[TO THE EDITOR OF THE FORUM]
Dear Sir,—With due deference to your valued journal, the article of Claude Bragdon, Learning to Think in Terms of Spaces, in your August number, is essentially illogical. The writer thus introduces his subject: “A point, moving in an unchanging direction, traces out a line; a line, moving in a direction at right angles to its length, traces out a plane; a plane, moving in a direction at right angles to its two dimensions, traces out a solid. Should a solid move in a direction at right angles to its every dimension, it would trace out, in four dimensional space, a hypersolid.”
Now this may pass current in blackboard geometry, but does not hold good in the abstract. The physical point is indeed extended to represent the line, and the physical line, to represent the plane, etc. But these concrete objects are not to be conceived as true geometrical figures, which are not movable, for motion presupposes sensuous experience. Only matter is movable. The true geometrical line is not the extension of the point, nor is the cube formed by the extension of the plane. When a point “moves” it is no longer a point, and when a cube “moves” it becomes annihilated.
“Student,” in a letter upon the same subject, speaks of a division of a cube into smaller cubes. But when a part of a geometrical figure is conceived the first figure is of necessity annihilated.
Mr. Bragdon, after expatiating upon the vastness of the firmament,
makes this extraordinary conclusion: “Viewed in relation to this universe of suns, our particular sun and its satellites shrink to a point. That is, the earth becomes no-dimensional.” The last word is in italics. Now this is manifestly a misconception, since the most minute atom, notwithstanding its insignificance in proportion to the universe, cannot be considered as an abstraction, which a point really is. Those who are not satisfied with the intuitive evidence of the limitation of space to three dimensions, solely because no logical proof can be adduced of this limitation, would do well to read the essay of Schopenhauer on The Methods of Mathematics, in which is cited as an instance of the undue importance of logical demonstration the controversy on the theory of parallels. The eleventh axiom of Euclid “asserts that two parallel lines inclining toward each other if produced far enough must meet,—a truth which is supposed to be too complicated to pass as self-evident and thus requires a demonstration…. It is quite arbitrary where we draw the line between what is directly certain and what has first to be demonstrated.” (The italics are mine.)
I believe with Schopenhauer, who quotes Descartes and Sir W. Hamilton in support of his contention, that the science of mathematics has no cultural value. Far from affording “a new way of looking at the world,” as Mr. Bragdon tries to convince us, “its only direct use is that it can accustom restless and unsteady minds to fix their attention.” That such mental concentration may be woefully misdirected is instanced in the cases of Swedenborg and Madame Blavatsky, reference to whom by Mr. Bragdon is alone sufficient to cause a sniff of suspicion.
Indeed your author himself, while evidently well versed in bookish mathematics, has been unable to free his mind of its limitations. Upon a basis of phrases devoid of significance he builds his extravagantly mystical speculation, which dissolves in the light of reason, “into air, thin air.”
Philip J. Dorety, M. D.
Trenton, N. J.