In any case, the sense in which Platonism says that numbers (etc) are independent of particular minds, is simply based on the observation that they are the same for anyone capable of counting. But at the same time, they're only perceptible by a rational intelligence. So they're 'intelligible objects', or the objects of reason, which are fundamental to the operations of rational thought, and indeed to science itself.

I mentioned in a thread the other day, that Einstein said in dialogue with Tagore that 'I cannot prove scientifically that Truth must be conceived as a Truth that is valid independent of humanity; but I believe it firmly. I believe, for instance, that the Pythagorean theorem in geometry states something that is approximately true, independent of the existence of man.' However, I argue, what this overlooks is that, to our knowledge, that is something that can only be known to h. sapiens. In fact, arguably, it is what makes humans sapient in the first place.

Now this insight has been preserved almost nowhere outside neo-thomism, but that may only be because Thomism itself was the last vestige of a real metaphysics in the Western tradition. In other words, it's not due to any particular intellectual superiority of Catholicism per se, but just that this is one of the only intellectual milieu in which the insights of Greek philosophy were kept alive (the other being Orthodoxy). Elsewhere, the effects of enlightenment rationalism and positivism were such that a 'presumptive realism' became ascendant, along with the 'reign of quantity' and neo-Darwinian materialism, which is fundamentally self-defeating in its outlook due to its denigration of reason.

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? — Metaphysician Undercover

Only consistently defined objects can be part of the mathematical world.

Axioms are properties of an object (also called axiomatic system). Axioms like "The continuum hypothesis is true" and "The continuum hypothesis is not true" would be contradictory if they were properties of the same object but they are not contradictory if they are properties of different objects.

In any case, the sense in which Platonism says that numbers (etc) are independent of particular minds, is simply based on the observation that they are the same for anyone capable of counting. But at the same time, they're only perceptible by a rational intelligence. So they're 'intelligible objects', or the objects of reason, which are fundamental to the operations of rational thought, and indeed to science itself. — Wayfarer

If they are only perceptible by a rational mind it doesn't necessarily mean that they are inside the rational mind. They may be perceptible only by a rational mind and still be outside the rational mind - that's what Platonists/realists are saying. They are saying it because it seems inconceivable that truths about numbers didn't hold before someone perceived them or will stop holding when there is no one around to perceive them. Maybe a more accurate word would be "to infer" than "to perceive", since abstract objects like numbers don't seem amenable to sensory interaction. But to infer the existence of an object doesn't mean that the object only exists in the mind that is doing the inference. Scientists make inferences (predictions) about the external world and test them.

They may be perceptible only by a rational mind and still be outside the rational mind - that's what Platonists/realists are saying. — litewave

Indeed - that's the point. And you're right in saying they're not 'perceived' in a literal sense, but are 'grasped by reason' - seen by the mind's eye, so to speak.

to infer the existence of an object doesn't mean that the object only exists in the mind that is doing the inference. — litewave

But inference relies on reason which is, on the one hand, internal to the operations of thought, but, on the other hand, makes real predictions. Reason is strictly the relationship of ideas, but at the same time, rational inference can be used to predict novel discoveries about the world - which is pretty close to the meaning of Kant's synthetic a priori.

In any case, the sense in which Platonism says that numbers (etc) are independent of particular minds, is simply based on the observation that they are the same for anyone capable of counting. — Wayfarer

And? The unarticulated premise here is that this mutual recognition entails 'independence'. But this is clearly nonsense. That people recognize words - and especially non-referential words like 'and' and 'is' - does not entail any ostentatious, overblown metaphysics of Platonic Is's and And's- it simply entails that we've learnt how to use a language, like the slightly-less-than-dim animals that we are.

As for neo-Thomism being the last refuge of Greek philosophy - all the better that it's drawing its last, dying breaths. The sooner we forget about hot mess that is Greek philosophy and it's intellectual spawn, the better.

The unarticulated premise here is that this mutual recognition entails 'independence'. But this is clearly nonsense. — StreetlightX

Doesn't the fact that basic mathematical truths are the same for anyone who can count demonstrate that they're independent of particular thinkers? What is nonsensical about that?

Doesn't the fact that basic mathematical truths are the same for anyone who can count demonstrate that they're independent of particular thinkers? — Wayfarer

It's like saying: wow, look at all these various languages that have nouns! Guess Nouns must be Platonic Entities. It's reasoning made for and by idiots.

It's like saying: wow, look at all these various languages that have nouns! Guess Nouns must be Platonic Entities. It's reasoning made for idiots. — StreetlightX

There is an undeniable phenomenon of convergence. But there are two mistakes one could make regarding such phenomena.

The mistake that animates modern naive empiricism is to explain the phenomenon of convergence -- such as the discovery of laws of nature, or of general logico-grammatical features shared by (most) natural languages -- as a result of the faithful (or approximate) reproduction, as contents of our mental representations, of the structure of an independently existing empirical ('external') reality. This mistake is almost indistinguishable from the mistake of reifying intelligible ideas, where such ideas are being ascribed a role similar to the role being played by raw 'sense data' for the empiricist.

