Re: Fourier Spaces

Omega (
Sat, 01 Feb 1997 19:23:14 -0800


Thanks for the constructive criticism, so let's see if I can
answer the points you brought up.

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

Maybe, maybe not, I really don't know what would be the right way to
frame the semantics on this. I don't want to misuse existing jargon,
but on the other hand what I am describing has a very definite meaning
which, to say the least, is not widely recognized.

Worse yet, what I'm describing is "intuitively" known in a very poorly
defined way within our culture, leading to a constant confusion between
the perspectives from which a truth may or may not be able to be beheld,
and the actual truth (or lack thereof) itself. This is an attempt to
define in a more precise way this distinction between truth and perspec-
tive and to ground that distinction in solid mathematical principles.

In any case physicists themselves have also set a precedent in this
by calling momentum space just that, a space. I strongly feel that
on this tangent we have a case of physicists not taking their own
theories and descriptions seriously enough.

> > In math these are the spaces that are defined by the myriad Fourier
> > waveform families.
> Do waveforms come in families?

Yes, and the families can be totally arbitrary as Nick Herbert describes
in pages 82-89 of his book _Quantum Reality_. The impulse and sine wave
families of classical Fourier analysis are simply those waves that are
kin and conjugate to the Cartesian aspect of reality that we are localized
within. An infinite number of families are possible, even if only these
two conjugate families seem to be needed for basic Fourier analyis.

> > 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.
> Hmm...

Interesting isn't it.

> > 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.

This is very true, "Fourier space" is a freshly minted term, I could have
made that more clear in my initial post on this. Going back to your
original question, about calling it a space, the Cartesian coordinate
system very clearly defines what we ordinary think of as a space; the
space known more formally in mathematics as the Euclidean space.

Now higher level spaces (like the Hilbert space) do exist in mathematics,
but the problem is that unless we generalize our concept of what a
space is at this lower level we automatically assume that the Euclidean
space has an ontological reality greater than it actually does; thus I
coined the brand new phase "Fourier-space" to reveal how much our fam-
iliar Euclidean space is just an ultimately arbitrary perspective upon
a more general reality no matter how important it is to us locally.
Again, I strongly feel that this arbitrary perspective is the reason
why QM seems so mysterious in general.

> One applies a Fourier transform to a function to
> yield another function. The functions themselves live in a space called a
> Hilbert space.

I certainly slipped a cog when writing this by forgetting that a gen-
eralized set of these Fourier spaces is called a Hilbert space. It's
amazing how I can get so focused on one line of thought that I start
forgetting stuff like this. It's also true that some mathematical
techniques especially in the area of matrix math actually deal with
this beast in a systematic way. Unfortunately, I've never heard of
any metaphor that even comes close to allowing us to picture Hilbert
space. Any one got any ideas?

> 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.)

This is not my meaning. It's very clear that functions can arbitrarily
be moved from one Fourier space to another, IF one is willing to use an
infinite number of terms. A quick review of what I actually said:

> > 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 cor-
> > relations that can exist within mathematical reality.

This is the main point. The functions might well be totally equivalent,
but no finite brain is going to solve the problem if it is trying to define
a function in a Fourier-space that requires a near infinite number of terms
to approximate that function well. What we have here is a very fundamental
issue regarding the usability of information as a "function" of which
Fourier-space that infomation is being represented in.

> 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?

Thus this is a non-issue as my point is not that they can't be
transformed, so much as how many terms will be necessary to effect
an accurate transformation.

> 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?

IMO, it is simply an error to extend the Turing theorem across the
boundaries of Fourier-spaces. A high precision tunable multi-channel
analog processing unit attached to a Apple II with an appropriate
digital to analog control interface, and a fast latching analog to
digital data readback interface could easily solve in seconds certain
classes of problems that would give a Cray indigestion for hours at a
time. Analog computers might be out of fashion for a number of reasons,
but they are immensely powerful in their native realm. The problem
with "general purpose Turing machines" is that there will always be
classes of problems that any given class of machines will choke on.

> > 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).

I agree totally, I never meant my assessment of what contributes to
the "idiot savant" nature of computers to be exhaustive.

> 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.

Actually I wouldn't expect that, in the limited sense of showing that
differing Fourier spaces can hold different patterns of information
with varying degrees of ease and localization, there would be any
problem with making this rigorous.

> 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
> domain.
> 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.

This is not my meaning. It's not just simply an instruction set issue, since
by hardwired, I mean that the hardware would have built in analog frequency
generators and analog computer logic. It is my contention that if someone
could solve the practical problems of fusing digital logic with analog logic
into a seamless computational device then we would have machines that would
be qualitatively more intelligent on a wide range of problems.

Ultimately this would lead to a computer that could change, on the fly, the
ratio of digital to analog in its computation so as to surf through the
general set of Fourier spaces to find a concise representation of whatever
problem is at hand. A process that we might call meta-problem solving.

> >
> > 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.

My hypothesis is that the known physiology of the human brain is that of a
combined digital and analog device as described above. A combination that
gives us much flexibility in viewpoint, but can also leave us to talking
past each other if the people involved can't or won't match viewpoints.
The relevance of Fourier spaces to this is that formally, this is what
would be involved when describing the characteristics of this kind of
digital/analog combination.

In the Ecstatic Service of Life -- Omega