PM (Pure Mathematician): Let alpha be a real number.

NJ (Me): What does that mean?

PM: Surely you are joking. What do you mean by such a question? Everyone uses this phrase all the time, probably you also.

NJ: I used to, but now I am not so sure anymore what it means. In fact I suspect it is nonsense. So I am asking you to clarify its meaning for me.

PM: No problem, then. It means that we are considering a real number, whose name is alpha. For example alpha = 438.0457897416622849… .

NJ: Is that a real number, or just a few decimal digits followed by three dots?

PM: It is a real number.

NJ: So a real number is a bunch of decimal digits followed by three dots.

PM: I think you know full well what a real number is, Norman. You are playing devil’s advocate. Officially a real number is an equivalence class of Cauchy sequences of rational numbers. The above decimal representation was just a shorthand.

NJ: So the real number alpha you informally described above is actually the following: {{32/141,13/55234,-444123/9857,…},{-62666626/43,49985424243/2,7874/3347,…},{4234/555,7/3,-424/55,…},…}?

PM: Well obviously that equivalence class of Cauchy sequences you started writing here is just a random collection of lists of rational numbers you have dreamed up. It has nothing to do with the real number alpha I am considering.

But now that I think about it for a minute, I suppose you are exploiting the fact that Cauchy sequences of rationals can be arbitrarily altered in a finite number of places without changing their limits, so you could argue that yes, my real number does look like that, although naturally alpha has a lot more information.

NJ: An infinite amount of more information?

PM: If you like.

NJ: What if I don’t like?

PM: Look, there is no use you quibbling about definitions. Modern pure mathematicians need real numbers for all sorts of things, not just for analysis, but also modern geometry, algebra, topology, you name it. Real numbers are not going away, no matter what kind of spurious objections you come up with. So why don’t you spend your time more fruitfully, and write some papers?

NJ: Have you heard of Wittgenstein’s objections to the infinite shenanigans of modern pure mathematics?

PM: No, but I think I am about to.

NJ: Wittgenstein claimed that modern pure mathematicians were trying to have their cake and eat it too, when it came to specifying infinite processes, by bouncing around between believing that infinite sequences could be described by algorithms, or they could be defined by choice. Algorithms are the stuff of computers and programming, while choice is the stuff of oracles and slimy intergalactic super-octopi. Which camp are you in? Is your real number alpha given by some finite code or by the infinite musings of a god-like creature?

PM: I think you are trying to ensnare me. You want me to say that I am thinking about decimal digits given by a program, but then you are going to say that that repudiates the Axiom of Choice. I know your strategy, you know! Don’t think you are the first to try to weaken our resolve or the faith in the Axioms. Mathematics has to start somewhere, after all.

NJ: And your answer is?

PM: Sorry, my laundry is done now, and then I have to finish my latest paper on Dohomological Q-theory over twisted holographic pseudo-morphoids. Cheers!

NJ: Cheers. Don’t forget to take alpha with you.