More on Numbers

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.

“C” Circumference:

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…

Numbers?

Difference being discrete, and our definition of Existence (as opposed to the more unattainable “Wu”), then, can be counted. In my first post I said “Wu” was zero.

Well, the first difference from Wu is one, so we could write:

0-1=-1

But ‘negative one’ is just a place holder, an expression of the absence of one. That absence is ‘felt by’ the Wu. But we are not the Wu, we are the 1, so in a weird way:

0-1=+1

Or, more comfortably: 1-1=0.

But that ‘absence’ is not to be ignored! As a matter of fact, it’s crucial.

Look at astrophysicists racking their brains at what Dark Matter could possibly be.

It’s only ‘Dark’ because they can’t measure it or observe it – and yet they noticed it – because the behaviour of our Universe was too strange without it – an absence of information is still information. -1 says “there used to be a one there” and we know how big it was too.

Our Existence, defined because of difference, is the reference point from which to measure Wu. Sure, we may not be able to measure all of Wu (is it infinite?), but we can certainly measure between here and over there.

We need to start thinking of ‘negative space’ like the artists do – the ‘pauses’ in music, the material removed from marble sculptures, the unoccupied space of the canvas. We don’t know what Wu is, but we know it’s there, in the gaps. But it’s also here, the ‘stuff-of-us’.

Difference being something discrete gives rise to the Natural Numbers (but let’s not forget zero!). We also have the Integers and Rationals. But what about the Real and Complex numbers? Quaternions and Octonions and so on?

Well, mathematicians state that the Real numbers are uncountably infinite, so infinite that they are infinitely continuous. Hmmm. Sounds a lot like Wu to me.

The continuity of Real Numbers pretty-much covers any and all dimensions. Wait. What? So Time is continuous? That’s pretty mind-boggling right there. Time, a sequence of events or changes, is continuous?

It’s hard to imagine a continuous flow of changes – especially since we just said that Difference is discrete!

Sure, compared to “Wu”, something is discretely different, but Time (being the observation of change) has to be discrete, because any change has to be discretely different from its previous state (it’s either different or it isn’t). So for a continuous yet discretely changing time-line, changes are continuous, but their “differences” are infinitesimally discrete. Isn’t that the same as ‘continuous’?

I’ve thought myself into a corner with this one… or I’ve come full-circle. Yes, Time is continuous. Or instead, Time doesn’t exist. Only change is continuous.

Is rate-of-change constant or varied? Without Time, one would be tempted to say ‘rate’ of anything is impossible. But that would be false. A rate is the ratio of the number of events occurring compared to another number of events.

One set of changes could be ‘faster’ than another (8 changes for every 4 of the other means it’s twice as fast). Well, by that comparison, we’ve just realized that rate-of-change is varied, by definition. Obviously – if everything changed at the same ‘rate’, then there would be no ‘rate’ to speak about.

Well, enough of that for now – on to more thinking. Leave a comment if this made you ponder…