Nature doesn’t even know about mathematics.
In a conversation yesterday my friend Elizabeth Jeffery commented that to nature, all of science, from physics to psychology, is the same. I added “and nature doesn’t even have mathematics!” That comment deserves explanation.
We humans understand the world principally through analogy. To investigate this claim, let us consider an analogy. What will happen if I throw a drinking-glass at my housemate?
Let’s start with the physics. The particular throw I’m contemplating I’ve never seen; I don’t know how it will turn out I could argue that even after seeing it I still don’t know how it turned out, only how my brain processed photons that entered my eye—or how I remember how my brain processed photons… But that’s a matter for a different post. . But I can compare that particular throw to other times I’ve seen drinking glasses in motion and other things I’ve thrown. I expect this particular throw will be like most throws in the general trajectory of the glass. I expect the water will fly out of the glass as it flies because that’s happened with other drinking receptacles I’ve seen in motion. I also have a reasonable understanding of how the glass will shatter because I’ve seen glasses shatter before and guess this one will be like those.
Likewise with the personal elements. I’ve never seen my housemate react to a thrown water glass—if I had, that would be a very good analogy for this contemplated throw though still only an analogy since this is a different throw. But I can draw analogies between this scenario and other scenarios I’ve seen, like how when I threw other things at him or when he spilled water on himself or when other things broke in loud, violent, and unexpected ways.
Some things make better analogies than others. Mathematics is one of humanities efforts to create very versatile analogies. There may not be any such thing as a “3” nor such an event as a “+” in the world, but I can compare all kinds of things to numbers like 3 and operators like +. I can compare my mass to a number and eating to addition, and that analogy tells me a lot about eating. Or I can compare ground water to a number and rainfall to addition and that analogy tells me a lot about weather. In fact, I have some difficulty coming up with something I cannot profitably describe by analogy to numbers. Even something innumerate like love can still benefit from descriptions like “the act of serving someone adds to your love for them.”
Mathematicians are people who specialize in this almost-universal analogy. Where most of us use numbers just briefly as tools for understanding our world, mathematicians seek to discover how much there is to know about this invented world. The assumption behind this is that the analogy will continue to hold: if addition is a good analogy, hopefully Gröbner bases is a good analogy too. Like any fans of invented world, there are mathematicians who stop caring about the analogy and just delight in the mathematics itself. Still, one of the testaments to the value of this particular analogy is that even very convoluted and involved observations about the invented world often turn out to be directly analogous to at least some aspects of the real world.
In a way, mathematicians are much like professors of literature: Note to self: tell the mathematicians and professors of literature this and observe their reactions. both make it their business to understand invented analogies in a depth and detail that seems quite out of balance with the intended meaning of the analogies.
In my experience, two things are true. First, every analogy can be taken too far, the analogous world stretched to a point where the analogy ceases to hold. Second, there are many situations where the analogy of mathematics never fails, no matter how far you take it. These two are obviously at odds with one another. And one of the things mathematics tells me is that when two things are at odds, they can’t both be true.
There are certainly things that I cannot meaningfully compare to mathematics for long. Love, as I’ve already mentioned, is one. And yet somewhere, deep inside, I expect that this is just because I haven’t made the right analogy between love and mathematics. And somewhere else, deep inside, I also am certain that something, be it love or something else, transcends mathematics entirely.
One day in the eternity to come I will learn from an omniscient Teacher exactly where, if anywhere, the limits of the mathematical analogy come into play. But for now, mathematics seems to be one of the best analogies for domains where individual choice doesn’t come into heavy play.