A Case Study in Strict Finitism

Faculty of Arts and Philosophy Centre for Logic and Philosophy of Science

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, considering 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 paragraph 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 philosophy 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 mathematician 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. The human-scale mathematical practice would remain a finite enterprise. 6I.e. with its special status of course, but a number nonetheless, with which we can reason and calculate. 7We do not want to impose our intuitions on other readers, but we think these are pretty universal... 8See [6]. 9See e.g. [15].

5 we could pour a finite volume of paint in the horn (hence painting the inside), but we couldn’t paint the infinite surface with it. But even more puzzling10: say we consider a 1 second, smaller trumpet, e.g. by rotating the graph of f(x) = 2x around the x-axis and say we shove this second one in the first one, which is filled for three quarters with paint. Can we, or can we not, consider the second trumpet “painted”? Its outside surface is clearly entirely in contact with the paint inside the first trumpet11, but as a Torricelli trumpet itself, its surface cannot be painted with a finite amount of paint. Yet another example are the various equivalences of the axiom of choice12, some of which are very intuitive (e.g. the axiom of choice itself), and some are not (e.g. the well-ordering theorem). A consequence of the seemingly innocent axiom of choice is the famous Banach- Tarski paradox:

[T]he Banach-Tarski paradox is one of the most celebrated paradoxes in mathematics. It states that given any two subsets A and B of R3, which are bounded and have non-empty interior, it is possible to ‘cut’ A into a finite number of pieces which can be moved by rigid motions (translations and rotations) to form exactly B. This has many amusing consequences; it is most commonly stated as the assertion that a pea can be cut into a finite number of pieces which could then be reassembled to form the sun. Intuitively, of course, this is absurd. Rigid motions are supposed to preserve volume. The Banach-Tarski paradox seems to completely contradict this. This is why it is called a paradox and not a theorem. This is also why many people have tried to argue that it must be false. Since the logic of the proof is unquestioned, they argue that one of the underlying assumptions must be flawed. The assumption that they pick is called the axiom of choice.13

All these cases, where our intuitions diﬀer from the mathematics, make use of the inﬁnite.

1.1.2 Naturalism

Ye takes a naturalist position, more precisely scientiﬁc methodological naturalism, in the sense of the Stanford Encyclopaedia of Philosophy14. As Ye puts it:

[...] we accept our scientiﬁc knowledge as our starting point in philosophy [of mathematics]. (p. 6)

For Ye, the most important consequence is that mathematics (as a mental process) is a neural process in a physical mind. As such, the study of mathematical practices is the study of these neural activities and physical interactions in human environments. Ye extends this “radical naturalist” point of view to the mind itself. He gives us this evocative story:

Naturalism is not the view that scientiﬁc methods are the best methods for a nonphysical ‘subject’ to know objects in a world ‘external to the subject’. A naturalist philosopher may

10I thank professor Van Bendegem for the suggestion. 11 R ∞ 2 Its volume equals π 1 (1/2x) dx = ... = π/4, filling up the remaining quarter in the first trumpet. 12For infinite sets. However, Professor Van Bendegem suggested that the finite version of the Axiom of Choice might give rise to paradoxes as well. Take e.g.: From the finite set {a, b}, choose a if Goldbach’s conjecture holds and b if it doesn’t. 13See [14]. 14See [9].

6 start her philosophical thinking with a tacit and vague assumption that she is a ‘subject’ herself and scientiﬁc methods are the best methods for her to know things in a world ‘external’ to her. However, after accepting the mainstream scientiﬁc theories, she should admit that she is a physical system herself and there is no ‘subject’ dwelling inside her brain and trying to know a world ‘external to the subject’ by utilizing her brain. She should admit that human cognitive subjects themselves are natural objects, namely, physical or biological systems, and that cognitive processes are natural processes. Then, human mathematical practices are literally neural activities inside human brains and physical interactions between brains and the environments. (p. 8)

