coming from here
I
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.
II
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.
But.
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.
III
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.
Or:
IV
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
sam[ ]zdat 
I’m still not quite convinced. Humans can form patterns because of magically evolved intelligence. Math is just what we call the collection of these patterns abstracted away from reality. Many of these abstractions (like pi, above) are simple enough that they reappear in disjointed parts of reality, like a table and a house both using the same kind of nails. So I’m not what is the puzzle there?
Further, reality is just all this stuff we can touch — if there’s something we have no interaction with at all, why do you think it’s real? — so it makes sense that we can touch it and that our abstractions can touch it. And we can touch anything at all because humans don’t have pure cartesian minds, but are monkey-minds grown out of the world where anyone without the ability to interact got got/selected by others, naturally.
Well this is how I resolve the unreasonable effectiveness question anyway: Evolution did the hard work of letting us interact with the world at all (and giving us intelligence/abstraction power), then we did that and whatever worked is what we call math/science. And stuff works beyond its domain because these are basic patterns that repeat in different parts of reality.
I feel I’m approaching this naively, what am I missing? At any rate, I’m interested in where part IV is going!
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I think the problem with your reasoning is that it has a ton of implicit assumptions. In no particular order:
That there is such a thing as reality distinct from what we think about it, and that mathematics (patterns, in your wording) inheres in it.
That the empirical method works, in that it brings actual connection rather than the Humean notion of the stage-play of consciousness.
That evolution, which is the practical application of the mathematical-logical idea “that which has the nature of becoming-more will become more,” is valid. This requires prior concessions to mathematics.
None of these are bad reasons, per say, but they all are so steeped in existing assumptions of the validity of mathematics and mathematical reason that they fail to serve as reasons to believe in mathematics. Instead, they serve as explanations of mathematics. The difference here is that one is before the fact (a priori) and the other is after the fact (a posteriori). This is, practically word-for-word, Kant’s introduction to the Critique of Pure Reason. If you want to learn more about this issue, I highly, highly recommend you read that book. In many sections he is incredibly arcane, but in a few places (usually introductions) he backs off the technical language which he’s been inventing on the fly and gives ordinary-language explanations of why he’s doing what he’s doing.
I’m gonna be a little rude and step all over Lou’s toes here, but his big reason (and likely Lou’s big reason) is that if we simply accept math and our insights thereof to be true reflections of the nature of the universe, with no caveats about their limitations, we get into a state where we can apply erroneous and culturally steeped observations with all the force of fact, which incidentally is the force of a hobnailed boot to the skull. Ask Orwell; he did the math on that one. This was happening all over the place in Kant’s time, and is still happening in the present day. Not coincidentally, this is also the kind of thing that the super duper science/math fanboys do most often. On the other hand, if we simply nix the validity of math, we get insufferable and legitimately harmful levels of cultural/moral relativism. Look at the generic group of university leftists to see why this might not be the greatest. Kant’s big plan was to find a way between these two, to give math its own particular, explicable, and definite sphere of validity, where it could be guaranteed to work and guaranteed not to exceed. Unfortunately, not everyone really got the memo, and so despite Kant giving some incredible arguments, the problem persists into the present day.
I hope this helps get to the root of the issue, and show what’s really at stake here.
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I agree, it’s tempting to reduce the “difficult question” to an extreme case of the problem of induction. As usual I don’t have a clear picture of all this myself, but I’ll try to make some points in favor of math’s weirdness:
From everything we can tell empirically, the universe runs on math. That is, if we look closely enough at the empirical data for some phenomenon, we find not only that it behaves comprehensibly– which, as you say, might be expected as a result of evolution– but that it conforms 100% precisely to some mathematical formula. What’s more, it’s very often a simple formula– e.g. a combination of a few terms raised to integer exponents. I don’t see any reason to expect this based purely on evolution. If you’re saying that math’s effectiveness is unsurprising because math describes the universe and we evolved to understand the universe, that strikes me as a circular argument. It doesn’t really explain why math in particular should be so effective.
