The Problem of Modelling the Mathematical Mind
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- Опубликовано: 28 июл 2024
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Speaker: Sir Roger Penrose, Mathematical Institute, University of Oxford
License: Creative Commons CC BY-NC-ND 3.0
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Following Alan Turing's ground-breaking 1937 paper, which introduced his notion of the Universal Turing machine, he suggested, in 1939, generalizations based on ordinal logic and oracle machines, these being apparently motivated by attempts to model the mathematical mind in a way that could evade the apparent limitations presented by Gödel's incompleteness theorems. In this talk, I introduce the idea of a "cautious oracle" as a more human version of Turing's oracles. Nevertheless, I show that even this fails to capture the essence of the full capabilities of our understanding.
I raise the issue of possible physical processes that would appear to be needed in order to circumvent these Gödel-type restrictions. At the end of the talk, I report on some startling new experiments, which appear to point to new insights into the possible physical processes underlying conscious brain activity, and I speculate on how this might relate to the power of human understanding. 
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The most vocal arguments were " the brain cannot produce quantum coherence because the brain is too warm and too wet"...from Krauss and Tegmark.
Recent discoveries show plants use QM for photosynthesis, so much for too warm and wet.
Penrose is not a theist but he doesn't allow atheism to cloud his common sense.
Krauss and Tegmark will have to eat their own rash words.
Yes, I’m with you. I’ve been saying for a long time that the next step beyond q physics and relativity will be accompanied by a new understanding of consciousness in physical terms.
Personally, my speculation is that our universe will be seen to have a pseudo 4 spatial dimensional structure with qcd responsible for the 3D shadows we can observe and consciousness as a higher dimensional analogue of light.
It's always a pleasure to hear from Professor Penrose
I wish smart ppl ditched such Hercules nonsense by looking deeper into meanings of operations for the universe and actually focused on how it works, stayed true to it and explained that with more clearness, density and precision
@@GEMSofGOD_com send me an email for my speculation: schutzer@jhu.edu
I wish Penrose and others focused on 51:05 - ~1:07:00 and ditched everything else in fact. It's all computable in distributions with human-scale representations. Probabilities are tracks from certain sources in reality, and it's all determined at the largest scale. The question is - what's this skeleton of the whole universe? This we want the most, denser, clearer, more precise.
We don't know if quantum mechanics is ultimately computable. Our current understanding and interpretation is incomplete. Penrose mentioned that Schrodinger himself, by way of his famous cat, understood his equation to say things that are counterintuitive, even absurd.
Modern lights at the cutting edge like t'Hooft believe, as Einstein, that Q.M. is incomplete (Bell's Theorem notwithstanding) -- or that it is an emergent phenomenon of something more fundamental. If it's determined to be computable, that still doesn't address the Godelian limitation on formal systems. It may be that our language (or any Godel-motivated expansion) is unable to fully characterize physics. If that's the case, then modeling the brain with language (i.e., programs) will always fall short. Ultimately, the gap may not be significant in practice, but that remains to be seen.
Another point: It's amusing to see people back-peddle on claims of quantum coherence in biological systems after recent experiments have proved it's existence. This was their biggest criticism of Penrose, and they reveled in sneering him into the crackpot corner over it. Now they completely ignore this aspect and claim that already known mechanism are enough to explain consciousness, although they haven't quite worked that out yet.
For me, I don't know. I do know that before Godel, if there were mechanical theorem provers (computers), they could never have proven Godel's result (they would have been happily working their way through Russel and Whitehead's Principia forever). But Godel transcended that by expanding the language, rules, and perspective sufficiently. The real question is: How was that done? If you can't answer that, what exactly are you modelling?
Simply good..
Penrose's none conventional presentation of Godel's incompleteness theorem is simply enlightening. 27:30 "understanding" isn't it Searle's "semantics" in the Chinese Room argument?
To be conscious or be aware involves being mathematical, and in that sense is conducive to selection.
Observing the tendency, that I usually check the remaining duration of videos to see when its over.. With penrose I check to see if there is enough time remaining for him to get to the point
Oddly enough, Gödel's let-out makes little sense. We need not know that a particular R is the one representing human intelligence in order to be able to believe G(R): we only need to meet R and trust it. What could make it impossible for us to meet that R? And if we meet it, how could we not trust it if it consists of the same principles we use in thinking? R could be very complicated but it's sure finite, so that there's no logical impossibility for us to peruse and trust it.
