The way this is usually presented is - that we can simulate physics on a computer. So what is to stop us eventually simulating your whole body including your brain? And if so, is it not just a matter of time, and increasing computer power before we have exact simulations of humans as computer programs? Programs whose behaviour is indistinguishable from humans?
This is a staple of many science fiction stories of course. But some logicians, philosophers and physicists think there are flaws in this argument.
We know the laws of physics are incomplete. Could there be physical processes which for some reason are impossible to simulate using a computer program? And could processes like that go on in a human being?
This is what Roger Penrose thinks.
Roger Penrose, professor at Oxford university and theoretician in quantum gravity - who gave detailed arguments for the view that human brains can't be simulated with a programmable computer.Author of "The Emperor's New Mind" and "Shadows of the Mind" and other publications where he explores these ideas.
His work follows on from work by Kurt Gödel himself
Kurt Gödel who proved the famous "incompleteness theorems" showing that maths can never be completely described in a fixed set of axioms and deduction rules - he thought that humans could not be simulated using finitely programmable machines.He explores these philosophical ideas in "Some basic theorems on the foundations of mathematics and their implications" -
Then later by the philosopher John Lucas who also thinks that human understanding of maths can't be reduced to the actions of a finitely describable computer program.
John Lucas, author of ‘Minds, Machines and Godel’, in which he argues that an automaton cannot represent a human mathematician.
The main thing Roger Penrose contributed, as a quantum physicist, was to suggest possibilities for physical processes involved in the brain, and to present a much more detailed argument from Godel's result to his conclusions.
I'm not going to talk so much here about his arguments in detail, as rather, what the implications are if he turns out to be right - whatever you think about the actual arguments used for the conclusions. But I'm going to say a bit about the arguments themselves as well.
SIMPLE SEEMING TASKS THAT CAN'T BE SIMULATED BY A COMPUTER PROGRAM
This claim is not as wild as it might seem because there are some quite simple seeming tasks that can't be simulated by a computer program. The one all the arguments here focus on is, recognizing mathematical truths.
You might think it was easy enough to program a computer to prove the same things a human can prove, for instance about numbers. To check our proofs and come up with new proofs of its own.
HUMAN "OUT OF THE BOX" THINKING
But turns out - that it's harder than you think. You can set out a simple set of axioms and deduction rules - but the thing is that humans can always "think out of the box" and that out of the box thinking lets you see that some things have to be true, which a computer program caught in the box of those rules you wrote down can never find out.
And - it's not enough to make a program that is self modifying - that also has limits because at some level you have to specify how it does its self modifications - and that then puts limits on its creativity, it is still incapable of "out of the box thinking" about its own self modification rules.
To duplicate this "out of the box" thinking that humans are always capable of, a program needs to be endlessly creative adding new axioms and continually tweaking its program in a way that can never itself be programmed.
That's basically what Godel proved in his first incompleteness theorem.
So - that's a kind of plausibility argument that no finite computer program can understand truth in the way a human does, because all such programs are limited in some way - and how could there be some similar limit on what humans can see to be true?
I'm not saying yet that this is rigorous. But - irrespective of whether we can make this into a rigorous, watertight argument - what if it turns out that the conclusion is true?
This happens all the time in maths. When you try to prove something, your first few attempts may fail - but still - that doesn't mean at all that your conclusion is false.
So, what if there is indeed some non computable process going on here, when humans understand mathematical truth (or indeed, any kind of a truth)?
What would that say about physics and artificial intelligence?
POTENTIAL FOR PHYSICS THAT WE DON'T KNOW ABOUT WITHIN LIVING CREATURES
Of course this is no use if you think that we already fully understand all the laws of physics that are going on at the level of living creatures. Because then our behaviour would still be capable of a computer simulation no matter what we think is going on.
You might think at first, that the only scope for new fundamental physics would be in the likes of black holes, neutron stars, and the like, or high speed particle collisions such as the LHC, ultra low or ultra high temperatures and so on.