A further point… the level at which math describes the universe isn’t one where we’d expect evolution to do much for us. If I’m an expert at playing a video game, does that imply a good understanding of how the game is implemented in the code? Not really. In that sense, the fact that evolution gave us the cognitive tools for understanding how the world works “all the way down” comes as a surprise.
Finally– and I assume this is what Kant is getting at– doing math just feels so different from normal interaction with the world. One can make a case for axioms (rules of addition, planar vs. elliptic geometry, etc.) being empirically determined, but beyond that, math doesn’t use empirical data at all. I can’t think of any cases where new math was developed to fit an empirical phenomenon, but I know tons of cases where existing “pure” math was found to fit an empirical phenomenon. The most impressive cases are the ones like non-Euclidean geometry and Penrose tiles (https://en.wikipedia.org/wiki/Quasicrystal) where no one involved had the slightest suspicion that the results might describe real-world objects. And yet they did.
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For cases where new math was developed to fit empirical phenomena, don’t geometry, trigonometry, and calculus all fit the bill? Take pi, for instance. Pi was worked out entirely from physical phenomena, involving measurements of existing objects until a decent approximation could be made. The Pythagorean Theorem, similarly, was worked out as proof to the existing knowledge regarding the ratios between the sides of a triangle. The derivative and integral were developed because Newton/Leibniz wanted to be able to properly represent different degrees of acceleration in formulaic terms, and the relevant functions didn’t exist prior to that even though people had been recording the positions of the planets for something like millennia. I think all of those situations fit in pretty nicely, but I’m not sure if they’re what you mean.
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You’re right. Calculus is a particularly good (and reasonably modern) example of what I had in mind. I stand corrected.
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Thanks. And, yeah, I mean. I don’t think I’ll convince you until we’re half-way through it if ever.
You’re right that the critical movement is us being a part of – and evolving in tandem with – some broader world-structure. The question is, in some sense, rhetorical. As I said: I don’t expect us to stop being able to do physics. The point is that reflection and proof start to map out the particularly way this structure works with us (as you noted, that’s where part IV is leading to), which allows us to begin to understand other epistemological problems.
@hnau brings up the critical point, which is that math functions better than it should, and it’s uncertain why it should be this thing we call math that actually describes things. There’s a whole lot of human psychology geared towards survival, but most of it doesn’t do very well at dealing with the word. Take, say, pareidolia, which is certainly an evolutionary trait but terrible at mystery-unraveling. Again: the point is not math, per se, but to recognize that there’s a unique claim it has, which makes it a helpful focus for epistemology.
A somewhat silly way to put it is: if math was so critical to survival, why can so few people do high-level math? In other words, we expect selection to occur from pretty subconscious, instantaneous decisions, and so why are so many results of mathematics counter-intuitive, non-obvious? Or, hnau’s point about game-designers vs. players.
@Sam Reuben brings up the much broader point, which is that – if you wanted to be super careful about this – almost all of our physical theories are already based in mathematics. In some sense, it becomes a kind of ur-question that’s particularly compelling for us, and also pretty deeply necessary. There’s a somewhat obnoxious tic among the continentals (kind of? you’ll know who I mean) to talk about science and religion and “faith in science.” These are different kinds of faith, it’s normally a gotcha argument, etc.
But. There really is something weird about math, and I think a lot of those papers really are trying to point out that it’s super non-obvious why math works the way it does, and that any explanation for that which relies on later scientific discoveries winds up begging the question.
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I’m kind with Flat Earth Theorist on this, tho I’ll admit I’ve traveled thru the weird valley your posts describe on my way to that position.
From my current position, this claim is really weird. I submit that it only seems sensible from a naïve understanding of the origins of mathematical knowledge. As Sam Reuben pointed out above, math certainly did arise from intelligent people reasoning about the world. It’s most definitely not something that just spontaneously appeared, or even something that a single person developed in an afternoon.
