Spencer-Brown: Cataphatist

George Spencer-Brown (GSB) did some wonderful work for a system of logic that has us consistently painting ourselves into a corner. That’s the problem. His work contributes to a system of reasoning that is consistent but incomplete.
What he doesn’t see (and few who I’ve encountered who understand his work) is that he is stuck within a cataphatic mode of thought. When he says “Let us take the form of distinction for the form.” he effectively merges the boundary between things with one of the spaces that boundary ‘encloses’.
So he takes ‘the mark’ as being indicative of what it encloses. This is wrong – or maybe it’s better that I say it’s ‘too eager’. Continue reading “Spencer-Brown: Cataphatist”

The Law of Existence – Part 2

Law of Existence – continued

In my previous post, I promised to illustrate how the Laws of Thought, as they are used today, emerge naturally from the Law of Existence. Time to make good on my promise.

In a comment left by reader SelfAwarePatterns he mentioned that he didn’t see how the Law of the Excluded Middle (LEM) was rendered false by the Law of Existence. He’s absolutely right – it’s not false, and indeed it’s very naturally present already within the Law of Existence. What is false however, is the ‘closed’ interpretation of the LEM which says that a Thing only ever is or (exclusive or) is not. That is clearly false, and to use the example he gave:
Continue reading “The Law of Existence – Part 2”

The Deep Symbolism of the Mobius Strip

The Mobius Strip

If ever there was something which merited the name “God” in my eyes, it would be the Mobius Strip. But I don’t believe in a personal, let-alone sentient, god. I’d be far more inclined to call it “Tao” instead. Buddhists might call it “Om” (or “Aum”). Mathematicians should call it “i” (the square root of negative one), but there are even more examples in Mathematics (the involution, the half-rotation, inconsistency, contradiction, “not” or the symbol ¬). Electronics circuits represent it as the inverter whose ouput feeds back into its input. Philosophers might call it “contradiction” or more formally the “paradox of self-reference” epitomized in the Liar Paradox:

“This statement is False.”
Continue reading “The Deep Symbolism of the Mobius Strip”

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…