coming from here
Pythagoras assigned cyclical motions to the planets. Circles are eternal, and thus the motion most suited for the motions of the heavens. This essay is about circles, as well, albeit the more homely human kind. It’s about racing so far in one direction that you wind up back at the get-go.
All theories have assumptions, all assumptions lead to their own conclusions. Inconsistency is not bad for the sin of pride, it’s bad because it makes you wreck yourself in conversations. Worse is inconsistency with power for reasons that are too obvious to lay out, [Goya etching here], etc.
This blog has recently been focused on the epistemology of mathematics. It has interesting and far-reaching consequences, but it’s often ignored as meaningless specialist nonsense and/or ivory tower shit.
Those consequences are the real interest, and I’ve explicitly stated that the end is modern phenomenology. But to get to [anything modern] you need Kant, to get to Kant you need Hume, to get to Hume you need Idealism, to get Idealism you need Plato.
Platonism (in math) is, essentially, the position that mathematical objects are real. They are as “out there” as a planet is “out there” (just not in space-time, spoiler alert). Because it’s hard to really precise this, here’s (hilariously) an entire appendix of people defining it.
Naive versions of Platonism are astoundingly common when it comes to the epistemology of mathematics. These aren’t “wrong” per se, they just lead to consequences counter to what we tend to want. I’m pretty sure this is because mathematics is secure enough that it’s the very last metaphysical “thing” we want to deny. The denial also leads to tricky questions about the physical sciences, i.e. the point of this series. Thus, we’re a lot more willing to grant ontological primacy to mathematics than we are to, say, “beauty” or “virtue.”
But also: Plato himself is a necessary nightmare to talk about. He’s a great example of why one should read primary sources, because “platonism” is historically sideways. This is bad enough that I have to write two separate articles. This one is on “Platonism.” The next will be on Plato.
When we talk about Platonism now, we’re not actually talking about a 4th century BC philosophical school. We’re talking about a 20th century one. Godel absolutely stomped the early analytic schools, and everyone wandered in a daze looking for a new position. Kind of, this is bad history, don’t @ me. I’m not going to get into that because [long] and [besides the point], but it’s consistent that Godel himself was a devoted Platonist.
It’s quite popular, so note that any criticisms I make will 100% have objections and counter-arguments. Platonism is the plurality position by this survey (PDF) of philosophers. (Q: Abstract objects: Platonism or nominalism? Results: Abstract objects: Platonism 39.3%; nominalism 37.7%; other 23.0%.) Since it says “abstract objects” rather than “mathematical objects”, that probably confounds full-blooded Platonism (“all abstract concepts exist”) with mathematical Platonism (“at least mathematical objects exist”), but I’ve yet to meet someone who thought that the abstract concept “beauty” is real but numbers are not. In other words, that 39.3% almost certainly covers all mathematical Platonists.
If I get around to talking about the analytics (way later), I’m going to have to return to Platonism, i.e. this is incomplete. I’m much more interested in arguments mustered for naturalism on Platonic grounds, both as a personal preference and for subsequent articles. Less in arguing for or against Platonism than in showing some of the consequences, and for those we basically assume it’s true. After all, this series begins with the question: “Why does math work in reality?” and Platonism is an answer to that question. It works because math is real, it doesn’t matter how frail the human mind is, somehow we frailed our way into the Truth of the World, take it and run.
Still, there’s a reason that a shocking number of otherwise-impartial descriptions of modern platonists use phrases like “bite the bullet” to describe their admissions. The consequences of the argument are wild, and for that one actually can turn back to Plato. It matters less whether he himself believed it than it does that he develops some of the results and, even if ironically, these went on to have some super weird consequences.
You might ask why start with Platonism, then. Long story short: [history] happened, modern Platonism is enough like what pre-modern philosophers were responding to that it’s basically fine. There was a long historical bit here, but it’s been banished to an appendix for taking up space without moving the argument forward.
I’ve praised the virtues of careful philosophical argumentation. In an act of stunning hypocrisy, I will now write a very reckless article about Plato and Platonism.
This is because I want to.
This series began with the question how do we know anything at all? It presupposes that – at least in the realm of math – we do know things. This is basically how Plato starts, something I pointed out with the Meno: “It’s uncertain if we do know anything, but suppose that we do. What would it look like for that to be possible?”
Platonism, as a school, gets extremely confused extremely quickly. It was originally an idealist philosophy, but now it’s commonly invoked as a semi-defense of empiricism and naturalism. And, again, it has very little to do with Plato. For the purpose of this essay I’ll pretend it does, next essay I’ll explain why that was a bad thing to pretend.
If you know about Plato, you know about the forms. This is the “naive” variant. I’ll break my own rules and give you the Stanford (read the Phaedo instead) if you want a careful overview.
Here’s the short version: The world of coming and going – material – is complete and utter darkness. It’s quite literally Heraclitean Flux. We can have no certainty about it due to its changing nature, which means if “knowledge” is a thing we have, it cannot come from empirical study alone (or at all, depending). Every horse is uniquely imperfect in some particular way, it deviates from the ideal “horseness.” Still, it partakes of “horseness” just enough that we can correctly identify the flawed versions of horses as things-which-participate-in-horseness. “Horseness” is a Form, and one could also phrase it as “The Form of Horseness.” True knowledge, then, is knowing “The Form of Horseness” rather than “this particular horse.” Further, forms are normally understood as epistemological and ontological. They’re the reason the intelligible world exists – were there not the stability of forms, there would be no world to make sense of – while being the reason we can understand the world. If we did not have access to forms in some way, we’d fail to identify the common traits of horseness that allow us to distinguish horses from mountains.
