Mathematics

Platonist Mathematics Goedel Theorem



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Note: this article has been shortened for Helium - longer version available upon request.

The Sum Of All Parts:

A Dialogue On Consciousness Between Platonism and Quantum Physics.




Whilst being a highly respected philosopher whose work one is inevitably required to study before moving on to any form of philosophical training (if such a thing is possible at all), Plato is often regarded with the same mild amusement with which we treat all the seemingly risible ideas of the founders of our disciplines. "It's alright", we say, somewhat patronisingly for people who would be unable to locate Plato's cobbler, let alone step into his shoes, "they didn't have science then. They were laying the foundations; they didn't have the background to be able to develop as good an understanding as we have today." This is, I believe, a viewpoint popularly held about Plato's World of Forms. The idea that there may be perfect Forms of everything around us, in some other realm outside of our comprehension, seems ludicrously far-fetched and phantastical. Nonetheless, we will gladly accept the possibility of a Divine Other; perhaps even a whole spiritual realm; without subjecting it to the kind of academic rigour employed by Plato in his writings. Yet recent developments in science - physics in particular - seem at least to some to be compatible with a Platonist viewpoint. Could it be that the prodigal son is finally returning to the father, having spent years herding swine and eating from their swill? In this article, I intend to conduct a treatment of Goedelian Platonism in the light of Roger Penrose's two major psychophilosophic works.

'Quantum physics deals only with aggregations, and its laws are for crowds and not for individuals'[I], wrote Einstein in his Evolution of Physics, just three years after the Austrian physicist Erwin Schroedinger devised his cat paradox. Quantum physics was at that point (and still is) a relatively young discipline, having only been under the microscope of scientific speculation since the late 1830s. One of the fundamental principles of quantum physics seems to boil down to observation; the laws of physics as we know them do not tally with the alien world of quanta. As John Gribbin so adeptly puts it, 'what quantum mechanics says is that nothing is real and that we cannot say anything about what things are doing when we are not looking at them'[II]. Physicists of the 1930s Schroedinger included were understandably outraged by the notion of the laws of physics being not lawful at all. Indeed, John Gribbin quotes him as proclaiming on the subject: 'I don't like it, and I'm sorry I ever had anything to do with it.'[III] To show the apparent absurdity of quantum theory, Schroedinger conducted a thought experiment in which he imagined a cat being placed in a box with a phial of poison, and arranged in such a way that if radioactive decay occurs - which has a 50/50 chance of happening - then the cat dies. According to the laws of physics as they had been understood, the cat is either dead or not dead, and we find out which is the case by opening the box. If, however, quantum theory is correct, neither of the two possibilities has a place in reality unless it is observed. 'The atomic decay has neither happened nor not happened, the cat has neither been killed nor not killed, until we look inside the box to see what has happened.'[IV]



So, observation is everything; or so it would appear. What, then, happens to people? Am I only me when I am observed? Is my mind only my mind when it is being watched by some other entity? Such an idea would be problematic; as the saying goes: 'I've been a neurologist for many years, but I've never seen a single thought.' Such entities - if they can even be so called - as mind, thought, sense, and so on are sensitive subjects when discussing observation. How do we know if we 'see' someone's mind? There is enough debate as to the nature of mind in modern and postmodern philosophy as it is, without adding to the equation the need to observe something in order for it to be real. And what about mathematics? Numbers as we understand them are surely not entities to be observed; yet we generally have a clear concept of two plus two equalling four. As Penrose points out, however, the 'reality' of numbers is in itself questionable. We call numbers real, he argues, because 'they seem to provide the magnitudes needed for the measurement of distance, angle, time, energy, temperature, or numerous other geometrical and physical quantities.'[V] Thinking more deeply on this point, however, we come to the realisation that numbers cannot be as 'real' as we believe them to be, for there will always be the mathematical possibility of a number, however small, in between two other numbers. This is just not the case with physical distances. If the goblin stands two centimetres away from the pixie, then the goblin is two centimetres away from the pixie. If the goblin rather likes the pixie and feels that he may be up for a bit of pixie action, he shuffles a centimetre closer. The pixie, if she is in the mood, glances at him coyly and wiggles towards him, closing the final centimetre-wide gap between them. The goblin and the pixie are now standing literally side-by-side. There is no gap between them. In this context, mathematics falls short. The mid point between zero centimetres and two centimetres is, of course, one centimetre. The mid point between zero and one centimetre is half a centimetre. The mid point between zero and half a centimetre is a quarter of a centimetre. And so it goes on and on. By this point, however, the goblin and the pixie have become bored by all this talk and wandered off to narrow the gap between them even further. That is just the point, though: they can't. However close they get, there comes a point when they cannot physically get any closer; the mechanics just don't work. A system of mathematics, however, 'has the property that between any two of them, no matter how close, there lies a third.'[VI]