But - some physicists think there may also be new physics at a more ordinary level. For instance, some think that this could happen whenever you have enough mass intereacting - a Planck's mass worth (about 1019 times the mass of a proton, it's 21.767 micrograms) - that's around the same as the mass of a human eyelash.
Measure the mass of an eyelash with a DIY microbalance
Enthusiast measures human eyelash at about 35 micrograms. The Planck mass - amount of mass needed for "spontaneous collapse gravitational" is 21.767 micrograms, so around the same mass as an eyelash.
Roger Penrose thinks that as soon as you have as much mass as this in an indeterminate state - that it has to "collapse" - has to "make up its mind" where it is - and that when that happens, he thinks something non computable happens, something that can't be simulated with a computer program.
This suggestions comes as a result of the problem of an observer in quantum mechanics. The idea that everything is somewhat fluid until you observe it. An electron can be in a "superposition of many states" until you make an "observation" - and then when that happens then it "collapses" into a single state.
In the classical "two slit" experiment, if you observe which of two slits an electron goes through, it always goes through one or the other. But if instead you look to see where it hits a screen, the other side of the two slits - then repeat this experiment many times, you get a diffraction pattern effect, that can only be explained by saying that all the electrons go through both of them at once.
So - who or what does this "observation" that collapses the electron into some particular state? There is no properly accepted explanation in quantum mechanics, and possibly if we get an answer to this, it might take us into new physics.
There are many ways of attempting to solve it.
ORCHESTRATED REDUCTION - ROGER PENROSE'S IDEA
Roger Penrose's idea - shared by other quantum physicists - is that you get a spontaneous collapse whenever the mass of interacting particles is large enough. This makes it an example of an objective collapse theory for quantum mechanics.
So in this case then by observing the electron we couple the entire mass of ourselves, experimental apparatus etc to the state of the electron - which then forces it to go into one or other state.
Well - if that happens - then there would need to be new physics to explain in detail how, and when, this collapse happens - and what leads to it collapsing one way rather than another. This then could be something that is going on all the time, quite ordinary - not in neutron stars or black holes or particle accelerators - that we don't yet understand.
Then the Planck's mass - it's the amount of matter needed to form a mini black hole, if you compressed all that matter to a single point. So - as best I understand it - the idea is that - when you get enough matter involved in superposition of states so that if it was all in one point it would form a black hole - at that point some "decision" has to be made about where it is. So you get a spontaneous collapse into one of the numerous possible states, and this is the part of the process where something non computable would enter in through the interaction of gravity with quantum mechanics.
Then the particles start interacting together again until you get another Planck's mass of interacting matter - at which point it has to collapse again and so it continues.
Roger Penrose proposes to tie this to quantum superpositions going on inside neurons - and also quantum states spanning several neurons - to explain how the brain is able to go beyond the limitations of computable physics.
He ties this to quantum superposition in microtubules - these are tiny structures within individual neurons. He thinks that an individual neuron is vastly more complicated and plays a more important role in our thinking than the simplified "neurons" of neural nets.
Staining of microtubules in a cell fixed by means of anti -beta - tubulin antibodies
These patterns show how microtubules may engage in some form of "computation" using processes similar to the way cellular automata like Conway's game of life work.
It is a controversial theory. First, it's based on the idea of objective collapse which is just a theory at present which we are a long way from being able to verify or disprove, when we can't yet even entangle more than a few qubits, never mind 1019 of them. Also though, at firt many thought that it was impossible that there are any large scale quantum processes going on in warm conditions such as our brains. But - there is a fair amount of evidence now of different types of "warm quantum effects". See for instance Evidence that photosynthesis efficiency is based on quantum mechanics
For the latest on their theory, a paper they wrote earlier this year lead to several news stories: see Discovery of quantum vibrations in microtubules inside brain neurons corroborates controversial 20-year-old theory of consciousness and Discovery of quantum vibrations in 'microtubules' inside brain neurons supports controversial theory of consciousness
This is not the only way that new physics could arise on the ordinary level of the human brain, though. I don't think the idea of non computable physics in the brain should stand or fall depending on whether Orchestrated Reduction is confirmed or disproved. It's just one way it could happen. It could be that we just don't know enough yet to work out the physics of what is going on. Maybe it doesn't even involve quantum mechanics, maybe something else.
