STRICT FINITISM AS A FOUNDATION FOR MATHEMATICS Submitted for the degree of PhD JIM MAWBY 0003528 SEPTEMBER 2005 ABSTRACT The principal focus of this research is a comprehensive defence of the theory of strict finitism as a foundation for mathematics. I have three broad aims in the thesis; firstly, to offer as complete and developed account of the theory of strict finitism as it has been described and discussed in the literature. I detail the commitments and claims of the theory, and discuss the best ways in which to present the theory. Secondly, I consider the main objections to strict finitism, in particular a number of claims that have been made to the effect that strict finitism is, as it stands, incoherent. Many of these claims I reject, but one, which focuses on the problematic notion of vagueness to which the strict finitist seems committed, I suggest, calls for some revision or further development of the strict finitist's position. The third part of this thesis is therefore concerned with such development, and I discuss various options for strict finitism, ranging from the development of a trivalent semantic, to a rejection of the commitment to vagueness in the first instance. ACKNOWLEDGEMENTS Many thanks to the staff and post-graduates at Glasgow University during the period 2001-5, many of whom have helped to shape this research. In particular, thanks to Bob Hale, for his instructive supervision and continual support, and of course to my wonderful wife, Helen. Contents Introduction ....................................................................................... PART ONE: A DEFINITION AND EXPLANATION OF THE PHILOSOPHY OF STRICT FINITISM Chapter One An Introduction to Finitism in Mathematics 10 ................................................ Chapter Two Anti-Realism Constructivism 25 and ........................................................... Chapter Three Reliable Construction 'Surveyability . 42 and ................................................. Chapter Four Complexity and the Problems Notation 59 of ................................................. PART TWO: THE PROBLEMS FOR STRICT FINITISM ADDRESSED Chapter Five The Surveyability Dilemma .................................................................. 77 Chapter Six Vagueness General Problem -a ............................................................. 92 Chapter Seven Sorites Surveyability and ................................................................... III Chapter Eight Weakly Finite and Weakly Infinite Totalities ............................................ 127 PART THREE: ALTERNATIVE LOGICS AND FORMULATIONS FOR THE STRICT FINITARY ACCOUNT Chapter Nine Alternative Logics ............................................................................ 143 Chapter Ten Higher-order Vagueness ..................................................................... 160 Chapter Eleven Alternative Fonnulations .................................................................... 177 Conclusion .............................. , ..................................................... 196 fNTRODUCTION The aim of this research is to provide a proper account of strict finitism as a foundation for mathematics,and to provide a robust defenceof the theory against numerousobjections which have been thought to deflate the theory as a seriousor tenableposition. I shall also aim to presenta model of strict finitism which, while it differs from the traditional account,will, I suggest,solve many of the problems associatedwith such an account. An outline of the project Strict finitism has been discussed variously in the literature surrounding the philosophy of mathematics, but proponents of the theory are rare. For this reason, perhaps, there is no reliable, general account of the theory, and I hope to go some way towards offering such in the opening chapters of this thesis. In particular, the notion of surveyability, which seems central to the strict finitist's account, is used in a variety of contexts to mean anything from 'is possible to physically reproduce' to 'is recognised and understood in the mind of the surveyor'. I shall try to offer an account of surveyability which clearly marks the bounds of the criterion. There are also many objections to the theory to be found in the literature, and a large focus of this thesis will be to addressthese issues. Not least, there have been several attempts to show that the theory is internally inconsistent, and I shall argue that no such thing is established. Lastly, after a thorough consideration of the various options for the formulation of a strict finitary model, including a detailed examination of the plausibility of a trivalent semantics,I shall offer a form of finitism that I have labelled 'Fanatical Finitism', because,if anything, it takes the fundamentalideas of strict finitism more seriously than a traditional account,which I contend proves robust to the various remaining obstaclesfor the position as a whole. Fanatical finitism is, I 4 further suggest,the preferred model for the strict finitist - and one worthy of study both in itself and for the potential consequencesits failure might have on the finitist programme as a whole. On the subject of numbers Although I shall offer some qualification in the opening chapter as to why I shall find it convenient to focus on the objects of mathematics - in particular on numbers themselves - rather than on the statements or proofs which must also ultimately be the concern of the philosopher advancing a coherent theory for the foundation of mathematics, I would also like to be clear from the start about the central concern of this work. Firstly, I should reiterate that numbers are of paramount infinite, become importance and interest to me - my suspicion of the which will apparent, is a suspicion that it is not a genuine number at all. Furthermore, in general the objects of study will be the natural numbers; again, this is largely for the sake of simplicity, and to ensure that the examples are intuitively accessible in what is, after all, for many people a very counter-intuitive model for the foundation of mathematics. However, where appropriate, I shall also include some discussion of the other species of number (in relation to finitism), and will try to offer enough remarks to indicate how what I suggest may be successfully applied in a wider context. I would also like to make somepreliminary remarks about what I take the term number to apply to, in an ordinary sense.(I do not mean here how far I think the term 'number' may be usefully applied 'up the number line', as it were; this will be the subject of a good deal of the work that follows. By 'ordinary' sense,I mean I should like to explain what I understandby the term '2 is a number', where 2 is perfectly acceptableas a number, finitistically speaking). There is an apparentconfusion betweennumbers and their physical representations, the symbols (or string of symbols) used to represent the numbers; and I have been surprised to learn that many philosophers think the distinction is not an important one. It seems to me that numbers are distinct from numerals in a very important sense. A numeral may be canvassed and not understood, and indeed this is 5 precisely the casefor all children before they come to understandthat a particular representationstandsfor a particular number. A number, on the other hand, is a concept that one has a possessionof, irrespective of the presenceof a representation of it. Indeed, it is plausible that a child might know some of the very early numbers in a relational context (e.g. 'there are two applesover there') without knowing or recognising which particular numeral representsthat number. Moreover, we needn't think that understandingof a particular notation (including its component symbols) is an understandingof every number in that notation - in fact, I think quite the opposite. When children learn to count, they may at first learn simply the sequenceof words cone' 'two', 'three', But if before, they understandthat these , and so on. soon, not will words refer to numbers,which they understand,and understandin the sensethat they recognisethat the number 'three' (for example) correspondsto a collection of three things. We do not teach numberspurely by the introduction of symbols, but by explaining that the number is exemplified in this or that particular case,and then that such a number is representedby this or that symbol. Two very young children might both mechanically be able to count to ten, but while one will recognisea collection of ten objects as exemplifying the number, the other may have no understandingof the conceptand henceno ability to use the number appropriately. And this seemsto hold true if they are both taught to write the symbol '10' for 'ten'. I maintain then that understandingof a number is more than simple recognition of the numeral. Children usually learn the symbol for one hundred before they have sorted out all the numbersbelow it, and indeed when a child who has only really graspedthe conceptsof the first few numbersis askedhow big numbers get, one hundred is a good candidate,or one thousandperhaps, or even a million. But no genuineunderstanding accompanies the assertion,even though, as I assert,a genuine understandingdoes accompanythe numberswith which the child can already operate, and apply, like 'two', or 'five'. And this is not becausethey will recognise or can write down the symbols '2' and '5' but not '100' since in many casesthey will be able to do both.