As entertaining as all this talk of pixies and goblins may be, how does it leave us in mathematical relation to Platonism? 'It has been said that in Plato the intuitive mode of receiving knowledge is accepted implicitly'[VII], states Wild regarding the Platonist mind. Michael Lockwood also puts it interestingly: 'acquiring language or concepts may be more akin to progressively finer tuning of an instrument with a vast number of strings than it is to learning a set of rules.'[VIII] He goes on to paraphrase Simmias' comparison of the psyche to the attunement of a lyre in order to show that Plato's ideas on consciousness are in fact rather similar to the Chomskian view of 'concepts [e.g. grammar] being appropriately attuned to the functional architecture of those parts of the brain that are involved in thought and other cognitive tasks.'[IX] Could it be that knowledge - or conceptual consciousness - is at least hypothetically infinite? After all, it is well known that we use but a tiny percentage of our brain, and 'the architecture of the cortex has a complexity of astronomical proportions.'[X]



This is all very well; and we can see that parallels may be drawn between the potential infinity of consciousness and the point made earlier by Lockwood concerning the 'reality' of numbers. But Platonism stretches to more than just consciousness. For Plato, mathematics was central to philosophy; like Pythagoras before him, who claimed that 'the ultimate nature of reality is number'[XI], he 'stripped the material world of all qualities except the severely mathematical'[XII], and pointed towards an ideal realm in which abstract entities existed, outside of space and time. 'Plato held that such Ideas or Forms are the only things that really or wholly exist, on the ground that it is only of them that we have absolutely certain knowledge, namely, in mathematics.'[XIII]

Thus mathematics was for Plato the central point of knowledge. Indeed Russell accuses Plato of being 'under the influence of the Pythagoreans' to the extent that he 'assimilated other knowledge too much to mathematics'.[XIV]
Plato argued that there is no knowledge to be acquired through the senses; rather, that 'the only real knowledge has to do with concepts'[XV]. One would imagine, therefore, that Platonism could not fit in with quantum theory; for the latter holds observation as the key and conceptual possibility as its antonym, whilst Plato exalts concepts as the only real knowledge, thus apparently rubbishing that which appears to us through the senses. One just has to read his cave analogy to understand that Plato is quite open to the thought that our perceptions are nought but delusions. In equating persons in society with those in the cave, he imparts an allegory that leaves the reader in a state of wonderment concerning what reality is, or potentially could be. He has Socrates painting for Glaucon the picture of a cave, in which men have lain since childhood. They are chained to the wall behind them and therefore prevented from looking anywhere except at the parallel wall directly in front. Further back, towards the entrance of the cave, there burns a fire, and between the fire and the prisoners is a road. The fire would cast the shadows of the people on the road onto the wall opposite the prisoners, thus creating a kind of human puppet show. Socrates then goes on to explain the allegory further:




'They are like ourselves', [he elaborates]. 'For in the first place do you think that such men would have seen anything of themselves or of each other except the shadows thrown by the fire on the wall of the cave opposite to them?'

'How could they,' he said, 'if all their life they had been forced to keep their heads motionless?'

'What would they have seen of the things carried along the wall? Would it not be the same?'

'Surely.'

'Then if they were able to talk with one another, do you not think that they would suppose what they saw to be the real things?'

