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From: Tim May <tcmay.domain.name.hidden>

Date: Sat, 30 Nov 2002 11:45:23 -0800

On Friday, November 29, 2002, at 02:44 AM, Marchal Bruno wrote:

*> Stephen Paul King <stephenk1.domain.name.hidden> wrote:
*

*>
*

*>> I agree completely with that aspect of Bruno's thesis. ;-) It is the
*

*>> assumption that the 0's and 1's can exist without some substrate that
*

*>> bothers me. If we insist on making such an assuption, how can we even
*

*>> have a
*

*>> notion of distinguishability between a 0 and a 1?.
*

*>> To me, its analogous to claiming that Mody Dick "exists" but there
*

*>> does
*

*>> not exists any copies of it. If we are going to claim that "all
*

*>> possible
*

*>> computations" exists, then why is it problematic to imagine that "all
*

*>> possible implementations of computations" exists as well.
*

*>
*

*> But then you need to explain what "implemention" are. Computer
*

*> scientist
*

*> have no problem with this. There are nice mathematical formulation of
*

*> it.
*

*> Tim would say that an implementation is basically a functor between
*

*> categories.
*

*> You seen to want a material preeminent level, but this is more a source
*

*> of difficulty than an explanation. What is that level?
*

Bruno is right that I would emphasize the mathematics over the "COMP"

aspects. Computations are kinds of mathematics: mappings, iterations,

theorem provings, even topological operations of various kinds. Not all

mathematics is easily implemented on computers, but the principle is

clear.

I suppose I am partly a Platonist, in that I believe there's more to

mathematics than merely symbol manipulation (the Formalist) school.

Computers are exciting because they give us another way to make real

(or reify) the abstractions of mathematics. I believe, for example,

that categories (e.g., HILB or VECT) are in some sense "real," that we

can send our minds and our computers as robot explorers into these

"scapes" of Platonia, into the ideosphere, into noespace, or whatever.

(Sorry for waxing poetic...)

There is a sense in which the Platonist point of view is consistent

with the Chaitin/Wolfram notion that mathematics will become largely

explorational. Arguably, this has been what mathematics has _always_

been, that the process of discovering truths is not about proving

theorems from postulates, at least not exclusively. Even geometry got

its start not from considering abstractions out of a pure ideosphere,

but from issues of measuring the earth (geo-metry), of building

pyramids, of dividing farmlands, of measuring grain storage, and so on.

Later mathematics was also guided at least partly by the practical,

whether the study of differential equations or elliptic functions. Of

all of the possibly-provable truths, laid out like stepping stones in a

vast marsh of as-yet-unproved and possibly-unprovable truths, which

stepping stones are followed, and are laid in the marsh as new proofs

are obtained, is often shaped by engineering and physics

considerations. Even in the purest areas of mathematics, such as number

theory. The Chaitin argument that computers will be used increasingly

to explore this landscape is, I think, certainly correct. (Personally,

I am tremendously excited to think about what future versions of

Mathematica, for example, will look like when running on computers 100

or 1000 times faster than my current Mac and running with immersive VR

graphics systems. At Moore's Law rates of progress, I'll have this here

in my home within the next 10-15 years or so. This is my main interest,

more so than speculating on whether the universe runs on a computer or

not. But Everything issues touch on this...)

OK, so which is it, really, Platonism or Formalism? Paul Taylor makes a

good case in "Practical Foundations of Mathematics" that category

theory in general and topos theory in particular provide the

unification of these two points of view. Mathematical objects live in a

universe of categories, with certain rules for moving between

categories, and that various universes exist as toposes. We as humans

can manipulate these rules, learn how these objects behave, and thus

explore these spaces.

Now whether it makes sense (or "is really the case") to say that

Reality is some kind of computer program is not all clear to me. Like

many others, I have problems with the notion that reality is a program

running on some kind of metacomputer. Perhaps computation is woven into

the fabric of spacetime at a deep enough level, and perhaps there are

alternative "state machine" rules which could be imagined in other

universes (or even in different parts of our universe, e.g., changes in

rules at very high energies, or near singularities, etc.

I'm not--at this time--much engaged by the "universe as a computer

program" idea. A useful hypothesis to have--the

Zuse/Fredkin/Lloyd/Schmidhuber/Wolfram/etc. thesis, in its various

forms--but a long, long way from being established as the most

believable hypothesis. To me, at least.

(I think Egan gives us a fairly plausible, fictional timeline for

figuring this stuff out: a workable TOE by the middle of this century,

i.e., within our lifetimes. That is, a theory which unifies relativity

and QM, and which is presumably also brings in QED, QCD, etc. Perhaps

involving a mixture of string/brane theory, spin foams and loop

gravity, etc. Lee Smolin has some plausible speculations about how

these areas may come together over the next several decades. This TOE

is of course not expected to be truly a theory of everything, as we all

know: the phrase TOE is mostly about the unification of the two major

classes of theories noted above.

