“Phenomena of the external sense are examined and set forth in physics; those of the internal sense in empirical psychology. Pure mathematics considers space in geometry and time in pure mechanics. To these is to be added a certain concept, intellectual to be sure in itself, but whose becoming actual in the concrete requires the auxiliary notions of time and space in the successive addition and simultaneous juxtaposition of separate units, which is the concept of number treated in arithmetic.”

— Immanuel Kant

Kant's Inaugural Dissertation (1770), Section II On The Distinction Between The Sensible And The Intelligible Generally

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Immanuel Kant 200

German philosopher 1724–1804

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“The concept of space, therefore, is a pure intuition, being a singular concept, not made up by sensations, but itself the fundamental form of all external sensation.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section III On The Principles Of The Form Of The Sensible World

“The arithmetization of mathematics… which began with Weierstrass… had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics.
These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought.”

Tobias Dantzig (1884–1956) American mathematician

p, 125
Number: The Language of Science (1930)

“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)

“In the most general mathematical sense, a space is a set of elements which conform to certain postulates.”

James Grier Miller (1916–2002) biologist

Source: Living Systems: Basic Concepts (1969), p. 51

“Classical mechanics is mathematical not only in the sense that it makes use of mathematical terms and methods for abbreviating arguments which might, if necessary, also be expressed in the language of everyday speech; it is so also in the much more stringent sense that its basic concepts are mathematical concepts, that mechanics itself is a mathematics.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: The mechanization of the world picture, 1961, p. 499

“Minkowski… supposed that this fourth dimension of time was not detached from and independent of the three dimensions of space. He introduced a new four-dimensional space to which ordinary space contributed three dimensions, and time one; we may call it 'space-time'…. The succession of positions which a particle occupied in ordinary space at a succession of instants of time would be represented by a line in space-time; this he called the 'world-line' of the particle…. Newton's absolute space and absolute time fell out of science, and they carried much with them in their fall. First to go was the concept of simultaneity…. It now became necessary to find a way of treating gravitation which should not involve simultaneity. Einstein found through the medium of his 'Principle of Equivalence'.”

James Jeans (1877–1946) British mathematician and astronomer

The Growth of Physical Science (1947)

“The concepts which now prove to be fundamental to our understanding of nature—a space which is finite; a space which is empty, so that one point [of our 'material' world] differs from another solely in the properties of space itself; four-dimensional, seven- and more dimensional spaces; a space which for ever expands; a sequence of events which follows the laws of probability instead of the law of causation—or alternatively, a sequence of events which can only be fully and consistently described by going outside of space and time—all these concepts seem to my mind to be structures of pure thought, incapable of realisation in any sense which would properly be described as material.”

James Jeans book The Mysterious Universe

Source: The Mysterious Universe (1930), p. 122, 1937 ed.

“In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system.”

Charles Proteus Steinmetz (1865–1923) Mathematician and electrical engineer

New York Times interview (1911)

“We must… maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.”

Hans Reichenbach (1891–1953) American philosopher

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

“Geometry can in no way be viewed… as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I… realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Forward, as quoted by Mario Livio, Is God a Mathematician? (2009)
Ausdehnungslehre (1844)

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