akc lecture 1 plato, penrose, popper - king's college … · akc lecture 1 plato, penrose,...

AKC Lecture 1Plato, 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 almosteveryone knows the ‘right answer’ and almost nobody is willing toreconsider.

Unfortunately people do not agree about what the right answer is.

Almost everything that I will say would be regarded as contentious bysomeone.

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Penrose

Popper

Introduction

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Penrose

Popper

Introduction

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Penrose

Popper

Introduction

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Penrose

Popper

Introduction

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Plato, Aristotle

Penrose, Godel, Ward

Popper, Atiyah

World Views

A world view (or metaphysical belief) is a set of fundamental beliefs aboutreality used to evaluate a number of other, more particular, beliefs.

We have already seen examples this term in the Jewish and Islamic worldviews, but another is the view that reality can be completely described bya set of mathematical equations.

Yet another is the view that everything has a cause, if one looks deeplyenough.

World Views

A Proposition to be Discussed

A mathematical truth (theorem) is discovered rather than invented. It istrue before its proof has been found, before it is formulated, and would betrue even if the human species had never existed.

Alternatively mathematical theories are human social constructions, whichcan only exist because our brains have a particular form.

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Some set of mathematical laws completely controls everything thathappens in the universe.

Some set of mathematical laws completely describes everything thathappens in the universe.

Physical phenomena have a degree of regularity, which we can modelapproximately using a variety of mathematical laws.

A Proposition to be Discussed

A mathematical truth (theorem) is discovered rather than invented. It istrue before its proof has been found, before it is formulated, and would betrue even if the human species had never existed.

Alternatively mathematical theories are human social constructions, whichcan only exist because our brains have a particular form.

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Some set of mathematical laws completely controls everything thathappens in the universe.

Some set of mathematical laws completely describes everything thathappens in the universe.

Physical phenomena have a degree of regularity, which we can modelapproximately using a variety of mathematical laws.

The Reality of the Past

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 tosee them existed.

Alice in Wonderland had five toes on each foot.

What is/was Platonism?

Plato was born in Athens around 427BC and founded a School there laterin his life. He could be said to have founded academic philosophy. Hewrote about a theory of ideal forms, and took great support from itsapparent applicability to mathematics.

Long after his death a religious/philosophical movement calledNeoplatonism developed, in which a notion of an infinite ideal source of allreality was central. This later influenced Augustine and hence thedevelopment of Christianity.

The reality of ideal forms

Plato took ideal forms to be more real than particular objects. Thusbeauty or justice were more central than particular instances of them.

God created only one essential Form of Bed in the ultimatenature of things, either because he wanted to or because somenecessity prevented him from making more than one; at any ratehe didn’t produce more than one, and more than one could notpossibly be produced . . . And I suppose that God knew it, and ashe wanted to be the creator of a real Bed, and not just acarpenter making a particular bed, decided to make the ultimatereality unique. [Plato. The Republic, Book Ten, Theory of Art.]

The reality of ideal forms

Plato took ideal forms to be more real than particular objects. Thusbeauty or justice were more central than particular instances of them.

God created only one essential Form of Bed in the ultimatenature of things, either because he wanted to or because somenecessity prevented him from making more than one; at any ratehe didn’t produce more than one, and more than one could notpossibly be produced . . . And I suppose that God knew it, and ashe wanted to be the creator of a real Bed, and not just acarpenter making a particular bed, decided to make the ultimatereality unique. [Plato. The Republic, Book Ten, Theory of Art.]

Platonism and Darwin’s theory of evolution

Religious objections to Darwin’s theory were partly based on the notionthat a species was a natural kind, an ideal form ordained by God.

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, allcomprising an infinite set of possibilities. We are back to thePlatonic world of pure forms, pure possibilities. But how canmere possibilities exist? One must be logically ruthless, and saythat either there are really no possibilities or that they exist insomething actual. [K Ward, God, Chance and Necessity, p. 36]

That something actual turns out to be the mind of God.

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, allcomprising an infinite set of possibilities. We are back to thePlatonic world of pure forms, pure possibilities. But how canmere possibilities exist? One must be logically ruthless, and saythat either there are really no possibilities or that they exist insomething actual. [K Ward, God, Chance and Necessity, p. 36]

That something actual turns out to be the mind of God.

Mathematical Platonism

I will next consider mathematical Platonism as described below. It is morethan the statement that the mathematical consensus about somethingmay be unrevisable in the assumed context.

Theorems are supposed to be true statements about timeless entities, andto be true whether or not they have ever been or will ever be formulatedby human beings.

In this view proofs are merely our way of ensuring that our dim perceptionof the truth is not misleading us.

Roger Penrose

He has played a major role in the theory of quasiperiodic tilings,black holes and twistor theory.

Roger Penrose

When mathematicians communicate, this is made possible byeach one having a direct route to truth, the consciousness ofeach being in a position to perceive mathematical truths directly,through this process of ‘seeing’ . . . The mental images each onehas, when making this Platonic contact, might be ratherdifferent in each case, but communication is possible becauseeach is directly in contact with the same externally existingPlatonic world! [Penrose, The Emperor’s New Mind, p. 428.Oxford Univ. Press.]

In 1931 Kurt Godel stopped the efforts to provide a firm foundation formathematics by proving that in any sufficiently rich formal system theremust exist a statement that cannot be proved or disproved within thesystem.

