“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

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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Joshua Girling Fitch6

British educationalist 1824–1903

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term. The formal and abstract mathematical theory has an independent and sometimes lonely existence of its own. But when an undefined mathematical term such as random is given a definite operational meaning in physical terms, it takes on empirical and practical significance. Every mathematical theorem involving this mathematically undefined concept can then be given the following predictive form: If you do so and so, then such and such will happen.”

Walter A. Shewhart (1891–1967) American statistician

[Shewhart, Walter A., Deming, William E., Statistical Method from the Viewpoint of Quality Control, The Graduate School, The Department of Agriculture, 1939, 18]
Economic Control of Quality of Manufactured Product,1931

“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

“[The process of system design is]… consisting of the development of a sequence of mathematical models of systems, each one more detailed than the last.”

A. Wayne Wymore (1927–2011) American mathematician

A. Wayne Wymore (1970) Systems Engineering Methodology. Department of Systems Engineering, The University of Arizona, p. 14/2; As cited in: J.C. Heckman (1973) Locating traveler support facilities along the interstate system--a simulation using general systems theory. p. 43.

“If a lack of empirical foundations is a defect of the theory of logical probability, it is also a defect of deductive logic.”

David Stove (1927–1994) Australian philosopher

The Rationality of Induction, Oxford: Clarendon, 1986. Page 176, last paragraph.

“So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Introduction to Mathematical Philosophy (1919), Ch. 18: Mathematics and Logic

“P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert’s sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory…. In the absence of a precise notion of what it means to be evident, either in the intuitive or in the abstract realm, we have no strict proof of Bernays’ assertion; practically speaking, however, there can be no doubt that it is correct…”

Paul Bernays (1888–1977) Swiss mathematician

Kurt Gödel (1958, CW II, p. 241) as cited in: Feferman, Solomon. " Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program* http://math.stanford.edu/~feferman/papers/bernays.pdf." dialectica 62.2 (2008): 179-203.

“Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.”

David Hilbert (1862–1943) German prominent mathematician

Quoted in Hilbert's Die Grundlagen der Mathematik (1927)

“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 (1891–1953) American philosopher

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)

“Primitive religions consisted mainly in the worship of the powers of nature.”

Walter Rauschenbusch (1861–1918) United States Baptist theologian

...the essential thing in religion was not morality, but the ceremonial method of placating the god, securing his gifts, and ascertaining his wishes. He might even be pleased best by immoral actions, by the immolation of human victims, by the sacrifice of woman's chastity, or by the burning of the first born.
Source: Christianity and the Social Crisis (1907), Ch.1 The Historical Roots of Christianity the Hebrew Prophets, p. 4

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Joshua Girling Fitch6

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