However, one can ask the question if the two mutually exclude each other. There certainly is a conscious “self” inside my brain, I cannot help but noticing it and a lot of other people probably feel the same. However, in the meantime, this “self” might very well realise that all her thoughts and indeed those thoughts of self-consciousness themselves, are taking place as neural, material processes inside this very brain. It might be far-fetched then to reduce15 the study of mathematical practice to the study of changing brain states. We know matter is governed by quantum laws. But we do not consider this whilst having a sandwich. We will never appreciate its taste by considering the sandwich as the sum of its quantum states. We know music is produced by drawing a bow across a ﬁxed string and the air resonating with that particular pitch. But we do not consider this when enjoying a piece of classical music. We will never appreciate Mozart’s Requiem by systematic analysis of all the instrument-manipulations. It is an explanation at another level of description. In the same analogy: we know mathematics takes place as changing excitation states of neurons in (socially interconnecting) brains. But we do not consider this when going through a proof of a theorem. Information is not simply transferred form one level of description to another: mathematics might be the sum of all ink stains we call publications and the sum of all neuron states in the minds of those people we call mathematicians, but that does not mean we can study that sum by merely analysing its addends. Besides, we might never have access to these addends as well. We encounter this Level of Description problem further throughout the chapter and give it a name for further reference: the LoD-problem. Ye also points out that naturalism in this sense implies nominalism, a point which we think is correct: if one reduces mathematical concepts to physical brain states, these concepts will have no existence as abstract concepts. However, as will become clear in section 1.2.1, Ye appears to soften this hard line nominalist position. Also, we think that the other implication holds as well: (his) nominalism implies (his) naturalism. Or more to the point: his nominalism needs a certain kind of naturalism. As Burgess and Rosen16 point out:

Reconstructive17 nominalists generally agree in giving priority to reconstruing mathematically formulated science and scientiﬁcally applicable mathematics, so that the theories to

15Using the term “reduce” might look like a grave accusation, however we see no other possible interpretation. 16See [4]. This work gives a general overview diﬀerent nominalist approaches, both on the philosophical and technical front. 17“Reconstructive” means positive, contemporary nominalism, as opposed to the earlier, negative nominalists, who mostly deﬁned their position by refuting what they didn’t recognise.

7 be dealt with generally involve just observable and theoretical physical entities on the one hand, and abstract mathematical entities on the other. (p. 16)

But in doing so, we might indeed end up with mathematics being a subject with no object... Popular opinion seems to be that mathematics is all about abstract entities. How to solve this?

Many reconstructive nominalists do go beyond nominalism to a more general materialism (often euphemistically called physicalism), involving denial of minds and spirits. (p. 16)

Indeed, this solution is the one favoured by Ye as well: adopting a physicalist perspective.18 For Ye the existence of mathematical concepts as abstract entities is denied and moved to the realm of physical reality, namely the constellation of concrete brain states making up these mathematical concepts.

1.2 The Applicability of Mathematics Under Naturalism

1.2.1 Naturalizing Reference, Truth and Validity

In the spirit of section 1.1.2, Ye naturalises some abstract notions to concrete brain states. The collection of neural states that make up e.g. “rabbit” in someone’s brain is called a realistic concept. No problem here: there is a one-on-one link between the physical token (the rabbit) and the type “rabbit”. The collection of neural states that make up e.g. “2” are called abstract concepts, as they only make sense when paired with a realistic concept, to make “2-rabbit”. Mathematical concepts belong to the latter category. As both are interpreted as the neural states which make them, both concepts are naturalised. We want to address two issues. Firstly, this is an instance of the LoD-problem we met in the last section. There is no reason why we should take the concept of “rabbit” to be merely a collection of neural states. It might be subtly paired with other mental associations, emotional excitations, etcetera. Moreover, it might be a diﬀerent collection of states in 7.109 diﬀerent brains, so we end up with as many concepts referring to the same rabbit. This point is more of a remark, as it bears no fundamental critique of Ye’s position. Secondly, by calling the latter concept “abstract”, we are confused. If Ye rejects the existence of abstract entities (nominalism), what are these “abstract concepts” he introduces? Clearly, they only make sense when paired with or applied to a realistic concept (2-rabbit), but why then can we talk about “2” in itself, 2 in the sense of “two-ity”? Ye’s answer would be that these are naturalised concepts, because they correspond to a physical brain state. But why can people and mathematicians talk about this amongst each other, seemingly independent of their brain-state corresponding to “2”? They should be able to talk about it, or we have no mathematics left. This needn’t to be incompatible

18We refer to the Stanford Encyclopaedia of Philosophy, see [5]. We think it correct to say that physicalism is the ontological position that corresponds with methodological naturalism. Ye only mentions the latter, but we think he endorses both.

