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Mathematics is metaphysics. I'd like your refutation to center on any nuance which might separate the two disciplines other than the sad fact that many philosophers don't learn much maths.

Logic is metaphysics: https://philpapers.org/archive/ALVLIM-3.pdf

Logic is mathematical, logic is categorical (structured in a deep way amenable to category theory), maths is categorical: http://math.ucr.edu/home/baez/rosetta.pdf https://ncatlab.org/nlab/show/internal+logic https://ncatlab.org/nlab/show/topos

Ontology is categorical: https://arxiv.org/abs/1102.1889v2 https://arxiv.org/abs/1706.00526

The working programmer is an ontologist and a taxonomist, but this fact isn't well-understood.

I'd like to draw a distinction between non-physical and metaphysical. The opposite of a thing is not the same thing as the meta of that same thing. The meta attempts to unify the topic and it's reverse. Metaphysics wants to kill the distinction between physical reality and non-physical reality. Non-physical disciplines such as mathematics wants to distill the non-physical essence of things

This is a really nifty insight. Mathematical objects aren't physical, but that doesn't mean that there aren't objects that are neither physical nor mathematical. I wonder what those objects might be like!

This kind of response really grinds my gears about these discussions of philosophy (more often than not related to metaphysics). You're referencing the mostly unobjectionable features of the field to defend the objectionable features, when there is very little similarity between the two. You can't defend ontology language games by appealing to the truth of logic or mathematics, as they share nothing of relevance in common. If you want to give an actual defense of the things people find objectionable about metaphysics, you have to defend the features being attacked, not tangentially related things that happen to be categorized under the same broad label.

>Logic is metaphysics:

Maybe this is true, but doing logic isn't doing metaphysics. One does not need to take a stance on the metaphysics of logic to study and create formal systems.

>The working programmer is an ontologist and a taxonomist, but this fact isn't well-understood.

Not in any substantive sense. There is a surface-level connection between what a programmer does and a metaphysician does, but this is the extent of the connection. The ontologies studied in metaphysics attempt to make true substantive statements about what exists in the actual world. The programmer is merely stipulating the basic objects and relationships in a fully contingent domain of inquiry. The programmer is doing metaphysics only if you remove anything of substance from what we take the metaphysician to be doing.

And this kind of response really grinds my gears; I am relatively certain, given the time scales involved and the nature of your response, that you didn't read my links, and you haven't provided any of your own. Nonetheless!

Doing logic is doing metaphysics. Suppose 1+1=2 for addable numbers, that you have oranges and can add more oranges, and also that the number of oranges you have is an addable number. Now you know, as a matter of plain old philosophical handwavey logic, that if you have 1 orange, and you add 1 more orange, then you'll have 2 oranges. Easy, right? This generalizes to any topos and it's known as "internal logic". The slogan we have is, "a topos is a place for doing logic".

Doing metaphysics is doing logic. Want to know what's impossible? If it can be characterized purely by mathematical structures, and those structures' existence leads to contradiction, then it's impossible. Thus, "models" in the first sense of the original article, free unconstrained unicorn metaphysics models, are actually logically (and thus mathematically) constrained by "models" in the second sense, in the mathematical sense. A powerful example of this is M-theory, borne from string theory; right now, the "swampland" cleanup is sweeping through string theory and helping refine our sense of which particle physics are possible.

An ontologist in today's postmodern world surely knows that the (inherently mathematical!) structures that they are building and studying are created, not discovered; subjective, not objective; narratives, not truths. Mathematicians know this formally, via Tarski's Undefinability. Just like an ontologist tries to find models, a programmer tries to find models, searching for the database schemata and the class hierarchy that will match their problem.

(Indeed, what is "anything of substance"? Is it physical?)

>I am relatively certain, given the time scales involved and the nature of your response, that you didn't read my links, and you haven't provided any of your own

The content of the links didn't seem relevant to the objection I offered. Perhaps that's incorrect, but you didn't make the case.

>Doing logic is doing metaphysics. Suppose 1+1=2 for addable numbers... that if you have 1 orange, and you add 1 more orange, then you'll have 2 oranges.

I could object to doing math being an instance of metaphysics, but setting that aside, the issue is that the connection between logic and metaphysics is the "mostly unobjectionable" subset of metaphysics that I mentioned previously. So again, you can't substantiate the objectionable parts with arguments that only apply to the unobjectionable parts.

>An ontologist in today's postmodern world surely knows that the (inherently mathematical!) structures that they are building and studying are created, not discovered; subjective, not objective; narratives, not truths.

Here you seem to be admitting that metaphysics is unsubstantive! Metaphysics purports to discover what exists, not subjective notions of preferred taxonomies. But it is precisely the charge that nothing objective is being discovered that underpins the argument against metaphysics.