'Necessarily.'

'Then what if there were in their prison an echo from the opposite wall? When any one of those passing by spoke, do you imagine that they could help thinking that the voice came from the shadow passing before them?'

'No, certainly not,' he said.'Then most assuredly,' I said, 'the only truth that such men would conceive would be the shadows of those manufactured articles?'

'That is quite inevitable.'[XVI]





Plato goes on to discuss the man who is taken from the cave into the world outside; he passes the fire, sees the light of the sun, the heavens, the water, the people; and at first refuses - or is unable - to believe what he sees. Eventually, however, the man realises that the world he is now seeing is more 'real' than the world he has left; and, returning to his fellows in the cave, exhorts them to accompany him outside, and so the allegory continues.

Glaucon and Socrates then go on to discuss puzzles such as the contradiction of the mind when processing entities which can be both X and the opposite of X simultaneously; 'Socrates is now taller than Theaetetus, who is a youth not yet full grown; but in a few years Socrates will be shorter than Theaetetus. Therefore Socrates is both tall and short.'[XVII] This flows swiftly on to a treatment of mathematics, with Socrates proclaiming:




'Furthermore it strikes me now that we have mentioned the study of calculation, how elegant it is and in what manifold ways it helps our desires if it is pursued not for commercial ends but for the sake of knowledge.'

'In what way?' [replies Glaucon]

'In this way which we have just been mentioning. It powerfully draws the soul above, and forces it to reason concerning the numbers themselves, not allowing any discussion which presents to the soul numbers with bodies that can be seen or touched.'[XVIII]




Thus Socrates leads Glaucon through a process of deduction towards the conclusion that mathematics is the ultimate nature of truth. How, though, does this underpin the quantum theory of consciousness? In Shadows of the Mind, Roger Penrose outlines the implications of quantum physics for the problem of conscious minds. His chapter on Consciousness and Computation specifies four potential viewpoints that may be held regarding the matter of computation and conscious thinking. These are:




'A. All thinking is computation; in particular, feelings of conscious awareness are evoked merely by the carrying out of appropriate computations.




B. Awareness is a feature of the brain's physical action; and whereas any physical action can be simulated computationally, computational simulation cannot by itself evoke awareness.




C. Appropriate physical action of the brain evokes awareness, but this physical action cannot even be simulated computationally.




D. Awareness cannot be explained by physical, computational, or any other scientific terms.'[XIX]





It is now important to step away from the above and glance in the direction of the Goedelian case; or, more specifically, the Goedel-Turing conclusion to Goedel's theorem. In 1899, David Hilbert posited a mathematical system of axioms and rules through which it would be possible to decide the truth or falsity of any mathematical proposition correctly formulated within it.[XX] Goedel, however, goes on to disprove Hilbert's proposal by means of the creation of an arithmetical proposition which is not proven within the system, shown in Penrose as Pk(k). The proposition has been constructed to mean that 'there is no proof, within the system, of the proposition Pk(k)'.[XXI] Thus there cannot be any proof of the arithmetical proposition; for, if it were proven to be true, then the meaning that it asserts (i.e. that there is no proof of it within the system), would be a falsity. Therefore, we arrive at the conclusion that it must be the case that there is no proof of Pk(k) which is exactly the point of the proposition! So we arrive at a true arithmetical proposition which has no place within the system. Looking at its negation~ Pk(k) we can deduce that it must be false; but within Hilbert's system, we are not supposed to be able to prove a falsity. Goedel's theorem is hereby established.[XXII] This has long been accepted as an integral contribution to the foundations of mathematics, but Penrose goes one step further than this, arguing that it also paves the way to advancement in the philosophy of mind.

How so? As Penrose explains:




'Among the things that Gdel indisputably established was that no formal system of sound mathematical rules of proof can ever suffice, even in principle, to establish all the true propositions of ordinary arithmetic. But a powerful case can also be made that his results showed something more than this, and established that human understanding and insight cannot be reduced to any set of computational rules. For what he appears to have shown is that no such system of rules can ever be sufficient to prove even those propositions of arithmetic whose truth is accessible, in principle, to human intuition and insight whence human intuition and insight cannot be reduced to any set of rules.'






