„If we wish to express our ideas in terms of the concepts synthetic and analytic, we would have to point out that these concepts are applicable only to sentences that can be either true of false, and not to definitions. The mathematical axioms are therefore neither synthetic nor analytic, but definitions.... Hence the question of whether axioms are a priori becomes pointless since they are arbitrary.“

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Hans Reichenbach
1891 - 1953

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„Every definition implies an axiom, since it asserts the existence of the object defined.“

—  Henri Poincaré French mathematician, physicist, engineer, and philosopher of science 1854 - 1912
Context: Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted. Part II. Ch. 2 : Mathematical Definitions and Education, p. 131

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Hans Reichenbach photo

„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 American philosopher 1891 - 1953

Étienne Bonnot de Condillac photo

„We shall not … begin this logic by definitions, axioms, or principles; we shall begin by observing the lessons which nature gives us.“

—  Étienne Bonnot de Condillac French academic 1715 - 1780
The Logic of Condillac (trans. Joseph Neef, 1809), "Of the Method of Thinking", p. 3.

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„All acts of mathematical reasoning may... be considered but as applications of a corresponding axiom of quantity“

—  William Stanley Jevons English economist and logician 1835 - 1882
Context: Aristotle's dictim... may then be formulated somewhat as follows:—Whatever is known of a term may be stated of its equal or equivalent. Or, in other words, Whatever is true of a thing is true of its like.... the value of the formula must be judged by its results;... it not only brings into harmony all the branches of logical doctrine, but... unites them in close analogy to the corresponding parts of mathematical method. All acts of mathematical reasoning may... be considered but as applications of a corresponding axiom of quantity...

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„Mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity. [Provisional definition. ]“

—  George Holmes Howison American philosopher 1834 - 1916
"The Departments of Mathematics, and their Mutual Relations," Journal of Speculative Philosophy, Vol. 5, p. 164. Reported in Moritz (1914)

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