Multi-dimensional geometry for a multi-dimensional world?

"Imagine that I draw a `straight line' on a black board with a piece of chalk. What a complicated thing is this `line' compared with the `straight line' defined by geometry! In the first place, it is not a line at all, since it has a definite breadth; even more than that, it is a three-dimensional body made of chalk, an aggregate of many small bodies, the chalk particles... All the same, we do know that the exact idealized conceptions of pure geometry are essential tools for dealing with the real things around us. We need these abstract concepts just because they are simple enough that our minds can handle them with comparative ease.

"Attempts have been made to construct geometries in which no `infinitely narrow' [unidimensional] lines exist but only those of definite width [multidimensional]. The results were meager because this method of treatment is much more difficult than the usual one. Moreover a strip of definite width is only another abstraction no better than a straight line, and is really more complicated.

"All theoretical constructions, including geometry, which are used in the various branches of physics are only imperfect instruments to enable the world of empirical fact to be reconstructed in our minds. But I do not believe that there is any other way to achieve progress in science than the old method: to begin with the simplest, i.e., the exact theoretical scheme, and to extend and improve it gradually. I aim at the construction of a rational theory, based on the simplest possible exact concepts, one which, although admittedly inadequate to represent the complexity of real processes, is able to reproduce satisfactorily some of their essential properties" (p. 7-8).

"An exact identity between theoretical premises and real conditions is not required, but only a similarity which makes a successful application of the theory to empirical data possible" (p. 145).

Richard von Mises (1957) Probability, Statistics and Truth. London: George Allen and Unwin.

Courtesy of Mark Stone

It is sometimes thought that, since reality is always multi-dimensional, it will be more easily understood if conceptualized as multi-dimensional. But multi-dimensionality confuses rather than clarifies thought. Whenever we try to understand more than one thing at a time, we understand nothing.

James Clerk Maxwell, the greatest physicist of the nineteenth century, never lost sight of the phenomena to be explained, nor permitted himself, as he said, to be drawn aside from the subject in pursuit of "analytical subtleties"; on the other hand, the use of mathematical methods conferred freedom on his inquiries and enabled him to gain physical insights without committing himself to a physical theory. This blending of the concrete and the abstract was the characteristic of almost all his researches.
James R. Newman, Science and Sensibility, Vol 1 p.140 New York:Simon and Schuster 1961.

It would be a mistake to suppose that a science consists entirely of strictly proved theses, and it would be unjust to require this. Only a disposition with a passion for authority will raise such a demand, someone with a craving to replace his religious catechism by another, though it is a scientific one. Science has only a few apodeictic [necessarily true] propositions in its catechism: the rest are assertions promoted by it to some degree of probability. It is actually a sign of a scientific mode of thought to find satisfaction in these approximations to certainty and to be able to pursue constructive work further in spite of the absence of final confirmation.
Sigmund Freud, Introductory Lectures on Psychoanalysis III. London: Hogarth Press, 1963 p.51

Quotations. … Rasch Measurement Transactions, 1994, 8:2 p.366

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