“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."”

— Hans Reichenbach

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.
The Philosophy of Space and Time (1928, tr. 1957)

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Hans Reichenbach 41

American philosopher 1891–1953

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“There is no pure visualization in the sense of a priori philosophies; every visualization is determined by previous sense perceptions, and any separation into perceptual space and space of visualization is not permissible, since the specifically visual elements of the imagination are derived from perceptual space. What led to the mistaken conception of pure visualization was rather an improper interpretation of the normative function… an essential element of all visual representations. Indeed, all arguments which have been introduced for the distinction of perceptual space and space of visualization are base on this normative component of the imagination.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“Perceptual space is not a special space in addition to physical space, but physical space which we endow with a special subjective metric. …apart from the definition of congruence in physics and that based on perception, there is no third one derived from pure visualization. Any such third definition is nothing but the definition of physical congruence to which our normative function has adjusted the subjective experience of congruence.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“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 (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“The axioms which Tertium Organum embraces cannot be formulated in our language. If we attempt to formulate them in spite of this, they will produce the impression of absurdities. Taking the axioms of Aristotle as a model, we may express the principal axiom of the new logic in our poor earthly language in the following manner:
A is both A and Not-A.
or
Everything is both A and Not-A.
or,
Everything is All.
But these axioms are in effect absolutely impossible. They are not the axioms of higher logic, they are merely attempts to express the axioms of this logic in concepts. In reality the ideas of higher logic are inexpressible in concepts. When we encounter such an inexpressibility it means that we have touched the world of causes.”

P. D. Ouspensky book Tertium Organum

Source: Tertium Organum (1912; 1922), Ch. XXI

“It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. …the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection… analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". …If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels.”

Hans Reichenbach (1891–1953) American philosopher

The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface.
The Philosophy of Space and Time (1928, tr. 1957)

“We can… treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“[The special properties of visual perception of data]… is the visual means of resolving logical problems.”

Jacques Bertin (1918–2010) French geographer and cartographer

Source: Graphics and graphic information processing (1981), p. 16 as cited in: Riccardo Mazza (2004) Introduction to Information Visualisation http://www.dti.supsi.ch/~mazza/infovis_introduction.pdf

“We are frequently faced with the necessity of looking for the picture required for the visualization of an object, not in the perception of this particular object, but in a different perceptual image. …we can assert the discrepancy between the perceived picture and the objective state. This discrepancy… proves absolutely nothing against the fact that all visualizations are merely sense qualities of the perceptual space. …If the parallelism is …to be visualized, we must supplement our assertion by the description of certain qualities with which we are familiar from perceptual space.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“The use of the intellect in the sciences whose primitive concepts as well as axioms are given by sensuous intuition is only logical, that is, by it we only subordinate cognitions to one another according to their relative universality conformably to the principle of contradiction, phenomena to more general phenomena, and consequences of pure intuition to intuitive axioms. But in pure philosophy, such as metaphysics, in which the use of the intellect in respect to principles is real, that is to say, where the primary concept of things and relations and the very axioms are given originally by the pure intellect itself, and not being intuitions do not enjoy immunity from error, the method precedes the whole science, and whatever is attempted before its precepts are thoroughly discussed and firmly established is looked upon as rashly conceived and to be rejected among vain instances of mental playfulness.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section V On The Method Respecting The Sensuous And The Intellectual In Metaphysics

“Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.”

Joshua Girling Fitch (1824–1903) British educationalist

The Fourth Dimension simply Explained. (New York, 1910), p. 58. Reported in Moritz (1914); Also cited in: Howard Eves (2012), Foundations and Fundamental Concepts of Mathematics, p. 167

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