This is full of holes, and I’m not going to defend it or the idea that Plato meant it seriously. It’s a little less ridiculous than what I pointed out, because he’s often talking about genuine abstracts. Thus, in the Meno, Plato asks what virtue is, and Meno replies with this:
Let us take first the virtue of a man-he should know how to administer the state, and in the administration of it to benefit his friends and harm his enemies; and he must also be careful not to suffer harm himself. A woman’s virtue, if you wish to know about that, may also be easily described: her duty is to order her house, and keep what is indoors, and obey her husband. Every age, every condition of life, young or old, male or female, bond or free, has a different virtue: there are virtues numberless, and no lack of definitions of them; for virtue is relative to the actions and ages of each of us in all that we do. And the same may be said of vice, Socrates.
To which Socrates replies that Meno has just given him a “swarm” rather than Virtue, and there must surely be some form of Virtue by which all these specific acts are called and considered “virtuous.”
Either way, Plato tends to pivot to mathematics whenever he brings up the Forms, and he does so in really weird ways. The most obvious is (also) the Meno, where to prove inborn knowledge of the forms he has a slave-boy perform geometry. More explicit is [everything in the Republic]. In both of these cases, souls are regarded as immortal due to [Platonism things]. See the appendix for deets.
The most extreme is the Timaeus. Timaeus is a highly detailed monologue (by Timaeus) on the creation of time, the heavens, and the earth, while also being the origin of the myth of Atlantis (not joking). It starts with a distinction between eternal things and temporal things, and moves into an account of how the Demiurge sets up all of it. Critical is the fact that underlying the world of flux is mathematical harmony. The four elements are the Platonic Solids, meaning that matter is constructed according to geometric patterns. The planets are set in specific proportions from the earth, and these proportions just happen to be the Pythagorean Intervals. So on, so forth. The implication of the Timaeus, of course, is that the “true” world, is ordered according to mathematical forms of which “matter” is a kind of degradation. Philosophers then use mathematics to get closer to understanding the true reality which underlies the visible world, and thus the divine and/or the good.
It’s more complex than this, I don’t want to get into it, look back at the title of this piece to understand why.
I’ll throw caution to the wind to point out a few further things: Plato is famous for banishing poets from his Republic, and for only allowing extremely specific musical patterns. That comes directly from Timaeus-style Pythagoreanism. The harmony of the spheres, if interpreted as a relation to the eternal soul, must be maintained, i.e. math is ethics, because the soul must be “tuned” to partake in the beauty of The Good. This is also a really common starting point for Natural Law and moral realism (i.e. “moral facts exist objectively”), for reasons that ought to be pretty clear.
A less naive reading that I’m still unsure has anything to do with Plato is the following:
If you interpret Timaeus et al. figuratively, and you make a really careful and complex argument I don’t care to make right now, you can almost arrive at a doctrine resembling modern physics. Matter is not a unique, indivisible substance, but a composition of various elements which are, critically, mathematical in nature. Thus, mathematics can be used to determine physical laws which govern the universe, and given the eternal truth of math, these will be more “real” than mere speculation. Moreover, the emphasis on ratios within the heavenly bodies shows that they too are subject to these physical laws, which heavily implies that they’re composed similarly to the earth. This is only kind of a stretch, and other philosophers had proposed similar.
Either way, that still leaves us with a Platonic implication of mathematics: it is real, it is “embedded” in the universe somehow while not really being a part of space-time. Also: the naive reading was the most popular for a very long time.
Given the Form of the Horse thing, it’s probably going to sound odd when I say that Platonism is a really common assumption. When people talk about math as a human discovery – we pieced it together from real parts of nature, and it’s not merely an approximation – the Platonic argument is underlying that. It’s common enough that I have to be careful when discussing modern versions of this, because there are Platonic and non-Platonic versions of arguments that look ridiculously similar. I’m going to try and separate them piece by piece, but this is a blog and not a dissertation.
Recap: Mathematical Platonism means the following, a) math is independent of humans; b) pretty clear from (a), but mathematical objects exist; c) these are abstracts, as in “mathematics is an abstract concept.” Normally included in this is an argument that mathematical objects exist outside space-time. (Note that other options are worse: there are ideal triangles floating somewhere in physical reality, and apparently they cast geometry-beams into everything. Personally, I find this a bit forced. That being said, I’m pretty sure that New Age websites use this to explain how “Quantum Mechanics” proves that True Material Existence is One True Consciousness of God on a different dimension or whatever.)
There are actually good arguments. I’m going to stick with Putnam’s work with Quine’s Argument from Indispensability. I have no idea why, but the Internet Encyclopedia of Philosophy has a better description than Stanford. Read that for rigor, my description won’t have it.
The argument goes something like this: The descriptions of the universe that best match the phenomena are all based on the physical sciences, and those descriptions are the things we ought to afford existence, i.e. electrons exist in a way that phlogiston and the Four Humors do not. It would be strange for any necessary predicate of a theory to not “exist” in the same way, that is to say, it would be odd for an electron to exist but a neutron not to, or to be merely as reliable as phlogiston. If a scientific theory is a “real” description of the physical world, then things which are indispensable to that theory should also be assumed to be real, for the obvious reason that, without them, we would not have the theory, they’re part of it, it would be like allowing for cars but rejecting combustion. Mathematics is indispensable to the natural sciences. Therefore, mathematical objects are real.
This is, more or less, a variant on naturalism. Putnam argues that not accepting it is tantamount to grave intellectual dishonesty.
