[FRIAM] signal and noise

[Steve Smith]

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.
>>
>