The best explanation of ‘why math works’ I’ve encountered is that it’s simple. ‘Math’ is the body of knowledge of the simplest results about the simplest possible systems. So the simple abstract systems people study almost inevitably correspond to simple concrete systems we observe in the world. And given that complexity, always and everywhere, is composed of some (usually large) number of simpler systems, we should expect everything to ultimately cash out on a level that’s simple enough to correspond nearly perfectly with an abstract system we can derive from a small number of simple rules.
In a very real sense, almost everyone can do high-level math. Throwing a baseball, or a rock or a spear, requires an effective solution to a pretty complicated set of differential equations. There are several results from studies of people’s mathematical abilities demonstrating that people can in fact do ‘high-level math’ if the problems are posed to them in ‘natural forms’, e.g. kin relations, or politics.
But in a larger sense, ‘math *doesn’t work’, and we shouldn’t expect it to either. Lots of systems – probably almost all systems – are so complex (or large, or both) that the most efficient means of determining their future behavior is to just observe them. That’s a clear defeat for the effectiveness of math.
I strongly suspect that ‘why does math work’ is a ‘confused question’ in the sense that it would be ‘dissolved’ by a better understanding of why our minds consider it mysterious.
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Superb logos, here. The only thing I think it’s missing is a proper mythos to go along with it – the pattern and echo of the Gorgias, in fact. You mention Lovecraft, but I think in order to make a fairly obscure (to most) point about mathematics really stick, you need to really dredge up his horror. In fact, that’s probably why Nietzsche is more popular than Kant, even though Kant’s the better philosopher (shots fired). Of course, that’s me arguing in favor of demagoguery for style points, so take it with a grain of salt.
Then again, I think there’s probably a fifty-fifty that you’re about to do just that. Either way, splendid prologue to transcendental idealism, assuming that’s where you go from here. Otherwise, I’m interested to see you develop past Kant.
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Thanks. Disagree on Kant/Nietzsche, but I can respect the call.
Current model has the Plato sectioned organized around the Timaeus.
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The perfect story is that of bleem, the hidden integer between three and four.
http://strangehorizons.com/fiction/the-secret-number/
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I’d personally say the perfect remedy is reading lots of Lovecraft, but Lovecraft has been so thoroughly fetishized that he’s lost a lot of his meaning to contemporary readers.
That was a lovely little story, though. Thank you for linking it!
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It’s just an intuition, not an explanation, but I can’t help but feel that the answer must be somehow related to how very complex behaviors sprig up from simple systems in math. You see it eg. in how nearly everything doing some kind of computation is Turing complete or how only extremely simple logical systems can’t express arithmetics and are decidable (that’s of course Gödel theorem). Which is actually the same thing — there’s this thing called the Curry-Howard correspondence that’s in a sense just a notational tautology almost by construction, but it gives an insight how to look on a typed computational system and see a logic system or vice versa. So maybe one can be convinced that it’s not surprising that the world can be fruitfully described with math and that people can model math with their brains because having some analogon of function application/modus ponens is an awfully low bar to clear.
If this is what is happening than the relation between math and universe/humanity is of secondary importance to the real question: why math is so rigid. While with everything subjective there are lots of moving parts and not much feels predetermined, with math there’s an inescapable quality of discovery and touching something that’s already there. It’s really mysterious and indeed makes platonism seem not completely crazy. But is “why is math so rigid?” actually a distinct question than “why is math effective?” or just passing the buck?
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Math is ‘rigid’ because that’s what ‘math’ means (to the relevant community of people). But note that the practice of mathematics is almost totally uninterested in almost all mathematical knowledge. One can pretty easily write a computer program to generate a Vast number of theorems – all of them true statements based on some set of axioms! But effectively none of those theorems are interesting. The theorems we find interesting, and the systems we study as part of the activity we consider ‘mathematics’, are all in some sense simple, e.g. X is (in some strict sense) the same as Y, ‘assume these five axioms are true’.
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No. No no no no no.
I appreciate your attempts to address the subject, but you’ve got the whole thing ass backwards.
