Quotes from Roger Penrose

It is important to bear in mind that this is a property of single photons. Each individual photon must be considered to feel out both routes that are open to it, but it remains one photon; it does not split into two photons in the intermediate stage, but its location undergoes the strange kind of complex-number weighted co-existence of alternatives that is characteristic of quantum theory.

I am not so much concerned, at this stage, with how individual mathematicians might differently approach a mathematical problem, but more with what is universal about our understandings and our mathematical perceptions.

It would seem that it is in some kind of combination of top-down and bottom-up organization that we must expect to find the most successful AI systems.

do not see how natural selection, in itself, can evolve algorithms which could have the kind of conscious judgements of the validity of other algorithms that we seem to have.

In order to decide whether or not an algorithm will actually work, one needs insights, not just another algorithm.

It is only with very large masses indeed that light-cone tilting can be directly observed; whereas its actual presence in very tiny amounts in bodies as small as specks of dust is a clear-cut implication of Einstein's theory.

Support for the Platonic viewpoint (as opposed to the formalist one) was an important part of Godel's initial motivations. On the other hand, the arguments from Godel's theorem serve to illustrate the deeply mysterious nature of our mathematical perceptions. We do not just 'calculate' in order to form these perceptions, but something else is profoundly involved-something that would be impossible without the very conscious awareness that is, after all, what the world of perceptions is all about.

The viewpoint is that it is simply the logical structure of the algorithm that is significant for the 'mental state' it is supposed to represent, the particular physical embodiment of that algorithm being entirely irrelevant.

string theory is completely different from this. Here there appear to be no results whatever that provide it with experimental

Thus the robot is incapable of knowing that it was constructed according to the mechanisms M. Since we are aware-or at least can be made aware-that the robot was so constructed, this seems to tell us that we have access to mathematical truths, e.g. Omega (Q(M)), that are beyond the robot's capabilities, despite the fact that the robot's abilities are supposed to be equal of (or in excess of) human capabilities.

Whereas I reject mysticism in its negation of scientific criteria for the furtherance of knowledge, I believe that within an expanded science and mathematics there will be found sufficient mystery ultimately to accommodate even the mystery of mind.

Perhaps it is conceivable that, in the future, some different kind of 'computer' might be introduced, that makes critical use of continuous physical parameters-albeit within the standard theoretical framework of today's physics-enabling it to behave in a way that is essentially different from a digital computer.

WHAT PRECISELY IS an algorithm, or a Turing machine, or a universal Turing machine? Why should these concepts be so central to the modern view of what could constitute a 'thinking device'?

Though it indeed seems reasonable to rule out space-time geometries with closed timelike lines as descriptions of the classical universe, a case can be made that they should not be ruled out as potential occurrences that could be involved in a quantum superposition.

Another example of a class of well-defined mathematical problems that have no algorithmic solution is the tiling problem. This is formulated as follows: given a set of polygonal shapes, decide whether those shapes will tile the plane; that is, is it possible to cover the entire Euclidean plane using only these particular shapes, without gaps or overlaps?

There is even a view, not uncommonly expressed, that might best be regarded as a combination of A and D (or perhaps B and D)-a possibility that will actually feature significantly in our later deliberations. According to this view, the brain's action is indeed that of a computer, but it is a computer of such wonderful complexity that its imitation is beyond the wit of man and science, being necessarily a divine creation of God-the 'best programmer in the business'!

Although it might well be possible for a sufficiently cleverly constructed such system to preserve an illusion, for some considerable time (as with Deep Thought), that it possesses some understanding, I shall maintain that a computer system's actual lack of understanding should-in principle, at least-eventually reveal itself.

Any complicated activity, which may be mathematical calculations, or playing a game of chess, or commonplace actions-if they have been understood in terms of clear-cut computational rules-are the things that modern computers are good at; but the very understanding that underlies these computational rules is something that is itself beyond computation.

In general, when we consider an object in a superposition of two spatially displaced states, we simply ask for the energy that it would take to effect this displacement, considering only the gravitational interaction between the two. The reciprocal of this energy measures a kind of 'half-life' for the superposed state. The larger this energy, the shorter would be the time that the superposed state could persist.

Embracing the subjective interpretation quickly leads us to assertions that are patently absurd, underscoring the independence of mathematical knowledge of any human activity Just take me there...

G* No individual mathematician ascertains mathematical truth solely by means of an algorithm that he or she knows to be sound.

Nevertheless, the Geroch-Hartle work does indicate the clear possibility that non-computability may have a genuine role in whatever quantum gravity theory finally emerges as being physically correct.

In the present chapter, we tried to pinpoint the place in the brain where quantum action might be important to classical behaviour, and have apparently been driven to consider that it is through the cytoskeletal control of synaptic connections that this quantum/classical interface exerts its fundamental influence on the brain's behaviour.

Does awareness play some kind of role as a 'bridge' to a world of Platonic absolutes.