There's a lot to cover, here. But I want to focus on the two "hypers". No, Wolpert's
reference to hypercomputation is unrelated to the idea that hyper[meta]graphs might be more
expressive than other "languages". To the best of my abilities, hyper[meta]graphs merely
allow for the expression of multiple edge and node types, expanding it, as a graphical system, into
the domain already expressible by algebraic languages. I could easily be wrong.
Hypercomputation, which Wolpert only mentions in passing, is simply an attempt
to talk about things we can't talk about ... effibility, I suppose. Again, to
the best of my knowledge, every claim to hypercomputation has been thoroughly
debunked. Wolpert only uses it to help talk about things we don't know how to
talk about.
But *if*, by some miracle, anyone wants to talk about the 12 questions Wolpert
asks, I'd love it. Here it is again:
https://arxiv.org/abs/2208.03886v1
For convenience, the 12 questions are:
1. Why is there a major chasm with the minimal cognitive capabilities necessary
for survival by pre-Holocene hominids on one side, and on the other side, all
those cognitive capabilities that Kurt Gödel, Albert Einstein, and Ludwig van
Beethoven called upon when conjuring their wonders?
2. Restricting attention to what are, in some sense, the most universal of
humanity's achievements, the most graphic demonstrations of our cognitive
abilities:
Why were we able to construct present-day science and mathematics, but no other
species ever did? Why are we uniquely able to decipher some features of the
Cosmic Baker's hands by scrutinizing the breadcrumbs that They scattered about
the universe? Why do we have that cognitive ability despite its fitness costs?
Was it some subtle requirement of the ecological niche in which we were formed
— a niche that at first glance appears rather pedestrian, and certainly does
not overtly select for the ability to construct something like quantum
chronodynamics? Or is our ability a spandrel, to use Gould and Lewontin’s
famous phrase — an evolutionary byproduct of some other trait? Or is it just a
cosmic fluke?
3. Are we really sure that no other species ever constructed some equivalent of
present-day SAM? Are we really sure that no other apes — or cetaceans or
cephalopods — have achieved some equivalent of our SAM, but an equiva- lent
that we are too limited to perceive?
4. If the evidence of the uniqueness of our SAM is the modifications that we,
uniquely, have wreaked upon the terrestrial biosphere, should the question
really be why we are the only species who had the cognitive abilities to
construct our SAM and were able to build upon that understanding, to so
massively re-engineer our environment? To give a simple example, might some
cetaceans even exceed our SAM, but just do not have the physical bodies that
would allow them to exploit that understanding to re-engineer the biosphere in
any way? Should the focus of the inquiry not be whether we are the only ones
who had the cognitive abilities to construct our cur- rent SAM, but rather
should the focus be expanded, to whether we are the only ones who had both
those abilities and the ancillary physical abilities (e.g., opposable thumbs)
that allowed us to produce physical evidence of our SAM?
5. Ancillary abilities or no, are we unavoidably limited to enlarging and en-
riching the SAM that was produced by our species with the few cognitive
abilities we were born with? Is it impossible for us to concoct wholly new
types of cognitive abilities — computational powers that are wholly novel in
kind — which in turn could provide us wholly new kinds of SAM, kinds of SAM
that would concern aspects of physical reality currently beyond our ken?
6. Is possible for one species, at one level of the sequence of {computers run-
ning simulations of computers that are running simulations of ...}, to itself
simulate a computer that is higher up in the sequence that it is?
7. Is the very form of the SAM that we humans have created severely con-
strained? So constrained as to suggest that the cognitive abilities of us hu-
mans — those who created that SAM — is also severely constrained?
8. Is this restriction to finite sequences somehow a necessary feature of any
complete formulation of physical reality? Or does it instead reflect a lim-
itation of how we humans can formalize any aspect of reality, i.e., is it a
limitation of our brains?
9. In standard formulations of mathematics, a mathematical proof is a finite
sequence of “well-formed sentences”, each of which is itself a finite string of
symbols. All of mathematics is a set of such proofs. How would our per- ception
of reality differ if, rather than just finite sequences of finite symbol
strings, the mathematics underlying our conception of reality was expanded to
involve infinite sequences, i.e., proofs which do not reach their conclu- sion
in finite time? Phrased concretely, how would our cognitive abilities change if
our brains could implement, or at least encompass, super-Turing abilities,
sometimes called “hyper-computation” (e.g., as proposed in com- puters that are
on rockets moving arbitrarily close to the speed of light [1])? Going further,
as we currently conceive of mathematics, it is possible to em- body all of its
theorems, even those with infinitely long proofs, in a single countably
infinite sequence: the successive digits of Chaitin’s omega [69]. (This is a
consequence of the Church — Turing thesis.) How would mathe- matics differ from
our current conception of it if it were actually an uncount- ably infinite
collection of such countably infinite sequences rather than just one, a
collection which could not be combined to form a single, countably infinite
sequence? Could we ever tell the difference? Could a being with super-Turing
capabilities tell the difference, even if the Church — Turing thesis is true,
and even if we cannot tell the difference?