There are other good arguments for mathematical Platonism, but Quine’s is the most common (even if not explicitly stated) and it neatly fits into the “Platonism somehow became a defense of naturalism” theme. It’s also implicitly invoked much more often than the more careful, Fregean arguments.
It should be pretty clear that Platonism is also necessary for all manner of wild beliefs, see the appendix, but arguing against sacred geometry is ugly in the way that beating up a child is ugly.
First, the indispensability argument is good. Getting into the details of arguments for and against it will take time, be boring, and, it’s an axiomatic certainty that they must therefore come in a temporally later and more boring essay. I’ll note two points, neither of which disprove Putnam/Quine but both of which reframe it.
My favorite is not a good argument, but it is funny: Certain branches of mathematics are not currently indispensable for the physical sciences, even if they come from axioms that earlier principles use. To his credit, this is why Quine tossed out a bunch of set-theory. But it’s not clear that some of those branches won’t be useful some time in the future – plenty of higher branches of mathematics were pure until someone found a use for them. You’re left in the position of declaring existence or non-existence based on temporality, which is almost certainly wrong inasmuch as things exist or do not exist, stop Schrodingering at me.
This isn’t a very good argument: science is well willing to recalibrate which things exist and which don’t. The problem – and this is implicit in the phrasing of the argument – is that it makes the argument a suggestion for what things we should believe exist, which leaves the door open for all sorts of things to exist-non-exist.
It’s this next part that leads to phrases like “Quine bit the bullet and…” (Edit: A friend points out that I should be a little more cautious here to avoid angering the dead. So, to make explicit what’s implicit above and expanded on below: the Putnam/Quine argument has taken on a life of its own to the point where it’s hard to tell exactly how closely it relates to their work, if at all. Putnam and Quine would both, likely, disagree with what comes below, but the indispensability argument as it gets used is open to them.)
First, if mathematical objects are “real”, then they’re certainly not physically real, i.e. they exist outside space-time. It’s not exactly clear where they are, but equations are most definitely not physical things. Frege, who admits and develops this part of the argument, quite literally refers to a “third place” of existence, which is the math-realm.
Platonism isn’t necessarily wrong, as in it’s not obviously false, you can formalize it and the argument will be valid. But, in arguing for the primacy of the sciences and the import of mathematics, our Platonic certainty destroys materialism. This is normally phrased as an argument against it, but one can consistently hold the position so long as they admit the following: metaphysics is real.
This opens the door to some unfortunate consequences. I’ll phrase them as questions:
1) If there is a metaphysical math-plane, then how does it interact with the physical world? Is there an ether? It’s impossible for us to empirically investigate a metaphysical medium, so we can let this one linger. Resolving it normally means we try to logic at it until we understand some kind of medium (the Timaeus proposes a very strange third-medium-thing to resolve it), but that takes us to the next problem.
2) If we’ve already opened up the metaphysical plane, then it’s not immediately clear why we should stop with math. If we take the indispensability argument as a should rather than an is, i.e. we should provisionally accept the existence of certain branches of math, we’re probably worse off here. That should is not a killer argument against undesirable metaphysics.
This is doubly true if one takes the position that logic and math are the same thing, and opens up logical argumentation of non-mathematical abstracts (say, human freedom, virtue, God) to the rigor of mathematical work. Aristotle vaguely tries this: one reading is that he deals with Platonic forms by making them in the “formal cause” of matter, but that’s how you wind up with things like the Prime Mover, i.e. a thinking thing thinking itself into existence because it thinks about thinking.
3) Mathematical laws are seemingly eternal, so how do humans have access to them? Is there a soul that communicates through the ether? I get that this was assumed by the argument, so it’s kind of unfair. The point is this particular consequence is what leads to Pythagorean arguments for resurrection and eternal consciousness.
4) Implicit in the prime mover reference: it’s ridiculously unclear where mathematical objects come from. You can’t rely on the Big Bang for their creation, because the Big Bang needs math to Bang, and they can’t presuppose-themselves-for-their-own-creation like weird autogenerative algebraic deities (unless they are deities).
Those four points have responses, some more convincing than others. They’re not reserved to mathematical Platonism; similar issues arise when one postulates any metaphysical reality. Platonism in math just happens to be particularly appealing, because mathematics is “secure” in a way that theological arguments are not, and yet mysterious in a kind of similar way. Still, one has to respond in some way to the issue of math and reality, or else give up on the task of knowledge forever (coming up soon).
Still, that last one is where everything falls to pieces.
Quine and Putnam bite the bullet, but they aren’t the circle I was referring to. I have no problem with careful beliefs which are different from my own. The issue is with a lack of careful beliefs.
It’s at this point of the essay that I’m going to start getting accused of straw-manning. Coincidentally, it’s also the point I began writing when another round of STEM vs. Humanities broke out. I have no side in the fight, I like both, I get that by talking about philosophy I’ve already sided in a few minds. I’m not going to convince them, who cares, [insult here].
Look, even if I have a habit of defending religion, I’m deeply sympathetic to the materialist project. This is a labor of love, and the point of it is the following: these aren’t strawmen, they’re materialist arguments that accidentally eat themselves, slippery Platonic arguments that result from trying to make too broad of a case. Without some other argument, these are wide-open to all manner of metaphysical speculation, which is especially bad when most are framed as arguments against metaphysics.
I’ll take A Universe from Nothing as an example, though it’s not the only one. When released there was an (inevitable) flare up in the atheist/theist debate. Almost the entire argument was over (4). Now, I actually liked that book, but the more flamboyant aspects of its debate were a series of ourobouros arguments.
