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Noticed an interesting article
on an unfathomable and completely pointless dispute among mathies. The 'foundational' types seem to be puzzled about the distinction between countable infinities and the continuum. This leads to some kind of irreconcilable thingamajig which must be resolved by either of two mutually exclusive and perfectly unfathomable thingamajigs.
The article mentions an alternate approach, first formulated by Aristotle, that makes sense. Just ignore infinity and continuity. Operate as if everything is countable. We don't actually handle infinity as an object or a number; real math works with limits and never tries to resolve the limits. We simply think of 'really really big numbers' and leave it there. Aristotle usually gets things right, while Plato usually leads to genocide.
Much as I hate to differ with Aristotle, I'm going to try. I'd say the simpler solution is the other way around. Set theory is useless. It's not the foundation of anything, though you can pretend it's a foundation.
Counting is not the basis and springboard of useful math; counting is just a trick that intelligent animals have mastered.
Why did we animals need to count, and what were we counting before we improved the technique? We were counting Nature's wrapped-up packages.
Fruits, trees, flowers, cows, wolves, our own eggs or larvae or babies. In short, present organisms or future organisms.
MUCH later we expanded counting to non-organism things like rocks or houses, and even later (within written history) to abstractions like hours of labor or acres of land. Some tribes still haven't made the first step of abstraction, still have distinct counting methods for living and non-living.
Counting is thus an important product of life and an important tool of life, but not crucial to survival. Counting is life recognizing life.
Counting helps us to keep our own eggs or babies safe from predators, helps us to locate the best field to find nectar or the best field to plant alfalfa.
But counting doesn't get us anywhere close to the underlying substance for all living and non-living things. The substance is energy and waves, not particles.
These packages we see, whether fruits or bees or rocks, are made of waves gathered in even smaller packages that we call atoms. We can do useful stuff like chemistry and cooking by counting these packages, but even there we aren't really counting. We're using balances and ratios to measure quantities by mass or volume.
Life uses waves from the sun and gradients of gravitational or magnetic or static force to drive its infinitely complex processes. We have many built-in methods of measuring these waves and gradients: ears, eyes, electrostatic senses, gravity senses. From those we have created a wide variety of differential comparator tools for making refined external measurements, but these measurements are NOT counting. We can paste a list of numbers onto a stick to give us a speakable and writable record of a measurement, but we were able to handle these comparisons quite nicely before we pasted the numbers. Fold the rope in half, fold it again in thirds, ....
Well, what about this activity that I'm doing at the moment... or rather SHOULD be doing instead of maundering about numbers? I should be developing courseware using DIGITAL computers. Aha! Gotcha! DIGITAL means counting! And why SHOULD I be doing this? Because I have the probably false belief that my bankrupt publisher will send me money in return for my labor. Aha! Gotcha! MONEY means units and coins!
Really? The programming, which claims to be about basic countable numbers, is deeply about ratios, with some arbitrary packages involved in the measurement of ratios. I'm doing a lot of trig to manipulate lines and shapes on screen. Pure ratios. I'm using loops to determine when a file has been fully read. Ratios again, with an arbitrary linguistic marker (newline) as the list pasted on the stick. I'm figuring grades as percentages. Pure ratio, turned into linguistic markers for portability. And how does the computer turn ratios into linguistic markers? Balanced ratios of electrostatic charge in CMOS chips, arbitrarily grouped into 1's and 0's for portability.
And what about the MONEY that I may receive if JPMorgan
doesn't decide to keep it all because JPMorgan's purpose on earth is to own everything? Coins are countable, aren't they? Yes, but they weren't countable to begin with. Coins arose
as a measurable and balance-able weight of gold, serving as a portable carrier of labor. They went through a long phase as strictly countable arbitrary markers, but in the last few decades they've been reverting back into their natural continuity, into balanced ratios of electrostatic charge in CMOS chips, grouped into 1's and 0's for portability.
No matter where you look, you find that COUNTING is a wonderfully convenient trick, invented by life to help us remember and transfer ratios and gradients of energy, but the basic stuff, the universal stuff, is always ratios and gradients of energy. It's all continuum.
= = = = =
Sidenote: There's a parallel to randomness/entropy. In this case the Platonist mathies insist that Number was the primary ingredient of the universe; Number existed before humans came along and decided to use it. Nope, backwards. Number is strictly a product of life. In the randomness case,
Platonic physicists insist that the distinction of order vs disorder is a natural part of the universe, existing before humans came along to use it. Nope, backwards. Order is strictly a product of life.
Labels: Danbo, Grand Blueprint