Now I understand why you see a close connection between what a programmer does and what the metaphysician does: they're both theorizing about contingent domains of inquiry, with no external objective facts at stake. But now I don't get why you object to saying metaphysics is unsubstantive.

It wouldn't change much even if there were 10 metaphysics papers for every mathematics paper. The main difference is that math is rigorous: there is an agreed upon standard for correctness that is falsifiable in two ways. Firstly, a proof of a theorem can be falsified by finding a mistake, and what is a mistake is agreed upon. Second, and more importantly, the system of proof itself is falsifiable. If a theorem says that an equation p(x,y,z) = 0 has no solutions, and a proof without mistakes is found, then it better have no solutions. If somebody comes along and demonstrates that x=43,y=23,z=809 is a solution, then the system is in trouble. This ensures that math is not a formal game of arbitrary axioms that is internally consistent but ultimately meaningless. That is important, because metaphysics could develop internal correctness criteria that everybody agrees upon, but it could still be meaningless.

> Second, and more importantly, the system of proof itself is falsifiable. If a theorem says that an equation p(x,y,z) = 0 has no solutions, and a proof without mistakes is found, then it better have no solutions. If somebody comes along and demonstrates that x=43,y=23,z=809 is a solution, then the system is in trouble. This ensures that math is not a formal game of arbitrary axioms that is internally consistent but ultimately meaningless.

I don't see how what you've said is anything other than the criterion of formal consistency - that a thing and its negation are not both provable. And a system may certainly be consistent, yet not particularly useful.

It should be noted that mathematics has not always been as rigorous as people assume. Attempts to ground math in formal logic didn't really get off the ground until the late 1800s. And this isn't a trivial point; the "rules of the game" were up for debate, and often debated, before then (and, to a lesser extent, since). People disagreed about which numbers exist; people disagree about which proof methods are acceptable; and for example, calculus - a highly practical branch of mathematics - was invented before its formal justifications were found.

It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers. You just need the numbers a,b,c,n of the counterexample and a calculator. Mathematics thus makes predictions about the world (in this case about the behaviour of calculators, human or electronic) in a somewhat similar way that physics makes predictions about the world. Or as Arnold famously said: "Mathematics is the part of physics where experiments are cheap". You wouldn't have that if math was an arbitrary symbol pushing game with arbitrary axioms, with the only requirement that they be consistent.

[If a counterexample to Fermat's last theorem is found it probably isn't because ZFC is inconsistent, but because Wiles' proof has a mistake, which would show that the checking of the proof was inadequate and that mathematics needs to use a higher standard for what is considered a valid proof. So the system of proof is falsifiable in a very broad sense: it can expose a bad proof checking culture within mathematics as well as inconsistencies in the formal system. I don't think that metaphysics has anything equivalent. If somebody comes up with a counterexample a,b,c,n then Wiles would immediately admit that his proof has a mistake, even though he wouldn't even know where in the proof the mistake is. Try convincing a metaphysicist that their argument is wrong or meaningless... That said, there are certainly also stubborn mathematicians who insist that their proof is right even though other mathematicians have pointed out that a particular step isn't clear. However, if the mathematician who came up with the proof cannot clarify that step, ultimately down to the axioms of ZFC, then the proof isn't accepted.]

> It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers.

This follows from consistency (and is one reason why consistency is important). If there were some whole number solution to a^n+b^n=c^n for n>2, then this would be provable in Peano Arithmetic (a much simpler system than ZFC, that axiomatizes natural number arithmetic). ZFC extends PA (every theorem of PA is a theorem of ZFC, interpreted correctly; this is relatively easy to show); thus, anything ZFC proves is not falsifiable in PA, otherwise we'd have a contradiction provable in ZFC (& it wouldn't be consistent).

It doesn't follow from consistency in itself. It is possible to imagine a formal system that is consistent but nevertheless doesn't predict the behaviour of calculators. If you presume that ZFC correctly models the behaviour of calculators, or that PA models calculators and inconsistency of PA implies inconsistency of ZFC, then sure it follows from consistency of ZFC, but that was the point, namely that there is a difference between a purely formal system that only exposes itself to inconsistency tests, and a formal system that exposes itself to external tests. Numbers aren't the only way that math does that. Geometry makes predictions about the behaviour of rulers and compasses, for instance.

Contrast this with a discipline in which there is jargon and arguments about that jargon. It might be the case that the jargon and the basic ways of arguing about the jargon forms a consistent system in some sense, even though the whole enterprise is meaningless.

The general idea you're missing here is that it's very hard -- perhaps impossible -- to develop a clear, bright-line method for distinguishing between fields that deal with concepts we can't empirically test, in such a way that it will put mathematics into the "good, useful, keep doing it" side and metaphysics into the "bad, silly, stop doing it" side.

