Numbers don’t lie. Do they?

I realize my posting has been pretty thin of late – still tackling home projects, still stalling on another video. And then I go off on these tangents that don’t help at all, but I want to put them down while they’re still fresh, so here we are.

While working on the post about Ultrafinitism, I had read that it was not gaining as much acceptance as the form/realm that permitted infinity, because it had no axioms or distinct rules, and while I pointed out that in and of itself, it disproved one or more of those axioms, I still toyed with the idea that there was a simple addition or change to mathematics that would clarify the matter and prevent some of the unprovable and undemonstrable nonsense that pervades math, or at least these advanced axioms. And for giggles, I started working them out – and then abruptly realized that not only was it far simpler than even that, but I’d been voicing the ideas multiple times in posts without distinctly pinning them down. So try this:

1) Numbers are not real. They have no existence in and of themselves, and do not define anything. Rather, they are defined by that to which they are applied, specifically the scale or increment. For instance, one may have three apples, or a mass of 390 grams, or a volume of 540 cubic centimeters – but none of these might accurately compare to the others, meaning each apple does not automatically mass 130 grams. The scale defines the numbers that apply.

2) By extension, creating any given number does not imply any application for it until it can be assigned to objects, scales, or measurements. While any number can be imagined to any value/degree of whole or decimal places, this has no meaning without application, any more than listing letters creates a new word; it does not exist as such until it has a definition, preferably agreed-upon by multiple people at the minimum.

Anyone can, as they wish, scribble down a shape or symbol and then claim that it means something, or has certain properties – but that doesn’t translate to any such properties actually existing. While a great deal of mathematics applies quite well to describing and comparing real, physical traits, this does not mean that, by extension, whatever lengths you might take the math to will have application as well. Beyond physical constraints, you are only dealing with make-believe.

Most especially, what these two axioms do is eliminate a very large amount of ‘theoretical’ (these aren’t in any way theories in the scientific sense) and ‘infinite’ progressions. As a ground rule, it doesn’t alter Set Theory, or Zermelo-Fraenkel Choice axioms if you prefer, save for a) eliminating the idea of infinity, and b) stipulating that all sets abide by application. As yet, I have not found a place where this breaks down, but again, I’m no mathematician, nor even close, so we perhaps know what end I’m talking out of.

But it didn’t stop there. Prompted by a communiqué with The Manatee (outside of the comments on that post,) I started wondering why humans, as a whole, have spent so much time on inapplicable math, and now strongly suspect that it ties in with other aspects that I’ve wondered and talked about. Evolutionary psychology is a pet hobby of mine, pondering what makes us (any animal) think and react in the way that we do; an awful lot of it comes down to, we evolved to think this way because it worked better than the available alternatives. We humans like to believe that we carefully consider most of our decisions, but it’s likely far less than we believe, and we’re prompted towards certain decisions or biases by inherent traits that assign greater importance to certain factors. The big three – avoiding death, eating, and reproduction – are undeniable, but there are plenty more that are likely there, just not recognized too often. The way we respond positively to large eyes in a small face, a trait of infants, or negatively by the crying of the same. The nonsensical fascination with sports to such a huge degree (tribal instincts.) Making posts about things we think we might have figured out…

And I came back to one I’ve long known: we are pattern-seeking animals. This on its own likely had a lot to do with our progress towards industrialization, since we recognized properties of certain things, like how to start fires and make metals and so on, but it also drives us too far, believing in superstitious things like gambling ‘streaks’ and gods that are angry because we didn’t feed the volcano or ate meat on Friday. Way back in the long-ago, pattern recognition helped us survive, in two ways. First, most animals actually have patterns to their coloration, shapes, and even behavior, so this would let us spot both potential prey and potential predators from the ‘chaos’ of natural foliage. And by extension, we can see a pattern to this very chaos, in that leaves grow in certain clumps and branches taper to specific ratios, so we can spot the differences within. Additionally, asymmetry is often an indication that something is wrong with a species, perhaps signifying a lack of health or a poor choice to mate with.

