coming from here
Let’s start this with an obvious question: “Why are you so concerned with math?”
It has to be answered for any description to make sense. Looking back, it does appear that my interest is just because of Kant’s interest. No, and:
It’s not good that our root instinct is a lazy Cartesian skepticism, but it does make my job easier. That was something earlier writers had to inflict; it being the default lets us move quicker.
Ask a college kid what Truth capital T is and they’ve already absorbed the right lessons: truth is a construct determined by your culture’s valuations and epistemic suppositions, therefore we’ll never be able to actually arrive at the Truth. “It’s subjective.” Moreover, logic is dependent on language, which is dependent on culture; reason needs categories to manipulate, which are dependent on [relative thing]; trying to interpret the empirical world is a problem because you can never step outside yourself, data isn’t “there,” it must be uncovered, hence the flaws of naive science.
I say this like I’m mocking the arguments, but I do really mean this is the correct way to start. Not only are we in the cave, that cave is in a metacave. Don’t trust yourself to trust the outside world, reason splinters out of a bunch of psychological flaws and biases, inner fickledom influences outer fickledom, there is a great gnashing of teeth.
Nietzsche, who I will quote this once and then avoid until [distant post]:
Once upon a time, in some out of the way corner of that universe which is dispersed into numberless twinkling solar systems, there was a star upon which clever beasts invented knowing. That was the most arrogant and mendacious minute of “world history,” but nevertheless, it was only a minute. After nature had drawn a few breaths, the star cooled and congealed, and the clever beasts had to die.
One might invent such a fable, and yet he still would not have adequately illustrated how miserable, how shadowy and transient, how aimless and arbitrary the human intellect looks within nature.
We should be extremely skeptical that the human intellect can touch on anything True, much less even vaguely functional. I do mean that as a starting point: get as deep into skepticism as you can, or else what follows won’t be as baffling as it should be.
If human reason were worth a damn, there would be no question about the status of math. Only in light of its complete and total vapidity do we realize how baffling it is that Leibniz can think about geometry real hard, Maxwell can apply that hard thinking, and now we have computers.
You’re not convinced.
Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences (PDF), is quick and good. Here are the first and last paragraphs.
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate, The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is π.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”
A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. […] Furthermore, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called the “ultimate truth”. The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence their accuracy may not prove their truth and consistency. […]
Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning.
Wigner’s joke is not a joke, it’s the most reasonable response in the world and we only disregard it because we’re so accustomed to this world. The issue isn’t “why science” where science=comparison of empirical phenomena and extrapolation of patterns from the same. I mean, that is a problem, it just isn’t ours at the moment. “Uncertain experience” does not mean we can’t compare “uncertain phenomenon A” to “uncertain phenomenon B”. Experimentation makes (some) sense even in a radically skeptical world, one can understand taxonomy yielding insights about the specific phenomena at hand. Descartes’ demon may be fruitfully compared to other demonic incursions.
Once we get into mathematics there’s commerce between the subject and the object in a way that ought to counter easy skepticism. Math seems to be an entirely intellectual activity, an autogenerated thought-bubble, it ought to be part of that closed-off-subjectivity we love. Why should the circumference of a circle bear any relation to an empirical phenomenon that is not, in any obvious way, a circle? Moreover, it’s not even related to a circle – it’s related to an abstracted form of “circles” measured using subjective numerical tools. There are no degrees in nature, presumably.
The response to lazy skepticism tends to be lazy empiricism, and the question of math resists this, too. It’s gauche to scream about scientism, so I’ll argue from authority instead. There’s a reason that these physicists throw their hands in the air when they try to deal with the math/physics relationship. Other empirically derived facts are temporally restricted, i.e. all swans are white until you see a black swan, so you change priors after. You cannot progress without new phenomena. Math is not this way: you (apparently) derive it once, and that’s enough to figure out the rest even before seeing any proof of other mathematical objects. Moreover, it’s weird that it should not be culturally relative. Ethics and culture (by this scenario) are also empirically derived, but compared to basic arithmetic their conclusions are radically different. That arithmetic is common is extremely strange. Neither of those are arguments, they’re just observations.
Much closer to, thought not yet, an argument: while it’s possible that we derive the laws of mathematics from the empirical world, it’s not really clear why we should be able to reapply them, nor is it apparent why they should have predictive power. If you make the argument that Euclidean geometry comes from empirical observation, and that’s the only way we got it, then non-Euclidean geometry should be aggressively, directly anti-empirical, caustic to human phenomenology, cf. Lovecraft’s favorite adjectives.
In the past history of the subject, progress has almost always come from such experimental hints, but there always has been an alternative way forward, that of pursuing connections to mathematics as a very different sort of guidance. One can argue that Einstein’s successful development of general relativity was an example of this. Little help came from experiment, but a great deal from mathematicians and the powerful new formalism of Riemannian geometry.
This last quote comes from a recent paper (PDF again) by Woit gratefully affirming Wigner’s hope that math and physics continue to work.
“Why” is still a question. So is, one imagines, “what,” as in, “what is math?”
[Tedious list of philosophical schools here], and I’m going to ignore all of them for now. Our man on the street reports: “Math is just a system of logic based on axioms.” Fine and all, but stop prolonging the issue. Axioms are things you accept, the Greek means “something worth [allowing],” what makes them worth that? Did you accept those axioms a priori or were they empirically derived? To make it more interesting: why should one be able to change one (technically a postulate, same point, deal with it later) and still wind up with physical results, i.e. why does non-Euclidean math work? This collapses into the more obvious and general question, that of Wigner and Woit above, which is “How is mathematics so uniquely effective?”