The Status of Proof

The claim that one can make a logical distinction between the truth of atheorem and the existence of a proof of it is not self-evident, but it followsfrom Godel’s theorems provided one believes in the absolute Platonicexistence of mathematical entities.

Natural numbers and Set Theory

Platonists believe that the infinite set of all natural numbers actually existsand has objective properties.

This is quite different from adding the existence of this set to one’smathematical framework as a deliberate choice or convention.

It is often considered that Cantor and then Frege laid the foundations of asystematic theory of infinite objects, but the consistency of this theory isnot known in spite of enormous efforts to resolve this problem between1890 and 1930.

Natural numbers and Set Theory

Platonists believe that the infinite set of all natural numbers actually existsand has objective properties.

This is quite different from adding the existence of this set to one’smathematical framework as a deliberate choice or convention.

It is often considered that Cantor and then Frege laid the foundations of asystematic theory of infinite objects, but the consistency of this theory isnot known in spite of enormous efforts to resolve this problem between1890 and 1930.

Merits of Platonism

It corresponds to the way many mathematicians feel about theirsubject.

It explains why everyone eventually agrees whether a theorem is trueor false.

It sanctions the reference to infinite entities as if they were as real asthose that we have direct physical experience of.

It fits in with the fact that mathematics seems to underlie all of ourmost successful physical theories.

Merits of Platonism

Weaknesses of Platonism

Research has shown that the way people feel their mental processeswork bears no relationship with how they actually work.

There are innumerable cases in which mathematicians have agreedabout some theorem only to have to admit that they should not have.

No mechanism by which a Platonic world of mathematics couldinfluence the physical world has ever been outlined.

An alternative approach

There is an alternative philosophy due to Aristotle, in which the physicalworld takes priority over the mathematical world, and in whichmathematics arises by the process of abstraction, i.e. by our mentalactivities, individual or collective.

According to Aristotle the truly infinite does not exist. What does exist isthe possibility of extending certain procedures indefinitely, while rejectingcontemplation of the completed process.

It has been said that for over two thousand years philosophy has been acontinued debate about the merits of the outlooks of Plato and Aristotle.

Prime numbers

It is often said that Euclid proved that there is an infinite number of primenumbers, 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 primenumbers

or, in contemporary language, ‘given any finite list of prime numbers, thereis another prime number not in that list’.

Prime numbers

It is often said that Euclid proved that there is an infinite number of primenumbers, 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 primenumbers

or, in contemporary language, ‘given any finite list of prime numbers, thereis another prime number not in that list’.

Michael Atiyah

Mathematics is an evolution from the human brain, which isresponding to outside influences, creating the machinery withwhich it then attacks the outside world. It is our way of trying toreduce complexity into simplicity, beauty and elegance. It isreally very fundamental, simplicity is in the nature of scientificinquiry – we do not look for complicated things. I tend to thinkthat science and mathematics are ways the human mind looksand experiences – you cannot divorce the human mind from it.Mathematics is part of the human mind.

Karl Popper

Popper was a twentieth century philosopher who spent much of his life atthe LSE. His scientific legacy was to demolish the idea of certainknowledge in science, and to replace it with the idea of constant testingwith the possibility of refutation.

Perhaps even more important is the notion of a domain of applicability. Atheory may be approximately true, or useful, within a certain context,whose boundaries need to be determined.

Karl Popper

Popper was a twentieth century philosopher who spent much of his life atthe LSE. His scientific legacy was to demolish the idea of certainknowledge in science, and to replace it with the idea of constant testingwith the possibility of refutation.

Perhaps even more important is the notion of a domain of applicability. Atheory may be approximately true, or useful, within a certain context,whose boundaries need to be determined.

Popper’s three worlds

World 1 – the world of physical entitiesWorld 2 – the world of mental statesWorld 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 byindividual men. [Popper, K. R. and Eccles, J. C. (1977). TheSelf and Its Brain, An Argument for Interactionism, Chap. P2and p.38.]

World 1 – the world of physical entitiesWorld 2 – the world of mental statesWorld 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 byindividual men. [Popper, K. R. and Eccles, J. C. (1977). TheSelf and Its Brain, An Argument for Interactionism, Chap. P2and p.38.]

I am an opponent of what I have called ‘essentialism’. Thus, inmy opinion, Plato’s ideal essences play no role in World 3. (Thatis, Plato’s World 3, though clearly in some sense an anticipationof my World 3, seems to me a mistaken construction.) On theother hand, Plato would never have admitted such entities asproblems or conjectures – especially false conjectures – into hisworld of intelligible objects. [Popper and Eccles, loc. cit., p.43.]

Summary

Platonism is not the only way of understanding the world, and a moremodest acceptance of the fallibility of all human knowledge fits oursituation better.

One can survive without claiming to know how all the pieces willeventually fit together, or even whether they will.

Mathematics is no more special than language in general. It is merely thename of our best current way of understanding certain aspects of theworld around us.

It does not explain ethics, our subjective consciousness and will not evenenable us to predict the weather a month ahead.

It does not provide irrefutable evidence of the existence of ideal objectsoutside space and time, in spite of the fact that many puremathematicians are Platonists.

Summary

akc lecture 1 plato, penrose, popper - king's college … · akc lecture 1 plato, penrose,...

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