If the platonic realists are right, the name of that junkyard is the Platonic realm of forms. — fdrake

Right, but I'm asking if there is a human junkyard of abandoned math, whether constructed or discovered. Because the argument turns on most of math being a junkyard. I'm asking whether this is a hypothetical, or actually historical.

Well no. There's nothing like what there would be if all the mathematical forms instantiated in the same way. Which really raises the question - when we recognise an instantiation of a mathematical object, with our platonist goggles on, are we seeing the world conforming to mathematics or mathematics conforming to the world? Sometimes we will see mathematics conforming to mathematics, but that's of no interest here.

We already have to filter out the junk in the platonic realm to obtain something resembling the collection of mathematical objects we are familiar with; what if this filter was internal to mathematics as a field of study, and roughly demarcated its contours of relevance and topics of interest?

Under that presumption, the realm of mathematical entities sure does look a lot more like its current self.

There's nothing like what there would be if all the mathematical forms instantiated in the same way. — fdrake

So Rovelli's argument summarized in the OP is that Platonism would be full of useless math instead of just the math we're interested in.

But what is the argument justifying this claim? Are there examples of useless math of interest to no one? What makes the case that Platonism would lead to this? Because other creatures would develop maths we wouldn't care about? Is that actually true? I'm thinking human mathematicians would actually quite interested in how much farther than us the Jovians had developed their geometry, and I'm guessing Jovian mathematicians would be quite curious about arithmetic.

Which brings about a second question. Why is utility an important criterion for math? Certainly applied math is important for various fields, but mathematicians also are interested in math for it's own sake.

Trying to give examples of 'non-mathematical' math from the canon of mathematics is a poisoned well. But I did try to give some demonstratively weird objects in my response to litewave. There are also some suggestive examples. EG, every number we're ever going to see as a digit based representation is computable, but computable numbers are a measure 0 set in the real line which is their home. We have a rich mathematical theory about continuous functions, augmented it with continuously differentiable and smooth functions, regardless generic continuous functions are nowhere differentiable. Generic functions themselves are nowhere continuous. Ways of associating elements of sets with elements of other sets generically are not functions, either.

If we have that derp face from before as a constant symbol, it satisfies the axioms of a group on a single element. We could make the same generalisation with any addition of a black pixel somewhere in the plane to the image, and we have countably infinite isomorphic copies of the group on one element that are literally just derp faces. If you were to write all of them down and establish the isomorphisms between the different representations, that's more mathematics than will ever be written solely devoted to the stupid application of an algebraic structure to the derp face.

All of these are in the elemental plane of mathematics. But they too are relatively well behaved.

Given the tiny frequency of tame objects in the vast sea of batshit lunacy that are models of some collection of axioms, it would be very hard to claim that generic mathematical objects typically are interesting or resemble/are related to the ones whose study constitutes what could be topics of mathematical study. We don't just not care about them for reasons of utility, we don't care about them because we have a standard of intelligibility which automatically excludes them from our mathematical discourse.

Yes yes, the Platonist will insist, they're still there, even though they can't be instantiated in our mathematics. But that still concedes enough for them to hang themselves with; this is our mathematics, it's mathematics for us by us, it's not an independent realm of existing objects at all, it's a conceptual web whose contours are delimited by our standards of intelligibility. All of that is mathematics, and no more.

Edit: previously I had an example of adding different multiplicative 0's to our usual arithmetic in this post, which satisfy the property of 0 like a*x = x*a=a for all other numbers x, where a is now the extra 0, but if I took another 0, b, I would have that a=a*b=b through the definition. This was based around a vague recollection from university but I obviously didn't remember it properly and I can't figure out how to set it up exactly again. In lieu of that, if we have a less restrictive 'left 0' so that a*x=a for all x, I can add any number of leading extra 0's. If you are reading this and have already read my unedited post, sorry for the flub.

But how mathematics looks as a category theorist is quite a lot different from how it looks like under the aspect of set theory. Say, to a category theorist, natural numbers don't look like the names of individual objects, they look like isomorphism classes of sets. Set theory was built out of intuitions about composite objects of multiple elements, category theory was built from intuitions of transformation and symmetry. — fdrake

Yes. This is how I understand it: more general (more abstract) mathematical objects are instantiated in more specific mathematical objects (e.g. "geometric object" is instantiated in "triangle") and ultimately in concrete mathematical objects (e.g. in concrete triangles), which are not instantiated in anything else. (Those objects that can be instantiated in other objects are also called properties.) All concrete objects are concrete collections, that is, collections of concrete objects, so all mathematical objects are ultimately instantiated in concrete collections. This fact is used in set theory, where every mathematical object is represented as a collection (set) and that's why set theory can be a foundation of mathematics.