PLAUSIBILITY ARGUMENT - THAT OUR NEURONS MUST AT LEAST BE MORE COMPLEX THAN "NEURAL NET" NODES
Recently there was a news story: "Supercomputer models one second of human brain activity" about a super computer that modeled a neural net thought to be as complex as the human brain in terms of the number of neurons. It took it forty minutes to simulate one second of neural activity.
However, a single cell creature such as an amoeba moves around, has complex decisions to make, do I go here or there, eats food etc.
Amoeba Eating - and rejecting food
If you modeled the behaviour of an amoeba with a neural net you would need thousands of neurons. But it doesn't have any.
So - this is just a plausibility argument - it's not proof, and just my own suggestion.
But - would like to suggest that it would be absurd if our brains did not take advantage, at some level, of these decision making and problem solving capabilities of a single cell that all the neurons have.
If our brains were as simple as these neural nets suggest - then a creature with a single neuron that takes advantage of its internal structure would be easily outsmarted by another creature with a brain with many thousands of neurons which it treats as just nodes in a neural net.
NOBODY HAS PROVED THAT ALL THE LAWS OF PHYSICS ARE COMPUTABLE
Whether or not you think Roger Penrose has succeeded in this proof - certainly nobody has proved in the other direction that all the laws of physics have to be computable.
So - until someone comes up with a totally convincing proof either way, we are free to take either possibility as a hypothesis. There is no a priori reason to suppose that all physics has to be computable.
So, let's start there, by looking at the consequences of non computable physics, and especially non computable processes going on in us..
SO, WHAT IF THIS IS RIGHT, THAT THERE ARE NON COMPUTABLE PHYSICAL PROCESSES GOING ON IN LIVING ORGANISMS?
The first consequence would be, that we will never simulate physics completely with a programmable computer machine.
That includes also, quantum computers if they act according to the currently known laws of physics with entangled qbits - as well as hardware neural nets, and machines that use probabilistic methods to solve things (e.g. probabalistic annealing), or massively parallel computers. All of those are logically equivalent to a Turing machine and can't introduce anything essentially new. All they can do is to speed things up, make our computers faster.
So - then none of those types of computers would be able to completely simulate the workings of a human being.
COMPUTER PROGRAMS CAN STILL BE BETTER AT MANY TASKS THAN HUMANS
Obviously computers are better at calculations. They can also beat humans at chess, now, since Deep Blue won against Kasporov. And may in the future be better at driving cars than us. We may rely on them more and more for various aspects of our lives.
But the computer program that beats you at chess has no idea what chess is, or what a chess piece is. It couldn't discuss the match with you, or recognize a chess game in a photograph. It doesn't understand anything. All it can do is follow instructions.
Our programs so far are good at many things, and far better at us at quite a few things, but I think fair to say, that they don't really "understand" anything in the way that humans understand them.
If Roger Penrose is right, then no programmable computer can ever understand mathematical truth. If so - perhaps arguably they can never really understand anything at all, just follow the rules programmed into them.
WHAT DOES NON COMPUTABLE MEAN HERE
It might help to have an example of some non computable functions. Because it is easy to think that our computers can simulate everything, if we can just make them fast enough. That's not true though.
There are many problems in maths that our computers have no hope at all of solving, because they are non computable problems.
A simple example is to calculate the tiling function. This is a function that given any finite set of square tiles, with various indentations around the sides that constrain how they fit together - tells you a value k for the largest k by k square pattern of tiles it can tile.
No computer will ever be able to evaluate this. That includes quantum computers, artificial neural nets, and machines that incoroporate random processes. It's because of the existence of Wang tiles which can be used to simulate the behaviour of any Turing machine.
