Marc, list,
The heck with the train. I'll do chores today instead.
I should add to that which I said below, in order to respond to Marc's remarks
a bit more specifically.
Insofar as any sensory form of mathematical objects will have some sort of
"flavors" in whose terms the senses sample the world, it would actually be kind
of restrictive to have a sensory modality for mathematical objects per se. The
point of mathematics is the transformability, the rationally supported and
constrained imaginative metamorphizability, across sensory & senselike
information modalities as well as across particular concreta. In a sense, we
already have a sensory/intuitive modality or two for maths -- the cultivated
sense for space(s) and the cultivated sense for symbols. There'd be no point to
regarding one or the other as the one true general model the mathematical
reality in itself. Mathematicians will tend sooner or later to try to get
beyond that set of flavors or hues or etc., that specialized model.
I do certainly agree that the human mind is limited such that there are, very
likely, intelligences next to which we're canine or much lower than that. At
least, it's hard to disbelieve that there could be and that the possibility is
there. But the simplest meanings of this in turn are that our imaginations,
intellects, senses, and commonsense perceptions are limited, and that all of
them require & invite cultivation and extensions in mathematical or scientific
research -- and in many other things as well. Now, one can easily suppose these
cognitive powers to become so increased that they would be rather unlike
anything which we have experienced. But I see no reason to suppose that they
would _necessarily_ become comparatively more sense-like than imagination-like
or commonsense-perception-like or etc. My guess is that a mind so strengthened
would have increased freedom to employ all those modalities variously,
integratively, etc.
It seems likewise to me that the simplest meaning of the ascribing (I don't
mean the limiting) of reality to all established subjects of research, in their
full range including maths, is the ascribing of capacities to discover & learn
about reality to cognitive modes in _their_ full range -- rather than some
squeezing of all levels of reality into the subject matters of the sensory
modalities, out of a narrow interpretation of "reality" and a somewhat
questionable association of sensory modalities with concrete singulars rather
than with the qualities & flavors in terms of which the modalities sample &
taste the world. Yet I think that that idea -- sensory faculties for everything
-- actually has some foundation to it as well. For instance, the intuitive
sense of a thing's meaning or value as, for instance, a symbol of something
else, is a kind of sense-doing-the-job-of-intellect. An intuitive sense of a
thing's validity, legitimacy, or soundness as a kind of observational p!
roxy for something else, is a kind of sense-doing-the-job-of-imagination. But
this sort of thing is only to the extent that the full range from mathematicals
to flavors, tendencies, & kinds of appearances, to concrete individual
things/events, can be "squeezed in" as subject matters of _any_ of those
cognitive modes as employed as scientific/mathematical roads to truth -- (Level
IV) imagination (universals), (Level III) intellect (universes, total
populations, etc.), (Level II) sensory & related intuitive faculties
("flavors," "non-universal generals," qualities, etc.), and (Level I)
commonsense perception (singulars embedded in their concrete historical
tapestry -- singulars not as constituting a universe or gamut such that it is
supposed that nothing else exists -- instead, singulars among more singulars).
For my part, I doubt that platonic entities undergo real change, but they're so
rich that they might as well change -- finite minds like ours will never
exhaust them, or at least I tend to suppose not.
Anyway FWIW that's my story and I've been sticking to it, so far.
Best, Ben Udell
----- Original Message -----
From: "Benjamin Udell" <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Friday, January 27, 2006 8:17 AM
Subject: Re: Mathematics: Is it really what you think it is?
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: [email protected]
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"
