Faculty of Arts and Philosophy Centre for Logic and Philosophy of Science A Case Study in Strict Finitism Feng Ye's Strict Finitism and the Logic of Mathematical Applications Nigel Vinckier Postgraduate of Logic, History and Philosophy of Science Academic year 2015 { 2016 Prof. Dr. Jean Paul Van Bendegem Graduational dissertation of the postgraduate LHPS Contents Preface 2 Acknowledgements3 1 Introduction4 1.1 A Naturalistic Philosophy of Mathematics..................4 1.2 The Applicability of Mathematics Under Naturalism............8 1.3 A Logical Explanation of Applicability.................... 11 Conclusions on Chapter 1 14 2 Strict Finitism 16 2.1 The Formal System SF for Strict Finitism.................. 16 2.2 Doing Mathematics in Strict Finitism..................... 19 2.3 Sets and Functions............................... 22 Conclusions on Chapter 2 25 3 Calculus in SF 26 Conclusions 27 Preface In this dissertation, we will discuss a concrete proposal of an elaborate strictly finitist calculus. Several authors, and from different perspectives, denote the concept of infinity as problematic in mathematics. However, no concrete and elaborate strictly finitistic mathematical theories have been developed from this philosophical position. That is, until Feng Ye's Strict Finitism and the Logic of Mathematical Applications was published. One can argue on a philosophical level about the necessity or feasibility of strict finitism, but new elements emerge when discussing a concrete proposal. We will take Ye's work as a case study. We would love to compare different systems for strict finitism, with different philosophical and technical/mathematical approaches and see how the philosophical and technical arguments mutually influence each other. Unfortunately, we know of no other system as concrete as Ye's to compare it to... 2 Therefore, we will not be caught in the trap of generalising from one data point, con- sidering this book as the elaboration of strict finitism. We will however assess both Ye's philosophical and technical/mathematical contribution, whilst simultaneously clarifying our own position on the subject. Some conventions. Unless mentioned otherwise, emphasis in citations is by Ye. A para- graph refers to one block of text separated by a blank space, a section is a piece of a chapter, numbered with double or triple numbers. For easy reference, our sections have the same numbering and titles as in Ye's work. Apart from the acknowledgements, we honour the somewhat strange (and sometimes perceived as pedantic) tradition of writing in the first-person plural format. Poetic quotes are from The Prelude, by the romantic poet William Wordsworth. The author gives his permission to make this work available for consultation and to copy parts of the work for personal use. Any other use is bound by the restrictions of copyright legislation, in particular regarding the obligation to specify the source when using the results of this work. Acknowledgements I want to thank several people without whom strict finitism and, more generally, the phi- losophy of mathematics, would have remained obscure and far removed from my personal life. It's safe to say that the postgraduate of LHPS expanded my world. Bliss was it in that dawn to be alive. Firstly, there is Albrecht Heeffer, who recognised my desire to learn more about the history, logic and philosophy behind mathematics, beyond the mathematics themselve. He informed me about the very existence of this postgraduate and introduced me to it. I would like to thank the lecturers of the postgraduate who motivated me to continue research in logic and the philosophy of mathematics. In particular, I am grateful to professor Van Bendegem, who piqued my curiosity for strict finitism and was willing to be the supervisor of this graduational dissertation. I also thank him for granting me the opportunity to work at the VUB for a year, enabling me to prepare articles and a PhD proposal. And finally, I thank my girlfriend, who helped me to get through difficult moments. My love for her is the only instance of actual infinity I know of in this world. 3 1 Introduction In this chapter, the general philosophical position of the author is established, leading him to the conclusion that a finitistic mathematics is needed for the application to real, physical problems. We will not make a systematic evaluation of Ye's philosophical position in se, but we will address a few key points and contradictions. 