And in fact mathematics is a great place to dive in, because one very important question -- namely, whether mathematical realism is true¹ -- is really a metaphysical question, and one which can never be answered by empirical observations about the world.

I'd also caution against too much scorn on philosophers going "beyond the reaches of their knowledge base"; historically, philosophy's successes have spun off into entire fields whose origins we now like to forget or at least overlook.

(and, well, I've had a successful career in programming, and also hold a degree in philosophy, and I think the basic observation that lots of people do metaphysics who we don't normally think of as doing metaphysics is a sound one, regardless of how you feel about the paper's specific examples)

---

¹"Mathematical realism" asserts that truths of mathematics are necessary and discovered, rather than contingent and created, and thus have some sort of existence independently of human minds and human culture. The exact sense in which these mathematical truth-entities would exist is, of course, a metaphysical question, as is the question of what mathematics really is, if mathematical realism is false.

>And in fact mathematics is a great place to dive in, because one very important question -- namely, whether mathematical realism is true¹ -- is really a metaphysical question, and one which can never be answered by empirical observations about the world.

The important distinction between math and metaphysics is that the substantive content of the field of mathematics does not turn on whether mathematical realism is true. Mathematics tells us something about the world by telling us what can be true about the world. We know that there are prime numbers because we know that no such world exists where prime numbers are finite. But this is just saying that a system with certain axioms have certain relationships, or necessarily exclude certain relationships (e.g. whatever axioms of arithmetic you take).

On the other hand, there is no substantive content in metaphysics (at least as far as ontology goes) if the statements do not pick out real entities.

On the other hand, there is no substantive content in metaphysics (at least as far as ontology goes) if the statements do not pick out real entities.

You've just committed the fallacy of begging the question -- you assumed the truth of mathematical realism in this line, and arguably in a couple others.

If mathematical realism is false, various parts of your argument start to crumble.

I fail to see how the quoted portion of my comment assumes mathematical realism. I could see how other parts of my comment might suggest I'm assuming MR. Assuming you meant to reference the portions where I talk about how the results of mathematics tells us something about the world, modal structural realism[1] is an underappreciated interpretation of mathematics.

[1] https://plato.stanford.edu/entries/nominalism-mathematics/#M...

Isn't the equivalent metaphysics all those moments we have a theory of how to examine the world, so as to come to an accurate conclusion?

Metaphysics is required to make an inference -- it's the philosophical underpinning of things like science. Hence "meta-" "physics": the study of the study of physics.

Metaphysics is the glue that takes the language of mathematics to ontologies of physics. Mathematics is the modern rigorous language of metaphysics.

Assorted marketing campaigns have made the topic murky.

> Yeah, that's not metaphysics; it's pretty much just regular taxonomy (and, by the way, the term of art is labeling).

That literally is metaphysics. It's okay there's another word (or two?), but words are not exclusive in the way your logic implies.

> That literally is metaphysics.

The claim that toddlers figuring out what shapes fit in what wooden holes are doing metaphysics is seriously off-base. Metaphysics deals with first causes, ontology, modality, free will, and so on. Let's not mince words. Taxonomy is not metaphysics.

> No, they do.

I think the burden of proof is on you. You're making fantastical claims but I don't see any connection between a TensorFlow model and any metaphysical model whatsoever.

> This is why people get frustrated: you don't seem to understand the link, squawk about how it doesn't exist, and then insist philosophy must be useless because you never took the time to understand where you do it naturally.

I studied philosophy.

> what shapes fit in what wooden holes

> ontology

We must be talking past each other, or something.

For context, for other readers --

> Traditionally listed as the core of metaphysics, ontology often deals with questions concerning what entities exist or may be said to exist and how such entities may be grouped, related within a hierarchy, and subdivided according to similarities and differences.

https://en.wikipedia.org/wiki/Metaphysics#Ontology_(Being)

> We must be talking past each other, or something.

I guess we are. According to you, when my cousins go to Six Flags, they're doing fluid mechanics. Forgive me if I don't buy that.

It's literally ontology -- they're discovering the idea of shape as a concept, which is an equivalence of structural forms, which is an ontological concept.

I mean, I get that you don't like that because it proves my point, and that's why you're making really silly comparisons.

But to answer -- yes, most children at water parks also are figuring out fluid mechanics at a simple level, eg drag or interference.

> It's literally ontology -- they're discovering the idea of shape as a concept, which is an equivalence of structural forms, which is an ontological concept.