In other words, Goedel's theorem proves that within any system of mathematical rules, it is not possible even hypothetically to establish all true arithmetical propositions, however simple they may be. Therefore, even if human consciousness could be reduced to computation - which is in itself disputable - this would still be impossible to correctly simulate.

Remarkable indeed.

One common criticism of the view that Goedel's theorem could be applied to human intuition in order to show that the latter has no specific 'rule book' to follow is that it forces us into a form of mysticism, thus being unpalatable to the mathematician's philosophic taste buds.[XXIII] The reason for this rests on the four potential viewpoints of computation and conscious thinking mentioned earlier. Why? If Goedel's theorem can be correctly applied to consciousness, then neither A
nor B can be true. A, the view that 'all thinking is computation', is incompatible with a Goedelian framework of consciousness because if 'human intuition and insight cannot be reduced to any set of rules', then all thinking cannot be reducible to computation. B also does not hold, as a structure of rules must be in place in order for computational simulation to take place. This leaves us with C or D. The latter has to be discounted if we are discussing consciousness from a physical viewpoint, due to its inclusion of 'scientific terms' in the list of means by which awareness cannot be explained.[XXIV]

Thus we arrive at C: 'Appropriate physical action of the brain evokes awareness, but this physical action cannot even be properly simulated computationally.'[XXV] Less mystical than D, perhaps, yet still seemingly unsatisfying to the die-hard 'science must be made up of that which is computable' school. As Penrose goes on to discuss, C has implications of a Platonist nature for our view of consciousness. We have seen throughout the essay how Plato's view of Forms, as shown by his cave allegory, led into a further discussion between Socrates and Glaucon concerning mathematics as the higher truth; the guide that 'powerfully draws the soul above'[XXVI]
and allows it to perceive the world beyond the dingy, echoing, shadow-infested terrain of the cave's innards. Plato's world of Forms is distinct from the physical world -or the world which we know to be physical - yet through it, the world we perceive can be understood. 'It lies beyond our imperfect mental constructions; yet, our minds do have some direct access to this Platonic realm through an 'awareness' of mathematical forms, and our ability to reason about them.'[XXVII] Having a Platonist conception of the world does not mean that computation is rendered invalid. On the contrary, the latter may aid us in our understanding and explanation of the world around us. After all, Plato's point in Book VII of The Republic is not that the cave does not exist at all, but that it is not all that exists. The sole mistake made by the bound prisoners was that they did not posit a world beyond the cave, preferring to believe themselves and that which they saw directly to be the only truth. Who can blame them? Much of the time we can catch ourselves doing the same thing in day-to-day life; conveniently 'forgetting' that there is a world outside the little bubble of our selves and our respective circles of extended subjectivity; that there exist other persons, other peoples, other entities, other concepts; and perhaps even other things for which we do not yet have a name. Plato showed that mathematics can be compatible with an uncertain world view. Quantum physics has been doing the same for the physical sciences since its initial development in the early 1800s.

The sum of all the parts, therefore, when regarded through the seemingly incompatible magnifying glasses of Platonist mathematics and Goedelian theorem, reflects the light of truth onto the ant of the quantum without burning it to a pulp; for in this world, there is both no truth and every truth; the cat is both dead and alive; the Platonist is both a member of this world and cognizant of the Forms; and the debate can be laid to rest with the final words of a poem by Conrad Aiken:




'How can you know what here goes on

Behind this flesh-bright frontal bone?