Going yet further, what would mathematics be if, rather than countable
sequences of finite symbol strings, it involved uncountable sequences of such
symbol strings? In other words, what if not all proofs were a dis- crete
sequence of well-formed finite sentences, the successive sentences being
indexed by counting integers, but rather some proofs were contin- uous
sequences of sentences, the successive sentences being indexed by real numbers?
Drilling further into the structure of proofs, what if some of the “well-formed
sentences” occurring in a proof’s sequence of sentences were not a finite set
of symbols, but rather an infinite set of symbols? If each sentence in a proof
consisted of an uncountably infinite set of sym- bols, and in addition the
sentences in the proof were indexed by a range of real numbers, then (formally
speaking) the proof would be a curve — a one-dimensional object — traversing a
two-dimensional space. Going even further, what would it mean if somehow the
proofs in God’s book [5] were inherently multidimensional objects, not
reducible to linearly ordered sequences of symbols, embedded in a space of more
than two dimensions? Going further still, as mathematics is currently
understood, the sequence of symbol strings in any proof must, with probability
1, obey certain con- straints. Proofs are the outcomes of deductive reasoning,
and so certain sequences of symbol strings are “forbidden”, i.e., assigned
probability 0. However, what if instead the sequences of mathematics were
dynamically generated in a stochastic process, and therefore unavoidably
random, with no sequence assigned probability 0 [106, 32, 44]? Might that, in
fact, be how our mathematics has been generated? What would it be like to
inhabit a physical universe whose laws could not be represented unless one used
such a mathematics [39, 53, 54]? Might that, in fact, be the universe that we
do inhabit, but due to limitations in our minds, we cannot even conceive of all
that extra stochastic structure, never mind recognize it? As a final leap,
note that all of the suggested extensions of the form of cur- rent human
mathematics just described are themselves presented in terms of ... human
mathematics. Embellished with colloquial language, I de- scribed those
extensions in terms of the formal concepts of uncountable in- finity,
multidimensionality, Turing machines, and stochastic processes, all of which
are constructions of human mathematics involving finite sets of finite
sequences of symbols. What would a mathematics be like whose very form could
not be described using a finite sequence of symbols from a finite alphabet?
10. Is it a lucky coincidence that all of mathematical and physical reality can
be formulated in terms of our current cognitive abilities, including, in par-
ticular, the most sophisticated cognitive prosthesis we currently possess:
human language? Or is it just that, tautologically, we cannot conceive of any
aspects of mathematical and physical reality that cannot be formulated in terms
of our cognitive capabilities?
11. Are there cognitive constructs of some sort, as fundamental as the very
idea of questions and answers, that are necessary for understanding physical
re- ality, and that are forever beyond our ability to even imagine due to the
limitations of our brains, just as the notion of a question is forever beyond a
paramecium?
12. Is there any way that we imagine testing — or at least gaining insight
intowhether our SAM can, in the future, capture all of physical reality? If
not, is there any way of gaining insight into how much of reality is forever
beyond our ability to even conceive of? In short, what can we ever know about
the nature of that which we cannot conceive of?
On 9/9/22 11:33, Steve Smith wrote:
glen wrote:
Jochen posts more questions than answers. Even EricS' conversations with Jon
about the expressive power of hypergraphs shows an impetus to circumscribe
what's computable and what's not. I mentioned a Wolpert paper awhile back,
wherein he gives some air to hypercomputation, to which nobody on the list
responded. And you've even defended brute force computation by highlighting the
progress and efficacy of techniques like Monte-Carlo simulation.
I remember your reference and waded as deeply into it as I could before my cerebro-spinal fluid got saturated with lactic acid (or depleted of ketones?) and
remember hoping/trusting that someone with fresher fluid (or more of it) would pick up the discussion and help me take a go at it with more parallax or maybe
only once-rested. I'm not clear on how/if you mean that EricS/JonZ's "expressive power of hypergraphs" relates directly to Wolpert's cogitations on
"hypercomputation"? I *do* connect hypergraph thinking to Simon's "nearly decomposable" systems and think if there might be a specific
link between the two hypers (graph/computation) it might be in the definition (and relevance) of "nearly"? This refers back to the "nearly
random" or "nearly noise" or "mostly noise" or "irreduceable limit" to noise.
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