Then perhaps several centuries of very little progress, as the energies

to get to the Planck energy are enormous (e.g., compressing a mass

about equal to a cell to a size 20 orders of magnitude smaller than a

proton). Egan plausibly describes an accelerator the length of a chunk

of the solar system, using the most advanced "PASER" (the solid-state

lasing accelerators proposed recently), to accelerate particles to the

energies where discrepancies in models (computer programs??) might show

up. In one of his novels ("Diaspora") he has this happening a few

thousand years from now. This sounds about "right" to me. (I'll be

happy to give some of my reasons for "pessimism" on this timetable if

there's any real interest.)

Of course, breakthroughs in mathematics may provide major new clues,

which is where I put my efforts.)

I take the "Everything" ideas in the broader sense, a la Egan's "all

topologies model," a la the "universes as toposes" (topoi) area of

study, etc. My focus is more on logic and the connections between

topology, algebra, and logic. It may be that we learn that at the

Planck scale (approx. 10^-35 m) the causal sets are best modeled as

computer-like iterations of the spin graphs. But this is a long way

from saying consciousness arises from the COMP hypothesis, so on this

topic I am silent. As Wittgenstein said, "Whereof one cannot speak, one

must remain silent." Bluntly, don't talk if you have nothing to say.

Which is why I have little to say about the COMP hypothesis. I'll be

excited if evidence mounts that there's something to it. If the COMP

hypothesis has engineering implications, e.g., affects the design of AI

systems, this will be cool.

*>
*

*>> Could we not recover 1-uncertainty from the Kochen-Specker
*

*>> theorem of QM itself?
*

*>
*

*> Probably so.
*

*>
*

This seems to be assuming the conclusion. Gleason's Theorem and

Kochen-Specker are about the properties of Hilbert spaces. But the

reason we use the Hilbert space formulation for quantum mechanics, as

opposed to just using classical state spaces, is because the Hilbert

space formulation (largely of Von Neumann) gave us the "correct"

noncommutation, uncertainty principle, Pauli exclusion principle, etc.,

things which were consistent with the observed properties of simple

atoms, slit experiments, etc. In other words, the

Planck/Einstein/Heisenberg/Schrodinger/Bohr/etc. results and successful

models (e.g., of the atom) gave us the Hilbert space formulation, which

Gleason, Bell, Kochen, Specker, etc. then proved theorems about.

I don't think it would be kosher to assume reality has aspects of the

category HILB and then use theorems about Hilbert spaces to then prove

the Uncertainty Principle.

(My apologies if this was not what was intended by "recover

1-uncertainty.")

This is a good example, by the way, of how the physics applications of

Hilbert spaces incentivized mathematicians to study Hilbert spaces in

ways they probably would not have had Hilbert spaces just been another

of many abstract spaces. Gleason had many interests in pure math, so he

probably would have proved his theorem regardless, but Bell, Kochen,

and Specker probably would not have had QM issues not been of such

interest.

--Tim May

Received on Sat Nov 30 2002 - 14:50:41 PST

Date: Sat, 30 Nov 2002 11:45:23 -0800

On Friday, November 29, 2002, at 02:44 AM, Marchal Bruno wrote:

Bruno is right that I would emphasize the mathematics over the "COMP"

aspects. Computations are kinds of mathematics: mappings, iterations,

theorem provings, even topological operations of various kinds. Not all

mathematics is easily implemented on computers, but the principle is

clear.

I suppose I am partly a Platonist, in that I believe there's more to

mathematics than merely symbol manipulation (the Formalist) school.

Computers are exciting because they give us another way to make real

(or reify) the abstractions of mathematics. I believe, for example,

that categories (e.g., HILB or VECT) are in some sense "real," that we

can send our minds and our computers as robot explorers into these

"scapes" of Platonia, into the ideosphere, into noespace, or whatever.

(Sorry for waxing poetic...)

There is a sense in which the Platonist point of view is consistent

with the Chaitin/Wolfram notion that mathematics will become largely

explorational. Arguably, this has been what mathematics has _always_

been, that the process of discovering truths is not about proving

theorems from postulates, at least not exclusively. Even geometry got

its start not from considering abstractions out of a pure ideosphere,

but from issues of measuring the earth (geo-metry), of building

pyramids, of dividing farmlands, of measuring grain storage, and so on.

Later mathematics was also guided at least partly by the practical,

whether the study of differential equations or elliptic functions. Of

all of the possibly-provable truths, laid out like stepping stones in a

vast marsh of as-yet-unproved and possibly-unprovable truths, which

stepping stones are followed, and are laid in the marsh as new proofs

are obtained, is often shaped by engineering and physics

considerations. Even in the purest areas of mathematics, such as number

theory. The Chaitin argument that computers will be used increasingly

to explore this landscape is, I think, certainly correct. (Personally,

I am tremendously excited to think about what future versions of

Mathematica, for example, will look like when running on computers 100

or 1000 times faster than my current Mac and running with immersive VR

graphics systems. At Moore's Law rates of progress, I'll have this here

in my home within the next 10-15 years or so. This is my main interest,

more so than speculating on whether the universe runs on a computer or

not. But Everything issues touch on this...)