0:11:29 Note that for Gödel's theorem to apply, R must be consistent. So, if R is inconsistent, there's no G(R) which we must believe and R can't prove, because an inconsistent R would prove every sentence in its language, also G(R).
So, he's assuming we can't believe in the validity of an inconsistent system, and this seems empirically false: Frege, Church, Quine... proposed inconsistent systems.
The point could be, however, that no inconsistent system can represent human mathematical intelligence. On that asumption, for each R purportedly representing human intelligence and which we trust, there would be a G(R) (e.g. 'R is consistent'), which we can in principle believe and R can't, and this would show R doesn't represent human intelligence after all.
So, if an R exists that represents human intelligence, then it is inconsistent or unable to be trusted by human intelligence. I admit both alternatives seem unlikely.
+Laureano Luna
However unqualified, I cannot help making the following conclusions. Godel's statement begins with the paradox, like the liar or the barber's, in which the opposite is included into the premise, that then turns out to be inconclusive/incomplete, rather naturally, as expected.
What I find of value is called self-reference, that enables Turing to prove 'halting problem', explained by Penrose, as the result of finite axioms, whereas infinite axioms brings completeness (in the case of the natural numbers).
How all these makes sense, beats me.
I still have problem with inclusion of opposites in the premise, that is bound to be confusing.
Inclusion of opposites, was first made sense by Hegel (although in a different context) that went on to enable the discovery of positron by Dirac, and the following:
At the big bang dark matter and dark energy were produced with opposite properties, followed by the production of particles and anti-particles, followed by production of matter and anti-matter.... and on to hot and cold, male and female, up and down etc.
Hegel himself had no idea of anti-gravity or positron etc., yet his 'unity of opposites', enabled to give modern physics all the rational explanations of reality, and does not lead to any confusion, surprisingly.
Naimul Haq You're a bit confused, I'd say. Gödel's formula has nothing paradoxical or contradictory to it. Whether a formal system S proves or fails to a certain string of symbols is a mathematical definite problem that cannot be paradoxical. Imagine a formal system E generating English sentences. The sentence 'E does not generate this sentence' expresses a mathematically definite state of affairs and can't be paradoxical. In contrast, truth is not mathematically definite and this is why 'this sentence is not true' can be paradoxical.
Laureano Luna In other words, all knowledge is undecidable/paradoxical.
Naimul Haq This is too strong a thesis. And a counterintuitive one. I know I exist, I know two plus two equals four and a bunch of similar propositions. But I assume you mean something more concrete, don't you?
Laureano Luna
Indeed, duality is the essence of nature that enables self-organizing property of matter, making it possible for us to dig for truth, not incompleteness/undecidable.
The human brain employs quantum computation (doing many functions at the same time) producing consciousness and awareness over evolutionary scale, enabling us to do mathematics, just like a tiger chasing a deer calculates how fast it should move/run or what angle to take while chasing or to accelerate or decelerate.
When Turing solved the halting problem, he inadvertently incorporated 'understanding' into his computing machine.
But that computation is not done with the awareness of a mathematician. Animals and humans doing natural things which require spatial awareness do not conscientiously calculate using newthons laws, but rather have a unconsious sense of how the world works and how to get the expected outcome.
Understanding is the ability to judge that the mapping between reality and the abstract model of reality is sound.
And when it’s not sound? Then what?
What a lovely talk, deep and exhaustive within limits. The comments in the comment section doesn't do justice to Penrose's efforts. Most didn't seem to understand what he meant going beyond computation to grasp consciousness/awareness. What he precisely said is we may never know the 'infinite axiom algorithm' of quantum computing functions, which could let us have a peep into reality. Since we cannot deny consciousness, Penrose indicated the existence (without any proof or any inkling when he said 'there s more to the story') implying indirectly the existence of cosmic consciousness (carefully avoided by physicists). Elsewhere Penrose admits mathematics is grounded on 'faith and belief'. Personally I believe in divine design and purpose. Human consciousness is entangled with the 'mind of god', which only knows the algorithm, ensuring 'determinism'.
