[ExI] dicovery of irrational numbers

[Anders Sandberg]

On 2013-12-19 01:10, spike wrote:
>>>It's even worst than that, at least with PI (and e) there is a infinite
> series that produces it so you can find a decimal that is as close to PI
> as you like, but most irrationals, nearly all in fact, are not like
> that, no infinite series or anything else can produce them, they're not
> computable, so you can't even get good approximations.  We can't even
> point at most numbers.
>
> Oy, good point.  My son is reacting much like scientists do when we keep
> finding out there isn’t really a simple consistent underlying principle,
> or if so, we still don’t know it.

Well, you could reassure you that mathematicians are on top of the game. 
We do understand irrationals, transcendentals and uncomputable numbers 
fairly well.

The real lesson is that there are some deep principles but most everyday 
approaches only approximate them. Natural numbers need negative numbers 
as an extension, and fractions require the rationals. Anything 
continuous will bring in irrationals, but *which* irrationals depends on 
what kind of continuous stuff you use (classical geometry works in a 
particular field lacking cube roots, for example). When you just make an 
intuitive jump and generalize, like the real line, you might introduce a 
lot of unexpected subtleties and unwanted extra stuff. But the game of 
mathematics is to figure out principles that get it under control: group 
theory, field theory, extension theory and so on.

Fun thought I had with Ben Goertzel: in a different world we might have 
invented mechanical computers before calculus and a counterpart to 
Turing proved computability. In that world we might regard computable 
reals as "real", and uncomputable reals as weird special cases for 
mathematicians to talk about. Most of calculus would be unchanged, but 
the mean value theorem and a few others would only be true for the 
"uncomputable real line" and likely be regarded as esoteric.


> Ja, that part is OK, but math is clean and precise.  Even I agree that
> irrationals are ugly things.  I love them anyway; all numbers are my
> friends.  But pi is ugly.  It has inner beauty with that way cool series
> expansion.  Actually it has a number of expansions, but my favorite is
> pi= 4/1-4/3+4/5-4/7…  oh so cool is this.

I love continued fractions. Although pi is a bit messy there too. 
Incidentally, continued fractions is the reason the golden ratio is the 
most irrational number in a strict sense.


-- 
Anders Sandberg
Future of Humanity Institute
Oxford University