Roger Penrose Quotes

Ordinary photons do have spin, they have a notion of helicity so they spin around their direction on motion.

This book is about physics and its about physics and its relationship with mathematics and how they seem to be intimately related and to what extent can you explore this relationship and trust it.

Q5. Have not I merely shown that it is possible to outdo just a particular algorithmic procedure, A, by defeating it with the computation Cq(n)? Why does this show that I can do better than any A whatsoever?

The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will know of Euclid's argument that there is no largest prime number. This, also, is a reductio ad absurdum. Euclid's argument is as follows. Suppose, on the contrary, that there is a largest prime; call it p. Now consider the product N of all the primes up to p and add 1:

N=2*3*5*...*p+1.

N is certainly larger than p, but it cannot be divisible by any of the prime numbers 2,3,5...,p (since it leaves the remainder 1 on division); so either N is the required prime itself or it is composite-in which case it is divisible by a prime larger than p. Either way, there would have to be a prime larger than p, which contradicts the initial assumption that p is the largest prime. Hence there is no largest prime. The argument, being a reductio ad absurdum, does not merely show that a particular prime p can be defeated by finding a larger one; it shows that there cannot be any largest prime at all. Likewise, the Godel-Turing argument above does not merely show that a particular algorithm A can be defeated, it shows that there

I was indeed very slow as a youngster.

My own way of thinking is to ponder long and I hope deeply on problems and for a long time which I keep away for years and years and I never really let them go.

Consciousness ... is the phenomenon whereby the universe's very existence is made known.

So what I'm saying is why don't we think about changing Schrodinger's equation at some level when masses become too big at the level that you might have to worry about Einstein's general relativity.

If we try to make general inferences about the theoretical possibility of a reliable computational model of the brain, we ought indeed to come to terms with the mysteries of quantum theory.

These are deep issues, and we are yet very far from explanations. I would argue that no clear answers will come forward unless the interrelating features of all these worlds are seen to come into play. No one of these issues will be resolved in isolation from the others. I have referred to three worlds and the mysteries that relate them one to another. No doubt there are not really three worlds but one, the true nature of which we do not even glimpse at present.

The reason that I have concentrated on non-computability, in my arguments, rather than on complexity, is simply that it is only with the former that I have been able to see how to make the necessary strong statements. It may well be that in the working lives of most mathematicians, non-computability issues play, if anything, only a very small part. But that is not the point at issue. I am trying to show that (mathematical) understanding is something that lies beyond computation, and the Godel (-Turing) argument is one of the few handles that we have on that issue. It is quite probable that our mathematical insights and understandings are often used to achieve things that could in principle also be achieved computationally-but where blind computation without much insight may turn out to be so inefficient that it is unworkable (cf. 3.26). However, these matters are much more difficult to address than the non-computability issue.

Q7. The total output of all the mathematicians who have ever lived, together with the output of all the human mathematicians of the next (say) thousand years is finite and could be contained in the memory banks of an appropriate computer. Surely this particular computer could, therefore, simulate this output and thus behave (externally) in the same way as a human mathematician-whatever the Godel argument might appear to tell us to the contrary?

While this is presumably true, it ignores the essential issue, which is how we (or computers) know which mathematical statements are true and which are false. (In any case, the mere storage of mathematical statements is something that could be achieved by a system much less sophisticated than a general purpose computer, e.g. photographically.) The way that the computer is being employed in Q7 totally ignores the critical issue of truth judgment. One could equally well envisage computers that contain nothing but lists of totally false mathematical 'theorems', or lists containing random jumbles of truths and falsehoods. How are we to tell which computer to trust? The arguments that I am trying to make here do not say that an effective simulation of the output of conscious human activity (here mathematics) is impossible, since purely by chance the computer might 'happen' to get it right-even without any understanding whatsoever. But the odds against this are absurdly enormous, and the issues that are being addressed here, namel

But I think it is a serious issue to wonder about the other platonic absolutes of say beauty and morality.

What right do we have to claim, as some might, that human beings are the only inhabitants of our planet blessed with an actual ability to be "aware"? ... The impression of a "conscious presence" is indeed very strong with me when I look at a dog or a cat or, especially, when an ape or monkey at the zoo looks at me. I do not ask that they are "self-aware" in any strong sense (though I would guess that an element of self-awareness can be present). All I ask is that they sometimes simply feel !