At the heart of it is the claim that physics disproves God by neatly resolving the question of the universe’s origin. “Why is there something rather than nothing?” and the answer is “mathematical laws.” This is satisfying only if one assumes a Platonic position, because those are, ontologically, parts of the universe. There’s no more question, you don’t have the deep concern that math is merely describing perceptions. Again, I’m personally quite satisfied with perceptions. They just aren’t arguments against theological stances, and trying to make them so is dishonest.
However, if one takes the Platonic position, things are likely worse. You’re left with the fact that “mathematical laws” are ontologically a thing, which means they came from somewhere, can’t presuppose themselves to exist, and so why are they there rather than not being there? Hilariously, it makes some kind of God look pretty attractive: at least it resolves the question of mathematical ontology, and allows us to use that as a solid base for the natural sciences. Though one might be tempted to view this as a deistic, hands-off God, (3) might be interpreted in the following way: we can commune with the metaphysical plane somehow, which means we have privileged access to the divine realm and it clearly has some “place” in our souls, which means…
The non-Platonic position means that mathematics is merely descriptive, it’s a contingent-on-human-knowledge way to understand reality. Thus, it has absolutely nothing to say about what’s ontologically “real” or true, i.e. you lose the ability to claim that it resolves the question or says anything certain about reality. This may be unsatisfying, but at least it doesn’t lead to self-destruction.
The point is to destroy metaphysics. Stop letting it in to own the libs.
Coming clean: I hate Platonism. If it’s to be extracted, that takes some effort.
So begins a general theme of this series: you cannot have your cake and eat it too. Or, well, the only people who can have their cake and eat it too are mystics and absolute skeptics. This makes sense. If you doubt the principle of non-contradiction, then there really isn’t a reason that you can have a not-eaten and eaten cake at the same time.
Broadly, it’s about the way certain conceptions of mathematics sneak into our daily language. The point of this isn’t to straw-man [someone], it’s to point out just how embedded in our language Platonism is, that everyday defenses of mathematics and physics have somehow all begun to assume Platonism, and that this is really bad. If you want to move into any discussion of psychology or ethics or politics from a materialist platform, you can’t have third-space-objects floating around implying an eternal soul.
Last essay I implied that some of these arguments would fall, but I couldn’t get around to explaining why without an extended discussion of Platonism and [everything above].
So: We want to answer the question, “Why does mathematics work?” and we want to be precise about it. Here are some problems with the easy answers. To avoid straw-manning, I’ll offer the non-Platonic versions after the criticisms, with the recognition that these leave us in a state of total confusion regarding the original question.
1) The simple evolutionary answer goes: Our environment is mathematical, and so any selection pressure for [bundles of years] meant that brains which “understood” the math of the environment were best able to interact with it. Thus we arrive at the human capacity to understand math as well as an explanation for why that math works with the physical environment. Platonism is implicit here: mathematics exists independently within nature, or else it could not operate as a selection pressure on conscious organisms.
The implication here – a bad one – is that “truth” is in any way related to evolution, when [absolutely everything] should tell you that it isn’t. The genetic legacy is brute utility over truth: badgers don’t use trig, jocks beat up nerds, something something signalling. That we adapt to our environment does not mean we adapt to understand it, fuck this up and you’ve reproduced the Catholic Church’s favored view of evolutionary complexity, no I am not joking.
This actually isn’t an argument against that view, inasmuch as the guided evolution principle can be maintained, i.e. by Catholics. It’s just a consequence that (most) people who want to make the argument definitely don’t want.
This is not to say that evolution plays no part in mathematics’ relationship to the environment. It almost certainly does. The critical difference is whether we evolved for the purpose of solving physical tasks rather than metaphysical ones. We see the color blue does not mean that light waves “are” blue, it’s an interaction between our eyes and the light. The color provides useful information – these are lightwaves of a type that we visually categorize as blue – and it’s not opposed to some inner working of the world. Colors aren’t arbitrary, they are analogous to light-waves-of-such-and-such-frequency. They simply aren’t lightwaves-in-themselves.
I get that this seems like a minute distinction, but positing access to the thing in itself, and making that a mathematical relationship, results in [everything above].
I feel like I should bold this, but I’d rather see who attacks me for denying Darwin: evolution plays pretty strongly into my preferred view of mathematical epistemology.
2) The Anthropic Principle is originally a response to the question: “Why is there something rather than nothing?” It replies that if this world did not exist in exactly the way it does – that is to say, with the perfect and precarious conditions necessary for conscious life – there would be no creatures to ask the question. Ignore the fact that it’s tautological and answering a different question than it thinks it is (“how” rather than “why” for those following at home), we’ll save it for another day. The problem comes when the principle gets transmuted into an epistemological argument. So rather than answering “Whyhow are we here?” it’s addressing “Why do we live in an intelligible universe?” This is, as you might have noticed, not a question the principle was designed to answer.
In the original formulation, it doesn’t much matter how we understand the universe. It simply has to exist in its current state that we might be here to throw rocks at glass houses. In the epistemological formulation, we understand the universe because the particular order it has is the one we know, namely, a mathematical order a la our current understanding of force-laws. The ontological foundation of the universe is then, quite specifically, a mathematical one that must prefigure consciousness (so exists independent of our cognition).
This is a really similar problem with (1). Order and logic and mathematics are not the same thing, but they’re getting conflated. The universe certainly has some kind of order (if it was pure chaos, nothing would be here), but that does not mean we should understand it. A->B, fine, why does that entitle you to understand the “->”. Accounts are human, they can be wrong, they can overfit the data. I’m going to reserve the word “order” here for [all kinds], and “logic” for how humans perceive order.
