“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

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

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Immanuel Kant200

German philosopher 1724–1804

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

“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)

“Reason perceives the general and proceeds by logical operations, whereas Intellect perceives the principial − the metaphysical − and proceeds by intuition.”

Frithjof Schuon (1907–1998) Swiss philosopher

[2006, Gnosis: Divine Wisdom, World Wisdom, 36, 978-1-933316-18-5]
Human being, Intellect

“To prove cannot mean anything other than to bring the other person to my own conviction. The truth lies only in the unification of “I” and “You.” The Other of pure thought, however, is the sensuous intellect in general. In the field of philosophy, proof therefore consists only in the fact that the contradiction between sensuous intellect and pure thought is disposed, so that thought is true not only for itself but also for its opposite.”

Ludwig Feuerbach (1804–1872) German philosopher and anthropologist

Z. Hanfi, trans., in The Fiery Brook (1972), p. 75
Towards a Critique of Hegel’s Philosophy (1839)

“Physics is essentially an intuitive and concrete science. Mathematics is only a means for expressing the laws that govern phenomena.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

From Lettre à Maurice Solvine, by A. Einstein (Gauthier-Villars: Paris 1956)
Attributed in posthumous publications, Albert Einstein: A guide for the perplexed (1979)

“A physical theory, like an abstract science, consists of definitions and axioms as first principles, and of propositions, their consequences; but with these differences:—first, That in an abstract science, a definition assigns a name to a class of notions derived originally from observation, but not necessarily corresponding to any existing objects of real phenomena, and an axiom states a mutual relation amongst such notions, or the names denoting them; while in a physical science, a definition states properties common to a class of existing objects, or real phenomena, and a physical axiom states a general law as to the relations of phenomena; and, secondly,—That in an abstract science, the propositions first discovered are the most simple; whilst in a physical theory, the propositions first discovered are in general numerous and complex, being formal laws, the immediate results of observation and experiment, from which the definitions and axioms are subsequently arrived at by a process of reasoning differing from that whereby one proposition is deduced from another in an abstract science, partly in being more complex and difficult, and partly in being to a certain extent tentative, that is to say, involving the trial of conjectural principles, and their acceptance or rejection according as their consequences are found to agree or disagree with the formal laws deduced immediately from observation and experiment.”

William John Macquorn Rankine (1820–1872) civil engineer

Source: "Outlines of the Science of Energetics," (1855), p. 121; Second paragraph

“Indeed, it is not intellect, but intuition which advances humanity. Intuition tells man his purpose in this life.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

Source: Attributed in posthumous publications, Einstein and the Poet (1983), p. 103

“There are no solids. There are no things. There are only interfering and noninterfering patterns operative in pure principle, and principles are eternal. Principles never contradict principles.... The synergetic integral of the totality of principles is God, whose sum-total behavior in pure principle is beyond our comprehension and is utterly mysterious to us, because as humans — in pure principle — we do not and never will know all the principles.”

Buckminster Fuller (1895–1983) American architect, systems theorist, author, designer, inventor and futurist

From 1980s onwards, Critical Path (1981)

“Only through intuition does a work rise above more or less subjective expression. Different periods produce different feelings and conceptions, and in each period men differ. Consequently different art expressions even in a single period are not only logical but a tribute to the general development of art. Intuition always finds the way of progress, which is continuous growth toward a clearer establishment of the content of art: the unification of man with the universe.”

Piet Mondrian (1872–1944) Peintre Néerlandais

'A New Realism', p. 17
1940's, A New Realism', 1943-1945

“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 (1714–1780) French academic

The Logic of Condillac (trans. Joseph Neef, 1809), "Of the Method of Thinking", p. 3.

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