Because the halting problem is non computable, then there is no computable function that, given a finite set of Wang tiles, can put any bounds on the size of the largest region it can tile.
That is to say - to make this clearer - we can simulate Wang Tiles in a computer of course. But if some process in nature was able to take any finite set of Wang Tiles as input - and somehow output a tiling of the largest region it can tile - that is something our computers can never simulate.
For more examples, see
Non-computable but easily described arithmetical functions
Roger Penrose thinks that when we understand the notion of mathematical truth (and so presumably also when we understand truth generally) we are making a similar non computable leap that our programmed computers will never be able to achieve.
WHAT ABOUT BIOLOGICAL "COMPUTERS" AND COMPUTERS BASED ON PHYSICS WE DON'T UNDERSTAND YET?
What if we use actual biological neurons in our computers? There is already work underway on this, using biological neurons for computing.
This goes back to 1999 - this computer is made out of neurons taken from leeches - and it can add up - a very primitive computer - the neurons are in a petri dish, each neuron represents a different number, and when connected together they can "add up".
I don't think any simple neural net made of actual neurons in place of hardware neurons would do something non computable. But - what about use of actual neurons in future with better understanding of how they work and how they communicate, maybe based on Roger Penrose's theory or some other future version of it?
Or slime moulds - there is research right now into using slime moulds for computers.
Computing with slime: Logical circuits built using living slime molds
Or - we might create a computer that we think is a quantum computer - but nobody is quite sure how it works. Indeed we have one of those already. The D-Wave Systems quantum computer.
Quantum Computing - D-Wave's quantum computer, which seems to work, and is very fast also, but nobody seems to be entirely clear how it works.
Nobody quite knows how it works, but it does seem to, in independent tests.
Now chances are it is just a quantum computer - so faster - but essentialy doing the same sorts of things as ordinary programs can do - but massively multi-tasking basically.
But what if some future D-Wave systems type company constructs a computer that they think is a quantum computer - but instead it is operating according to whatever principles work in the human brain - some kind of underlying layer of operation that they don't understand, and didn't build in intentionally -but it is there and helps their computer work.
Such a machine, might, just possibly, be able to understand truth and falsity, and be in some way aware.
MORE THAN NON COMPUTABILITY NEEDED
Just triggering something non computable wouldn't be enough - hard though that is. It then has to start acting in a coherent sentient way. It is surely far more likely to be random, chaotic, or patterned in some way that is intricate but makes no sense.
So something more is needed, surely. But what?
Perhaps if we somehow come to learn how this works in the brain, maybe we can then apply that to make machine sentience?
NON PROGRAMMABLE ARTIFICIAL INTELLIGENCES
Anyway, whatever method we use, if they use non computable physics - that means we can't "program" them in the ordinary sense - and can't reduce their operations to a computer program.
This is the example of the "artificial intelligence baby".
One day, it "Wakes up" with an understanding of truth. But - what would our world be like to it?
UPLIFTED CREATURES
Another example of this could be the process of "uplift" as explored in David Brin's novels and his Uplift Universe.
Perhaps we can do this at an earlier stage than the other ideas - as we don't need to really understand how the brain works or is organized in detail, or how DNA leads to the brain. Rather figure out what in our DNA leads to particular structures and features in the brain that let us understand speech (for instance) - and then find a way to incorporate those into other animals such as dolphins.
In that way we could maybe end up with dolphins and other creatures able to speak, and understand human speech - without us ever truly understanding how it works.
OUR AI BABY GROWING UP
So, if this was ever possible, I think it means that an AI, whether a computer based on non computable physics, or a neural net using actual neurons, or a slime mould computer, or an uplifted creature, would be more like a baby learning things for the first time, with no way to program "prior knowledge and understanding" into them. Because what is going on is something that can't be programmed.
So then, that raises major ethical dilemmas. If we do manage to create physical systems that can truly "understand truth" - then I think when they "wake up" they will be confused and not understand how to make sense of their situation.