This is probably obvious, but I’ll lay it out anyway: that math works because it’s “supremely logical” implies that logic has some unique claim to the natural sciences. This is an assumption, not an explanation.
All sloppiness in argumentation above and below may be attributed to the necessity of clarifying what exactly we’re doing here before getting into specifics.
It’s hard to express just how important mathematics is to the history of philosophy. An earlier draft of this essay just had a series of quotes from philosophers all using geometric proofs as examples of human reason, but that failed to get at why they were using those. Nearly all of them use geometric examples to show that human reason is capable of making fundamental judgments about the outside world, i.e. Enlightenment style “we can think it out,” but that hides a whole lot of implicit argumentation. Basically everyone can agree that 2+2=4; no one can agree what this says about the world around us.
The unreasonable effectiveness of mathematics is a phenomenon that we start with (and/or have started with), it’s postulated, it’s the “well that’s interesting,” and it wants an answer. We start with math because it is so easy to point out the frailties of human reason, how poorly our intellectual sensibilities align with the physical world, because all of that is also assumed and then suddenly something breaks the pattern.
Mathematics occupies a unique position both because it is perfect – fuck off for now Gödel – and because it says something about the world outside while still appearing to be the product of human thought. That is to say, there are, presumably, numbers that no human being has ever before observed. But, if one wanted to, they could count to that number and then go outside and find it. It exists, in precisely the way that all sorts of things we think do not. That’s probably the most convoluted way possible of rephrasing the quotes above: while appearing to come solely from human thought, math has predictive power on the world outside.
Since this is the case, there’s at least some connection between human reason and reality. I cannot emphasize how important this is, nor how strange it is.
Hume provides some strong reasons to be skeptical of geometry and physics, but these aren’t threats to science. I want to be extremely clear on this point: the problem has never been “science will stop working if we don’t figure this out.” At stake is everything else. Answering the question means making an epistemological claim, and the implications of that claim have direct consequences on the rest of human knowledge. While Kant explicitly states that his purpose is “saving the sciences,” that’s not really what he does. His “positive” results didn’t really provide new confidence to physicists. They, instead, fucked with our capacity to make (certain) metaphysical claims.
If mathematics says something about the world, and if logic is related to it, then one might assume that human reason can get us to The Truth. Historically, this was the proof of metaphysics. Recently, it was also the proof of metaphysics, and it gives rise to a host of metaphysical claims that are much harder to argue against than they appear. Denying them outright leaves one without any explanation for mathematics and the sciences; admitting these obvious consequences is unacceptable.
The ontological argument for the existence of God is an infamous example, which everyone agrees is silly and irrelevant and clearly comes from Christianity up until the part where it doesn’t. We’ll bring back Gödel, because he made a 20th century version of the ontological argument, and made it logically impeccable (…mostly). That doesn’t come from his (supremely heterodox) Christianity, but from his stance on math.
Gödel sat down with all the ontological issues embedded in [everything above], thought about it, and revived Platonism. This is less insane than it appears, and understanding why it’s less insane than it appears will take you a long way towards understanding the stakes.
Some bad history, mostly metaphor, and definitely not a hill fit for a grave. Still, worth considering.
There are very few cultures that don’t have some tradition of astronomy, even if that’s just naming constellations. And there are very few cultures that don’t name those constellations in accordance with deities. The stars are numinous, divine, godly. We should be so lucky.
Take all my hedging – this is merely a metaphor – but one wonders what it was like to realize that those gods could be understood. Human intellect, some aspect of it anyway, could apply the same geometric formulas used by workmen to get at the habits and motions of the gods. And, further, if that could be understood, then why couldn’t our reason reach even higher? If reason can grasp movement, then it should be able to get at their nature. If it can understand the deities’ nature, then it should understand Being.
It’s telling that priestly castes are almost uniformly astronomers, and it’s suggestive that without this obvious application of reason we’d lack any sort of proof of human reason. But it only takes one.
Ptolemy had to introduce the epicycle because he insisted on circles, but he insisted on circles because they’re fitting for the divine. His explanation for what math allows him to do comes closest to understanding why mathematics got tied up with metaphysics and epistemology and, really, most branches of human reason in the first place:
From all this we concluded: that the first two divisions of theoretical philosophy [physics and theology] should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakable knowledge to its devotees, provided one approaches it rigorously. […] For [mathematics] alone is devoted to the investigation of the eternally unchanging. Furthermore, it can work in the domains of the other [two divisions] no less than they do. For this is the best science to help theology along its way, since it is the only one which can make a good guess at [the nature of] that activity which is unmoved and separated… As for physics, mathematics can make a significant contribution. For almost every peculiar attribute of material nature becomes apparent from the peculiarities of its motion from place to place.
In other words, that we have this certainty allows us to expand into non-mathematical domains. Its utility, then, is not merely description of the cosmos, nor of the word around us. It implies a great deal more about what makes humans human. “Ethics.” Sure, but who cares about ethics? I mean stuff like the meaning of art.
At stake in the question is how far we can be certain, and at stake in that is most of what interests us when we discuss human beings. Cause->Effect, until someone disproves that.
top image from Häxan by Benjamin Christensen