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Feb5-12, 10:51 PM
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I never understood what is meant by the Platonic world. I always considered abstract objects like mathematical objects as mental stuff. So when Penrose writes this quote for me this seems more an argument for mathematical objects being innate and mental stuff:

There is perhaps something mysterious, however, in the fact that we do seem to know instinctively what the natural numbers actually are. For as children (or adults) we are provided with just a comparatively small number of descriptions as to what 'zero', 'one', 'two', 'three', etc., mean ('three oranges', 'one banana', etc.); yet, we can grasp the entire concept despite this inadequacy. In some Platonic sense, the natural numbers seem to be things that have an absolute conceptual existence independent of ourselves. Notwithstanding such human independence, we are able intellectually, to make contact with the actual natural-number concept from merely these vague and seemingly inadequate descriptions.
I mean except for being quite specific, this isn't different than the way internalists like Chomsky treat linguistic concepts:
The cognitive revolution of the 17th century also led to inquiry into the nature of concepts, with important contemporary implications, also insufficiently appreciated. Aristotle had recognized that the objects to which we refer in using language cannot be identified by their material substance. A house, he pointed out, is not merely a collection of bricks and wood, but is defined in part by its function and design: a place for people to live and store their possessions, and so on. In Aristotle’s terms, a house is a combination of matter and form. Notice that his account is metaphysical: he is defining what a house is, not the word or idea “house.” That approach led to hopeless conundrums. The ship of Theseus is a classic case that may be familiar from philosophy courses; Saul Kripke’s puzzle about belief is a modern variant. With the cognitive turn of the 17th century these questions were reframed in terms of operations of the mind: what does the word “house” mean, and how do we use it to refer. Pursuing that course we find that for natural language there is no word-object relation, where objects are mind-independent entities. That becomes very clear for Aristotle’s example, the word house, when we look into its meaning more closely. Its “form” in the Aristotelian sense is vastly more intricate than he assumed. Furthermore, the conundrums based on the myth of a wordobject relation dissolve, when viewed from this perspective, which I believe has ample empirical support...In all such cases, there is no mind-independent object, which could in principle be identified by a physicist, related to the name. As we proceed, we find much more intricate properties, no matter how simple the terms of language we investigate. As Hume and others recognized, for natural language and thought there is no meaningful word-object relation because we do not think or talk about the world in terms of mind-independent objects; rather, we focus attention on intricate aspects of the world by resort to our cognoscitive powers. Accordingly, for natural language and thought there is no notion of reference in the sense of the modern philosophical tradition, developed in the work of Frege, Peirce, Russell, Tarski, Carnap, Quine, and others, or contemporary theorists of reference: “externalists,” in contemporary terminology. These technical concepts are fine for the purpose for which they were originally invented: formal systems where the symbols, objects, and relations are stipulated. Arguably they also provide a norm for science: its goal is to construct systems in which terms really do pick out an identifiable mindindependent element of the world, like “neutron,” or “noun phrase.” But human language and thought do not work that way.
It's not surrprising that Chomsky thinks the two are related:
Nonetheless, it is interesting to ask whether this operation is language-specific. We know that it is not. The classic illustration is the system of natural numbers. That brings up a problem posed by Alfred Russell Wallace 125 years ago: in his words, the “gigantic development of the mathematical capacity is wholly unexplained by the theory of natural selection, and must be due to some altogether distinct cause,” if only because it remained unused. One possibility is that it is derivative from language. It is not hard to show that if the lexicon is reduced to a single element, then unbounded Merge will yield arithmetic. Speculations about the origin of the mathematical capacity as an abstraction from linguistic operations are familiar, as are criticisms, including apparent dissociation with lesions and diversity of localization. The significance of such phenomena, however, is far from clear; they relate to use of the capacity, not its possession. For similar reasons, dissociations do not show that the capacity to read is not parasitic on the language faculty.
Some simple evo-devo theses: how true might they be for language?

To be honest, the more I read on this stuff, the more I'm persuaded by both the internalist and nativist view.