to be the ideal.

 

Gilbert Keith Chesterton