Mathematics deals with properties and ideas which are applicable to 'things' apart from feelings, emotions, or anything connected with them.
Whitehead sums up the nature of mathematics in the following statement: "the leading characteristic of mathematics [is] that it deals with properties and ideas which are applicable to things just because they are things, and apart from any feelings, or emotions, or sensations in any way connected with them. This is what is meant by calling mathematics an abstract science" (IM 2f).
The 'things' of mathematics, with individuality of their own, are non-temporal: eternal objects.
This analysis of mathematics seems to be the reason for Whitehead to attach the attribute of a "particular individuality" (SMW 229) to eternal objects in his later philosophy. By this he meant that "the [eternal] object in all modes of ingression is just its identical self" (SMW 229). The reasoning behind this statement is that, if the ideas of mathematics and their respective properties were dependent on diverse cases of application, they would not be able to maintain the same identity in every mode of application.
Whitehead does not think 'space-perception' is necessary to mathematical thinking. Entities can exist that do not exist in one space or any space.
The comparison between algebra and geometry led Whitehead to another conclusion, in a sense complementary to the arguments above. He states: "Space-perception accompanies our sensations, perhaps all of them, certainly many; but it does not seem to be a necessary quality of things that they should all exist in one space or in any space (IM 182).6 Here he does not only stress the independence of abstract mathematical ideas from any special application to nature, as described above, but extends their independence to the point where these abstract ideas are no longer bound to find application in nature, as perceived by our senses.
Whitehead does not show the relationship of mathematics to the humanities or the subjective side of reality.
"One of the areas that is not covered in IM is the relationship of mathematics to the humanities or, more generally, to the subjective side of reality. However, one can already find hints intimating the forthcoming emergence of that question."
Mathematical symbols are not more mysterious than everyday language; they are shorthand for the brain.
They are not the elements of a new language, distinct from our ordinary language, but simply represent a shorthand for our everyday speech, as pertaining to mathematics. Two statements underscore this understanding of symbolism. First, Whitehead emphasizes that the signs for numerals, letters, and mathematical operations are not the outward side of a language essentially different from, and more mysterious than everyday language, but that they are introduced to relieve the brain (see IM 39) and "to make things easy" (IM 40). For "by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which other-wise would call into play the higher faculties of the brain" (IM 41). Thus the function of mathematical symbolism is to help perform "important operations . . . without thinking about them" (IM 42).
After Principia Mathematica, Whitehead did not return to mathematics in a serious way.
"We examine Whitehead’s early philosophy of mathematics in this article because it was his only explicit philosophy of mathematics. After Principia Mathematica, Whitehead let major new mathematical developments pass him by, and he never returned seriously to a philosophy that considered those new directions in mathematics."
Whitehead was an empiricist. He believed mathematics is abstracted from experience.
"Whitehead’s theme [in his Introduction to Mathematics], begun in the first chapter and maintained throughout the book and, in our judgment, for the rest of his philosophy, is that mathematics begins in experience and as abstracted becomes separated from experience to become utterly general. "We see, and hear, and taste, and smell, and feel hot and cold, and push, and rub, and ache, and tingle" (IM 4). These feelings belong to us individually. "My toothache cannot be your toothache" (IM 4). Yet we can objectify the tooth from the toothache and so can a dentist who "extracts not the toothache but the tooth," (IM 4) which is the same tooth for both dentist and patient. Whitehead would give later in Process and Reality a metaphysical explanation of how we may objectify precisely an individual thing from vague feelings by his description of indicative feelings (PR 260).
Whitehead denied that mathematics deals with the emotional side of life. He failed to see the esthetic side of mathematical exploration.
"His mathematical research tended to two extremes: applications and foundations. The mainstream mathematical culture, which, regardless of ontological commitment, is driven as much by esthetics as by science, seems to have had little meaning for him. In spite of his great interests in esthetics generally, he had only a narrow sense of mathematics as, in the words of C. H. Clemens, "an esoteric art form," and even less sense of passion for mathematical adventure. Later, he would declare that mathematical form does not even admit emotional subjective form for its feeling (AI 251)."
Whitehead also failed to sense the process-relational implications of mathematics. It remains for others to develop a more robust philosophy of mathematics grounded in the wisdom of Process and Reality.
"Whitehead, a mathematician of note to his contemporaries but of small consequence to his successors, never scented a relational approach to mathematics. Perhaps functoriality had to await the further maturation of cross-disciplinary fields such as algebraic topology and algebraic geometry, but in light of Whitehead’s eccentric tastes, we doubt that fifty years would have made much difference. He seems implicitly to have accepted a condition of ontological stasis for the mathematical world. All the more remarkable, then, that Whiteheadian metaphysics explicitly countenances the occasions of actual entities through the dynamic, relational process of concrescence, a process remarkably similar to the dynamic evolution of mathematical forms. The holism of functoriality is the holism of process thought. We stand in amazement that Whitehead saw this so clearly in his adopted field of philosophy but not in his native field of mathematics."