Even if we don't build pain receptors into their systems (and we might well in order to help them respond to things likely to damage them) - they will have this idea of truth, and falsity - and a wish to know what is true. And that then will lead also to frustration - how do you know what is true and what isn't? So we would be creating a creature that can suffer. So we have a responsibility to it.
Same also with uplift - that if you modify say a dolphin, so that it can understand human speech - and communicate with us. It's a bit different, a dolphin already, I'd say, "understands its world" that it lives in - it knows what other dolphins are, what the sea is, in some sense it "knows things".
It's not just running a program. Surely if humans "truly understand" things - then dolphins also - and indeed other living creatures, even dogs, even ants, in a way - they are not just following a program, but, they are relating to things, in some sense they understand things, if in a dim way compared to ourselves. They understand e.g. what is food, they have friends, they know things about their world that they live in.
So, I don't think uplift here gives them an ability to understand that things are true or not.
But if you then modify its DNA and biology so it can speak to humans and understand as we do - and have similar capabilities to us - you end up with a creature that is part dolphin part human in its intelligence. And it's going to have many problems - though as well - maybe many insights as well.
ETHICAL RESPONSIBILITIES TO OUR AI BABIES
So - not saying we shouldn't do this ever. But if we do - it's again like having a baby - we would have a responsibility to the uplifted creatures, just as to the AIs and can expect them to have a difficult time as they mature and try to make sense of the world they are born into.
In David Brin's novels then the ETs that uplift other creatures speak for them and have responsibilities towards them in the galactic community. I imagine that it would be a bit like that - in the future, species that have been uplifted will know who did the uplifting (usually) and will have - maybe some gratitude, but their "species parents" as it were also have responsibilities towards them as well.
ARTIFICIAL INTELLIGENCES TAKING OVER THE WORLD
As for the problem of artificial intelligences taking over the world and doing harmful and malicious things - well I think that happens later.
If we do manage to give birth to these "AI babies" as a civilization - then also bring them to maturity - then they are our responsibility also, to give them as good an upbringing as we can. Just as with parents and children, some of our ideas and values will end up influencing them. But also - they'll be influenced by the whole civilization they grow up in.
If they have a problematical upbringing then they could become problematical AI adults - just the same as with humans.
If this ever happens we'd have got to know them well at this stage, and them us. And developed mutual trust (hopefully).
And if brought up well, then they may be valuable parts of our society as adults. And - hopefully if there are many of them they can help each other to mature as well, as some of them reach adult hood, they may help their less mature "newborns".
EXAMPLE OF A BABY AXIOM SYSTEM THAT A COMPUTER PROGRAM COULD NEVER COMPLETELY UNDERSTAND IF PENROSE, LUCAS AND GODEL ARE RIGHT
The maths we are talking about here need not involve abstract difficult to understand concepts. Godel proved his result for one of our simplest theories with just a few axioms, the axioms of elementary number theory.
Then later, it was found out that you can manage with an even simpler theory. All you need to have is an axiom system powerful enough to be able to add 1 to any number, has a 0, and lets you define addition and multiplication.
This "baby" system makes your mathematical proofs cumbersome - still - it's enough for Godel's results.
For the mathematicians amongst you, here are the actual axioms, of Robinson's Arithmetic - one of the simplest axiom systems of the type that Penrose and the others are talking about.
It's an axiomatization of the positive numbers (no negative numbers or fractions), and it goes
1. You can't get to 0 by adding 1 to any other number
2. If x+1 = y+1 then x = y
3. Every number other than 0 can be got by adding 1 to another number
4. x+0 = x
5. x + (y+1) = (x + y)+1
6. x*0 = 0
7. x * (y+1) = x*y + xThere, axioms 1 tells you what 0 is, and 2 to 3 tell you how the successor operation, of adding 1 works. Axioms 4 and 5 tell you how addition works, and axioms 5 and 6 tell you how multiplication works.
Amazingly - this alone, plus ordinary rules of "first order logic" so you can say things like "There exists x..." or "For all x ..." and use logical deduction - is enough to be beyond the grasp of our computer program, according to Penrose, and Godel indeed.
