infinity divided by infinity

You spoke of ‘infinity’ as if it were a number. It’s not. You may as well ask, ‘What is truth divided by beauty?’ I have no clue. —John Derbyshire, 2003

I have no clue myself, but these are snippets smartly writ:

Nonmathematical people sometimes ask me, You know math, huh? Tell me something I’ve always wondered, What is infinity divided by infinity?

I can only reply, The words you just uttered do not make sense. That was not a mathematical sentence. You spoke of infinity as if it were a number. It’s not. You may as well ask, What is truth divided by beauty? I have no clue.

I only know how to divide numbers. Infinity, truth, beauty those are not numbers.

What is a modern definition of analysis, then? I think the study of limits will do for my purposes here. The concept of a limit is at the heart of analysis. All of calculus, for example, which forms the largest part of analysis, rests on the idea of a limit.

Arithmetic: The study of whole numbers and fractions. Sample theorem: If you subtract an odd number from an even number you get an odd number.

Geometry: The study of figures in space points, lines, curves, and three-dimensional objects. Sample theorem: The angles of a triangle on a flat surface add up to 180 degrees.

Algebra: The use of abstract symbols to represent mathematical objects (numbers, lines, matrices, transformations), and the study of the rules for combining those symbols. Sample theorem: For any two numbers x and y, (x + y) (x y) = x2 y2.

Analysis: The study of limits. Sample theorem: The harmonic series is divergent (that is, it increases without limit).

Modern mathematics contains much more than that, of course. It includes set theory, for example, created by Georg Cantor in 1874, and foundations, which another George, the Englishman George Boole, split off from classical logic in 1854, and in which the logical underpinnings of all mathematical ideas are studied.

The traditional categories have also been enlarged to include big new topics geometry to include topology, algebra to take in game theory, and so on.

Even before the early nineteenth century there was considerable seepage from one area into another. Trigonometry, for example, (the word was first used in 1595) contains elements of both geometry and algebra. Descartes had in fact arithmetized and algebraized a large part of geometry in the seventeenth century, though pure-geometric demonstrations in the style of Euclid were still popular and still are for their clarity, elegance, and ingenuity.

Tidying up is a relative term… for some a Prime Obsession.