[In the following essay, Weinberg views Lovecraft's invented mathematics in several stories as a blending of science and fantasy.]

One of the strongest points in the Cthulhu stories by H. P. Lovecraft is the skillful blending of the unreal and the real. True and false are juggled together until one is undistinguishable from the other. Probably the most mentioned example of this work is Lovecraft's invention of a number of fictitious books complete with quotes, mysterious authors and histories. However, little attention has ever been given to the mathematics, or in reality, the pseudomathematics used in several of the Cthulhu tales. In this [essay], I hope to cover this area, however briefly.

In the period that Lovecraft did his writing (1920–1935) science was just emerging from the greatest traumatic period in all history. At the turn of the century, the Michaelson-Morley experiment had all but destroyed the notion of an all-pervasive ether. Einstein and Planck had completely disrupted all of classical physics with the theory of General Relativity and their restructuring of the physical universe. Heisenberg's Uncertainty Principle had completely reshaped the idea of what we know, and more important, of what we can learn. Research done by half-a-dozen famous scientists had determined the structure of the atom. For the first time in history, modern man was exposed to what is now being called “Future Shock.” Man tried, with little success, to grasp all that was occurring about him. Popular science articles in the Sunday papers were quite common, as well as many books proclaiming “Mathematics for the Millions” and “Relativity Made Easy.” The unfortunate fact is that without a strong background and training such concepts are not easy. Nor are they simple. Most simplistic views of the subjects were no more than a thin gloss of a much deeper idea.

Most of the writers in the fantasy-science fiction field of this time were not scientists. The few that were, were not associated with the fields of major advances. There were exceptions, of course, like John Campbell, but even he did not use a strong straight science background in his stories. Most of the tales in the period employed what can be loosely called pseudo-science. That is, made-up science which had very little or no relation to the real work of the period. The speed of light, which was just then being recognized as an absolute upper bound on speed, was ignored by every writer who used a FTL drive. Structural impossibilities in construction of super cities were common (and still are). Biological impossibilities (such as violations of the square-cube law) were the order of the day. Though the claim has been made that this was the period of SCIENCE fiction (i.e., the science emphasized while the fiction was not), this statement is not true. Anyone competent in the sciences could tell otherwise. It was a time of PSEUDOSCIENCE-fiction. In other words, a time when impossible science was emphasized. Not speculative science, with a possibility of reality someday (as in Ralph 124C41+), but just sheer nonsense masquerading as science. This is not to say that some of these stories were not entertaining, but just to point out that they were straight fantasy. Any speculation they contained was false and misleading. (A fine example of such work is Ray Cummings' stories about “The Girl in the Golden Atom.” These are fairly entertaining tales, but Cummings' basic premise that the structure of an atom is somewhat similar to a solar system is utter nonsense, and was known to be nonsense long before most of the series appeared.)

The revolution in mathematics had taken place sometime before the revolution in physics, though the two events are closely interrelated. Cantor's work with infinite sets was a major breakthrough from finite to infinite mathematics. Work in non-Euclidean geometry, showing that structure was possible as long as an axiomatic system was maintained, produced a minor breakthrough. Again, popular science articles tried, with little success, to convey the meanings of such breakthroughs to the public. Misconceptions immediately arose.

Lovecraft's misunderstandings in both geometry and quantum physics are therefore nothing uncommon for the time he wrote. Even if he had (or did) consulted various reference works of the period for information, it is doubtful that the references to mathematics in his stories would have been very different.

In nearly all of the Cthulhu stories, some mention is made to the alien geometry, encountered (as in “The Call of Cthulhu,” “The Shadow Out of Time,” and others). Instead of going over every story, I will attempt to note the pseudomathematics used in one story, and thus avoid repetition. The mistakes in one are common throughout all of the mythos stories (including work by others, such as “The Hounds of Tindalos” by Frank Belknap Long). The story I have chosen to study is “Dreams in the Witch House.”

The story is the one in which HPL makes his greatest use of mathematics. The protagonist of the story is Walter Gilman, a mathematics student at Miskatonic University. Gilman is very interested in non-Euclidean calculus, we are told, as well as quantum physics. Needless to say, no such subject as non-Euclidean calculus exists, nor does such a name make any real sense. While calculus does have a strong background in geometry, an in-depth study of the subject reveals the relative unimportance of the field in which the actual limit process takes place. The name of the subject sounds good, but means nothing. Quantum physics is a fancy name for the study of quantum mechanics, i.e., the motion of the universe as related by the theory developed by Planck and Einstein.

Lovecraft's pseudomathematics we might call pseudogeometry. That is, the notion that certain geometric shapes could be constructed that might not be entirely of this dimension. In the story itself, Gilman speculates on the possibility of creating a hole in the space-time continuum by a geometric construction, so that a person could step in through the hole at one place and emerge in another. It is much the same idea that was popular around the same time and after about a space warp. Space, we know, is curved. If, we are told by the pseudoscientists, we were able to bend space, then we could step from one spot in space to another without traveling the intervening distance. This concept of curved space is easily explained for the curious in the book entitled Sphereworld (the author I've forgotten).

This idea, unfortunately, is not quite true. The existence of higher dimensions should be of little concern to this world as any contact with such dimensions is impossible. Lovecraft's angles and curves that vanish into some other space is absolute nonsense. The reason for this is that geometry is a closed system. It is impossible (not unlikely or not yet possible, but impossible, actually shown to be never possible) to construct a higher dimension from a lower one. A quick reading of Flatland by Abbott will suffice to convince the reader. A two-dimensional being which lives on a flat surface cannot grasp the concept of “up” and “down.” Such flatworlders cannot understand the meaning of height, or depth, or thickness, as no such thing exists, nor can it exist, in their two-dimensional world. As I do not want to belabor this point, I would strongly advise all of those interested to read Flatland, a book quite easy to understand and readily available in most libraries.

The same facts, thus, also apply to a three-dimensional world. There is no way that we can construct a four-dimensional object. Nor can such a thing even exist in our world. Since our perceptions are only three-dimensional, we could not see the fourth dimension extension of the object even if it had one. As all of our building material is only three dimensional, it would be impossible to construct anything of a higher dimension out of it. A quick way to grasp this impossibility is this. A straight line segment defines one dimension. Put another line segment perpendicular to this first line segment (getting something in the form of an L) and you have defined two dimensions (length and width). A third line, sticking out from the paper at a right angle, defines depth. Now, to define a fourth dimension of measurement (as we are talking of instantaneous occurrences, we ignore time measurements in our argument) take another line segment and put it at right angles to all of the other three. This will give you a fourth dimension. This is also quite impossible in our physical universe.

We are also told in the story of Gilman stating that time could not exist in certain belts of space, so that one could live forever in such regions. This fact would surprise a number of scientists.

In conclusion, Lovecraft was a master craftsman who used whatever knowledge he could in the furtherance of his story. Unfortunately, while his grasp of science and mathematics might have been greater than the average layman, it was not strong enough to present a convincing picture to the careful reader. Further, Lovecraft made the cardinal mistake of speculation of the impossible. While to the non-scientist, this may not sound like much of a sin, it is the cardinal mistake of the uninformed.