Quotes from Roger Penrose

The judgement-forming that I am claiming is the hallmark of consciousness is itself something that the AI people would have no concept of how to program on a computer.

there seems to be something non-algorithmic about our conscious thinking. In particular, a conclusion from the argument in Chapter 4, particularly concerning Gödel's theorem, was that, at least in mathematics, conscious contemplation can sometimes enable one to ascertain the truth of a statement in a way that no algorithm could.

In view of the anomalous relation that consciousness has to the very physical notion of time, as was described at the beginning of this section, it seems to me to be at least possible that there is no such clear-cut 'time' at which a conscious event must occur.

Second Law of thermodynamics is not an equality, but an inequality, asserting merely that a certain quantity referred to as the entropy of an isolated system—which is a measure of the system's disorder, or 'randomness'—is greater (or at least not smaller) at later times than it was at earlier times.

is not easy to ascertain what an algorithm actually is, simply by examining its output.

Moreover, the slightest 'mutation' of an algorithm (say a slight change in a Turing machine specification, or in its input tape) would tend to render it totally useless, and it is hard to see how actual improvements in algorithms could ever arise in this random way. (Even deliberate improvements are difficult without 'meanings' being available.

It is a famous theorem first proved by the great (Italian-) French mathematician Joseph L. Lagrange in 1770 that every number is, indeed, the sum of four squares.

I'm pretty tenacious when it comes to problems.

People think of these eureka moments and my feeling is that they tend to be little things, a little realisation and then a little realisation built on that.

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.

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.

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.

Even an aardvarks think their offspring are beautiful

To make this condition mathematically clearer, it is convenient to assert it in the form that the space-time can be continued smoothly, as a conformal manifold, a little way prior to the hypersurface . To before the Big Bang? Surely not: the Big Bang is supposed to represent the beginning of all things, so there can be no 'before'. Never fear—this is just a mathematical trick. The extension is not supposed to have any physical meaning! Or might it Ã¢â'¬Â¦?

Objective mathematical notions must be thought of as timeless entities and are not to be regarded as being conjured into existence at the moment that they are first humanly perceived.

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, ....

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.

A scientific world-view which does not profoundly come to terms with the problem of conscious minds can have no serious pretensions of completeness.

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

The final conclusion of all this is rather alarming. For it suggests that we must seek a non-computable physical theory that reaches beyond every computable level of oracle machines (and perhaps beyond).

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.

All I would myself ask for would be that our perceptive interrogator should really feel convinced, from the nature of the computer's replies, that there is a conscious presence underlying these replies

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.

I argue that the phenomenon of consciousness cannot be accommodated within the framework of present-day physical theory.