On 9/9/22 1:28 PM, glen wrote:
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".
I thought maybe, but it was useful to have it explicated.
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.
I tend to assume a "graph" IS the supertype with something like "common
graph" or "simple graph" representing the familiar untyped or singular
edge/nodes.
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.
Yes, I did google the term to make sure I understood it to be what I
understood it to be... the "hyperbole of computation" perhaps? That
which is currently ineffable seems good as well.
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
Excellent and thanks I will take the bait (if not the hook) and read
through them as carefully as my current mental state (unknown even to
me) will support... and harangue them one-by-one perhaps in a separate
posting.
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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