Mathematics is, almost entirely, patterns. Ratios, repetition, progression, and so on. We recognize when a tree or flower is symmetrical, and not just prefer it, but try to define how and why. We can easily spot the sine wave of water, or the expanding circles of a splash, as well as the break of these patterns that indicate something else is acting upon them, perhaps right under the surface. Douglas Adams (in Dirk Gently’s Holistic Detective Agency) pointed out one that we barely even register despite its pervasiveness: our ability to not just recognize the consistency of a parabolic arc, but predict its path from mere milliseconds of observation and catch a flying ball, much less throw something accurately enough to hit a target, one of the many factors that allowed us to progress past the ‘caveman’ stage. And the application of math to physics is undeniable, as well as showing us how predictable many things are.

Yet this, to me, is where that search for patterns leads to a bias that works against us. Because we like the idea, because we have a desire to find something, doesn’t mean that it’s actually there. I’ve touched on this before when discussing phi/the golden ratio/the rule of thirds in art & photography, which doesn’t stand up to examination (and none of those three actually match anyway.) It’s far more a matter of confirmation bias, of selecting only those factors that fit, within a reasonable approximation, and ignoring all of those that don’t. And the bare fact that we can spot patterns in places, especially differentiating them from non-patterns, illustrates that nature/the universe isn’t actually all that good at mathematical precision.

Now we do, in fact, have mathematical formulae that underlie all of physics – of everything. The four fundamental interactions – the strong nuclear force, the weak nuclear force, electromagnetism, and gravity – dictate how everything behaves, describing the interaction within the entire universe, from neutron stars down to the friction of simply holding something in your hand. They can be expressed mathematically, certainly. Aha! See that? But I’m not arguing that math doesn’t have its place, only that too many mathematicians fail to recognize its limitations. It wasn’t mathematicians that predicted or defined these forces either, but meticulous observations and measurements by physicists.

And that led to part two of what may be behind a lot of this, what I tend to call the ‘puzzle drive.’ We are obsessed with solving things, figuring them out, finding the solutions, so much so that we do it for fun, and I find it hard to believe there isn’t, not just a positive reward system within our brains when we’re successful, however trivial the puzzle might be, but also a negative ‘punishment’ system as well, bolstering our frustration with failing to solve such things – again, no matter how trivial. We play games all the time, and only a few of them tie into other survival traits.

Given this, it’s easy to see that finding something interesting within math can lead one to suspect (or hope, perhaps) that they’d found some solution, something meaningful, perhaps even a new trait of physics or of the universe itself. It does not help that mathematical proofs do not have to apply to anything real or predict anything – they just have to “add up,’ as it were. Nice and neat. These traits might also help explain why there’s such a thing as ‘chaos theory‘ despite the fact that there is no application and no actual predictions from such. In contrast, most of the hard sciences don’t even mess about with proof; instead, weight is given to the evidence that exists and supports any particular idea, and most especially, how well it predicts future findings. It’s rarely nice and neat, and very often goes through modifications when dictated by the evidence we find in the real world.

There’s a distinction here, a difference that I think escapes attention far too often. Mathematics can be used to express many ideas, including most of physics. Indeed, we don’t have any other way to explain these if we didn’t use it. But that means it’s a tool, just like language itself. The universe doesn’t care about the numbers, doesn’t even know what they are, doesn’t have the concept of knowing. It just has properties. And while we may hope to gain insight into further properties by applying our knowledge of ratios, this can only work if they exist to begin with. Yet we can only know that if we skip the axioms and find the supporting evidence – math alone cannot dictate or fulfill our knowledge.

In some cases, a hammer is the right tool for the job, but this does not extrapolate to being appropriate for every job, despite anyone’s wishes or any proverbs which insist that it should.