On 27/01/2006, at 8:08 PM, Marc Geddes wrote:
Open question here: What is mathematics? ;)
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.
This notion sounds almost as outrageous as my recently retracted
notion that music is *heard mathematics*. OK then, I'll make one more
attempt - could music be the sensory modality by which we perceive
mathematical entities? Since Bruno is currently injecting our
thinking with a good dose of ancient Greek "wisdom" could we not also
mention Boethius in this context with his "harmony of the spheres"?
Boethius was actually a Roman patrician who was a super Greek scholar
of the 6th century. Actually, Pythagoras discovered (or invented)
this idea. A part of the cosmology of the Pythagorean is this
extraordinary theory of the "harmony of the spheres", which caught
later generations in the ancient world and the Renaissance.
It is generally accepted by scholars that Pythagoras himself was the
first to formulate that concept, which reflects the whole cosmic plan
and showed the intimate connection between the laws of mathematics
and of music.
Aristotle characterizes the Pythagorean as having reduced all things
to numbers or elements of numbers, and described the whole universe
as "a Harmonia and a number".
Aristotle continued: "They said too that the whole universe is
constructed according to a musical scale. This is what he means to
indicate by the words "and that the whole universe is a number",
because it is both composed of numbers and organized numerically and
musically. For the distances between the bodies revolving round the
centre are mathematically proportionate; some move faster and some
more slowly; the sound made by the slower bodies in their movement is
lower in pitch, and that of the faster is higher; hence these
separate notes, corresponding to the ratios of the distances, make
the resultant sound concordant.
Now number, they said, is the source of this harmony, and so they
naturally posited number as the principle on which the heaven and the
whole universe depended."
Well - it's surely not such a big jump from this notion to the main
topic of this list.
We could imagine some super-intelligence that possessed this
ability to directly perceive mathematics. What would this super-
intelligence 'see' ?
Or *hear* - the architecture of reality as a symphonic musical
texture with an infinity of voices/instruments all producing/
generating the laws of physics, the immediate revelation of which are
the heavenly bodies in their orbits. Music surely is a form of
computation - it has an origin in "pure thought" and an outcome as a
hard physical reality in the form of compression sound waves. A super
intellect might be able to hear mathematics as music, indeed the very
notion of the "harmony of the spheres" seems to suggest this.
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
You are now describing essentially a musical phenomenon. Music is
dynamic, it modulates and self-references and develops from state to
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)?
Music is of course "time-dependent". The canvas that music uses *is*
time. I see great room for discussion of your 3-dimensional time view
vis a vis musical time-frames. For example the "Art of Fugue" (Bach's
method of composition) stipulates 4 quasi-independent voices or
melodies which each exist in their own right and are perfectly
satisfying on their own, yet somehow create an emergent, holistic,
greater sense of unity when all combined ie. played simultaneously.
Each of these melodies is largely unaware of the other three yet
dovetails its own sense with the sense of the others. This is what
creates the sense of *depth* in musical textures. The experience of
listening to music correctly (without talking over the top of it or
munching on your dinner) gives the impression of a physical object
which you nevertheless cannot see. I would suggest that whatever this
object is has been described by the musical process and cannot be
perceived in any other way.
I hope nobody gets upset by my reintroducing this idea. Once more, a
bit of a provocation to "up the ante" of the discussion.
Regards to all
"People often confuse belief in a reality with belief in a physical
reality" - Bruno Marchal