It’s not hard to construct a thought equivalent to no computation. For any x, say though t is *about* x iff t asserts ‘there are some P’ and x is a P. Call a computation c normal iff there’s no thought *about* c and equivalent to c. Consider thought T asserting just ‘there are some normal computations’. T is exactly *about* all normal computations. Assume for a reductio that T is equivalent to computation C (then, of course, there's no thought with other content that's equivalent to C). If C is normal, T is *about* C and C is not normal. If C is not normal, T is not *about* C and C is normal. Contradiction in either case. Hence, T is not equivalent to a computation. We say T diagonalizes out of all computations. (From Luna and Small: philpapers.org/rec/LUNIAC)
Laureano Luna I'm not formally educated in the theory of logic, but this sounds a lot like Russell's paradox to me.
Question @ 1:17:45, example of awareness without understanding ?
Answer: Thermostat.
Is he saying that induction cannot be used to prove that Hércules will end with all heads, even if the logic behind the solution is essentialy that of induction (every cut creates an ordinal number of lesser magnitude, and thus will end at zero, or something along those lines.)
It definately sounds like something one could express in an inductive way.
Why wouldn't it?
If some kind soul that knows the answer to this could shed some light on the subject, I would feel very grateful.
You might like to read his popular paperback," The emperors New Mind" , the book that started it all. He destroyed artificial intelligence based on classic computers with a sweep of the pen.
His books are very accessible, I quit school at 15.
Lawrence Krauss is a physicist, atheist, who hates the idea.
He called it nonsense, the brain is too warm and wet he claimed.
There are lots of them and they all look like fools now.
Penrose's interpretation of Godel's theorem doesn't find any takers among mathematicians. And they know better. Penrose is simply wrong about this. And he's nearly the only one. There are similarly people who don't understand how some infinities are larger than others. It's a personal sticking point
An argument from authority (who cares what the majority think, Penrose is as good if not better at math than anyone else). I do think that the Lucus-Penrose argument needs to be substantiated before it will command universal assent.
Thankyou.
2. Hercules vs Hydra:
velocity
if hercules cuts hydra heads faster than heads grow back, he wins.
if heads grow faster than hercules cuts, he doesn't.
density
if heads grow on base that doesn't grow, they will get too dense for new heads to fit.
if heads shrink in size to fit on base, they will get so small you can't see them.
(Penrose's proof seems to be they'll shrink to zero)
out of the box
if hercules cuts the base, all heads die.
"if the fundamental laws of physics can be modelled, then the human mind can be too"
Penrose points out very clearly there are gaping holes in quantum physics, holes that not reconcilable. He states very strongly these gaps cannot be argued away so Q physics is incomplete as yet.
The fundamental law of classical physics cannot account for consciousness.
Its not that its unlikely, its not possible.
Neither Classical nor Quantum physics addresses consciousness
Skepticism is not open minded or unbiased. When it involves atheism, then ego will fight for its life, to the point of making people look very stupid.
Penrose's approach eliminated what wasn't possible first, that left QM as the viable possibility.
In his second book he remarked how much of the critique was vicious and personal, not scholarly or scientific.
He's just following the evidence, something others could learn to do if they allowed themselves free will.
You wonder if you and a robot would get out of the way of a 🏃♀️ runaway train and talk about the experience of you both survived. Should you carry a screwdriver as backup. 💕
to believe is to know you believe to know you believe is not to believe
3. Penrose's cautious oracle doesn't do probability.
Caution implies care for uncertainty. Probabilistic.
We have electrical activity relative to the structure of the brain forming chemical changes or reactions. Chemical energy is stored in the bonds that hold the atoms together. When the bonds form and break we have the continuous exchange of photon energy with the future unfolding relative to the electrical activity and the structure of the brain. Conscious awareness is formed by this electrical potential that is always in ‘the moment of now’ in the centre of its own reference frame. It is this personalization of the brain being in ‘the moment of now’ in the centre of its own frame of reference that gives us the concept of ‘mind’ with each one of us having our own unique personal view of the Universe and the uncertainty of life. Consciousness is always in the forefront of creation or the creative process therefore each individual is able to look back in time in all directions from their frame of reference at the beauty of the stars. It is a common believeth that we live and die over a period of time, but in this theory the atoms of the periodic table form their own space time as part of a universal process of continuous creation. The flow of Time as a process of continuous energy exchange and our consciousness as a continuous stream of unbroken ever-changing flow of ideas, feelings! dreams! hopes! perceptions! and emotions are interlinked. Good video, thanks for sharing!!!
