“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

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

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Hans Reichenbach 41

American philosopher 1891–1953

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Hans Reichenbach (1891–1953) American philosopher

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

“Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. …Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. …the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold.”

Hans Reichenbach (1891–1953) American philosopher

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

“We are frequently faced with the necessity of looking for the picture required for the visualization of an object, not in the perception of this particular object, but in a different perceptual image. …we can assert the discrepancy between the perceived picture and the objective state. This discrepancy… proves absolutely nothing against the fact that all visualizations are merely sense qualities of the perceptual space. …If the parallelism is …to be visualized, we must supplement our assertion by the description of certain qualities with which we are familiar from perceptual space.”

Hans Reichenbach (1891–1953) American philosopher

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

“The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space.”

Paul Carus (1852–1919) American philosopher

Science, Vol. 18 (1903), p. 106, as reported in Memorabilia Mathematica; or, The Philomath's Quotation-Book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), by Robert Edouard Moritz, p. 352