I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts ... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.

Well I didn't actually see the Matrix but I've seen other movies where with similar sorts of themes.

I'm pretty tenacious when it comes to problems.

It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other.

Some years ago, I wrote a book called the Emperor's New Mind and that book was describing a point of view I had about consciousness and why it was not something that comes about from complicated calculations.

We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.

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.

To me the world of perfect forms is primary (as was Plato's own belief)-its existence being almost a logical necessity-and both the other two worlds are its shadows.

There are considerable mysteries surrounding the strange values that Nature's actual particles have for their mass and charge. For example, there is the unexplained 'fine structure constant' ... governing the strength of electromagnetic interactions, ...

It is always the case, with mathematics, that a little direct experience of thinking over things on your own can provide a much deeper understanding than merely reading about them.

And these little things may not seem like much but after a while they take you off on a direction where you may be a long way off from what other people have been thinking about.

Some people take the view that the universe is simply there, and it runs along - it's a bit as though it just sort of computes, and we happen by accident to find ourselves in this thing. I don't think that's a very fruitful or helpful way of looking at the universe.

The perceiving of mathematical truth can be achieved in very many different ways. There can be little doubt that whatever detailed physical activity it is that takes place when a person perceives the truth of some mathematical statement, this physical activity must differ very substantially from individual to individual, even though they are perceiving precisely the same mathematical truth. Thus, if mathematicians just use computational algorithms to form their unassailable mathematical truth judgments, these very algorithms are likely to differ in their detailed construction, from individual to individual. Yet, in some clear sense, the algorithms would have to be equivalent to one another.

It is hard to see how one could begin to develop a quantum-theoretical description of brain action when one might well have to regard the brain as "observing itself" all the time!

Our present picture of physical reality, particularly in relation to the nature of time, is due for a grand shake up

The basic theory in twistor theory is not to add extra dimensions.

What Godel and Rosser showed is that the consistency of a (sufficiently extensive) formal system is something that lies outside the power of the formal system itself to establish.

Quantum mechanics makes absolutely no sense.

As for morality, well that's all tied up with the question of consciousness.

Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules. Support for the Platonic viewpoint ...was an important part of Godel's initial motivations.

Intelligence cannot be present without understanding. No computer has any awareness of what it does.

Well, gauge theory is very fundamental to our understanding of physical forces these days. But they are also dependent on a mathematical idea, which has been around for longer than gauge theory has.

No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence
a duty and a duty alone
and no hint of magic or beauty of the subject might be allowed to come through.

We cannot say, in familiar everyday terms, what it 'means' for an electron to be in a state of superposition of two places at once, with complex-number weighting factors w and z. We must, for the moment, simply accept that this is indeed the kind of description that we have to adopt for quantum-level systems. Such superpositions constitute an important part of the actual construction of our microworld, as has now been revealed to us by Nature. It is just a fact that we appear to find that the quantum-level world actually behaves in this unfamiliar and mysterious way. The descriptions are perfectly clear cut-and they provide us with a micro-world that evolves according to a description that is indeed mathematically precise and, moreover, completely deterministic!

Thus, Godel appears to have taken it as evident that the physical brain must itself behave computationally, but that the mind is something beyond the brain, so that the mind's action is not constrained to behave according to the computational laws that he believed must control the physical brain's behavior.

If you didn't have any conscious beings in the world, there really wouldn't be morality but with consciousness that you have it.

The idea is if you use those two shapes and try to colour the plane with them so the colours match, then the only way that you can do this is to produce a pattern which never repeats itself.

I would say the universe has a purpose. It's not there just somehow by chance.

A point that should be emphasized is that the energy that defines the lifetime of the superposed state is an energy difference, and not the total, (mass-) energy that is involved in the situation as a whole. Thus, for a lump that is quite large but does not move very much-and supposing that it is also crystalline, so that its individual atoms do not get randomly displaced-quantum superpositions could be maintained for a long time. The lump could be much larger than the water droplets considered above. There could also be other very much larger masses in the vicinity, provided that they do not get significantly entangled with the superposed state we are concerned with. (These considerations would be important for solid-state devices, such as gravitational wave detectors, that use coherently oscillating solid-perhaps crystalline-bodies.)

There is a certain sense in which I would say the universe has a purpose. It's not there by chance.