Fourier Spaces

Lyle Burkhead (
Sat, 01 Feb 1997 16:33:13 -0500 (EST)

In his very provocative post of Thursday, Jan 30, Omega wrote

> While Godel defined one form of incompleteness, there is another
> form that is equally significant, and that is the incompleteness of
> one coordinate space for defining in a simple form all possible
> localizations and correlations that can exist within mathematical
> reality.

Coordinate space? How 'bout coordinate system? Do you really want
to put the coordinate systems themselves into a space of their own?

> In math these are the spaces that are defined by the myriad Fourier
> waveform families.

Do waveforms come in families?

> For those into music, this leads into the distinctions between the
> relative ease or difficulty that analog and digital synthesizers have
> in reproducing certain forms of music. For those in physics, this is
> the mathematical basis for Heisenberg uncertainty. But in both cases,
> we are looking at a basic mathematical principle manifesting itself
> in our reality.


> This is also the reason why computers are such idiot savants,
> they are hard-wired to the Fourier space defined by their RAM's
> linear address space.

Now wait a minute. First of all, non-mathematical readers should be
warned that "Fourier space" is not a standard mathematical concept.
Apparently it was freshly minted for the occasion, like my "deity space"
of a few weeks ago. One applies a Fourier transform to a function to
yield another function. The functions themselves live in a space called a
Hilbert space.

You seem to be saying that functions can be separated into equivalence
classes. The functions in one equivalence class (or "Fourier space" or
"waveform family") are interconvertible with other functions in the
same class, but not with the functions in other classes. This is news to
me. (Which doesn't mean you're wrong; a lot of things are news to me.
But I think some explanation is called for.) Any function that satisfies
the Dirichlet conditions (piecewise continuous and square-integrable)
can be Fourier-transformed. Nicht wahr? And the same sines and
cosines are used for all functions... so where do the equivalence classes
come from?

As for computers being "hard-wired to the Fourier space defined by
their RAM's linear address space" -- I have no idea what this means.

> Software can work around this to some extent, but without
> special hardware, no computer is ever going to operate as efficiently
> in other Fourier spaces as the default one defined by its hardware.

All computers are mathematically equivalent, aren't they?
One Turing machine can do anything another Turing machine can do.
Are you referring to the fact that different chips have different
instruction sets? What does this have to do with Fourier analysis?

> The human brain is so flexible because it contains the ability to
> look at things from radically different viewpoints.

Well, the essential thing is that the human brain can look at things
from a viewpoint. Computers are idiot savants because they have
no viewpoint. They have no general model of the world. They have
no organs of perception, and no language (in the human sense of
"language" -- C and Java are not languages in the sense that English
and Sanskrit are languages).

Nevertheless, in spite of all this criticism, I think you are onto
something. On an intuitive level, I think I see what you are getting at.
I'm not sure the concept of "Fourier space" can be made rigorous,
but it is a thought-provoking idea.

The essential ideas of Fourier analysis are
(1) Analyzing something into its components. A function is the sum of
a series of sines and cosines. This is analogous to saying that a point
(x,y,z) in 3-space is the sum of (x,0,0) + (0,y,0) + (0,0,z). Functions,
however, require an infinite number of "coordinates".

(2) Duality. Given a function of time, one applies a Fourier transform
to obtain a function of frequency; then one applies a transform to this
function, and the result is... the same function of time that we started
with. Thus, functions come in Fourier pairs. Alternatively, we can say
that the "same" function exists in the time domain and in the frequency

Now, the instruction set of a chip could be viewed as the "coordinates"
with which the chip represents problems, and the "Fourier space" would
be the set of programs that can be (simply or efficiently) written using
this instruction set -- is that what you mean when you say a computer
is "hard-wired to a Fourier space"? If so, I think it's confusing to bring
Fourier analysis into it.

Instead of using Fourier analysis as the basis of this whole discussion,
it would be better to start with a general idea of representing things with
coordinates, and then Fourier series would be a special case of this --
not necessarily the canonical case.

> Unless a person is both able and willing to switch to a different
> Fourier space, they literally can not understand what the other person
> is talking about.
> In fact they not only will not be able to answer the questions
> that are defined in that Fourier space, there's no guarantee that they
> will be able to even grasp the question.

Ok. Now your point is getting clearer. You are conceiving language
as based on semantic primitives, or coordinates, and the "Fourier space"
is everything that can be (simply or efficiently) expressed using certain
coordinates, i.e. a person's basic vocabulary. Again, I'm not sure that
trying to base this whole discussion on sines and cosines is the best way
to proceed.

But then --

> With this more or less fixed pattern of viewpoints, plus at least some
> amount of built in reluctance to surf though the myriad alternative
> viewpoints, together with an evolutionary requirement that our
> viewpoints harden as we age so as to selectively propagate the memes
> that got one to parenthood (and later on to the status of "wise elder")
> we find that we have defined, in a very natural way, the very basis for
> ego-identity along with its normally decreasing flexibility with age.

Maybe I should try being a wise elder instead of a cantankerous
old fart!