A program would have no trouble at all checking proofs that it can carry out with those 7 axioms, and proving new results.
But - turns out - that humans can look at this list of axioms and come up with new results about the numbers by reasoning "about the axioms and proof methods " - and a computer program programmed just with these axioms simply can't make that deductive leap and see that those things are true.
So in that sense, it doesn't really understand the axiom system, it is just following rules.
This is a general result for any theory that is rich enough to do Godel's proof construction. It doesn't have to have numbers in it as such, so long as we can find something in it that we can reintepret as numbers.
Now this result is not true of all theories. Ordinary Euclidean geometry is an example of a theory that is complete and decidable. It is a rich and complex theory - but not quite complex enough so that we can interpret arithmetic within it.
So - we could write a computer program to explore Euclidean geometry - and though "out of the box" thinking would help it to find shortcuts to proofs of the results - still, anything we can prove as humans can be put into a proof that our computer program can understand.
But once you have anything resembling numbers with addition and multiplication - then these "out of the box" Godel sentences arise.
See Gödel's Incompleteness Theorems (Stanford University of Philosophy).
RECOMMEND STANFORD ENCYCLOPEDIA FOR PHILOSOPHY OF MATHS
Note - that if you want to find out more about philosophy of maths - I would recommend the Stanford Encyclopedia of Philosophy - it is far better than Wikipedia in this area.
Wikipedia is good in formal logic - but when it comes to philosophy and for that matter the incompleteness theorems the Stanford Encyclopedia is currently far better, and Wikipedia tends to have inaccuracies as well. I expect this will change as Wikipedia improves - but - I think it's likely to be one of the harder areas for Wikipedia to cover.
SKETCH OF WHAT HUMANS CAN FIND OUT
Godel found a way to represent proofs, and theorems, as numbers within the theory. Let's just put them in single quotes, so 'q' is the number that encodes the theorem or proof q.
That's not so hard to understand nowadays. The text I'm writing just now is stored in the computer, not as letters, but as numbers. And - if I write out a proof, and then save that to a file - the file itself could be encoded as a single large number, easily, in many ways. S
Then - using various mathematical tricks, he managed to find a complicated arithmetical expression Provable('q'), a statement about the number 'q' just involving additions, multiplications, variables and logic, says nothing about proofs directly, but we can see that it is true if, and only if, q has a proof in our theory.
But then - here the maths gets really hard, it's not at all obvious that you can do this - but he manages by using clever mathematical tricks to show that you can always find a sentence
G
which is only true if Provable('G') is false.
It's important I think to realize - that here G doesn't actually say in the axiom system, or indeed in any language, that our arithmetical statement "Provable('G') is false". It is just some complex arithmetical expression that we, as mathematicians, can see has the value true if Provable('G') has the value false, and the value false, if Provable('G') has the value true.
And what's more you can actually show this equivalence within your theory. The only thing you can't do within the formal theory is to understand the implications of what you just proved.
I won't go into the details here - it is hard to explain it well, and the details don't matter here. For mathematicians and logicians, see Gödel's Incompleteness Theorems (Stanford University of Philosophy).
So - you can show in this rigorous way, that G is true if and only if it is not provable within the theory.
But then - going one step further - if it was provable in the theory - then it would be false, and what's more our theory would be plain inconsistent - you'd be able to come up with a particular, very large number that was a counter example to G, and also have a proof of G.
So - if our theory is consistent - and there are good reasons for supposing that arithmetic is consistent (though we can never prove it) - then G can't have a proof from the axioms. And so therefore it has to be true.
In this way - a human can see that certain things are true that no computer program, programmed with just the axioms and deduction rules of that theory could deduce.
WHAT IF WE ADD IN THE NEGATION OF G?