I think there's an important difference between the implicit, intuitive, relational knowledge that we build via direct interaction, and philosophical knowledge that the other poster is thinking talking about (and presumably, what ontology is about). You're equating the two without arguing why we should consider them the same. At best, our implicit model of the world knowledge informs our intuitions on some philosophical concepts, but that still doesn't entail this knowledge is the same as philosophical knowledge.

> I think there's an important difference between the implicit, intuitive, relational knowledge that we build via direct interaction, and philosophical knowledge that the other poster is thinking talking about

In the same sense that there's a distinction between counting and mathematics, or throwing things and physics, or conversations and psychology, or various other academic disciplines -- that is to say, a very artificial one.

In every topic, people think about them as a routine course of events -- they just form very simple effective theories on the subject, because their life doesn't require a more advanced one.

"Physics" the academic subject is just a very advanced version of the intuition you build about throwing balls, smashing rocks, walking on ice, and not jumping out of windows.

Similarly, your argument seems deeply confused:

The burden of proof is on the people asserting that an arbitrary distinction in what's clearly a spectrum of things is meaningful. In this case, it falls to you to show that there's a meaningful difference in kind between the sorts of thoughts a child is having about ontology (eg, that a positive version of a shape and a negative version of a shape are the same "shape") and whatever other example you want to use.

Each of you has clearly admitted that they're on the spectrum of ontological thought, because that's why you recognize it so clearly as an example, then just had special pleading of "Well, simple ontology shouldn't really count as ontology, because that would make me wrong."

So again, we see that philosophy is useful -- the "burden of proof" is a philosophical concept that lets me just reply, "No, how about you show that splitting ontological thought into 'true ontology' and 'not really ontology' is useful in the first place."

That playing with blocks is a precursor to theorizing about what exists does not justify a field that purports to make true statements about what exists. Similarly, throwing a football around does not justify a field that purports to describe the behavior of entities in the world. These practices need external justification. The point is that the substantive content of the field of metaphysics has no such justification.

Uh... but "shapes" existing is literally ontology. It's discussing the existence of non-physical entities in the world, and children first come into contact with that by matching the positive version of the "shape" (block) with the negative version of the "shape" (hole), abstracting the "shape" from its literal manifestation.

Discovering shapes is implicit ontology in the way throwing a ball is implicit physics.

Categorizing lumps of matter as different shapes is not theorizing whether shapes exist.

> In the same sense that there's a distinction between counting and mathematics, or throwing things and physics, or conversations and psychology, or various other academic disciplines -- that is to say, a very artificial one.

Except there's a very clear delineation in each case: proper knowledge is articulable, communicable and arguably formal, and in every case of proper knowledge, one can ask "why is that true?", and find a justification that is clear and unambiguous. I can communicate to you a formal understanding of a phenomenon that you had never before witnessed, interacted with or understood, and you can thereby gain a fairly deep understanding of it via this model and make predictions of it without ever witnessing, interacting with or experiencing it yourself.

This is simply not the case with implicit, intuitive "knowledge". The very process of elaborating, refining, and formalizing implicit understanding is what creates proper knowledge.

Implicit, intuitive "knowledge" is to proper knowledge, as seeds are to fruits when evaluating the question of what food can keep you alive. Seeds are by and large not digestible, not nutritious and won't sustain you, but a fruit will. Certainly there is a continuum between the seed to the fully formed fruit, but achieving the status of "fruit" that can sustain your life is definitely not arbitrary.

As I understand it, that is not the etymology of metaphysics.

In fact metaphysics means the subject matter of the books Aristotle wrote after (meta) he wrote The Physics.

https://www.etymonline.com/word/metaphysics https://en.wiktionary.org/wiki/metaphysics#Etymology

> Just consider volume: it's estimated that around a quarter of a million theorems are proved every year. That's real, measurable, progress; what's the metaphysics equivalent?

I'm not sure I disagree with your overall argument, but this point here is silly. First, I'd like to know where you got that number. (I must admit it sounds rather high to me, but I'm sure it depends how you count.)

But more importantly, measuring mathematical progress by number of theorems proved is absurd. I can write a computer program to prove theorems; it will almost surely prove more than a quarter-million a day (though eventually it will have to slow down). But they will not be _useful_ theorems. Nobody will read them, nobody will use them, nobody will build on them, nobody will find them beautiful, nobody will apply them to any practical problem. "Number of theorems proven" may be measurable, but it is not "real progress".

It boils down to the fact that Pure Mathematics has a well-tested (if sometimes rigid) methodology. Metaphysics may lack an adequate methodology. The role of methodologies in general is to determine what's true and what's false, and what's outright nonsense. Pure Mathematics may well be defined by its methodology.

I think that's close to hitting it on the head. The author makes it sound like metaphysics is distrusted because there is no clear way for it to put bread on the table, while in reality it is distrusted because there is no way to tell if it is true.