The Philosophy of Mathematics and Natural Laws

Another Copernican Revolution

Noel Curran

• Publisher: Routledge
• Year: 2019
• Print ISBN:
  9781138337831
• eBook ISBN:
  9780429808074
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

First published in 1997, this title is a sequel to Dr Noel Curran's first book The Logical Universe: The Real Universe (published by Ashgate under the Avebury imprint, 1994). The philosophy of mathematics in this book is based on ideas of Sir William Rowan Hamilton on the ordinal character of numbers, the real numbers, the measure numbers, scalar numbers and the extension to vectors. The final extension is to Hamilton's quaternions. This algebra is interpreted as the mathematics of spin. This led to a a new theory of time and space which is Euclidian. The motion of spin is absolute, no frame of reference is required. If time is assumed to have a beginning it would be asymmetric with an arrow. This concept is applied to the laws of nature, which are symmetrical. This is another Copernican Revolution in three aspects: absolute time is restored, time has an arrow - is asymmetric, and thirdly the theory is based on the motion of spin which is absolute and more fundamental than the motion of translation. This opens the way to the final unification of physics.

1 Introduction

The first problem to consider is how does one study the philosophy of mathematics. Is it necessary to be a professional mathematician to make a contribution or to understand what its philosophy is all about? The great names in the history of the subject, such as Husserl, Frege, Cantor, Peano, Hilbert, Whitehead and Russell, and in more modern times Gödel and Dummett, were all trained as mathematicians. The subject is usually called mathematical logic: therefore a general theory of logic is required, which is a fundamental part of the general subject of philosophy also. I think that a technical knowledge of mathematics is not essential for the philosophy of mathematics. All mathematical concepts can be expressed in language; thus, if a new concept is developed, a new language can be found to express it. A new word must be found for the new concept, usually from the Greek or Latin language. Therefore the study of the philosophy of mathematics can be an exercise in linguistic analysis. The fact that there are three schools of thought – the logicist, the formalist and the intuitionist – is sufficient evidence that the foundations of mathematics are not universally agreed. There is an interesting way of looking at the foundations of mathematics and comparing it with architecture. In constructing a building the foundations are laid first and the building is raised on top of that, reaching completion with the roof. In mathematics the procedure appears to be the reverse. There is an immense structure of mathematics built up over the centuries, but the very foundations of mathematics – what it really means – have not yet been provided.

In studying philosophy one must read the works of other writers, so as to understand their concepts on the subject. Yet there can be an overemphasis on this use of texts, and an almost endless series of comments on the texts can result, so much so that the fundamental problems of the subject are almost forgotten.

The fundamental concept of mathematics is that of number. Once that is established, one naturally has to use the texts of other people on the subject. One may agree or not agree with the concepts expressed in the texts of other philosophers, yet even when the views expressed in these works are the opposite of one’s own ideas they can be most valuable in helping to find the truth of the matter. After all, there are only two possibilities. What is said in the text is either correct or incorrect, or, in the ideas used in our own theory of the philosophy of mathematics, what is the case and what is not the case.

But finally you must develop a complete and coherent account of the philosophy. Therefore I must develop my own philosophy of mathematics, which of course will agree or not agree with the views expressed in the texts of other mathematicians or philosophers of mathematics. I have noticed a remarkable fact about the general theories of philosophers or scientists as expounded in their texts, Even in a complete book on the subject, the fundamental or central ideas are often expressed somewhere in one paragraph and often even in one sentence. This is of course most helpful in using quotations from the texts, which can agree or not agree with one’s own point of view.

Logic

Before one can begin a philosophy of mathematics it is essential to have a theory of logic. The foundation of any theory of logic is the judgement. For this purpose I propose to use the text of Frege’s famous paper of 1892, Über Sinn und Bedeutung. This paper is considered so important that the October 1992 issue of the philosophy journal Mind was entirely devoted to it, to commemorate its centenary. I have written an article on the paper which was worked out a considerable time before I had read the issue in question. This article was sent to Mind in November 1992 but was not accepted for publication. The following is a brief summary of my paper on Frege’s Über Sinn und Bedeutung. The standard translation of Frege’s paper is On Sense and Reference, but I prefer to use a different translation, On Meaning and Reference. The paper is essentially in two parts.

The first part

In this, different expressions or words have the same reference.

