AKC Lecture 1 Plato, Penrose, Popper

Transcription

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

2 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 / 26

3 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. E.B. Davies (KCL) AKC 1 November / 26

4 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 / 26

5 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. Plato Penrose Popper E.B. Davies (KCL) AKC 1 November / 26

6 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. Plato, Aristotle Penrose, Gödel, Ward Popper, Atiyah E.B. Davies (KCL) AKC 1 November / 26

7 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 / 26

8 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 / 26

9 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. Yet another is the view that everything has a cause, if one looks deeply enough. E.B. Davies (KCL) AKC 1 November / 26

10 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. E.B. Davies (KCL) AKC 1 November / 26

11 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 / 26

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

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

14 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 / 26

15 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 / 26

16 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. Alice in Wonderland had five toes on each foot. E.B. Davies (KCL) AKC 1 November / 26

17 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 / 26

18 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 / 26

19 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 / 26

20 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 / 26

21 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 / 26

22 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] That something actual turns out to be the mind of God. E.B. Davies (KCL) AKC 1 November / 26

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

24 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 / 26

25 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. In this view proofs are merely our way of ensuring that our dim perception of the truth is not misleading us. E.B. Davies (KCL) AKC 1 November / 26

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 / 26

27 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 Oxford Univ. Press.] E.B. Davies (KCL) AKC 1 November / 26

28 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 / 26

29 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 / 26

30 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. E.B. Davies (KCL) AKC 1 November / 26

31 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 E.B. Davies (KCL) AKC 1 November / 26

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

33 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 / 26

34 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 / 26

35 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. It fits in with the fact that mathematics seems to underlie all of our most successful physical theories. E.B. Davies (KCL) AKC 1 November / 26

36 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 / 26

37 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 / 26

38 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. No mechanism by which a Platonic world of mathematics could influence the physical world has ever been outlined. E.B. Davies (KCL) AKC 1 November / 26

39 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. No mechanism by which a Platonic world of mathematics could influence the physical world has ever been outlined. E.B. Davies (KCL) AKC 1 November / 26

40 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. E.B. Davies (KCL) AKC 1 November / 26

41 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 / 26

42 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. It has been said that for over two thousand years philosophy has been a continued debate about the merits of the outlooks of Plato and Aristotle. E.B. Davies (KCL) AKC 1 November / 26

43 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 / 26

44 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 / 26

45 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 / 26

46 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. E.B. Davies (KCL) AKC 1 November / 26

47 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 / 26

48 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 / 26

49 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 / 26

50 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 / 26

51 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 / 26

52 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. E.B. Davies (KCL) AKC 1 November / 26

53 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. E.B. Davies (KCL) AKC 1 November / 26

54 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 / 26

55 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. It does not provide irrefutable evidence of the existence of ideal objects outside space and time, in spite of the fact that many pure mathematicians are Platonists. E.B. Davies (KCL) AKC 1 November / 26