The continuum so conceived is only a collection of individuals ranged in a certain order, infinite to one another, it is true, but exterior to one another. This is not the ordinary conception, wherein is supposed between the elements of the continuum a sort of intimate bond which makes of them a whole, where the point does not exist before the line, but the line before the point. Of the celebrated formula “the continuum is unity in multiplicity”, only the multiplicity remains, the unity has disappeared. The analysts are none the less right in defining the continuum as they do, for they always reason on just this as soon as they pique themselves on their rigor. But this is enough to apprise us that the veritable mathematical continuum is a very different thing from that of the physicists and the metaphysicians.

Philotheo: All of this is true, and contradicts nothing we've said, rather to the contrary we've said that there are dissimilar finite parts in one infinity, and have offered considerations how this might be true. Perhaps it might be expressed proportionately, how one might have many continuous parts which form a unity, using the example and simile of liquid mud, which, though water is contiguous with water in every part, and earth with earth, smaller than we can apprehend sensibly, these are called neither discrete nor continuous, not water nor earth, but only a continuum of mud; another might like to say that since the atoms of water are not actually continuous with one another, nor earth with earth, but perhaps water with earth and earth with water; a third might disagree with both and say only mud is continuous with mud. Following these reasons it can be stated that the infinite universe is a continuum, in which discreteness is not created by the interposition of ether between the great celestial bodies, than it would be were air to be mixed and interposed among the dry and watery particles, the difference being only in the consistency of the smallest parts of the mud, beneath the level of our sensible apprehension, or in the size, greatness, and sensibility of the parts that make up our universe: and so, contrary and diverse moving parts cooperate and compose a single immobile continuum, where contraries converge to make up a single whole, achieve single order, and become one. Certainly, it would be inconvenient and impossible to imagine two infinities distinct from one another, since it would be impossible to imagine the dividing line between them, where one infinity would end and the other begin, or in what way each would terminate against the other. Moreover, it is extremely difficult to imagine two bodies which are finite and bounded on one side and infinite on the other.

The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.

Think of it: of the infinity of real numbers, those that are most important to mathematics—0, 1, √2, e and π—are located within less than four units on the number line. A remarkable coincidence? A mere detail in the Creator's grand design? I let the reader decide.

The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains.

The only real number is one, the rest are mere repetition

I shall conclude that there is in all of us an intuitive notion of the continuum of any number of dimensions whatever because we possess the capacity to construct a physical and mathematical continuum; and that this capacity exists in us before any experience, because, without it, experience properly speaking would be impossible and would be reduced to brute sensations. ... And yet this capacity could be used in different ways; it could enable us to construct a space of four just as well as a space of three dimensions.

... between the elements of the continuum [there is supposed to be] a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point.

Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers. Complex numbers are as intuitive for an electronics engineer as -100 is for the average person with an overdrawn bank account.

In recapitulation, the mind has the faculty of creating symbols, and it is thus that it has constructed the mathematical continuum, which is only a particular system of symbols. Its power is limited only by the necessity of avoiding all contradiction; but the mind only makes use of this faculty if experience furnishes it a stimulus thereto.

[T]he enigma of the continuum arises because language misleads us into applying to it a picture that doesn’t fit. Set theory preserves the inappropriate picture of something discontinuous, but makes statements about it that contradict the picture, under the impression that it is breaking with prejudices; whereas what should really have been done is to point out that the picture just doesn’t fit…

Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).

The introduction of numbers as coordinates by reference to the particular division scheme of the open one dimensional continuum is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations. The topological skeleton determines the connectivity of the manifold in the large.

"You mentioned . . . one of the two great labyrinths into which the mind is drawn. What . . . is the other?"
"The other is the composition of the continuum, or: what is space?"