Marc, Bruno, Russell, Hal, list,
First, a general note -- thanks, Hal, for the link to your paper on the Universal Dovetailer. I have gotten busy with practical matters, so I've gone quiet here. I hope to have time to pursue the UD soon.
As to a sensory modality for mathematical objects. The senses and related cultivated "intuitive" faculties are for qualities and relations that are not universal but merely general (i.e., they're not mathematical-type universals but they're not concrete particulars/singulars either). So to speak, the senses etc. are sample takers, they sample and taste the world. The senses and their cultivated forms and also their extensions (instrumental & technological), taking samples, lead to inductive generalizations, and the most natural scientific form of this process is in those fields which tend to draw inductive generalizations as conclusions -- statistical theory, inductive areas of cybernetic & information theory, and other such fields (I'd argue that such is philosophy's place, too). Mathematics is something else. Its cognitive modality seems to be imagination, or imagination supported and constrained by reason.
Edgar Allen Poe: "The _highest_ order of the imaginative intellect is always pre-eminently mathematical, and the converse."
http://www.eapoe.org/works/essays/a451101.htm first paragraph's, last sentence.
It is to be admitted that Poe counted mathematics as "calculating," but, on the other hand, he probably vaguely meant more by "calculating" than many of us probably would.
Imagination becomes the road to truth when the mind considers things at a sufficiently universal level. I.e., two dots in my imagination are just as good an instance of two things as any two things outside my imagination. The imagination along with its extensions (e.g., mathematical symbolisms, the imaginative "apparatus" of set theory, etc.), supported, checked, & balanced by reason, produces fantastic bridges, often through chains of equivalences, across gulfs enormously _divergent_ from a sensory viewpoint. It would all be indistinguishably universal but for abstractions (e.g., sets) whereby one can say that some of these universals are more universal than others, some are unique (as solutions to families of problems, etc.), and the world in its wild variegation (of models for mathematics) can be, as it were, re-created.
To say that mathematics is real doesn't imply that it consists of sensory qualities or of the concrete singulars cognized in their historical and geographical haecceity (or "thisness") by commonsense perception. It does imply that the kind of cognition which leads to mathematical truth is a cognition of a kind of reality, the reality, whatever it is, of which mathematical statements are true. Of course if we say that only singular objects are real, then there's no mathematical reality. But insofar as such objects are _really_ marked by mathematical relationships, mathematics has enough "reallyness" to count as reality, unless one wants to multiply "reality" words to keep track of syntactical level.
None of this is to say that the senses (& related "intuitive" faculties) have nothing in common with imagination. Both of them involve capacities to form creative impressions, to expect, to notice, and to remember. Both of them objectify & map, both of them judge & measure, both of them "calculate" or interpret, and both of them recognize & (dis)confirm. The mathematical imagination continually honors, acknowledges, and recognizes rules variously old and newly discovered of the "games" or "contracts" into which it enters soever voluntarily and whimsically.
Now I have to count on the subway's being on time -- if only I didn't have to work!
Best, Ben Udell
----- Original Message -----
From: Marc Geddes
To: everything-list.domain.name.hidden
Sent: Friday, January 27, 2006 4:08 AM
Subject: Mathematics: Is it really what you think it is?
Open question here: What is mathematics? ;)
A series of intuitions I've been having have started to suggest to me that mathematics may not at all be what we think it is!
The idea of 'cognitive closure' (Colin McGinn) looms large here. The human brain is not capable of direct perception of mathematical entities. We cannot 'see' mathematics directly in the same way we 'see' a table for instance. This of course may not say much about the nature of mathematics, but more about the limitations of the human brain. Suppose then, that some variant of platonism is true and mathematical entities exist 'out there' and there is *in principle* a modality ( a method of sensory perception like hearing, sight, taste) for direct perception of mathematics. We could imagine some super-intelligence that possessed this ability to directly perceive mathematics. What would this super-intelligence 'see' ?
Perhaps there's something of enormous importance about the nature of mathematics that every one has over-looked so far, something that would be obvious to the super-intelligence with the mathematical modality? Are we all over-looking some incredible truths here? Again, McGinn's idea of cognitive closure is that the human brain may be 'cognitively closed' with respect to some truths because the physical equipment is not up to the job - like the way a dog cannot learn Chinese for instance.
For one thing: Are platonic mathematical entities really static and timeless like platonist philosophers say? What if platonic mathematical entities can 'change state' somehow ? What if they're dynamic? And what if the *movement* of platonic mathematics entities *are* Qualia (conscious experiences). Are there any mathematical truths which may be time indexed (time dependent)? Or are all mathematical truths really fixed?
The Platonists says that mathematics under-pins reality, but what is the *relationship* between mathematical, mental (teleological) and physical properties? How do mental (teleological/volitional) and physical properties *emerge* from mathematics? That's what every one is missing and what has not been explained.
So... think on my questions. Is there something HUGE we all missing as regards the nature of mathematics? Is mathematics really what you think it is? ;)
--
"Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder's eye on the last day"
Received on Fri Jan 27 2006 - 08:23:42 PST