NOOOOOOOOOOOOOO it’s not strange, it’s by design. When two cultures meet to trade, you better fucking believe the numbers better be consistent to both. Two dozen apples for three shiny rocks is going to square no matter the language, creeds, or cultures of the respective parties—otherwise motherfuckers get stabbed.
The entire point of mathematics is communication of argumentation in a fashion as devoid of linguistic and contextual baggage as is possible; one of the most coveted demonstrations is that of isomorphism, i.e. that thing these French fucks are working with over there and that thing this Chinese math team is calling novel is actually the exact same thing as this goofy shit some Russian shut-in figured out 8o years ago. QED.
Yes, but you’re working at too high a level in some sense. At its most basic, mathematics requires algebra—not the cookbook shit you learned in high school, but algebra as structure. “Logic” is just a form of structure that allows for discerning equivalence between the “objects” in its domain. Of fucking course the universe has a degree of logical underpinning, because if it didn’t, you couldn’t differentiate between yourself and a baked potato. Thus by its very nature, the experiencing of an external universe dictates some degree of logic inherent to the greater system.
Lots of things have geometric analogs, and geometry is a popular “baseline” language because our experience of reality has a decidedly geometric flavor to it, thus allowing lived-experiences to serve as mnemonic devices. That’s probably why all the dead competent philosophers loved it: it offers reasonably accessible and accurate metaphors for the (inescapable) algebraic structure of any consistent language and argument.
That basic mathematical underpinnings are unused and uninteresting to most contemporary philosophers is indicative of how far the field has drifted away from actual truth seeking. Math shaves down the differences between arguments—sometimes into nothing, in the case of isomorphism—regardless of the languages in which the arguments are presented. I don’t just mean finding common ground in arguments given in English and Swahili, this also includes things like showing a specific geometric argument is equivalent to a specific set theoretic argument.
You can make that argument if you want to be wrong. It’s not that hard to arrive at structures isomorphic to Euclidean geometry starting from highly abstract axioms: in some sense, Euclidean geometry just happens to be a numerical “sweet spot” in a particular class of geometric spaces.
It’s just this thing in math, where you keep running into the same shit over and over again in contexts you don’t expect. To the point where it’s almost a game, e.g. “how can I jam the Fibonacci sequence into my dissertation?” Once you get past the hump of realizing it’s bloody fucking obvious mathematics plays a major goddamn role in a structured universe, and accustomed to the same little constants and sequences and structures popping up like Dennis the Menace, things like “the circle is intimately involved in the most useful description of population statistics” lose their holy shit impact.
Not that there isn’t mystery: Gödel showed that an axiomic system gave rise to “truths” which could not be proven in the closed system of those axioms. That means that even a bare-bones set of rules to a reality could very easily ensure the validity of rule-driven phenomenon absolutely unpredictable. even given perfect knowledge of these rules. The surprise isn’t that math itself works to describe the universe—that is an inevitability— the surprises are when what portions of mathematics work and how well, because—see Gödel—there are guaranteed to be times in which mathematics can not predict even a purely mathematical “phenomenon” (to abuse language).
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No, I don’t.
Your first quote needs context. I said those weren’t arguments, but observations. The specific observation being that it can’t be merely a cultural construct, given that cultures are different from one another in a way math is not.
Also: your argument – trade results in uniform systems – is pretty weird. Not only does that not explain why math should interact with the physical world, because then it’s just a useful tool for us, but… well, why should the pre-Colombian Americas have had the same system as Europe if it was merely convenient bartering tools? They certainly weren’t dealing with trade back then.
Isomorphism, yes. We agree. And?
You’re supposing that the universe has a form of causality analogous to human logic, which is not really obvious to anyone who’s thought about it. There are a whole lot of examples of things that are pragmaticaly necessary for humans, which definitely seem to be part of the “world” but which ultimately are not. Take, say, time. You find it obvious that the universe has a logical structure we can connect with, otherwise we couldn’t differentiate between ourselves and baked potatoes. It seems equally obvious that linear time is a necessary ontological structure of the universe – how else could we differentiate between me now, my corpse, and the nutrients that grow the spuds – and yet that’s completely false. You are presupposing things in this argument without realizing it.