The certainty of logic does not follow from the existence of order. Order is equally necessary for the existence of the Grand Canyon, inasmuch us there have to be consistent natural laws for water to wear it down into a pretty tourist trap, but the Grand Canyon has no claim on understanding why that’s the case. Needless to say, order is necessary for cats, automobile, and lightning. None of those things understand their existence by virtue of existing.
I once called extreme forms of the Anthropic Principle soteriological, and this is why. If the AP comments on evolution as a process of increasing truth-gathering, which is what the argument above leads you to, then you’re assuming a teleological principle indistinguishable from the guiding hand of God. If, further, you assume that consciousness and intelligence are good – say, biotechnology to enhance longevity is only possible with a measure of intelligence – then you’ve got a guiding principle of salvation, physical or no.
This is going to pattern match to a few comments on that last piece, but it’s not the same argument. Those were the non-Platonist version. Good arguments which incorporate the principle go:
- The universe must have some kind of order for anything to exist (AP).
- Mathematics is the best human system of order – whether it’s an approximation or not is uncertain, its existence as an ordered system is at least analogous to whatever kind of order we see in (1).
- Since mathematics is our best version of “order”, it makes sense that it best describes the ordered (1) universe, and thus has the strongest claims on physical utility.
This is a much more subtle and interesting argument. I’m not going to address it here, but it is worth considering.
For both of these, one can say that math is simply our best approximation of the order of the universe, but: a) that’s not Platonism, because it admits of the nominalist position that math isn’t “real” so much as descriptive; b) it takes us back to square one, as in “Man, math is super effective. What, uh, is it?”
It’s (b) which is the real problem. And so, we move back into the question: why does math work and how do we know it.
The current essay will not resolve that, sorry.
top image from Alphaville by Godard
Appendix on Plato
Given [everything above], it might sound weird to start with Platonism. I’m opening with 20th century arguments to get at earlier arguments right after talking about the necessity of history.
Here are four responses: 1) Plato himself becomes important for [everyone later], but he needs to be separated from Platonism; 2) The story of how Platonism became Platonism actually explains his historical influence; 3) There aren’t really any arguments for Platonism until recently. There are only arguments for neo-Platonism; 4) The theme of this series is “how does math work” and Platonism must be addressed.
All of these are true. The most important, though, is this: reading Plato is, itself, much more modern than anything else.
Here’s a not very careful history of what happened to Plato:
Pythagoras was the first person to use the word philosopher. By tradition, he discovered the Pythagorean theorem (no surprises) and the Platonic solids (foreshadowing). He (or his school) also discovered Pythagorean tuning, which led to their theory of the “harmony of the spheres.” All of this led them to believe that the world existed in perfect mathematical order, that they’d uncovered God.
When we moderns say something like “humans aren’t mathematical,” we mean to say, “humans have dignity beyond mere material, we aren’t bland deterministic objects, human feelings can’t be calculated so easily.” True or not, the point is that modern people view mathematical explanations as something lower, undignified, the work of stuff and things. It was the opposite for the Pythagoreans. Humans are fickle, chaotic, unpredictable. Euclidean geometry is divine. It’s consistent, predictable, and, critically, eternal. No one law will change, every circle is a perfect circle, every relationship holds true for forever. “Humans are mathematical” is precisely the opposite of the world of things – it’s an elevation into the realm of the eternal, into that spot wherein there is no change. Into truth.
It’s genuinely unsurprising that Pythagoras formed what may have been a cult (or may have been an early attempt at an academy or something entirely other. No one is sure.). Either way, its metaphysics was all geometric, based on the perfect order of the eternal truths. Notably, the Pythagoreans believed in the reincarnation of the soul. I’m relatively sure this came from the necessity of an eternal aspect of humans to explain how we can interact with eternal metaphysical things. See also: the Platonic myth of knowledge by reincarnation.
Pythagoras is the closest to what Platonism is normally interpreted as, and that’s not an accident.
Plato’s characters are normally real people, and the real Timaeus was a member of the Pythagorean school. This makes sense. If read literally, it comes off more like a Pythagorean Genesis than a work of philosophy.
There are a thousand reasons to doubt a literal reading of the Timaeus, not least of which is the fact that it conflicts with most other dialogues. This was, unfortunately, impossible to know until recently.
Stoicism was the most popular philosophical school in the Roman Empire, and at some point in there the Stoics had incorporated Timaeus into their physics and metaphysics. Whether or not this makes sense is irrelevant, the point is simply that it made Timaeus the most popular Platonic dialogue. Since the Stoics claimed to come from Socrates, and Plato claimed to come from Socrates, one imagines them branding it as the secret teachings of the great master.
The Neoplatonists come in around ~200 AD. They claimed to be the true followers of Plato, and argued that Stoic ethics missed the real heart of Platonism, which was essentially a mystical program to connect with the One. They more or less took Platonic myths (and forms) at their word, were heavily influenced by the Timaeus (which, again, they took to be the clearest explication of Platonic doctrine), incorporated Neopythagoreanism (itself undergoing a revival) and became the standard-bearers of “Platonic Philosophy.”
Flash to the end of the Roman Empire, and the only accounts of Platonic philosophy until the 13th century are: a) Neoplatonic treatises, b) Timaeus.
The end result of this is that the only thing anyone knew about Plato for a really long time is, actually, Pythagoras. Thus, most attacks on “Plato” had very little to do with what Plato thought, but a whole lot to do with Platonism.
