unnatural numbers – the short version
March 2, 2007
So the short version goes something like this: “natural” numbers (the non-negative integers we use for counting) are a man-made invention, and do not connect deeply with the fabric of our existence. There, I’ve said it.
I can’t tell you what a relief it is to finally say that out loud. So to speak.
Why do I emit such drivel? Well, it’s a rather long story, and I’ve been writing about it on again/off again for the past couple of months, and I have yet to sufficiently refactor my reasoning such that I am comfortable exposing it to public scrutiny. So I wanted to at least lead here with the short version so that I could briefly touch on some of the troubling consequences of this belief.
The biggest trouble is that I am now unable to map conclusions reached through any formal system – including, tragically, that old standby of geometry – back to the real world with any deep sense of certainty. If the axioms on which these formal systems are based don’t have a basis in the real world, how can we trust that the theorems we derive with them have any semblence of Truth when mapped back to the natural universe?
But surely, natural numbers must exist, you say. Why, look here: I have exactly 5 fingers on each hand. And I have exactly one car that I drive to work. And three quarters in my pocket. And so on. I’ll have to save my expanded reasoning for the long version of this story, but basically, I say that’s not true. In your mind, you have elected to partition a specific clump of spacetime and model it as “your car,” and for the purposes of reasoning (in your own mind) you find it convenient to think of all the components of the car as making up a single entity. But the idea that all of those components actually comprise a single, atom entity is a figment of your mind alone. The universe has no need to oversimplify patterns of matter such that they may more easily be manipulated by logic, and so doesn’t see things that way. I’ll look forward to digging into the reasons I think this is true later on.
For now, I’m just happy to have that zinger off my chest.
You are not alone.
Philosophers have struggled with this for centuries. Mystics have pondered it for even longer. The reason why it cannot be proved is fairly simple. The act of reasoning and judging is a core element of science and philosophy. The reason why the truth cannot be derived is that the very act of searching for this truth can only find what already exists in the minds of those thinking about this issue.
In order to really break free, you need to realize that you are really just a part of a really big organism which happens to be labeled as the universe. Much like a liver cell in your body has no sense of the body around it, we have no sense of the workings of the whole of this organism.
However, now that you have stumbled across this idea and have bravely posted your musings to the world let me be the first to congratulate you. Good job. You are well on your way!
The more you provoke your mind to think differently about this, the better things will get. It’s a rough patch at first but you’ll get the hang of it. Once you start looking for what is really going on, things will become a lot clear. Just give it time. You’ve got plenty of that.
Are you still thinking about numbers? Such thoughts have occupied me as well, and only recently have i learned about subitizing (from Latin: subito — sudden) which suggests that we and some other animals are born with an innate ability to deal with numerosity (up to about 4) and to form expectations about the results of addition and subtraction within this range. The important point is that this happens before the learned ability to count. So, just as we can tell that a small rock is heavier than a similarly sized egg, so can we tell that 4 is greater than 3, either as a collection in space or as a sequence in time.
shep[
shep – thanks for reminding me of this post. Recently I’ve been thinking about numbers again, because just recently my daughter (almost 4 years old) decided that her name was now “7″ and that I (“11″), my wife (“12″), and my son (“1″) were now numbers as well. For several days, she would refuse to respond to her given name, and even her morning ritual of waking us by calling out (“mommy! daddy!”) was replaced with (“number 12! number 11!”). There is obviously something very special about integers baked into all of we humans.
I hadn’t been aware of ’subtizing’ as a word, although the concept is familiar. I tend to think that the ability to immediately ‘feel’ how many objects are in a small set is probably due in large part to processing in the visual cortex, but it’s interesting to think that perhaps those cases of savants how can ‘count’ large numbers of items immediately might have some advanced wiring that does these kinds of things in parallel more efficiently than the rest of us.
I wonder if there might also be some relation to perfect pitch? People with perfect pitch can immediately know what note (frequency) they here, which, at its core, is just counting really quickly at a subconscious level.
In any case, thanks again for reviving this topic!
Speaking of wiring and savants, there’s a video online called “Extraordinary People – The Boy with the incredible Brain” — about a math savant who describes his mental “calculations” as colored shapes morphing and merging into new color/shapes (the solution!).
Speaking as a non-mathematician who cannot hold a tune, I am as amazed by people with perfect pitch as by those who can look at Andrew Wiles’ proof of Fermat’s last theorem, rub their chins, and say, “Ah, yes, I see.”