Your first argument for the cultural compatibility of math had to do with utility for human societies. There’s zero reason why we should expect that same purpose to apply to the natural world, i.e. ethical conversations are good for our own purposes, but we don’t really expect them to say much about the nature of the universe. After that you declared it obvious that this system should relate to the physical world because… “just this thing.” You get how bizarre and frail and flailing that sounds, right?
Our baseline reality does have a geometric flair to it, which is worth thinking about. There is some logic out there, but it’s not immediately obvious how it works. Also: I don’t think you really understood the Euclid thing. The point was that if the fifth postulate comes from empiricism and that’s the explanation for why it works with the physical world, then that cannot explain why denying it leads to other physical discoveries. In other words, it’s just a point about the go-to empiricist response. That we can Nash it into Euclidean space is irrelevant to the point at hand, level of abstraction is not what’s at stake.
Look: You and I don’t even disagree on half of this, but to actually get at a proper understanding of the problem you can’t just blithely assume all the things that appear obvious to blithely assume. You will trip yourself up, see above.
There aren’t any proofs above, nor much actual argumentation. This is the very first part of a much longer series that will go into all of that. I doubt I’ll do Gödel because everyone knows Gödel and it’s boring. Next is Plato, who very few people know.
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The fuck I am. I’m saying “human logic” is actually describing a class of structure so fundamental if the universe didn’t organize “logically” on a basic level, we couldn’t differentiate our selves from a baked potato.
No! You’ve Rorschach’d implications into my statements that don’t make sense, then admonished me for things I neither stated nor intended. My argument makes no points with regards to time, nor its “linearity”. You’ve presupposed I’ve discussed “causality” and “time” like I’m a goddamn chump. I’m not writing on a baby level here, I’m writing about mathematical philosophy, so start reading it like a mathematical work, i.e. maximally conservative in terms of what it is actually stating.
You have it backwards. Math has always been about working with or as close to possible simple inescapable truths, e.g. you are not a baked potato or two eggs provide more nourishment than one egg. Of course it transcends cultural grounds, because the problem of incompatibility is a translation problem: the number “1” still represents the same “thing” even if you’re a filthy Roman who writes it “I”. Math has always been about discussing the universe, or, more accurately, discussing the rules which even the universe cannot escape.
No! It only sounds “flailing” because you’re accustomed to weak arguments—and you don’t know any math. The most basic truths of human experience dictate some degree of consistency to the universe, and from there mathematics serve as the tools for working with logically consistent systems. Within logically consistent systems natural numbers arise almost instantly, and when you keep going you start to run into the same shit over and over again. Anyone mathematically inclined would understand what I’m talking about.
Here I’ll tell you what’s flailing: you’re so goddamn certain I don’t know what I’m talking about, you’ve projected weakness into my terminology which doesn’t exist, rather than draw into question your own understanding of my writing.
No, we’re not even speaking the same language. I have to deal with the enormous baggage that comes with the abysmal state of contemporary philosophy, and you still think that baggage is worth a damn. I didn’t trip myself up, those are dance steps. Try and keep up.
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I doubt this is going to be productive after a point, because we’ll just repeat things. I’m dead positive you’re not reading me, and you’re equally positive I’m not reading you. I tried to point out where we agree, you rejected it to make the same point, that assures me I’m in the right, I assume you feel the same.
The very first argument of your first comment was this:
That’s a claim about math coming from social utility. I recognize that you abandoned that claim later. I was fucking with you for opening with a bad argument. That’s not me making an argument about it not transcending cultural grounds, c’mon.
Sure, it’s a cheap shot or whatever. I’ll be careful then, since you requested it. This is what you’re saying, no snark added: the world must have some consistent structures for anything to exist, hence why we aren’t potatoes. Being things that experience things, we humans notice this. The best analyses of consistent structures are going to be ones that internally consistent, and mathematical axioms are the barest of bones from which to build as rigorous a representation as we can. Natural numbers fall out of that, and thus their obvious patterns all over the physical world. Is that an accurate description?