It should be noted that “Platonism” in this way also gave birth to: gnosticism, hermeticism, sacred geometry, (some kinds of) numerology, most esoteric socieites, and, from all of that, Philip K. Dick.
19 thoughts on “Platonism without Plato”
Feels relevant to give a contemporary example.
Max Tegmark’s Mathematical Universe Hypothesis is a kind of modern platonism (or mathematical realism?) that solves the “two worlds, why exactly two?” problem by flattening to just one big thing containing every possible mathematical object, with our physical universe just being one of these. But I don’t think it allows for any interaction between objects and is only meant to answer the anthropic question “why these physical laws?”: “because every possible set of physical laws!”
The fable is about foundations work (set theory/category theory/HoTT/etc.) from the perspective of the working mathematician: “As long as things keep working at a high enough level that I can use logic and numbers and stuff, I don’t care about the underlying machinery.” Thanks to the excellent responses I got last time, I can see that we’re not taking that approach (, duh). Gotta get into the nitty-gritty!
I have a clearer view on the importance and difficulty of the math question now, but I can’t help but think any resolution will either be some kind of logical trick (“math works because work is math!”), sneak in some further assumptions (like I did before), or be absurdly convoluted (like Russell’s classic proof of 1+1=2 in principia mathematica). It won’t detract much from the series if not, but I ask: Will there be a hardworking solution (or dissolution) of the “why does math work” question eventually?
Tegmark is a great example. He really is the guy most willing to take the Platonic view as far as it can go, to the point where even sympathetic observers are a little dismayed. I respect that quite a lot.
Thanks, and I’m glad it’s coming more into place.
Yes, actually. Or at least I think so. There’s, of course, some sense in which you can never really get the complete answer (probably for the same reasons that you can never really disprove solipsism, etc.), but Kant sets up a framework that is very satisfying and directly attacks the problem. Very few tricks at all, which is shocking, although the book is, uh. Well, it’s not convoluted (depending on who you ask) but definitely complex. It’s going to be interesting to try and write it all out. Of course, nothing is free, and some pretty weird consequences fall out of his system, too. They lead to the most interesting parts of modern philosophy, though, so it isn’t bad.
Also: he’s a little incomplete, and there are modifications. All in all, though, it’s an ingenious solution that retains science and mathematical efficacy but doesn’t require metaphysical epicycles.
I’m pretty sympathetic towards the idea that we live in a Tegmark 4 “Big World”, where physical objects are (very complicated) mathematical objects. Did you argue against this somewhere in your post? Or maybe this just isn’t very satisfying or something, and doesn’t answer the part about “how do we know it?”
Damn, pretty hard to grasp my mind around it, so the true nature of math is impossible to know without invoking some kind of philosophical problem or contradiction? So, what is the point to search for an answer, if you will never know the truth? Is kind of pointless… or I’m just a brainlet, still, why math works is an interesting question.
That was totally my own fault. When I published this I forgot to add a header noting it was part of a series. This is the first of a vague historical overview of certain schools before I go into the answers I think are genuinely satisfying. Admittedly, they’re just… well, they come later, and they have a few other weird results. They’re preferable consequences, though.
Don’t call yourself a brainlet. It’s a good question and a reasonable assumption.
Probably a worthwhile consideration for your second question: is it possible that the act of thinking about certain things, even if no absolute conclusion is reached, can open up wisdom about those things, about yourself, and about knowledge itself? For comparison, does contemplating the motivations of another person, even if you will never know exactly what they are thinking, grant you insight into them, into yourself, and into other people in general? If so, then can something similar apply to mathematics?
I mean, I’m making my position pretty clear, which is why I’m following this series.
From an editor’s note in my copy of the Theaetetus:
Greek mathematicians did not recognize irrational numbers but treated of irrational quantities as geometric entities; in this instance, lines identified by the areas of the squares that can be constructed on them.
This is in reference to Theaetetus telling Socrates what he’d just been taught in math, which is pretty difficult to parse on first read-through. The Greeks tackled the problem of squares and square roots from a somewhat different angle than what we have, and so our own mathematical expertise struggles a little to find a footing. This is comparable to how difficult it is to translate complex literary terms: do we really have a good translation for “hubris?” Could we really explain to a Greek the intensity of our cultural construct of sin? And yet, with mathematics, we have good reason to believe that in talking seriously with the Greeks, we would inevitably come to agree that we were just using different words for the same thing. That’s pretty hardcore. It also, once you’ve gotten good enough at literary analysis (the one true craft), shows Kant’s move in it: what is held as constant in this equation? Man, if only he’d been a film critic.
Nothing really new here, just taking pleasure in restating the old.
LikeLiked by 1 person
Shit, man, don’t give away the punchline. The next essay opens with a discussion of Theaetetus and the Greek view of irrational lines (literally: alogos) as a way of getting at why we shouldn’t trust the apparent Pythagoreanism of Plato’s myths.
Theaetetus, interestingly, was most famous for his work with irrationals. I can’t recall which ones, but a surprising number of the props in Elements X were attributed to him. Note also: he’s described as ugly and snub-nosed in precisely the same way as Socrates is.
My biggest question with this series is where or not to discuss the Third Critique.
Hey, I’m sorry! Merlin fucked with my RSS feed, and now I read things all out of order.