The collections referred to in set theory are not concrete collections though but abstract collections (generalized collections), because differences between concrete collections of the same kind are not relevant for mathematical purposes. So for example, set theory does not refer to concrete empty sets but to one abstract empty set (which is instantiated in all concrete empty sets).

The approach of category theory is not to represent mathematical objects as collections but to study similarities (morphisms) directly between mathematical objects themselves. Collections, then, are treated just as one of many kinds of mathematical objects.

I have no idea how you took the main thrust of my post to be about beauty or truth. The main thrust is simply that most mathematical objects aren't worthy of study, and agglomerating them all together; producing the final book and the final theorem, far from the ideal vision or ultimate goal of mathematics - produces a writhing mass of irrelevant chaos. — fdrake

Well, it depends on how you define mathematics. In the context of set theory, traditional mathematical topics can be extended to the study of all relational structures. This covers the whole relational aspect of reality and so is relevant for metaphysics/ontology.

Some categories aren't sets though.

But they can be instantiated in sets.

That's not how that works. There are more categories than the category of sets, Set. Only things in Set are categories which can 'be instantiated' in sets.

So you are saying that there are mathematical objects that cannot be represented as sets?

Yes.

I don't understand. Are there mathematical objects that cannot be expressed in set theory? I heard that set theory can express all mathematical objects - as sets. That's why it is considered a foundation of mathematics.

Except it isn't any more! There's more to mathematics than what can be represented through set theory.

What objects cannot be represented through set theory? And what does it have to do with your link to small categories?

Do some reading about it. Essentially small categories are those which have representations as sets and set relations/operations. Not all categories are small categories. So the link provides a starting point to start learning about under what conditions a category can be represented by a set!

Too technical for me, I am no mathematician.

Stanford Encyclopedia of Philosophy says:

"The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments. Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced in the Predicate Calculus from the axioms of set theory."

https://plato.stanford.edu/entries/set-theory/
(from year 2014)

Then it's wrong. You don't actually need to go to category theory for objects which don't exist in the ZFC universe, certain large cardinals need extra axioms to model.

If the platonic realists are right, the name of that junkyard is the Platonic realm of forms. — fdrake

Yes these kinds of objections to Platonism occurred quite early on in the form of questions such as "But is there a perfect form of the turd, or the pile of vomit?".

I should note though that by "set theory" I don't necessarily mean ZFC. As I clarified here, by set theory I mean all consistent versions of pure set theory. ZFC is just one version of pure set theory (it also appears to be consistent, though we may never know for sure because of Godel's second incompleteness theorem).

What all versions of pure set theory have in common is the concept of set as a collection of objects that can be defined by listing those objects (set members) or by specifying their common property. As it turned out, not every definition via a common property is consistent, and since it would not be very useful to define sets only via listing of members, one must also specify with axioms what common properties can be used to define sets. Which gives rise to uncountably many axiomatic versions of pure set theory.

I can't help but wonder how much of the motivation is driven by this:

The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather.

As Raphael Demos notes in Plato: Selections:

Whitehead, the distinguished English mathematician and philosopher, once said that "western philosophy is a series of footnotes to Plato." In so speaking of Plato, Whitehead was putting into memorable words the prevailing opinion of his professional colleagues. But times have changed and recent years [c 1927] have witnessed a powerful reaction against Plato; in the minds of a good many philosophers, reverence has been replaced by execration. Plato is now being seen as playing the role of the villain in the drama of philosophy, so much so that in some circles, merely to characterize a doctrine as Platonizing is to damn it.

which is a familiar sentiment around here.

As far as the essay itself is concerned, I think a lot rides on the qualification at the beginning of the paper:

Say we take a Platonic stance about math: in some appropriate sense, the mathematical world M exists. The expressions “to exist”, “to be real” and similar can have a variety of meanings and usages, and this is a big part of the issue, if not the main one. But for the sake of the present argument I do not need to define them—nor, for that matter, platonism—precisely.