Here’s my argument: I don’t disagree with that. What I do disagree with is the statement that it’s obvious we should be able to describe anything at all. It’s equally necessary for a goldfish that there be a consistent universe, but there’s no requirement that how goldfish conceive of that structure in any way cohere with it. They did not evolve to describe the universe, but to survive it.
Your presupposition is that we are different, that whatever consistent structures in the universe are adequately accounted for by what we call logic. It doesn’t matter how internally consistent mathematics is, you’re assuming that what we take to be “consistent” is the same as whatever noumenal shit is “real.” Hence, the points about time and causality. They were examples of our experience directing us at assumptions that are inaccurate but helpful for survival, in the same way that goldfish metaphysics are (presumably) inaccurate but helpful for it to survive.
You are making a metaphysical claim about the universe while trying to pass it off as an obvious result of algebra. This post was, quite literally, about that exact thing, so thanks. It was a helpful demonstration for subsequent readers.
At the limits of my charity I’ll assume you meant to describe why mathematics should be our own best description of the world. I don’t disagree, at all, which was pointed out.
I have zero idea who you are outside of this comment section, and you announced yourself in a pretty chumpish way. You showed up swinging dick in the funniest manner possible, i.e. being cool but mad in an obscure blog’s comment section while nodding at your Many Talents That Plebs Like Me Will Never Understand. Of course I’m not going to take you seriously.
The post was an obvious starting point, the idea being to point out common assumptions and step back to the point where pi is a weird thing before progressing to where it isn’t. You know, start from ignorance and all that. That apparently flew over your head, which is why I tried to be polite and said, “We agree on a lot more than you think,” and you responded with… I don’t even know, an attempt to Show Me the Truth, where Truth is apparently Anthropic Principle + Naive Logicism.
Thanks, I wasn’t worth the effort.
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That um, well, shoot. Re-reading OP I see I misinterpreted your tone and voice nigh-completely. My apologies.
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Hey, it’s cool. I know I’ve done the same far too many times. I appreciate the apology.
For what it’s worth, I think I probably could have been clearer about a few of the examples used and why I was using them. But I suspect you know that trying to write about philosophy of math (non-academically) is a balancing act between language too-loose-to-have-value and too-technical-to-serve-for-introductions. I may not have the acrobatic skill that I wish I did.
I also could have been less immediately snarky in comments here. It wasn’t conducive to clearing up any misunderstandings, nor was it particularly mature. So, I apologize as well.
If you do choose to continue reading, perhaps we’ll get to have a genuine debate some time. Thanks again.
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So…Math is basically magic, right? That is, all the analogies and experiences we apply to young adult novels about hormonal teenagers discovering the cheat codes to the universe are actually true of our primitive ancestors discovering math, to the point where we have real-world examples of people starting secret cults that were basically schools to impart hidden mathematical truths down from one generation to the next.
I don’t have a good answer for why humans should be able to magic, but I guess my best Hail Mary would be that there is something about the “math sense” that creates an evolutionary feedback loop. Meaning that once primates begin to abstract numbers of geometric patterns or whatever, the part of the brain that grows over successive generations of those guys being slightly more reproductively successful (would math-nerd monkeys be more successful? Their descendants don’t seem to be) is the part responsible for self-consciousness. That seems like something you could find out with an MRI, IDK, like I said, Hail Mary.
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In the universe there are
Tautologies
Coin flips. Not inconsistent with the above,but a point that not all outcomes can be accurately predicted.
As for why math works. I can give it a shot.
The way I think and communicate is in language.
But the linguists and cs guys and psychologists have found clear logical mathematical patterns in human communication.
And if there was not a pattern in communication, what beings recognized as “intelligent” could even arise?
In order to effectively understand this conversation, you are a mathematical pattern. (Probability, almost one)
Thus, these patterns then describe the patterns around them.
There is no gain to intelligence in a random void.
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Where randomness dominates (and thus a lack of mathematically described patterns) there is then a lowered utility gain to memory and intelligence.