Why do people think consciousness is so wierd? Its equally wierd to ask why any arbitrary object exists, well, its quantum states all the way down, so what?
His argument is: assume human mind is a Turing machine => some mathematical derivation with Turing machines and Gödel's theorem => contradiction. Therefore, human mind is not a computer, AI not possible.
Problem is: there are a lot of mistakes in the mathematical derivation. That's what ultimately is the source of his contradiction, not the idea that the human mind is a Turing machine. A lot of people are confused by the halting problem and with the Gödel's theorem, it's hard to get your head around these things, they are complicated. He presents the proof in a very hand-wavy, informal way, gets a bit confused about the details and arrives at a contradiction.
The mistakes have been pointed out in detail, but Penrose has essentially ignored them or papered over them.
See: citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.518.8666&rep=rep1&type=pdf
The manchester computer would equal a modern computer processor if it had enough memory and was not bound by time-constraints. Turing was correct in a sense.
Even with a working model of quantum consciousness, it still doesn't explain consciousness itself. The so called hard problem.
I like the idea that it is an emergent property, its more than the sum of its parts (processes).
To me... the mind taps into the universe like a radio.
A radio doesn't generate any music, it merely receives it.
Consciousness is not an emergent property if it is able to be transmitted and received, it would rather be a fundamental physical quantity.
Issues:
1. Penrose: Mathematicians aren't evolutionary logical.
Observe infants (admittedly, he may not have any). They vigorously test reality, test their perceptions, test their understanding. Discover new functional methods. They've got immense topological, tensor, curve derivative, and other "math" curiousity and intuition. Turing didn't observe kids enough either.
AH YES PENROSE
The audience questions are just embarrassing. Smh
A Rule can never be trusted because it's a relative measure of timing rates forming a dynamic constant that shifts in relationships with the universal connection.., another version of "incompleteness".
What? no atheists arguing "it can't be true" ?
jonesgerard Well, why should they (we)? It's not like he argues for a "god" as such, or entities like "spirits". He simply points to (quite compelling) arguments for properties that animal brains have that traditional computers seemingly can't have according to Gödel/Turing. His solution suggests QM (or possibly a domain that QM is a subset of). I'm not sure he is right, but he surely can be. The answer might, as some people have pointed out, lie more in the inconsistency of the system than the incompleteness. There are a lot of things to explore. I guess time will show actually. We are on the brink of being able to simulate parts of small animal brains (though far from in real time of course). I guess we should be able to get some minor results in the next dacades.
@@egilsandnes9637 There's an assumption that human intellect is an extension of animal intellect also encounters issues. As even trained higher primates in a controlled setting fail to demonstrate syntactic; relational conceptual ideas (Chomsky's syntax test) which is unique and only found in humans; and the most robust study on the matter by Herbert Terrace in 1974 the Chimapzees were able to master 125 signs, and could be trained to use those signs precisely as his trainer indicated, there was no evidence that they had any syntactical awareness, understanding, or control over the use of his signs. Terrace, et al, summarized their results as follows:
"...Unless alternative explanations of an ape’s combinations of signs are eliminated, in particular the habit of partially imitating teachers’ recent utterances, there is no reason to regard an ape’s multisign utterance as a sentence. … For the moment, our detailed investigation suggests that an ape’s language learning is severely restricted. Apes can learn many isolated symbols (as can dogs, horses, and other nonhuman species), but they show no unequivocal evidence of mastering the conversational, semantic, or syntactic organization of language"
@@egilsandnes9637 Also in his book he noted how viciously he was attacked by atheistic scientists primarily on a personal level rather than engaging in his arguments. So clearly the concept ruffled some Atheists feathers.
@@fiveadayproductions987 imagine quoting Chomsky's work on syntax in 2021. good lord. your age - and lack of self-motivation - is showing. not disagreeing (or endorsing) your point but uh you need to update your works cited lmao.
He was doing well until he got on to the 1939 paper. Then he just started pointing at quantum mechanics and appealing to ignorance.
I mean to say, there is no reason to think that quantum mechanics is even related to computation mechanics. This is evident when you realize that one can build a computer out of any system of distinguishments.
Your ignorance presumably.