That’s an interesting angle, though. My instincts would have been to lean hard on Plato’s overall weirdness, such as in the Parmenides, as a way to show that the dialogues aren’t even close to being treatises. Instead, to go into the alogical accounts… well, that’ll be fun. Plato’s arguments in general are a fun inversion of mathematics, though: rather than starting from certain axioms to prove things, he starts from uncertain axioms and disproves them – and everyone knows that a contradiction proves absolutely anything.
The Critique of Judgment, yeah? That would be swell. The more Kant, the better (except maybe the Groundwork – there’s still the mistake there of treating moral beings as empirical objects rather than things in themselves, which results in mathematical moral laws, when fact the moral laws should be transcendental i.e. aesthetic).
LikeLiked by 1 person
I recently wrote something in a similar vein – or perhaps I should say parallel rather than similar, since I take the problems with Platonism as givens rather than establishing them, and focus on “how do we think about numbers?” rather than “how is mathematical knowledge reliable?”
Part one is here.
I’m not a huge fan of the indispensability argument. In its bare bones, it goes “numbers are indispensable for making predictions about the world, therefore they’re Real,” which is so bare-bones it’s a complete non sequitur. But when you start filling it in to make all the things sequitur properly, it starts looking awful tautological. See, when you try to cash out “Real” or “existing abstract object” or “ontological commitment,” to figure out why such a thing should be implied by indispensability, you find yourself circling back, and it seems like the entire function of abstract objects is to be the things it is proper to use in human reasoning!
Of course, the platonists don’t see it that way, and would say that something being Real has important implications. But then I start making fun of them, because if the implications are so important, why not just tell which things are Real by those implications, rather than having to do this roundabout indispensability business? And now we’re back to the material of my blog post – why do we feel like some things exist?
LikeLiked by 1 person
So, you dismiss the evolutionary argument because it seems to lead to a God-like teleological principle — but how do we know there is no such principle? I mean, looking back at our best guess at the last 15 billion years of
primordial chaos –> planets –> bacteria –> multi-cellular life –> humans –> civilization
doesn’t it seem like there is some kinda teleological increasing-complexity thing going on? One doesn’t need to make any claims about the origin or exact nature of this phenomenon to observe it. You seem to deny this simply because it is similar to what Christians believe, therefore absurd. Which is not much of an argument.
I dismiss a naive evolutionary argument in the form above (“the environment is mathematical; we evolved by selection according to Euclid”) – not an argument using evolution. Also, I point out that it is a coherent argument, one just has to admit of Platonism and a whole bunch of other stuff. The point of that isn’t that therefore it’s wrong, but that most people who make the naive evolutionary argument tend to make it precisely to avoid any metaphysics. “What’s this dumb philosophy stuff? We have science now, duh, it’s Darwin,” and then their own system admits of very intense metaphysical claims.
“The descriptions of the universe that best match the phenomena are all based on the physical sciences, and those descriptions are the things we ought to afford existence, i.e. electrons exist in a way that phlogiston and the Four Humors do not. It would be strange for any necessary predicate of a theory to not “exist” in the same way, that is to say, it would be odd for an electron to exist but a neutron not to, or to be merely as reliable as phlogiston. If a scientific theory is a “real” description of the physical world, then things which are indispensable to that theory should also be assumed to be real, for the obvious reason that, without them, we would not have the theory, they’re part of it, it would be like allowing for cars but rejecting combustion.”
No. One of the key differences between philosophy and science is that philosophy is usually realist, but science must be nominalist. You can do good scientific work without understanding that, but proper scientific epistemology requires nominalism. If you understand nominalism, you understand the above is nonsense.
Take phlogiston. Phlogiston isn’t wrong. It’s a concept hypothesized to explain the numerical results of a large set of chemical experiments. It works out to mean, precisely, “negative oxygen”. Every place in a reaction equation where we would write “N molecules of oxygen” on one side, phlogiston theory instead writes “N molecules of phlogiston” on the other side.
It explains a lot of numeric results correctly. It ran into trouble because it required phlogiston to have negative mass, and because it implied things should be able to burn in a vacuum, and for other reasons involving experiments that measured things other than the object burnt. But–and this is very important–it is strictly inferior to oxygen theory, but it is not “wrong”.
By the same token, oxygen theory doesn’t require oxygen to “exist” any more than phlogiston theory required phlogiston to “exist” to get many predictions right.
P.S.–If you do science without a nominalist epistemology, you end up saying silly things like “Einstein proved Newtonian mechanics were false.” Then you angst about whether any theory can ever be proven “true”. Assuming realism thus brings you back into treating theories as if they belonged to the Platonist world of mathematical constructs, and of True and False.
The key words are “best” and “ought to afford existence.” The argument is saying that “being” or “existence” is a confused term, but if we have to use it, then we should use it to refer to our best current theories. In other words, it explicitly assumes a nominalist stance. Devil’s in the details, be careful about the construction of arguments before you respond with such confidence.
Anyway, and related, the bigger issue with your comment is that you interpreted that as my argument and thus this essay as an argument for Platonism. Or, at least, that’s how I’m interpreting the unqualified “No.” considering that’s the normal use. If you meant to qualify it with something like “Quine’s argument is also wrong for [reasons]” then say that. It was unbelievably clear that this was an essay against Platonism, both mathematical and in terms of abstract concepts, and Quine was invoked as the-best-I-still-oppose.
I honestly don’t understand how one can make that mistake and still comment with such self-assurance, but, like. Fine.
I know you’re doing “Platonism without Plato”, but look, he didn’t just found the entire universe in the harmony of math, he explicitly founded it all on the Form of the Good–and if that idea can resolve the problems you’re considering, it deserves to be treated with.