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So not only will a “random” universe have lower benefits for intelligent creatures, it’s static interference world also itself will probably lack the structure for a being to hold itself together in the first place.
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So to sum.
Effective communication and learning and memory requires an underlying consistent structure to communicate.( This one is big. Don’t want the brain to suffer from a lot of packet loss)
A random world is probably unlikely to generate intelligent life (why plan for the future)
World’s where randomness dominate are probably unlikely to have the structure for complex organisms to hold.
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An anthropic argument to explain a mathematical universe? Give me a break.
One, the reasoning is breathtakingly circular even by the standards of anthropic arguments.
Two, even if I grant the anthropic value of a mostly consistent universe, it’s a very long way from 99.9% consistent to 100%.
Three, how exactly do you visualize “random” vs. “non-random” “universes” being generated, if not with math?!?
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There is absolutely no way around a mathematical-logical argument.
All explanations can be described with mathematics.
The real question is why do we live in a universe with patterns and stable equations.
Your statement is simply mathematics in disguise. The cs-linguists have shown that concretely. It doesn’t obviously look like that….but there then becomes an quaint absurdity to that statement.
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You can’t. It’s simply logic and axioms.
There are information theorists who have answered this better than I have.
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The biggest issue here (as you seem aware) is making the problem itself actually evident. Of course, that’s usually the biggest issue, with any problem. This, and the “cave inside a metacave” comment reminds me of the remark that modern students have to be freed from a cave before they’re even IN Plato’s cave. Rather than knowing nothing, we are sure that we can’t know anything. It’s innocence wearing the clothing of jaded cynicism, if that makes sense. It’s only once people realise how deeply strange this problem is, that it IS a problem, and how poor are our seemingly solid explanations (that often explain in the sense of “explaining away”…), only then does the rest of the discussion open up.
Additional comment: I always love it when Platonism emerges in something I’m reading… like the third-act return of the presumed-dead hero, arriving when least (and most) expected.
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Well… no. The problem you address here is not a problem.
Considering Wigner’s essay: “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 wave and particle theories of light contradict each other. They’re still both useful theories.
The source of the problem is not in the world, but in the over-reaching desires of the enquirer–as shown in Wigner’s next sentences: “It would give us a deep sense of frustration in our search for what I called the “ultimate truth”. … fundamentally, we do not know why our theories work so well. Hence their accuracy may not prove their truth and consistency.”
Math has truth and proofs. Experimental theories have predictive accuracy.
We do know why our theories work so well. It’s because we construct them as statistical models of carefully-defined measurements. So they must work, regardless of how or whether what we’re measuring relates to “absolute reality”, a Platonic construct which you should have discarded when Einstein showed us how to operationalize the term “measurement”.
Statistics is the interface between math and empirical science, the thing you need to study rather than Kant to answer your philosophical questions. A statistical test is a function that maps an experimental claim and experimental data into a new claim, which IS proven, not for the real world, but for a hypothesized world embodying the statistical assumptions made in the analysis. And it is stated not on the direct level of “X results in Y”, but as a statistical claim: “The results seen would be produced less than 5% of the time in any hypothetical world in which X is uncorrelated with Y.”
Science really is correct and unproblematic if you operationalize your terms, and keep track of all the statistical assumptions made and all the hypotheses tested along the way. The problem is just that humans are, for the moment, computationally incapable of doing so. (Also, politicians are incapable of making decisions given statements phrased in proper statistical terms.)
“Why should the circumference of a circle bear any relation to an empirical phenomenon that is not, in any obvious way, a circle?”
The phenomenon was one of population dynamics, which takes effect by people interacting on Earth. People live on the surface of the Earth, which can be locally approximated as a flat surface. So effects spread in a circle.
Conclusion: Just because something isn’t immediately obvious to you doesn’t mean there’s a deep mystery at hand.
“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.”
No; this is extremely obvious. Math is culturally relative–it’s just that the only sophisticated math we have is Western math. It’s an artificial construct, a formal system. We define it. It doesn’t describe things in the world. It isn’t science.
