“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.

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James Jeans 54

British mathematician and astronomer 1877–1946

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“As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation”

Bernhard Riemann (1826–1866) German mathematician

can be compared with experience
Die partiellen Differentialgleichungen der mathematischen Physik (1882) as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book https://books.google.com/books?id=G0wtAAAAYAAJ (1914) p. 239

“The sequence of different positions of the same particle at different times forms a one-dimensional continuum in the four-dimensional space-time, which is called the world-line of the particle. All that physical experiments or observations can teach us refers to intersections of world-lines of different material particles, light-pulsations, etc., and how the course of the world-line is between these points of intersection is entirely irrelevant and outside the domain of physics. The system of intersecting world-lines can thus be twisted about at will, so long as no points of intersection are destroyed or created, and their order is not changed. It follows that the equations expressing the physical laws must be invariant for arbitrary transformations.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“Clocks are inherently four-dimensional instruments, since the endpoints of their unit distances are events. Measuring rods, on the other hand, are three-dimensional measuring instruments; their end points are space points and they can be changed into four-dimensional measuring instruments only if events are produced at their end points according to a special rule.”

Hans Reichenbach (1891–1953) American philosopher

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

“I started to introduce letters into my pictures [his first collage art]. These were forms which could not be deformed, because, being two-dimensional, they existed outside three-dimensional space; their inclusion in a picture allowed a distinction to be made between objects which were situated in space and those which belonged outside space”

Georges Braque (1882–1963) French painter and sculptor

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Source: posthumous quotes, Braque', (1968), p. 68

“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)

“We often resort to something called Hilbert space, which is described as n-dimensional—it’s like modern sex, any number may be played with.”

James Blish book The Quincunx of Time

Source: The Quincunx of Time (1973), Chapter 10, “Weinbaum on Sinai” (p. 116)

“Why is space-time doomed? There are many reasons, among which: In string theory we can change the dimension of space-time by changing the strength of the string force. Thus, the so-called II-A string theory, which semi-classically describes closed strings moving in ten-dimensional flat space for very weak coupling is dual for strong coupling to a theory, called M-theory, that at low energies is described by eleven-dimensional supergravity. By increasing the string coupling we can grow an extra dimension. How can the spatial continuum be fundamental if the number of spatial dimensions can be so changed?”

David Gross (1941) American particle physicist and string theorist

"Einstein and the Search for Unification", p. 11 https://books.google.com/books?id=rEaUIxukvy4C&pg=PA11, in The legacy of Albert Einstein: a collection of essays in celebration of the year of physics (2007)

“I must now explain what I mean by the quantity of action. A certain action is necessary for the carrying of a body from one point to another: this action depends on the velocity which the body has and the space which it describes; but it is neither the velocity nor the space taken separately. The quantity of action varies directly as the velocity and the length of path described; it is proportional to the sum of the spaces, each being multiplied by the velocity with which the body describes it. It is this quantity of action which is here the true expense (dépense) of nature, and which she economizes as much as possible in the motion of light.”

Pierre Louis Maupertuis (1698–1759) French mathematician, philosopher and man of letters

Histoire de l'Academie (1744) p. 423; Les Oeuvres De Mr. De Maupertuis (1752) vol. iv p. 17; as quoted by Philip Edward Bertrand Jourdain, The Principle of Least Action https://books.google.com/books?id=y3UVAQAAIAAJ (1913) p. 5.

“The triangles that we can measure are not large enough… to detect the curvature. Fortunately, however, we are, in a way, able to communicate with the fourth dimension. The theory of relativity has given us an insight into the structure of the real universe: …a four-dimensional structure. The study of the way in which the three space-dimensions are interwoven with the time-dimension affords a kind of outside point of view of the three-dimensional space… from this outside point of view we might be able to perceive the curvature of the three-dimensional world.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932)

“.. reality reveals itself by substantial, palpable forms, accumulated or dispersed in empty space.... these forms are part of that space and.... the space between them appears as form, a fact which evidences the unity of form and space.”

Piet Mondrian (1872–1944) Peintre Néerlandais

Source: 1940's, A New Realism', 1943-1945, p. 18

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