Another mistake would be to recognize the root of the convergence as a product of shared social practices but to view the fact of this convergence as a matter entirely of socially enforced conventions.

One might avoid both mistakes though recognizing that the phenomenon of convergence is a dynamical product of the enactment of the social practice of arguing for or against doing and/or believing things. The convergent phenomena are being constituted from within the enactment of dialectical reason, by social beings, rather than from without their historically situated lots of shared discursive practices.

One might avoid both mistakes though recognizing that the phenomenon of convergence is a dynamical product of the enactment of the social practice of arguing for or against doing and/or believing things. The convergent phenomena are being constituted from within the enactment of dialectical reason by social beings rather than from without their historically situated lots of shared practices. — Pierre-Normand

Totally agree. Insofar as we are (mostly) beings that count, and employ counting to engage in certain behaviors, it's simply unsurprising that our counting systems will tend to converge around certain invariants. And as Rovelli remarks, the fact that we are beings that count has itself a strong element of sheer contingency: "The development of the ability to count may be connected to the fact that life evolved on Earth in a peculiar form characterized by the existence of “individuals”. There is no reason an intelligence capable to do math should take this form. In fact, the reason counting appears so natural to us may be that we are a species formed by interacting individuals, each realizing a notion of identity, or unit. What is clearly made by units is a group of interacting primates, not the world".

I don't think it's a matter of coincidence as well that in the sphere of evolution, convergence is sometimes also used as an argument for theistic ends - the usual "wow, look at all these eyes across so many different species, must be God at work!"; well, no, eyes are the product of the dynamics involved with the properties of light and animals that move, many of whom share similar environments (with animals in the depths of the ocean evolving no eyes - what a surprise!). There's a very real parallel with those who think that convergent aspects of math similarly reflect some grand transcendent design decoupled from the material universe, rather than behavior and environment: it's nothing less than intelligent design shunted into the field of mathematics. And it's all just as bogus. Math is the last, shrinking, refuge of a God deservedly driven out of everywhere else.

Also totally agree with you about the critique of 'naive empiricism': as you said it's just the mirror image of Platonic Idealism, both of which are totally utterly blind to the evolutionary and historical dynamics of shared practices - in the case of math - and shared ecological and developmental dynamics - in the case of evolution.

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? — fdrake

I was alerted to the possibility of the distortion by a handful of scattered remarks on Plato versus Platonism by John McDowell. But I haven't pondered much on the historical roots of the distortion, nor do I feel equipped for tracing such roots anywhere earlier than the modern period.

I just did some literature search and found this paper, which makes similar remarks to McDowell's: Konrad Rokstad, Was Plato a Platonist? Analecta Husserlania. The Yearbook of Phenomenological Research, Volume CX, 2011.

Regarding what might be seen as a modern recovery of Plato's valuable insight about the dialectical nature of the constraining function of a priori 'forms' upon reason, I made this recent comment.

That's a convincing response.

Our culture is organized around an abstraction: money. That means that abstraction and metaphor are inextricable aspects of the world we are immersed in. It's part of who and what we presently are.

But money has no meaning pulled away from the world that embraces it. And in turn, that world will become nonsense if we try to understand it without the abstractions that organize it.

Mathematics and language are types of mental code (transformed, translated, or converted mental data) consisting of a mental vocabulary (symbol set) arranged according to a mental syntax (logical structure).

Mental codes require mental representations, and facilitate mental processing.

Numeracy (the ability to understand number arrangements, and perform numerical operations) and literacy (the ability to understand word arrangements, and perform verbal operations) are human faculties, hence; human universals.

Both faculties develop in parallel with mental maturation, personal experience, and social influences. Mathematical and verbal abilities develop subjectively, while mathematics and language develop intersubjectively.

"I don't care if it's real or not; I'm not interested in reality." Well, okay, then.

When your picture of reality has so little to do with reality then so much the worse for that picture. "But you're not dealing with reality!"; You can keep your two-bit reality.

Okay, so how does my picture have little to do with reality? We need something better than question-begging here.

I'm not the one playing with labels.

I pointed out that your argument is circular. Do you actually have a response?

All you dribbled off was some irrelevancies about 'abstractions not being real'; this being a phrase not even found in the paper. Come back when you've read it.

Ah! I see the problem here. Platonism is a stance toward the reality of abstractions, so argument against Platonism is argument against the reality of abstractions. Thus the reason for bringing up the reality of abstractions.

Right, so you want to have a debate over terms not employed by the paper... about the paper? Come back when you've read it and are prepared to actually do so.

Terminology is not terribly important. Instead of complaining that I'm being mean to the paper, why don't you respond to the argument?

Terminology is not terribly important — Pneumenon

"I want to have my own debate".

Have fun.

Do you have an argument?

I don't argue against irrelevancies that can't even get basic terms right.

I'll take that as a "no."

Well of course you would; you're demonstrably incapable of taking things in terms other than that which you've projected onto them from without.

Ah, so you do have an argument! What is it?

Et voila.

I'm afraid that wasn't an argument.