I’m still troubled by the notion of a number with relation to discrete vs. continuous, Rational vs. Real, etc.
Why? Because you can’t even talk about a number on a continuity which is infinite. There is simply no such thing, because there is no ‘thing’ upon which to attribute a number.
The mere fact of using a number implies, implicitly, difference from the whole – the continuity.
Consider a wrinkle. If we say that a wrinkle on a line – a crest or vertex – is different from the rest of the line (even though it’s ‘made of’ that line), then we can number it – count it.
However, when we want to talk about “the continuous number line,” i.e. the Real Numbers, then that is our first mistake because we are using numbers (implicitly discrete things) to describe a continuous thing. This is why we end up with infinite decimal expansions and transcendental numbers which go on forever. When we reach the ‘end’ of our decimal number, we’re merely wrinkling the continuity according to a well-established algorithm in order to obtain the next decimal in the infinite expansion!
When we compare a line segment (discrete), say, a circle’s diameter, to a continuous entity (the circumference) then of course we come up with the impossibly-finite ‘number’ pi! We’re comparing apples and oranges!
The two cannot be ‘com-paired’, paired-off against one another.
Let’s think about the sets to be paired-off:
“L” Line segment:
The ‘set’ which describes that segment is a single number – its length – which we take to be a countable set of wrinkles which are of a pre-established size.
The ‘set’ which describes the circumference, we assume, is also a number. That number is a countable set of wrinkles of the same pre-established size as the segment’s wrinkles.
When we pair-off these sets of wrinkles, we see that you can almost pair-off “L” to “C” three times, but just not quite. In the gap, we see we could fit another wrinkle which is ten times smaller than the wrinkles of the set “L”. But again, not quite. Now you can fit four more wrinkles which are a hundred times smaller than the wrinkles in “L”. So that’s 3.14, but again, not quite. And so on.
You cannot com-pair sets of the same-sized wrinkles! But by creating smaller wrinkles we’ve ‘broken the rule’ – our initial assumption.
Now that we’ve broken that rule, the problem can be turned the other way around: say “C”is discrete – let’s say there are exactly 24 wrinkles of a chosen size which fit around the circumference of our circle.
Well, since you can arbitrarily choose the size of wrinkles for your sets, then you can choose to say that the set “L”, the diameter, also has exactly 24 wrinkles… or 8-ish so that “L” is exactly 3 times smaller than “C”.
Heck, at this rate, you could even say “L” has only one wrinkle in it, a big one which exactly spans the diameter of your circle.
But you see the nonsense here? There are no rules. We’ve made assumptions which cannot hold, and we’re trying to compare two things of fundamentally different natures!
In this view, I understand where Norman J. Wildberger is coming from when he says that he doesn’t ‘believe’ in Real numbers. His sense of pure, mathematical Truth is offended by those ‘irrational’ infinite decimal expansions and I think it’s because we threw out consistency (fixed wrinkle size in my example) for the sake of making an answer to the wrong question.
Now, that being said, I also understand that we need a way to describe a circle, because, though continuous, they clearly exist, and there are things which can be said about them which we might use to describe them. In addition, I also seem to come up against continuous things which make it impossible to avoid comparison between these two opposing types. So unlike Wildberger, I’m not ready to throw the baby out with the bathwater just yet.
Definitely needs more thought though…