It is a good few years since I have written anything about the nature of number. It was in 2012 that I spoke about the realities of infinity in my post ‘Infinity, What Is It (And How To Find One Of Your Own)’ (1). What I didn’t speak about was how we arrived at number in the first place; I touched on this in another early post, about goats (2).
I want to assure you that these posts are extremely serious in nature, for all their glib veneer.
The number of mathematicians I have ever met who truly understand the nature of number are impossibly small. That doesn’t compromise their intelligence, it speaks of something else that is extremely subtle. Not to mention insidious. That is what this post is about to explore, and will be followed up with others that will discuss computer code generation and quantum physics. I assure you that they will be both readable, and to the best of my ability, understandable.
You all know that no two snowflakes are alike.
Ever, period. In the entire scope of time, no two snowflakes have ever been identical. No two leaves, blades of grass, goats, cats, dogs, wombats or even grains of wheat have ever been identical. Imagine a freight car filled with zillions of grains of wheat and you’ll begin to get the picture. That the stuff is sold by the pound, ton or shipload makes no difference to this simple fact: no two grains of wheat are ever the same twice.
There is one simple rule for nature: nothing ever happens twice.
The corollary to this is another simple rule – but that is reserved for my private blog. Not because of a paywall, but because most people cannot fathom the things I speak of publicly; those who can will find a quiet invitation to look deeper into the things that puzzle them. They are only puzzling because of the way we’re taught to think. Read on, because this is the topic of this post: albeit that it deals with the issue tangentially.
The Nature Of Number.
It has to be admitted that for all the individual oak leaves that have ever existed in creation, they are discernibly the leaves from an oak tree and not from an ash or a beech. Sure, you do need to be able to recognize an oak leaf, but it does mean that you can find two oak leaves. This will sound trivial, but I assure you that for all the excuses I have ever heard from mathematicians, this only tells me that they do not understand the nature of number.
Because to count two of anything means making an assumption: now it is true that two oak leaves are both from oak trees. Whilst they are never identical, they are identifiably oak leaves; you know that from their character. However, you’ll never get an oak leaf growing on a beech tree. Even the GM scientists can’t do that, not that they’d be interested in this aspect of GM nonsense anyway. Mainly because you can’t corner that market to squeeze the last pennies out of it before it dies. But this blog ain’t about profitability; it’s about reality. Furthermore, if you want a business that is profitable, you want one that accords with reality.
As mentioned, oak leaves all have a specific form and character, a nature that is recognizably consistent across all the oak leaves that have and ever will exist. Fair enough: we are dealing here with the reality of nature. My point here is that because people, and mathematicians in particular do not recognize this simple fact, they are unaware of the dangers they face if they step beyond the invisible bounds of reality.
The first step in this direction is to count, and count without counting anything. Like counting in the rote fashion: “One, two, three, four,” and so on. Again, this sounds pretty stupid as we all do this – but this only shows how far humanity has strayed from the consistency of reality. Nor have most humans strayed very far: the point is that they have strayed, and in that they do not know that they have strayed, they really are lost. Sure, it might only be the tiniest fraction of an inch that they have strayed, but in that they know nothing of it, that distance might as well be a million miles. How can you comprehend distance if you don’t know what you’re looking for? Trapped in a dark cellar, that tiny light could be the door, it could be a star seen through a window: there is nothing to guide you.
Hence: counting is an abstract concept. That is to say, it is ‘abstracted’ from the reality of the world we live in. There are several directions this post could take, having made this statement: these will be dealt with in forthcoming posts. If in the course of these posts, you notice something I have missed, please let me know via the comments.
Number As A Quality.
Having said that counting is abstract, the quality of the number itself is not. There are very real differences between two and three; four and five. If you cut an apple in half across its equator, you will always see that the seeds are distributed in five cavities. Never four, never six. This is one of the signs of the rose family of plants. A talisman, as it were. See ‘5’ and think rose (but not defunct German bank notes).
Fiveness itself is a little harder to describe; fourness is hard enough, but it is possible in terms of the human body in that we have forwards, backwards, left and right. But then, we also have up and down – thus I made an assumption about how to count things… but then, I am aware of most of the assumptions I make in my decision-forming life, where most people are are not. Again, the problem for those who can’t is that they cannot know… and in that they cannot know, they will tell me that it is impossible for me to know. All I can say to that is “don’t be so foolish”.
Because this isn’t hard, but it does take a little stretching of your brain to grasp the subtleties. Subtleties that are largely irrelevant.
Until it begins to dawn on you where our ignorance of them has led us. But that is the challenge of knowledge.
The number 5 is a prime number. That doesn’t affect its fiveness, prime numbers are only special in that they have fewer relationships with other numbers: a prime is a number that can only be divided by itself and 1. Division, however, is not something that is natural; division in its own way is a denial of the future. Anything that has a future is unitary: it is a seed in one form or other. Divide it in two and you have destroyed that potential future.
Back to primes: they have special qualities of their own, but lack the qualities that the more promiscuous numbers have. They are numerical unicorns, let us say. Their qualities are more those of absence than of presence: cicadas employ this because expressed over time, there is a prime number of years that pass between their emergence as adults from their subterranean hiding place where they dwell as grubs. If two species employed periods of time such as three and six years, there would be times when both insects emerged at the same time and food would be limited. The inequality of two prime numbers means that only one kind of cicada will emerge at any one time.
I want to conclude this post by asking a question I ask mathematicians: how many objects can you imagine in your mind? Most people I meet cannot imagine a number above eight; I will add that a particularly insightful friend said that she could imagine 64, which is for a human, a very large number indeed! So how many of you, my readers, can imagine 17 or 19 objects in your conscious mind?
Because if you cannot do this, how are you then to imagine the reality of their concomitant nature? The ‘nineteenness’ of things is far harder to describe than even the ‘fiveness’ of number. Then take the prime number 6037… and it is so abstract that I almost wrote it as 6023. What is ‘6037ness’? Does it have any meaning in the world around us? Or have we strayed beyond that invisible boundary and into the realms of fancy where anything goes just as long as you’ve said it does? (3)
As previously mentioned, this is too trivial for most sensible people. That only means that the dangers of not being able to do so go unseen. And in going unseen, can breed and infest in the way a cancer does until it is large enough that even a medical doctor cannot deny the existence of. Or dementia, for that matter. The problem for us is that by the time they can be seen they are life threatening. There is more to this than just prime numbers.
(3) I will discuss this in my forthcoming post that deals with the nature of quantum physics. Again, the concepts will be relatively simple.