The fact that the distinction between the expressions 'to be real' and 'to exist' are brushed off says a lot. Because the assumption that platonism really amounts to saying that a mathematical domain M exists begs the question as to the nature of its existence. Are numbers only real in the mind of the individual? The species? The culture? Or, as mathematical platonists argue, are they there to be discovered in the intelligible domain?

I think the significant thing about the Platonist view is that the nature of number is ontologically distinguishable from that of sensory objects; recall that this was the fundamental platonist distinction between doxa and pistis, on the one hand, and dianoia, on the other, given in the Analogy of the Divided Line, which is the central doctrine of Platonist epistemology. So, intelligible objects are real, in the sense of being 'the same for all who can count', but they're not sensory objects, because they're only perceptible by the rational intellect ¹ . They are known, and knowable, in a way that sensible objects cannot be; and at the same time, they enable the rational intellect to arrive at a foundational level of understanding, such as hypotheses expressed in mathematical terms, which could never be elicited by sensory experience. That is the source of the kind of reverence that the ancients had for rationality and logic, which is actually what gave rise to science itself.

So perhaps 'mathematical world, M', is really just a metaphorical depiction of the Platonist intuition of the nature of numbers. But then, it is 'the existence of M' that is thrown into doubt. But maybe this doesn't do anything more than show that this particular way of allegorising Platonism is what is at fault.

There's another thing. It there is indeed a 'perspectival error of mistaking ourselves as universal', then so much for the universalising claims of science itself. Reminds me of a remark by Tom Wolfe in his celebrated essay, Sorry, but your Soul just Died ² :

Recently I happened to be talking to a prominent California geologist, and she told me: "When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realise that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other."

Welcome to post-modernity. I guess.

The fact that the distinction between the expressions 'to be real' and 'to exist' are brushed off says a lot. — Wayfarer

The problem for Platonists is that they have failed to, and apparently cannot, explain in what sense the purported Platonic objects exist, or are real, in some way other than the familiar concrete (human mind independent) existence or reality of sense objects (or at least of whatever gives rise to them), and the familiar ideal (human mind dependent) existence or reality of the contents of thought, emotion and perceptual experience,

The problem for Platonists is that they have failed to, and apparently cannot, explain in what sense the purported Platonic objects exist, or are real, in some way other... — Janus

It's not difficult to explain, although it might be difficult to accept, or to understand. From Platonism in the Philosophy of Mathematics.

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences. Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects.

There's another argument, The Indispensability Argument for Mathematics, which I admit I haven't studied in depth. However, I would argue that the fact that the argument is necessary is significant.

Why do you think it might be necessary to state such an argument?

Why do you think it might be necessary to state such an argument? — Wayfarer

Although I don't believe in this, the indispensability argument suffers from the critical flaw of question begging.

As to how my cognitive dissonance can come to be is another question.

Rovelli begins with a simple definition of Mathematical Platonism, which "is the view that mathematical reality exists by itself, independently from our own intellectual activities." Now, he asks that we imagine a world M, which contains every possible mathematical object that could ever exist, even in principle. Not only does M include every mathematical object we have currently discovered (integers, Lie Groups, game theory, etc) it also includes every mathematical object we could possibly discover. M is the Platonic world of math. The problem, though, is that this world is essentially full of junk. The vast majority of it is simply useless, and of no interest to anyone whatsoever. — StreetlightX

Not only is this mathematical realm full of junk, but it's also full of contradictions. Go figure. Because of such contradictions, mathematics is clearly not logical. So, which is more reliable, mathematics or logic?

We don't just not care about them for reasons of utility, we don't care about them because we have a standard of intelligibility which automatically excludes them from our mathematical discourse. — fdrake

I think that's a point Plato would readily have acknowledged; and a reason why Plato may never have been a Platonist in the modern sense of the term. Modern mathematical Platonism likely is a distortion of Plato's thought, which distortion arose from taking his metaphors literally and misconstruing acts of the intellect -- themselves always portrayed by Plato as outcomes of strenuous and protracted dialectical effort -- as passive acts of contemplation of an independently constituted domain.

So perhaps 'mathematical world, M', is really just a metaphorical depiction of the Platonist intuition of the nature of numbers. But then, it is 'the existence of M' that is thrown into doubt. But maybe this doesn't do anything more than show that this particular way of allegorising Platonism is what is at fault. — Wayfarer

That's my suspicion too!