1.1 A Naturalistic Philosophy of Mathematics 1.1.1 Infinity and Nominalism Ye has a nominalist view on the ontology of mathematical objects. Nominalism generally denies the existence of abstract entities, but comes in many flavours.1 We will not get involved in the debates between nominalists and realists. However, we do want to address some points where Ye, in our opinion, contradicts himself. The problematic status of infinity for nominalists Infinity certainly is an abstract notion. Therefore, Ye { from his nominalist position { claims that infinity is (or should be) problematic for all nominalists. However, because one doesn't recognise the reality or the existence of an abstract entity, it is not entirely sufficient for rejecting the very notion altogether. One might reject the existence of god, but still be able to imagine him and even construct representations, discuss the notion of a god... Finitists should, in the same sense, in debates with infinitists, still be able to talk about the concept of infinity, be it as a “fiction”. Mathematics independent of physics Ye states that we do not want our philosophical account of human mathematical practices to depend on [...] assumptions about this physical universe. (p. 3) We think this is a correct point. It is of course true that different theories about the physical universe will need a different mathematics. But that does not imply that all our mathematical theories and practices would be blown to smithereens if physicists would be able to decide whether or not the universe is (in)finite or discrete/continuous. A verdict about the physical universe would confirm or refute the application of certain mathematical theories to physics, but it would not alter our human mathematical practices or the content of the mathematical theories themselves. However, throughout the book we encounter arguments for a strictly finitistic2 approach only with regard to the application to a finite reality. Although one might agree with the arguments, they are not sufficient for attacking non-finitist positions. A realist will indeed agree that most contemporary physics theories consider a finite universe, but might still have no problem with infinities or working with abstract concepts independent of this 1See [3]. 2SF further. Also for \strict finitism" as a noun. SF is to denote Ye's specific SF model. 4 physical universe. This realist will then have to assign a different task to the \bridging postulations"3 between reality and mathematical abstraction, avoiding the application of infinity to a finite or even possibly discrete reality. Hence, this application argument bears no objection to infinity in mathematics. But beware, these are no counterarguments against SF. On the contrary, we are very sympathetic to the project of SF, to such an extent we think it deserves an internal mathematical motivation, independent of the application4 or one specific position in the philosophy of mathematics. Our main reason why we think a SF mathematics should be developed, is not for its application to a finite reality. We think the finiteness of the work-space of the mathe- matician and the calculating power (human or non-human) at her disposal is decisive, a point which will be developed further in this dissertation.5 We will see that Ye's system SF is quite suited for our purposes. From this follows another reason why we think the infinite is problematic. If we treat the infinite as an ordinary number6 or we reason with infinite sets in the same way as we do with finite ones, certain basic intuitions are lost. Intuitions we get from mathematical reasoning in a finite setting, as denoted in the previous paragraph. One example is this. We have the following intuition:7 1. A proper subset of a set is \taking a (proper) part of the whole set". 2. Two sets are \equally large" if there is exactly one object in the second set for every object in the first set. This is mathematically translated as the existence of a bijection. These intuitions seem incompatible, but prove to be both true in the infinite case: N $ Z, but there exists an (even explicitly constructable) bijection N ! Z. A paradoxical fact which was taken by Dedekind as the very definition of an infinite set!8 Another example is the well-known Torricelli's Trumpet 9: a surface of revolution with a finite volume (namely π), but an infinite surface. This in itself is paradoxical enough, for 3See section 1.2.2. Postulate would probably be more correct, but we stick to the term used by Ye: postulation. 4In which infinity might be problematic because it is either applied to a finite reality or either because the calculating device (be it human or digital) has finite power. But we will get to that discussion later on, see the FRvsFC-problem. 5One could argue that the assumption of a finite work-space/means/place to conduct mathematics is already a consequence of the assumption of a finite universe. We argue that mathematicians have only finite ink stains and strokes at their disposal, have to communicate their findings in a limited social or public space and in a limited time, use computers with limited power and even have ideas that are the result of an arrangement of a finite amount of neurons. We think all this would not change if physicists were to decide in favour of the continuity and/or infinity of the universe.