Here are the world and god become

For all their depth a simple Sum.'[XXVIII]








































































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Bullock, Alan & Stallybrass, Oliver (eds.); The Fontana Dictionary of Modern Thought; Collins, 1977



Duncan, J. & Starling, S.G.; A Text Book of Physics; Macmillan, 1931



Einstein, Albert & Infeld, Leopold; The Evolution of Physics; Cambridge University Press, 1938



Feinberg, J.G.; The Story of Atomic Theory and Atomic Energy;
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Flew, Anthony; An Introduction to Western Philosophy; Thames & Hudson, 1971



Froud, Brian & Berk, Ari; Goblins; Chrysalis, 2004



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Gribbin, John; In Search of Schrdinger's Cat; Black Swan, 1991



Joad, C.E.M; Guide to Philosophy; Gollancz, 1955



Lockwood, Michael; Mind, Brain and The Quantum: The Compound I'; Blackwell, 1989



O'Connor, Timothy & Robb, David; Philosophy of Mind: Contemporary Readings; Routledge, 2003



Penrose, Roger; The Emperor's New Mind; Oxford University Press, 1989



Penrose, Roger; Shadows of the Mind; Oxford University
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~ 3,259 words








[I] Einstein, Albert & Infeld, Leopold; The Evolution of Physics; Cambridge University Press, 1938, p.302






[II] Gribbin, John; In Search of Schrdinger's Cat: Quantum Physics and Reality; Black Swan, 1991, p.2






[III] Gribbin, John; In Search of Schrdinger's Cat: Quantum Physics and Reality; Black Swan, 1991, p.V






[IV] Gribbin, John; In Search of Schrdinger's Cat: Quantum Physics and Reality; Black Swan, 1991, pp.2-3






[V] Penrose, Roger; The Emperor's New Mind; Oxford University Press, 1989, p.112






[VI] Penrose, Roger; The Emperor's New Mind; Oxford University Press, 1989, p.113






[VII] Wild, K.W.; Plato's Presentation of Intuitive Mind In His Portrait of Socrates; in Hooper (ed.); Philosophy; Vol. XIV No. 53; Macmillan & Co., 1939






[VIII] Lockwood, Michael; Mind, Brain and The Quantum: The Compound I'; Blackwell, 1989, p.122






[IX] Lockwood, Michael; Mind, Brain and The Quantum: The Compound I'; Blackwell, 1989, p.122






[X] Lockwood, Michael; Mind, Brain and The Quantum: The Compound I'; Blackwell, 1989






[XI] Stokes, Philip; Philosophy; Arcturus, 2006, p.10






[XII] Joad, C.E.M.;
Guide to Philosophy; Gollancz, 1955, p.322






[XIII] Bullock, Alan & Stallybrass, Oliver (eds.); The Fontana Dictionary of Modern Thought; Collins, 1977, p.476






[XIV] Russell, Bertrand; A History of Western Philosophy; Allen & Unwin, 1963, pp.171-172






[XV] Russell, Bertrand; A History of Western Philosophy; Allen & Unwin, 1963, p.163






[XVI] Plato; The Republic; Everyman's Library, 1935, pp.207-208






[XVII] Russell, Bertrand; A History of Western Philosophy; Allen & Unwin, 1963, p.164






[XVIII] Plato; The Republic; Everyman's Library, 1935, pp.219-220






[XIX] Penrose, Roger; Shadows of the Mind; Oxford University Press, 1994, p.12






[XX] Penrose, Roger; The Emperor's New Mind; Oxford University Press, 1989, p.153






[XXI] Penrose, Roger;
The Emperor's New Mind; Oxford University Press, 1989, p. 140






[XXII] Penrose, Roger; The Emperor's New Mind; Oxford University Press, 1989, pp.140-141






[XXIII] Penrose, Roger; Shadows of the Mind; Oxford University Press, 1994, p.50






[XXIV] Cf. Penrose, Roger; Shadows of the Mind; Oxford University Press, 1994, p.12






[XXV] Penrose, Roger; Shadows of the Mind; Oxford University Press, 1994, p.12






[XXVI] Plato, The Republic; Everyman's Library, 1935, p.220






[XXVII] Penrose, Roger; Shadows of the Mind; Oxford University Press, 1994, p.50






[XXVIII] Aiken, Conrad Potter; The Coming Forth By Day of Osiris Jones; in Bloom, Harold (ed.); Selected Poems; Oxford University Press, 2003, p.79

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