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Comment by 08-15 | Hacker News Reader

So much for "mainstream math". There is an alternative take called constructive algebra, which I believe to be more useful to a computer programmer. Briefly:

"There are only finitely many numbers."

That's trivially true, because numbers are man-made objects, and man has only had enough time to make finitely many of them. Integral and rational numbers can be made by writing them down, algebraic irrational numbers can be made by writing down an expression for them. (What language? Lambda calculus, obviously, because according to the Church-Turing thesis, nothing more powerful exists.)

Only countably infinitely many numbers can potentially be made this way, only finitely many will ever be made. Those non-algebraic transcendental numbers? Don't exist, because they can never be made. They may "exist" in a philosophical way, but they no impact whatsoever on the real world in which my programs operate.

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> numbers are man-made objects

That's plain wrong, animals can count too.

So you can't imagine a world in which our understanding of pi or e might impact or be useful to the real world?

More broadly, you can surely imagine how the study of transcendent numbers might inform results in other fields? I don't usually try to justify math by pointing at its applications, but for example, the transcendence of pi leads to a proof that squaring the circle is impossible. Even if things aren't constructable, they can still be valuable to think about.

Brilliant, just like the dimwitted TA at university who claimed full of conviction "NO amount of memory can represent pi!" Which is funny, because he represented it with a single character.

Pi and e are both algebraic numbers, there are expressions (algorithms) for both. And guys like you are funny: Faced with proof that almost all real numbers do not have a name, you immediately try to name an example.

You contradict yourself: any given real number is, in fact, given by its "name", which, as you said, is either a symbol or an algorithm (which, incidentally, are the same thing - a symbol can be seen as the name of an algorithm, or a function; the TA was probably referring to the fact that the algorithm would have to stop at some point so as to avoid overrunning the memory limit).

> You contradict yourself: any given real number

Err... I didn't say that. For good reason, as you point out. I know that you cannot "give" almost all real numbers. I'm merely pointing out that there are mathematicians who go the extra mile and decide that those don't even exist.

Symbol, name, algorithm; all of those are the same thing. And almost all real numbers (in ZFC aka "standard math") don't have one.

> the TA was probably referring to the fact that the algorithm would have to stop at some point

Nope, he wasn't. He specifically insisted that no amount of memory can represent pi, while any integer can be represented. He was technically correct, but I have an easier time working with pi (to any desired, even dynamically determined precision) than with Ackermann(4,4).

The dimwitted TA missed the point, and he did so, because standard (not constructive) math creates lots (continuously infinitely many) of infinities, which serve no use but to make irrelevant points about things that cannot occur in actual programs.

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