AKC Lecture 1 Plato, Penrose, Popper

AKC Lecture 1 Plato, Penrose, Popper E. Brian Davies King s College London November 2011 E.B. Davies (KCL) AKC 1 November 2011 1 / 26

Introduction The problem with philosophical and religious questions is that almost everyone knows the right answer and almost nobody is willing to reconsider. E.B. Davies (KCL) AKC 1 November 2011 2 / 26

Introduction The problem with philosophical and religious questions is that almost everyone knows the right answer and almost nobody is willing to reconsider. Unfortunately people do not agree about what the right answer is. Almost everything that I will say would be regarded as contentious by someone. E.B. Davies (KCL) AKC 1 November 2011 2 / 26

World Views A world view (or metaphysical belief) is a set of fundamental beliefs about reality used to evaluate a number of other, more particular, beliefs. E.B. Davies (KCL) AKC 1 November 2011 4 / 26

World Views A world view (or metaphysical belief) is a set of fundamental beliefs about reality used to evaluate a number of other, more particular, beliefs. We have already seen examples this term in the Jewish and Islamic world views, but another is the view that reality can be completely described by a set of mathematical equations. E.B. Davies (KCL) AKC 1 November 2011 4 / 26

A Proposition to be Discussed A mathematical truth (theorem) is discovered rather than invented. It is true before its proof has been found, before it is formulated, and would be true even if the human species had never existed. Alternatively mathematical theories are human social constructions, which can only exist because our brains have a particular form. Some set of mathematical laws completely controls everything that happens in the universe. Some set of mathematical laws completely describes everything that happens in the universe. Physical phenomena have a degree of regularity, which we can model approximately using a variety of mathematical laws. E.B. Davies (KCL) AKC 1 November 2011 5 / 26

The Reality of the Past E.B. Davies (KCL) AKC 1 November 2011 6 / 26

Truth, Existence and Language The diplodocus walked the earth many millions of years ago. E.B. Davies (KCL) AKC 1 November 2011 7 / 26

Truth, Existence and Language The diplodocus walked the earth many millions of years ago. The diplodocus had four legs. E.B. Davies (KCL) AKC 1 November 2011 7 / 26

Truth, Existence and Language The diplodocus walked the earth many millions of years ago. The diplodocus had four legs. There were rainbows a billion years ago, before any creature with eyes to see them existed. E.B. Davies (KCL) AKC 1 November 2011 7 / 26

What is/was Platonism? Plato was born in Athens around 427BC and founded a School there later in his life. He could be said to have founded academic philosophy. He wrote about a theory of ideal forms, and took great support from its apparent applicability to mathematics. Long after his death a religious/philosophical movement called Neoplatonism developed, in which a notion of an infinite ideal source of all reality was central. This later influenced Augustine and hence the development of Christianity. E.B. Davies (KCL) AKC 1 November 2011 8 / 26

The reality of ideal forms Plato took ideal forms to be more real than particular objects. Thus beauty or justice were more central than particular instances of them. E.B. Davies (KCL) AKC 1 November 2011 9 / 26

The reality of ideal forms Plato took ideal forms to be more real than particular objects. Thus beauty or justice were more central than particular instances of them. God created only one essential Form of Bed in the ultimate nature of things, either because he wanted to or because some necessity prevented him from making more than one; at any rate he didn t produce more than one, and more than one could not possibly be produced... And I suppose that God knew it, and as he wanted to be the creator of a real Bed, and not just a carpenter making a particular bed, decided to make the ultimate reality unique. [Plato. The Republic, Book Ten, Theory of Art.] E.B. Davies (KCL) AKC 1 November 2011 9 / 26

Platonism and Darwin s theory of evolution Religious objections to Darwin s theory were partly based on the notion that a species was a natural kind, an ideal form ordained by God. E.B. Davies (KCL) AKC 1 November 2011 10 / 26

Platonism and The Status of Possibilities According to Keith Ward, an eminent Oxford theologian Even if no actual universe existed, its possibility would exist, together with the possibilities of every other possible universe, all comprising an infinite set of possibilities. We are back to the Platonic world of pure forms, pure possibilities. But how can mere possibilities exist? One must be logically ruthless, and say that either there are really no possibilities or that they exist in something actual. [K Ward, God, Chance and Necessity, p. 36] E.B. Davies (KCL) AKC 1 November 2011 11 / 26

Mathematical Platonism E.B. Davies (KCL) AKC 1 November 2011 12 / 26

Mathematical Platonism I will next consider mathematical Platonism as described below. It is more than the statement that the mathematical consensus about something may be unrevisable in the assumed context. Theorems are supposed to be true statements about timeless entities, and to be true whether or not they have ever been or will ever be formulated by human beings. E.B. Davies (KCL) AKC 1 November 2011 12 / 26

Roger Penrose He has played a major role in the theory of quasiperiodic tilings, black holes and twistor theory. E.B. Davies (KCL) AKC 1 November 2011 13 / 26

Roger Penrose 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 this process of seeing... The mental images each one has, when making this Platonic contact, might be rather different in each case, but communication is possible because each is directly in contact with the same externally existing Platonic world! [Penrose, The Emperor s New Mind, p. 428. Oxford Univ. Press.] E.B. Davies (KCL) AKC 1 November 2011 14 / 26

Gödel In 1931 Kurt Gödel stopped the efforts to provide a firm foundation for mathematics by proving that in any sufficiently rich formal system there must exist a statement that cannot be proved or disproved within the system. E.B. Davies (KCL) AKC 1 November 2011 15 / 26