OK, so which is it, really, Platonism or Formalism? Paul Taylor makes a

good case in "Practical Foundations of Mathematics" that category

theory in general and topos theory in particular provide the

unification of these two points of view. Mathematical objects live in a

universe of categories, with certain rules for moving between

categories, and that various universes exist as toposes. We as humans

can manipulate these rules, learn how these objects behave, and thus

explore these spaces.

Now whether it makes sense (or "is really the case") to say that

Reality is some kind of computer program is not all clear to me. Like

many others, I have problems with the notion that reality is a program

running on some kind of metacomputer. Perhaps computation is woven into

the fabric of spacetime at a deep enough level, and perhaps there are

alternative "state machine" rules which could be imagined in other

universes (or even in different parts of our universe, e.g., changes in

rules at very high energies, or near singularities, etc.

I'm not--at this time--much engaged by the "universe as a computer

program" idea. A useful hypothesis to have--the

Zuse/Fredkin/Lloyd/Schmidhuber/Wolfram/etc. thesis, in its various

forms--but a long, long way from being established as the most

believable hypothesis. To me, at least.

(I think Egan gives us a fairly plausible, fictional timeline for

figuring this stuff out: a workable TOE by the middle of this century,

i.e., within our lifetimes. That is, a theory which unifies relativity

and QM, and which is presumably also brings in QED, QCD, etc. Perhaps

involving a mixture of string/brane theory, spin foams and loop

gravity, etc. Lee Smolin has some plausible speculations about how

these areas may come together over the next several decades. This TOE

is of course not expected to be truly a theory of everything, as we all

know: the phrase TOE is mostly about the unification of the two major

classes of theories noted above.

Then perhaps several centuries of very little progress, as the energies

to get to the Planck energy are enormous (e.g., compressing a mass

about equal to a cell to a size 20 orders of magnitude smaller than a

proton). Egan plausibly describes an accelerator the length of a chunk

of the solar system, using the most advanced "PASER" (the solid-state

lasing accelerators proposed recently), to accelerate particles to the

energies where discrepancies in models (computer programs??) might show

up. In one of his novels ("Diaspora") he has this happening a few

thousand years from now. This sounds about "right" to me. (I'll be

happy to give some of my reasons for "pessimism" on this timetable if

there's any real interest.)

Of course, breakthroughs in mathematics may provide major new clues,

which is where I put my efforts.)

I take the "Everything" ideas in the broader sense, a la Egan's "all

topologies model," a la the "universes as toposes" (topoi) area of

study, etc. My focus is more on logic and the connections between

topology, algebra, and logic. It may be that we learn that at the

Planck scale (approx. 10^-35 m) the causal sets are best modeled as

computer-like iterations of the spin graphs. But this is a long way

from saying consciousness arises from the COMP hypothesis, so on this

topic I am silent. As Wittgenstein said, "Whereof one cannot speak, one

must remain silent." Bluntly, don't talk if you have nothing to say.

Which is why I have little to say about the COMP hypothesis. I'll be

excited if evidence mounts that there's something to it. If the COMP

hypothesis has engineering implications, e.g., affects the design of AI

systems, this will be cool.

This seems to be assuming the conclusion. Gleason's Theorem and

Kochen-Specker are about the properties of Hilbert spaces. But the

reason we use the Hilbert space formulation for quantum mechanics, as

opposed to just using classical state spaces, is because the Hilbert

space formulation (largely of Von Neumann) gave us the "correct"

noncommutation, uncertainty principle, Pauli exclusion principle, etc.,

things which were consistent with the observed properties of simple

atoms, slit experiments, etc. In other words, the

Planck/Einstein/Heisenberg/Schrodinger/Bohr/etc. results and successful

models (e.g., of the atom) gave us the Hilbert space formulation, which

Gleason, Bell, Kochen, Specker, etc. then proved theorems about.

I don't think it would be kosher to assume reality has aspects of the

category HILB and then use theorems about Hilbert spaces to then prove

the Uncertainty Principle.

(My apologies if this was not what was intended by "recover

1-uncertainty.")

This is a good example, by the way, of how the physics applications of

Hilbert spaces incentivized mathematicians to study Hilbert spaces in

ways they probably would not have had Hilbert spaces just been another

of many abstract spaces. Gleason had many interests in pure math, so he

probably would have proved his theorem regardless, but Bell, Kochen,

and Specker probably would not have had QM issues not been of such

interest.

--Tim May

Received on Sat Nov 30 2002 - 14:50:41 PST

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