imaginary numbers are a fine and wonderful resource of the divine intellect, almost an amphibian between being and non-being.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?

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).

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.

Oh, but of course the story of Adam and Eve was only ever symbolic, wasn't it? Symbolic?! So Jesus had himself tortured and executed for a symbolic sin by a non-existent individual? Nobody not brought up in the faith could reach any verdict other than "barking mad". (Part 2, 00:30:25)

This doesn't mean you can picture a billion or a quintillion objects in your head—nobody can. But, with exponential notation, we can think about and calculate with such numbers. Pretty good for self-taught beings who started out with no possessions and who could number their fellows on their fingers and toes.

The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.

Adam and Eve probably had hundreds of children. They lived 800 years, and one could have a lot of children in 800 years.

Numbers exist in the void; it is impossible to imagine the destruction of numbers, even though you imagine the destruction of the universe.

The physicist ... engages in complex and difficult calculations, involving the manipulating of ideal, mathematical quantities that, at first glance, are wholly lacking in the music of the living world and the beauty of the resplendent cosmos. It would seem as if there exists no relationship between these quantities and reality. Yet these ideal numbers that cannot be grasped by one's senses, these numbers that only are meaningful from within the system itself, only meaningful as part of abstract mathematical functions, symbolize the image of existence.

Number was born in superstition and reared in mystery,... numbers were once made the foundation of religion and philosophy, and the tricks of figures have had a marvellous effect on a credulous people.

Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from arithmetic to another Science, (which yet is no science) viz. algebra.

Adam and Eve began to play romance when the latter was created out of the former's ribs and brought to him.

The familiar proposition that all A is B, and all B is C, and therefore all A is C, is contracted in its domain by the substitution of significant words for the symbolic letters. The A, B, and C, are subject to no limitation for the purposes and validity of the proposition; they may represent not merely the actual, but also the ideal, the impossible as well as the possible. In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accordance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power.