“If mathematics says something about the world… one might assume that human reason can get us to The Truth. ”
No. No. No.
Stop. You don’t need The Truth. You don’t need to be able to predict with perfect accuracy because there’s always a one in a trillion chance space aliens will evaporate the Earth tomorrow. You can’t get The Truth, but you can get 99.99% accuracy, and you’re better off spending your time adding another digit to that figure than worrying about The Truth.
“If it can understand the deities’ nature, then it should understand Being.”
Again, stop. “Being” is non-sense, a term not defined in terms of sensory data, and hence a term irrelevant to human needs. Don’t ask questions about it, because any answers you think you find must necessarily also be non-sense.
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I wrote: “No; this is extremely obvious. Math is culturally relative–it’s just that the only sophisticated math we have is Western math. It’s an artificial construct, a formal system. We define it. It doesn’t describe things in the world. It isn’t science.”
Of course, immediately after pressing “return” I realized the validity of this statement won’t be obvious to people who don’t habitually operationalize their terms.
You’re right that math seems to–let’s not say “describe things in the world”, but “give correct results when used to relate different numerical measurements of the world.”
This, however, is baked in before we even look at the world. When we say that “arithmetic describes the world”, we mean that if I measure the mass of one can of soda, and multiply it by 6, I’ll then get the same result as if I measured the mass of 6 cans of soda.
This, however, was inevitable, regardless of my interface to reality, because the operations I perform on my sense-data are defined by my mathematics. I have defined the number “6” and the operation of multiplication. I can observe and measure the mass of 1 can of soda, and multiply that result by 6, in which case my definition of “6” comes into play when performing the multiplication. Or, I can observe and measure the mass of 6 cans of soda–but how do I do that? How do I know when I am observing 6 cans of soda? My “observation” is itself a measurement, and some stage of that measurement must use my formal definition of “6” to construe an observation as being of “6 cans of soda”, extrapolated from an observation of “1 can of soda”.
So arithmetic doesn’t “describe the world”. It is a self-consistent formal system, and so it can take sense data–which is itself formal data–and manipulate it, and the results will be consistent. 6 * mass(X ) | is(X, soda) = mass(X) | is(X, 6*soda), not because the world knows about my math, but because my math is all in my head on both sides of the equation.
(I’m presuming there is a category “can of soda”, and that instantiations of that category will return the same values for the measurements of interest. I can presume this without believing math is objectively “true”. The validity of categorization is independent of the validity of my mathematics.)
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The mix of condescension and confusion here is pretty wild.
I don’t really know what to do with this besides say “read more carefully,” and “statistics uses math,” and maybe “are you the kind of person that argues with introductions because they haven’t yet made the case of the rest of the book”?
This was explicitly introductory, there were a lot of hypotheticals and thought experiments, you have some trouble understanding that. If I say, “One can imagine someone who assumes that x allows us to do y,” then you should not interpret that sentence as me saying “x therefore y.” Similarly, “x is not as obvious as we assume,” does not mean “x is unknowable.” Now go back and look at the context around the quotes you pulled. These are basic reading skills. Another reading habit that might help you: it’s generally safe to assume that a Nobel Prize winning theoretical physicist has considered wave/particle duality before. Using something a 7th grader parrots to ignore the fact that maybe someone knows something you don’t is a self-own of epic proportions. It allowed Wigner’s point to fly wayyy over your head.
Look, I’ll tell you the terrible secret that I repeated many times: the main interest is epistemology. Mathematics is just the best test-case for that. In other words, don’t tell me what “is” or “is not” useful if you fail to understand what I’m using it for. Jesus, dude.
Last: ignoring anything else about it, your explanation of the “obvious” cultural relativity of math proves no such thing. An equally valid interpretation is that human cognitive structures are designed to differentiate between “five” and “six” – regardless of the Platonic Truth of the matter – and different cultures attach different sounds, i.e. the sense-data is primordial, the construction later. That still doesn’t mean that “six” is a real thing in the world, but nothing you wrote proves it as a cultural construction.
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