cosmographer said:
who is this nature? and does it speak by itself? math is a tool for transforming realities. those realities don't preexist their making, as if the mathematician would shuttle back and forth between nature and humans to bring the holy word. it is much more mundane than that. it is a technology of thought that enables transformative work in and of nature, very useful indeed!
a great book on the invention of modern math is Reviel Netz' historical study of the emergence of formalist styles of reasoning in ancient greece. this review does a good job at extracting the significance of that invention from a rather complex book:
http://www.bruno-latour.fr/articles/article/104-NETZ-SSofS.pdf
Thanks, a very entertaining reference. To me it confirms that there is this key turn of the mind where we go from imagining reality as a process (developing towards ultimate limits) and as existent (the limits are now what have been achieved and so are "the real").
So when the Greek geometers drew diagrams in the sand with a stick, they made this leap from a reality seeking its perfection to a belief in the existence of the perfect forms themselves. The Greeks of course were not so willing to make the same leap when it came to ratios, incommensurability and infinity. Infinity was still a limit on the process of counting. But mathematics later fixed that with Cantor, etc.
And now we find ourselves torn between the two views. The developmental or process view is clearly "the real" as it is rooted in the "imperfect materiality" which is our world. There are only ever triangles as scrawled in the dust. But the ideal forms - the emergent limit states - also have a claim to reality because they "can't be imagined not to exist". What is more definite and concrete than an ultimate limit (a boundary concept like a triangle)?
This ontological confusion is then compounded by the usual epistemological one - mistaking the map for the terrain. The desire is to deal with the substance~form dichotomy (materiality and its boundary states) by assigning reality and unreality to an epistemological division. So the world (being out there) is real, the maths (being in our heads) is unreal. Yet clearly this does not work because the maths is still really out there in some sense - as the boundaries, the limits, the constraints. The maths is more than just a potential fiction, a social construction due to restricted cognitive technologies.
The way I sort out this nest of confusion is first to accept the epistemological division (I think the "modelling relations" crowd in theoretical biology - Rosen, Pattee, Salthe - do the best job here). So nature, reality, is a constructed view. Both our notions about its materials and its laws, its substance and its form, are "in our heads" and justified by a modelling relation (so it is a process with its own purpose, its own needs, not some dispassionate god's eye view).
So reality, as far as we can know it, is our invention. That applies to our mathematical ideas about it, but also our "physical impressions" too. It is all a map.
However on the whole, it is a very good map - as it has developed within the self-refining tradition of metaphysical abduction, scientific induction and logico-mathematical deduction (Peirce's pragmatic triad!).
And then the ontological bit of the story. We can see that limits only actually exist in the sense that they are the boundaries to what exists. They are how far a process of development can go in some direction before asymptotically tending to a limit. So in fact they are the ontically unreal. The boundary remains always infinitesimally just beyond where reality can reach (so as to be able to be seen to enclose it fully).
Yet boundaries also have a real causality. At least if you are a process thinker, a systems science, you believe that there is such a thing as downward causation and even final cause. So forms can act as constraints that actually shape materiality. Maths exists "out there" as something real in the sense that there are forms (of the kind maths can describe) which have a causal role in the realm of the real.
I guess we have to ask the question then whether the set of forms that humans can imagine is a superset or a subset of those that reality can express. Sometimes it seems our inventions are more fertile - we can elaborate to create more imagined things than can actually exist. Other times, that maths is in fact quite impoverished. It is a pretty crude map of the terrain. More subtle things are going on than we have captured so far.
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