The Math Thing
Alright, so it’s pretty obvious Lou Keep is never going to finish the series on epistemology and math (Platonism without Plato, On a Particularly Difficult Question, Euthyphro Dilemmas as Mathematical Objects) . This has been bugging me for quite awhile and I’d like someone to be able to direct me to some decent reading on it. It’s hard to explain this, mostly from Lou’s writing style, which is why I need to write this. But yeah, if you’ve read something about this topic before, please drop a comment and a link; this has been really bugging me for a few years.
What was Lou talking about?
Alright, science is true. We know science is true because, to paraphrase Richard Dawkins, we’ve flown to the moon. Our world and daily experiences are dramatically different than those of people 300 years ago thanks primarily to advances in science and technology.
Other fields like economics, psychiatry, even philosophy, can provide insight and help us understand complex and important things but they can’t prove things the way science can.
And the core of science is empiricism, testing real things and observing their results. You can’t prove something in science by thinking abstractly or through reason, you have to go perform an experiment and observe the results. Without experiments, without real empirical testing, you can’t prove things. That’s why there’s this big gap between the physical sciences and the “soft sciences” and why we almost always trust the results of physical sciences more than social sciences. It’s not a mindset issue or anything; we’ve tried importing physicists and their ilk into economics with moderate success but certainly nothing revolutionary.
So in general, we have this scale of things that are true with science near the top.
But at the top of this scale is math. And Lou points out how weird that is, because math is nothing like science. There not only is no empiricism in mathematics, there’s a categorical rejection of empiricism and a focus on logical abstractions. Measuring 10,000 triangles is no proof of the Pythagorean theorem and you’d be laughed out for trying. In fact, math pattern matches much closer to academic disciplines at the bottom of the “truth hierarchy” like philosophy. Both involve really smart people sitting around thinking hard, with no tests, no experiments, nothing but pure thought.
This gets worse because of how much science depends on math. I don’t want to say that you can’t have science without math, I lack the expertise to speak on it, but I wouldn’t want to try landing a man on the moon without using math. A lot of science depends on complex mathematical formulas, formulas proven logically without regard to physical reality, often by dead Greeks 1,000+ years removed from the scientists who put their abstract math to practical use.
This is weird. I want to divide this section, which I feel is on firm ground, from the mindblowing parts, which mostly involve speculation on why this is and how we can explain it. It’s worth stopping now and just noting “Huh, this is weird.” You’re free to move on without giving it further thought. In fact, this will put you in the comfortable company of Stephen Wolfram and, so far as I can tell, most mathematicians and scientists who notice this. Finding something unusual and not having an immediate perfect answer and preferring to focus on other issues is perfectly valid.
One other note, you might suspect that this is just philosophical sophistry, which has long been a concern of mine as well, until I read the following in Stephen Wolfram’s On the Quest for Computable Knowledge.
And by the end of the 1500s-with Galileo and so on-there was a notion that physical processes could be understood in the “language of mathematics”. The big breakthrough, though, was Issac Newton in 1687 introducing what he called “mathematical principles of natural philosophy”. Really pointing out that things in “natural philosophy” could be worked out not by some kind of humanlike reasoning, but rather by representing the world in terms of mathematical constructs-and then using the abstract methods of mathematics (and calculus and so on) to work out their behavior.
Why it worked wasn’t clear. But the big fact was that it seemed to be possible to work out all sorts of unexpected things-and get the right answers-in mechanics, both celestial and terrestrial. And it was that surprising success that has propelled mathematics as the foundation for exact science for the past 300 years.
I may still be ignorant and this still might be sophistry but at worst I am no more confused that Wolfram and that’s fine company to be in.
Lou’s primary point in these essays is that it’s really hard to explain math without some kind of metaphysics and once you allow any kind of metaphysics, things get weird.
Basically, if we reject some variant of the idea that math is just a mindset or a really precise language, then it has to be real. It’s kind of hard to define what real means in this sense but the closest I can come is that there’s something out there which fundamentally defines our reality that we do not, and almost by definition cannot, physically observe, measure, or interact with in any way beyond abstract thought.
That immediately leads to questions like “How do we interact with it?” and “Why can we perceive or identify this at all?” and “What other things beyond our physical reality define it?” Pretty soon you start wondering about this underlying reality shaping ours.
This sounds kind of magical, and it can be, most of the best arguments for some variety of God run through this, but it doesn’t have to be. The idea of humanity living in a computer simulation (“the Matrix”) is well known and would run off the same concept: that’s there computer code underlying all of our reality that determines the physical laws. This naturally leads to same kinds of questions, like “Are there bugs in this code?”, “Can we hack our own universe?”, and “Who wrote the code and why?”
If this sounds like drunken speculation, it kind of is, but I think the reason this has resonated with me for so long is because of the importance of math and science. Most mystical/computer simulation/religious etc speculation get rejected because there’s very little evidence for or against. After all, how would you prove or disprove you’re living in a simulation?
But once you accept,
A Science is critical to understanding the world
B Math is critical to science
C We don’t really know why math is so critical to science or how it works
, then you’re stuck because no one really wants to jettison either science or math but that leaves this giant unexplained gap between math and science and we’d kind of be forced to accept anything that could provably, or even plausibly, fill that gap. You can ignore it, and most people do, but it’s still there.
Lou’s primary concern isn’t answering this but forcing the argument into a clear binary and making sure people understand the consequences of each side. On one side, math is like language, rational thinking, or some other social construct; on the other, math is metaphysically real in a sense that brings all the mystical speculation. I buy this binary. There is some gray area, “mathy” things like the 80/20 rule or statistics/counting/accounting/etc which seem like arbitrary/observed applications of math without some deep fundamental truth behind them. But I’m not confident enough to dump all of math into that, I’m not even comfortable calling statistics just a social construct or mindset, and that leaves metaphysics. But science/modern thought rejects metaphysics altogether and that kind of eliminates the gray area; any metaphysical truth in math doesn’t just make metaphysics real, it makes it one of the most grounded and fundamental parts of our conception of the universe because of it’s ties from math to science to technology to most of the important material things in our lives.
So, in summation, Lou Keep proposed the following. If math real, then metaphysics, then magic. If math not real, then why math work?
I’d appreciate anyone else’s insights on the matter and especially any books or materials diving into more depth. Lou never finished this series and 3-4 years later it looks unlikely he ever will. I haven’t been able to find any other resources on this topic and the one’s Lou referenced are $80+ books, which is a bit too extravagant for my tastes.