[FRIAM] signal and noise

[glen]

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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