Now, you could say - that, so what? We don't need an actual example of a proof of G, to add in the negation of G as an axiom. We end up with a rather paradoxical theory - it is "omega inconsistent" - the negation of G - at least in classical logic, says that G has a proof - but no particular number can ever code up a proof. So we've added an axiom saying that a particular type of number exists, when we already know that there can never be any number with those properties. A theory like that is called "omega inconsistent" (omega is a mathematical symbol for the order type of a simple infinite sequence).
Still - there is nothing saying you can't built an "omega inconsistent" theory like that.
We can even work with inconsistent theories in paraconsistent logic.
So does it matter if the computer can't "see" that G is true? Is it really "true" in any absolute sense if we can consistently (but omega inconsistently) add in the axiom that it is false?
THAT DOESN'T MAKE ANY DIFFERENCE TO THE ARGUMENT HOWEVER
Well - that is beside the point here. The thing is - that our computer program, when presented with these proofs that we have come up with - if it doesn't have the additional understanding needed to understand Godel's proof - won't be able to deduce anything about G.
It can't deduce that it can add in the negation of G and get an omega inconsistent but consistent theory. It can't deduce either that it can add in G and get a theory that is both consistent and omega consistent.
As far as the computer program is concerned, our G is just a complicated arithmetical statement which it hasn't yet proved or disproved.
And it is true only if Provable('G') is false - but our computer program interprets " Provable('G')" here as just a complicated statement of numbers - so what? Doesn't mean anything of significance to it, just said that 'G' which it doesn't connect with G, is a number with some complicated properties.
So - however you interpret it - still humans understand something here that the program never can
WHAT IF YOU ADD IN THE GODEL SENTENCE PROOF METHODS
Now, you could program in all these leaps of deduction that a human can make - program in the rules for making a godel encoding and mapping formulae to numbers and proving Godel sentences - but that doesn't help
That's because, it turns out that whatever program you write like that - it just gives you a more complex axiom system for your computer to follow - and you can then find new results that it can't recognize within that axiom system.
And what's more - we aren't talking about abstract abstruse or questionable areas of maths (such as the "Large Cardinality Axioms" for maths geeks). We are just talking about ordinary arithmetic -a very intricate sentence, but basically it's not talking about anything more complicated or abstract than you do when you say that - for instance, there are infinitely many primes. Or indeed, just, that the numbers go on for ever.
GODEL'S TAKE ON IT
Godel's argument is not particularly convincing - and he says as much himself, but it shows that he thought himself that probably human minds can't be reduced to a finite machine (i.e. Turing Machine).
He looked instead at diophantine problems - simultaneous polynomial equations in several unknowns with only integer solutions. His result showed that no finite system of axioms and deduction rules is sufficient to solve all diophantine equations. In this quote, by "absolutely unsolvable" he means - not solvable by any human being.
"So the following disjunctive conclusion is inevitable: Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified"
He goes on to say that he finds the first of these two possibilites more plausible. See "Some basic theorems on the foundations of mathematics and their implications" - lots of missing pages in the preview, sorry.
The main difference is that Roger Penrose took this a lot further, applied it to physics, and supplied what he considered to be a proof that human minds can't be reduced to a program, using ideas of Turing machines combined with Godel's results.
PENROSE'S ARGUMENT
He presents a complex argument, and there are many details that need to be considered carefully. However the essence of it is that
- It is impossible for a Turing machine to enumerate all possible Godel sentences. Such a program will always have a Godel sentence derivable from its program which it can never discover
- Humans have no problem discovering these sentences and seeing the truth of them
So, therefore humans are not reducible to Turing machines
Here is a thoughtful essay by John Lucas, where he comments on the follow up discussion of Roger Penrose's ideas. The Gödelian Argument: Turn Over the Page
Also - Godel had platonist, maybe dualist ideas about mind and matter. While Roger Penrose is approaching this very much as a physicist, as a problem in physics, how can humans brains understand maths, when maths is not within the grasp of any computer program?
OBJECTIONS TO LUCAS'S AND PENROSE'S ARGUMENT
One simple objection made early on, is to say - that humans are also limited, there are things we can't assert too.
