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What is the relation between natural and mathematical knowledge?

The Grey Man

Denken ist schwer
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All thinges which are, & have beyng, are found under a triple diversitie generall. For, either, they are demed Supernaturall, Naturall, or of a third being. Thinges Supernaturall are immateriall, simple, indivisible, incorruptible, & unchangeable. Things Naturall are materiall, compounded, divisible, corruptible, and changeable. Thinges Supernaturall, are, of the minde onely, comprehended: Things Naturall, of the sense exterior, are able to be perceived. In thinges Naturall, probabilitie and conjecture hath place: But in things Supernaturall, chief demonstration, & most sure Science is to be had. By which properties & comparasons of these two, more easily may be described the state, condition, nature and property of those thinges, which, we before termed of a third being: which, by a peculier name also, are called Thynges Mathematicall. For, these, beyng (in a maner) middle, betwene thinges supernaturall and naturall, are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall: But are thinges immateriall: and neverthelesse, by materiall things hable somewhat to be signified.

—John Dee, The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara

"...somewhat to be signified."

What does John Dee mean when he says that natural things are "somewhat signified" by mathematical things? We might answer that, for example, a husband and wife number two, but are not the number two, or the silhouette of a mountain is triangular, but not triangularity, or, more generally, that though mathematical ideas are instantiated by natural phenomena, yet those phenomena are not identical to the ideas that they represent. We might say, in the traditional Platonic manner, that the ideas exist rather in the intelligible, supernatural World of Being rather than the World of Becoming, in which things are, as Agent 007 says, "compounded, divisible, corruptible, and changeable;" but this leaves open the question, If the world of ideas is distinct from the phenomenal world, how do phenomena instantiate the ideas?, which wants an answer like Aristotle's hylomorphism, which asserts an irreducible duality between the actual form of a thing and its dynamic potentialities, thus "immanentizing" the ideas.

What, then, is the relation between our natural and mathematical knowledge? What does it mean for us to live in a world of being which is at once a world of becoming? Are we the corruptible material things, the immortal ideas, or some third thing? Could mathematical intuition be a bridge between our souls and our bodies, a "middle," as Dee says, between the natural and the supernatural, between time and eternity?



Leibniz famously argued that space does not exist or, more precisely, that what we call space, contra Newton, is not something independent of the relations between things in space, but is precisely those relations. In other words, we do not perceive a single object which answers to the name 'space,' but pluralities of connections between natural things which are only recognized as belonging to the same species, and these species to the same genus, in abstracto. This explains why it took man millennia to produce a Euclid who would state some of the general principles of geometry in his Elements, even after he had relied on such principles to perform impressive technical feats in Egypt: he was never directly acquainted with mathematical ideas themselves, but only their instantiations in nature. One must do mathematics before one can do meta-mathematics. Our situation is comparable to that of a native speaker of a language who is only slowly and with great difficulty brought to an understanding the grammatical rules that his everyday speech takes for granted; and yet, just as I am no mean speaker of English despite lacking the abstract knowledge of a grammarian, so are we, as Delambre said, "in possession" of mathematical knowledge, however ignorant we may be, as Socrates memorably demonstrated with the assistance of Meno's slave.

Who, then, is this speaker whose nouns are the rivers and mountains of the Earth, his grammar the the mathematical rules that bind them together? What language is he speaking? Or is it more like singing?
 

Marbles

What would Feynman do?
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I lack words, Graymes. I told you I had a headache and wasn't in the mood, and now I find you in bed with my friends? Can't your philosophy remain flaccid for even a moment before you engage in filthy discourse on the internet? Tell me, Graymes.. Does my erudition fall short? Is my stamina unsatisfactory? Do my arguments lack potency? Or do you simply derive perverse satisfaction from these public group wrangles?
 

fishhead

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@Marbles too late, he's mine now

Who, then, is this speaker whose nouns are the rivers and mountains of the Earth, his grammar the the mathematical rules that bind them together? What language is he speaking? Or is it more like singing?

Singing, that's a pretty way to put it. It's adequate to say that the speaker himself is far beyond our wildest imaginations. The sounds he creates are scattered by the matter of our physical plane and only echos of it are received by humans. As mere newborns in this time and space, the best method we have of interpreting those echos is mathematics. As the angel said to Descartes, "The conquest of nature is to be achieved through measures and numbers."
 

sushi

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you have to prove the laws of nature and laws of mathematics are one and the same.

sometimes they do and sometimes they dont.
 

ZenRaiden

One atom of me
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Relationship between things are either visible or not.
Math has visible relationships. Math describes the relationships. Math is obviously not everything. Math is part of the descriptions of certain relationships.

What Liebniz says about space is not very important.
 
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