I think that's a point Plato would readily have acknowledged; and a reason why Plato may never have been a Platonist in the modern sense of the term. Modern mathematical Platonism likely is a distortion of Plato's thought, which distortion arose from taking his metaphors literally and misconstruing acts of the intellect -- themselves always portrayed by Plato as outcomes of strenuous and protracted dialectical effort -- as passive acts of contemplation of an independently constituted domain. — Pierre-Normand

This makes sense. I don't have the knowledge to bring out how Plato became distorted, though. What history are you tracing in this idea?

The moment an individual becomes self-aware the notion of unity (one) will naturally arise. Coming into contact with other such beings will automatically generate the natural numbers.


I'm assuming self-awareness is necessary for any bit of math a being can do. Am I right in this assumption? I heard bees can count and I don't see them asking existential questions.

Right, but I'm asking if there is a human junkyard of abandoned math, whether constructed or discovered. Because the argument turns on most of math being a junkyard. I'm asking whether this is a hypothetical, or actually historical. — Marchesk

I think this is a great question, and I think it's important to show that, if most of what could-be-math is junk, then we've come across that junk before. One example that I think fits the bill is John Wallis' proof that all negative numbers are - or rather can be construed as - greater than infinity (the reasoning is simple, and I summarized it in a previous post). The upshot of Wallis' proof is that the number line (which Wallis invented), which normally looks like this:

-∞ < ... < -1 < 0 < 1 < ... ∞

can look like this:

0 < 1 < 2 < ... < ∞ < ... < -2 < -1

The thing about this is that there's nothing particularly 'wrong' with this way of ordering the integers (here's a paper that fleshes it out in modern terms). The reason math doesn't opt for Wallis' construal of the number line - and his conception of infinity - is because Cantors' constural of it (now the canonial treatment of infinity) is much more productive. Wallis' number line is 'junk math'.

In a physicsforum post that discusses the paper, a commentator makes a point that's almost identical to Rovelli's with respect to Dirac that I quoted earlier: "when Dirac wrote in his book: 'principles of quantum mechanics' that the derivative of Log(x) should contain a term proportional to a so-called 'delta function' that he had just invented out of thin air a few pages back, was complete nonsense too."
(source); Compare Rovelli: "Dirac, in his book, is basically inventing linear algebra in the highly non-rigorous manner of a physicist. After having constructed it and tested its power to describe our world, linear algebra appears natural to us. But it didn't appear so for generations of previous mathematicians".

The reason it's not very easy to come up with examples of junk-math is precisely because it's... junk math. No one cares for it, and no ones cares to pursue it because its largely useless.

That said, another, perhaps less pertinent example might be geometrical definitions of infinity (i.e. definitions that rely on intuitions about physical space), which in turn relied on the fuzzy concept of the infinitesimal. Until the invention of non-standard analysis in the 60s (which provided a rigorous way of understanding the infinitesimals) mathematicians made a huge effort to understand infinity on a strictly arithmetic basis (i.e. without reference to physical space), because the logical foundations of 'geometrical infinity' were not considered to be secure. In that time before those foundations were secured, one could say that the concept of infinity teetered on the edge of 'junk math' - being 'saved', ultimately, because it's so damn useful.

The problem for Platonists is that they have failed to, and apparently cannot, explain in what sense the purported Platonic objects exist, or are real, in some way other than the familiar concrete (human mind independent) existence or reality of sense objects (or at least of whatever gives rise to them), and the familiar ideal (human mind dependent) existence or reality of the contents of thought, emotion and perceptual experience, — Janus

Yep. All you end up getting are these miserable negative non-specifications that are more than happy to specify what Platonic entities are not, all the while dodging the question by saying that 'oohh its so hard to explain'. It's a cheap rhetorical go-to that feeds right into the elitist, exclusionary, and cultish tendencies that span all of Plato's thought. No wonder that it ends up, in its later, Christian incarnations as negative theology and cultic mysticism: making a virtue out of intellectual failure (which also helps explains Neitzsche's diagnosis of nihilism which runs an intellectual line from Plato right through to Christianity: "man would rather will nothingness than not will.").

Yes these kinds of objections to Platonism occurred quite early on in the form of questions such as "But is there a perfect form of the turd, or the pile of vomit?". — Janus

This is a really nice historical point! Totally didn't think of that.

Plato has Socrates ask if there are forms of mud, hair and dirt.

Yeah, that was the passage I had in mind when Janus mentioned it. That, and, loosely connected, Voltaire's critique of Leibniz's best-of-all-possible-worlds.