Take for instance the sentence
"Lucas can't assert the truth of this statement"
That sentence is true, but John Lucas can't say that it is true.
or more generally
"I can't assert the truth of this statement"
- that's a sentence that applies to all of us. Nobody can assert that it is true - nevertheless seems that in some sense it is true.
A variation on the argument was brought up by Daryl McCullough in "Can Humans Escape Gödel? " where he uses a similar sentence "This sentence is not an unassailable belief of Roger Penrose."
However - Roger Penrose has answered this already. These are sentences in natural language, and we know that we run into difficulties in logic if we use unrestricted natural language.
However they don't seem to be problems with understanding maths as such. Especially such a simple area as number theory.
The thing is, that Godel's sentence - though it's construction was inspired by these sentences, is not in itself paradoxical. Indeed it is just a rather intricate expression about ordinary arithmetic involving nothing more complex than addition and multiplication - and what's more a sentence in a simple axiom system that we believe to be consistent.
So we are talking about a sentence about arithmetic that a human mathematician can see to be true, which the original computer program can't see to be true.
PENROSE'S "P SENTENCES"
To make this clearer - he uses a technical trick, to let him talk in a perfectly general way about mathematical truth - but to avoid all the problems of natural languages.
But this can be a bit confusing if you don't realize why he does this.
He made it into an argument about "P sentences" - sentences that are true if and only if some particular Turing machine can never halt.
Statements about mathematics can be put in that form, and many natural language statements can also - lots of things can. But these self referring paradoxical sentences can't be put into that form.
It's a bit confusing because - earlier on he is talking about Turing machines as computer programs that some people think could be artificial intelligences. While here he is talking about them in a different way - as a technical device to simplify discussion of what counts as a mathematical truth.
Anyway - I think best not to get too bogged down in this here. It's a helpful distinction but only if you are well up on formal logic, Turing machines etc. I thought I'd mention it as it is something that comes up if you read the literature on the subject.
MAIN THING IS - THAT WE CAN DO MATHS EVEN THOUGH NATURAL LANGUAGE IS PRONE TO PARADOXES
But - doesn't matter quite how you do it, the main thing is that we can still do maths even though natural language is prone to paradoxes. And we do that by being especially careful about the language we use when we do maths.
So - if you then require that humans and the computer program keep to acceptable "mathematical language" - then you no longer need to take account of these kinds of sentences.
And then - it turns out - the computer program still has things it can't see to be true.
And these are truths in simple areas of maths such as arithmetic.
WHAT IF WE HUMANS ARE ALSO LIMITED IN THE SAME WAY
But - what if we do have a Godel sentence ourselves also - but don't know it?
Most of the objections to his arguments are along these lines in one way or another. I.e. accept that the Godel sentences are true, and that every computer program does have such a sentence, that it can't see to be true - but they then argue, that we have sentences like that also, we just are not aware of them.
After all mathematicians make mistakes. But we are able to correct our mistakes.
Well - here the arguments get really intricate. Has he proved his case? Or is it logically possible that human beings also have "Godel sentences" that we are not aware of?
You can read some of the arguments and counter arguments here
Beyond the Doubting of a Shadow, A Reply to Commentaries on Shadows of the Mind,
Whatever you think though, whether you think he has proved his case or not, nobody, surely, has yet proved the opposite position - that all physics including human behaviour, can be simulated in a computer.
I'm not sure why so many are so sure of that conclusion - that a computer program can simulate a human - when nobody has proved it or has any idea yet of how to prove it.
Myself anyway - I expect that some time or another down the road, we will hit some truly non computable physics. Though whether in the form of Roger Penrose's Orch Or theory or some other idea - maybe not involving quantum mechanics at all - or even quantum gravity - I don't know.
And that then leads to the idea that if we ever do have Artificial Intelligence then it is likely to be impossible to program, in some intrinsic way because at its core, it is in some way doing something essentially non computable. And it would start off like a young thing or baby - or may be an "uplifted species" or similar.
What do you think?