Using a mathematical example:

The two expressions have the same reference, namely the number 9. That is, the expressions are equal – it is a theory of identity. It is in effect a declarative sentence – a judgement with one reference, the True. This is what an axiom means: a judgement which is necessarily true; its negation is nonexistent. These judgements are the same as Kant’s analytic a priori judgements: they are true of necessity – if they are axioms. His synthetic a priori judgements are also true of necessity, i.e. axioms. In a paper I wrote showing that Kant’s synthetic a priori judgements were true necessarily I gave a reference to his Prolegomena to any Future Metaphysics. But in Section 2 of the introduction to the Critique of Pure Reason Kant makes it very clear that it is the a priori character of the judgements, analytic and synthetic, which ensures that character of necessity, i.e. they are both axioms: judgements with one reference, the True:

Now in the first place, if we have a proposition which contains the idea of necessity in its very conception, it is a judgement a priori; if moreover it is not derived from any other proposition, unless from one equally involving the idea of necessity, it is absolutely a priori.¹

Husserl in Section 16 of Ideas I makes very clear and explicit reference to Kant’s synthetic a priori judgements:

If, despite notable differences in fundamental outlook which are not incompatible however with an inner affinity, one wishes to maintain approval of Kant’s Critique of the Reason, one has only to interpret the regional axioms as synthetic cognitions a priori and we should then have as many irreducible classes of such forms of knowledge as there are regions.²

Cognitions are of course judgements. Husserl’s interpretation of Kant’s synthetic a priori judgements is perfectly correct. He calls them axioms. The regional axioms are to be used for different regions of knowledge, such as mathematics, physics, biology, etc.

The first part of Frege’s paper is really a theory of judgements of identity – judgements which are true, which have one reference, the True.

Different words can have the same reference. Frege’s first example is:

This can be expressed in very simple symbolic form:

The second part

In this part of Frege’s paper he says that so far we have considered the meaning and reference of words or expressions. We now enquire concerning the meaning and reference of an entire declarative sentence. The declarative sentence is a judgement or proposition. But, as I explained in the previous chapter, the first part of Frege’s paper is really a judgement – a declarative sentence of identity – with one reference, the True, i.e. an axiom. He then goes on to give an account of the truth value of a sentence. Every judgement is true or false. It states what is the case and what is not the case. The judgement has two references – the True and the False. The judgements are contradictory, they are divided by a negation. They are not now judgements of identity, but judgements of difference. However, to say that this is the essence of a judgement or that all judgements have two references – the True and the False – is not correct. There is an important exception. Some judgements have only one reference – the True – i.e. axioms. Judgements with two references – the True and the False – can be falsified. Judgements with only one reference – the True – i.e. axioms, cannot be falsified – their negation is non-existent.

The judgement with two references can be expressed very simply:

The second form of the judgement is expressed as two poles: the judgements are polar opposites. Two contradictory states of affairs – what is the case and what is not the case – are polar opposites. This is the use of polar language. It is different from analogical language, where two states of affairs are partly the same and partly different – i.e. there is a proportion between the two states of affairs. With polar language there is no proportion: the two states of affairs are contradictory, they are divided by a negation; that is, the judgement has two references, the True and the False.

To have meaning in the fullest sense, a judgement must have two references – the True and the False. It must be capable of verification and falsification. There must be a state of affairs corresponding to its being true and its being false. There are therefore two types of judgements:

1 Judgements with one reference, the True – i.e. axioms.

2 Judgements with two references, the True and the False.

In formal logic proof is a fundamental concept. For proof theory you require true premises, i.e. axioms and valid arguments. Axioms are judgements with one reference – the True – their negation by definition is impossible and non-existent. Judgements with two references – the True and the False – have meaning. They have no axioms, for this is dialectical logic. Examples of judgements which are true and false can be very trivial: the red ball is on the table, it is not on the table. But when the distinction is applied to very general concepts such as whether space is flat or curved, time has a beginning or no beginning and even to whether there is a God or no God, problems of the most fundamental nature are involved. Equations in mathematics such as 7 + 5= 12 or 7 x 5 = 5 x 7 have validity Wittgenstein called them tautologies. They are of the same status as ‘Bachelors are unmarried men.’ He called them in his language sinnlos – i.e. meaningless – but they have validity.

In the two parts of Frege’s paper we have been concerned with the meaning of judgements. The meaning of words or expressions without reference to judgements has not been considered. In the first part of his paper Frege stated that different words and expressions can have the same meaning or reference. We now consider the important distinction that the same word can have different meanings – i.e. the opposite state of affairs. This is Wittgenstein’s famous dictum that the meaning of a word is its use in the language – implying that the same word can have different meanings or uses. Let us take as an example an expression from the first part of Frege’s paper – the morning star. This has at least two meanings:

An important example very relevant to mathematics is the word ‘number’, e.g.

1 Cardinal numbers, the...

Table of contents

• Cover
• Half Title
• Dedication
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Prolegomenon Frege – The Theory of Judgement
• 1 Introduction
• 2 The Concept of Number
• 3 Number Systems
• 4 Algebra – the Science of Pure Time
• 5 Geometry
• 6 Measurement and Numbers
• 7 Quaternions Versus Vector Analysis
• 8 The Unification of Physics
• References
• Bibliography
• Index