The Status of Proof The claim that one can make a logical distinction between the truth of a theorem and the existence of a proof of it is not self-evident, but it follows from Gödel s theorems provided one believes in the absolute Platonic existence of mathematical entities. E.B. Davies (KCL) AKC 1 November 2011 16 / 26

Natural numbers and Set Theory Platonists believe that the infinite set of all natural numbers actually exists and has objective properties. This is quite different from adding the existence of this set to one s mathematical framework as a deliberate choice or convention. It is often considered that Cantor and then Frege laid the foundations of a systematic theory of infinite objects, but the consistency of this theory is not known in spite of enormous efforts to resolve this problem between 1890 and 1930. E.B. Davies (KCL) AKC 1 November 2011 17 / 26

Merits of Platonism It corresponds to the way many mathematicians feel about their subject. E.B. Davies (KCL) AKC 1 November 2011 18 / 26

Merits of Platonism It corresponds to the way many mathematicians feel about their subject. It explains why everyone eventually agrees whether a theorem is true or false. E.B. Davies (KCL) AKC 1 November 2011 18 / 26

Merits of Platonism It corresponds to the way many mathematicians feel about their subject. It explains why everyone eventually agrees whether a theorem is true or false. It sanctions the reference to infinite entities as if they were as real as those that we have direct physical experience of. E.B. Davies (KCL) AKC 1 November 2011 18 / 26

Weaknesses of Platonism Research has shown that the way people feel their mental processes work bears no relationship with how they actually work. E.B. Davies (KCL) AKC 1 November 2011 19 / 26

Weaknesses of Platonism Research has shown that the way people feel their mental processes work bears no relationship with how they actually work. There are innumerable cases in which mathematicians have agreed about some theorem only to have to admit that they should not have. E.B. Davies (KCL) AKC 1 November 2011 19 / 26

An alternative approach There is an alternative philosophy due to Aristotle, in which the physical world takes priority over the mathematical world, and in which mathematics arises by the process of abstraction, i.e. by our mental activities, individual or collective. According to Aristotle the truly infinite does not exist. What does exist is the possibility of extending certain procedures indefinitely, while rejecting contemplation of the completed process. E.B. Davies (KCL) AKC 1 November 2011 20 / 26

Prime numbers It is often said that Euclid proved that there is an infinite number of prime numbers, but this is a Platonic gloss on his actual result. E.B. Davies (KCL) AKC 1 November 2011 21 / 26

Prime numbers It is often said that Euclid proved that there is an infinite number of prime numbers, but this is a Platonic gloss on his actual result. Proposition 20, Book 9 of Euclid s Elements states that Prime numbers are more than any assigned multitude of prime numbers or, in contemporary language, given any finite list of prime numbers, there is another prime number not in that list. E.B. Davies (KCL) AKC 1 November 2011 21 / 26

Michael Atiyah Mathematics is an evolution from the human brain, which is responding to outside influences, creating the machinery with which it then attacks the outside world. It is our way of trying to reduce complexity into simplicity, beauty and elegance. It is really very fundamental, simplicity is in the nature of scientific inquiry we do not look for complicated things. I tend to think that science and mathematics are ways the human mind looks and experiences you cannot divorce the human mind from it. Mathematics is part of the human mind. E.B. Davies (KCL) AKC 1 November 2011 22 / 26

Karl Popper Popper was a twentieth century philosopher who spent much of his life at the LSE. His scientific legacy was to demolish the idea of certain knowledge in science, and to replace it with the idea of constant testing with the possibility of refutation. Perhaps even more important is the notion of a domain of applicability. A theory may be approximately true, or useful, within a certain context, whose boundaries need to be determined. E.B. Davies (KCL) AKC 1 November 2011 23 / 26

Popper s three worlds World 1 the world of physical entities World 2 the world of mental states World 3 E.B. Davies (KCL) AKC 1 November 2011 24 / 26

Popper s three worlds World 1 the world of physical entities World 2 the world of mental states World 3 By World 3 I mean the world of products of the human mind, such as stories, explanatory myths, tools, scientific theories (whether true or false), scientific problems, social institutions, and works of art. World 3 objects are of our own making, although they are not always the result of planned production by individual men. [Popper, K. R. and Eccles, J. C. (1977). The Self and Its Brain, An Argument for Interactionism, Chap. P2 and p.38.] E.B. Davies (KCL) AKC 1 November 2011 24 / 26

Popper s three worlds I am an opponent of what I have called essentialism. Thus, in my opinion, Plato s ideal essences play no role in World 3. (That is, Plato s World 3, though clearly in some sense an anticipation of my World 3, seems to me a mistaken construction.) On the other hand, Plato would never have admitted such entities as problems or conjectures especially false conjectures into his world of intelligible objects. [Popper and Eccles, loc. cit., p.43.] E.B. Davies (KCL) AKC 1 November 2011 25 / 26

Summary Platonism is not the only way of understanding the world, and a more modest acceptance of the fallibility of all human knowledge fits our situation better. E.B. Davies (KCL) AKC 1 November 2011 26 / 26

Summary Platonism is not the only way of understanding the world, and a more modest acceptance of the fallibility of all human knowledge fits our situation better. One can survive without claiming to know how all the pieces will eventually fit together, or even whether they will. Mathematics is no more special than language in general. It is merely the name of our best current way of understanding certain aspects of the world around us. It does not explain ethics, our subjective consciousness and will not even enable us to predict the weather a month ahead. E.B. Davies (KCL) AKC 1 November 2011 26 / 26