And it seems like it can deal with a bunch of those problems! Take (1)-(4) above, your pet anti-Platonisms. We bite (2)’s bullet by bringing in the FotG; weird metaphysics, yes, but metaphysics is weird however you slice it. And once we’ve done that, (4) is easy to answer: the FotG is exactly the kind of thing that “presupposes-itself-for-its-own-creation.” Metaphysical bedrock. The remaining two are tougher, I guess, except not really, because it’s right there in the idea of the FotG that it’s metaphysically (1) and epistemologically (3) prior to everything that exists, justifying and creating and publicizing it all.
It solves other problems, too. Take evolution–jocks beat up nerds, and yet in the long term, nerds win. Evolution seems like it should most often be a race to the bottom, and yet we find ourselves at the top. (cf. Moloch.) You say “fuck this up and you’ve reproduced the Catholic Church’s favored view of evolutionary complexity.” Evolutionary complexity is still confusing as balls, last I checked; are you so all-in on Dawkins that this is a sufficient reductio? Why can’t we just say “yup, sounds great to me”?
Further on, you make a long argument against the sufficiency of AP to solve metaphysics; in order to be sufficient, it would have to imply/constitute a “teleological principle indistuingishable from the hand of God”/”guiding principle of salvation.” (i.e. foundational metaphysical goodness.) Why is such a principle a bad idea? There are all these questions you’ve raised–good ones, great ones, classic ones–and I’m missing why FotG/theism shouldn’t be the answer.
So… ctrl-F “Form of the Good” = 0; why? As someone strongly sympathetic to metaphysical Platonism, that seems like the weak point here. It seems like anyone who believes in a metaphysically foundational Goodness (i.e. Stoic God, or Christian God, or Enlightenment Providence, or whatever) can just endorse the conclusions reached above. Why modus tollens, and not modus ponens? Did I miss that part of your argument, or is (pseudo)theism deemed too outmoded to even engage with, or is a response still in the works?
Forgive me for replying so late. I hope I can get a response nonetheless.
LikeLiked by 1 person
This is a good comment. Thank you.
I should probably admit a certain trollishness up front – some of this essay was directed at unintentional Platonists, rather than commited mathematical realists, and that category tends to include a whole lot of naive materialists (hence the Krauss bit). I’m weak, and I couldn’t resist the temptation to twist the knife with some theism against committed atheists. As I said, though, I don’t think that’s an argument against Platonism at all – it only is if you’re Dawkins and, you know, let your desired ends determine the argument.
I kind of regret doing that, because the people I meant to twist the knife into didn’t get it (I’ve seen more than one “Well, the Ancients were just dumb” comment that appears to have ignore 90% of the essay), and it comes across a little bit too snarky to people whom I don’t mean to be snarky to.
I’m writing a piece right now about Plato himself. It’s, uh, hard.. I’m not super convinced that Plato is himself a Platonist (I don’t really think he is, actually, but you can get that from this piece), but I don’t have any problem with the Form of the Good. Or, at least, I’m not going to reject it out of hand. As you succinctly put it, all metaphysics is super weird. Since it isn’t possible to reject metaphysics out of hand, which is to say, there’s no non-metaphysical world, I’m fine with accepting most things so long as there’s a good argument. For whatever it’s worth, the only view of math I strongly reject is strict empiricism (I don’t even know how that works, but some people insist it does).
The one thing I do think is important, and which I’m glad you bring up, is the necessity of something like a FotG with mathematical realism. It can’t be rejected out of hand, and there are some good cases for its incorporation. “The only metaphysics is math, because I say so,” is bizarre and leads to serious inconsistencies. At the very least, I have no idea why they should stop there, and I’ve never gotten a good argument for it. It’s like allowing for neutrons but denying electrons, “That’s going a little far, now.” Accordingly, you’ll note that almost all my arguments here say: “It’s fine to accept this, but you need to allow for other possibilities.” Putnam, at least, was pretty consistent here. He forwards the indispensability argument, and he was a practicing Jew who wrote pretty extensively on religion and philosophy.
All of this is to say: I do take Platonism seriously. I take theism seriously too – not just the deist version, but the personal version. I’m not myself a believer, and my epistemological preferences are Kantian (or, well, modified forms of Kant), but that doesn’t mean much for the argument itself. I’ll be writing quite a bit more about math and Plato. Trying to argue about it at the moment might be difficult (I suspect that the theist/goodness issue might take more space than a comment section allows for), but I do take your objections seriously.
Again, thanks. Hope the Plato piece/rest of the math series clears some of this up. If it does not (it probably won’t), please let me know.
Yeah, looking back, that clarifies the argument a lot. And looking back, my comment was too aggressive; committed metaphysical Platonism (much less actual non-deist theism) isn’t really the norm in philosophy, so your audience is largely going to be people who need to hear exactly the points you’re making. So of course you’re not going to spend the most time considering thoroughgoing Platonism. And your argument is an excellent one, probably should convince a lot of people (assuming they parse it right, which apparently maybe not).
So, I’m sorry for coming in hot. I’m a committed Platonist, serious Christian, and current resident of the ivory tower. So I’m very used to hearing my beliefs condescended to, and I have to pick my battles, and just smile and laugh along the rest of the time. And I was upset to see that kind of condescension reproduced here, by someone whose thought I respect—although now I realize it was less reproduced than alluded to, since you’re writing to persuade exactly the kind of person who sneers at Forms while implicitly using them.
Mea culpa for being aggressive; I enjoyed the post quite a bit; and thank you for your response. Looking forward to seeing more!
LikeLiked by 1 person