8 with his nominalist naturalism: one might still talk about, reason or even calculate with these abstract entities in se, although they only bear meaning relative to their application to realistic concepts. But more clarity needs to be created on the subject. The very core of the problem is this: as we will discover later, Ye’s interpretation of naturalism doesn’t acknowledge the importance of discussing a diﬀerence in e.g. two-ity in the physical reality and the concept of two-ity in mathematics or in the brain: according to Ye mathematics is only about making realistic assertions about the neural states that make up the mathematical theory or concept. In section 1.3.1, we will characterise this problem as the application of naturalism or the AN-problem. Notions like truth, thoughts and logical inference rules are naturalised in a similar fashion.

1.2.2 Naturalizing the Applicability of Mathematics

Ye presents us a general scheme of reasoning:

bridging postulations realistic premises −−−−−−−−−−−−−−→ mathematical premises ⇓ (mathematical proof) bridging postulations realistic conclusion ←−−−−−−−−−−−−−− mathematical conclusion,

in which the bridging postulations link the realistic and abstract mathematical thoughts. One can discuss the speciﬁc roles one has in mind for the terms in the scheme, but no real drama at this point. However, further we encounter the following remarkable paragraph, explaining the core of the naturalisation of mathematics Ye has in mind.

An explanation of the applicability of mathematics means explaining why the realistic conclusions are true (in the naturalized sense) in ordinary valid mathematical applications in the sciences. This is similar to explaining why a physical property, for instance, a property about mass or energy, is present at the end of some physical processes (while it is neither present at the beginning nor preserved at the intermediate stages). It is a scientific question. It asks for an explanation of some regularity among a class of natural processes. Since it is a scientific question, an answer to it has to be a scientific answer. In particular, it has to consist of literally true scientific assertions about what really exist, for instance, about those neural structures, their functions in brains, and their naturalized representation relation with physical entities in the environments. The correctness or value of a putative explanation should be judged by ordinary scientific standards, like any other scientific explanation of natural regularity. The applicability of mathematics is thus naturalized. (p. 15)

It appears Ye considers the applicability of mathematics naturalised, because it is reduced to ﬁnding the link between the reality of physical world and the reality of the neural states that make up the mathematical theory. A remarkable position, which we discuss further in section 1.3.1.

9 1.2.3 Applicability as a Logical Problem

In this section, Ye explains how the application problem becomes a logical problem. He abstracts the inner representations of concepts to the syntactical structures of the linguistic expressions expressing them. He further abstracts this to the structure of a first-order language. Ye considers these reasonable assumptions for his specific purpose. However, we would say that the (rather big) step from inner representations to a FOL needs more explanation. Besides: what is a FOL for a nominalistic naturalist? There are a lot of questions to answer: where does it come from, where does it “live”? FOL can quantify over “all x’s”, which suggests an – at least in principle – infinite domain. These non-trivial questions are left unanswered.

For a speciﬁc application, he postulates sets Γr of realistic premises, Γm of mathematical premises and Γb of bridging postulations for that particular application. Application is then stated to be the purely logical inference

Γr ∪ Γm ∪ Γb ` ϕ.

Again, which rules then should steer this “purely logical inference”, and more importantly: where do these rules come from and why are they the right ones? Ye does not provide a waterproof explanation.

1.2.4 The Logical Puzzles of Applicability

Ye starts this section with a very true remark: realists do not offer a logical explanation of why their purely abstract inferences should preserve truth about physical things. Es- pecially when dealing with mathematical theories that use a concept of infinity, which is then to be reinterpreted to the real world. It is not impossible, but empirical verification is needed, justifying the use of the continuous/infinite approximation. Ye presents finitistic naturalism as a viable alternative, avoiding these altogether:

[...] we can perhaps eliminate infinity and transform the applications of infinite mathematics into logically valid deductions from literally true premises about finite physical things alone, to literally true conclusions about them. (p. 19)

As mentioned before, this is not our main reason for developing a SF system. Even more, we will denounce further that the application problem is in fact not guaranteed to be simpliﬁed at all when using a SF mathematics!19 On the other hand however, he does this only with the aim of providing a better mathematical description or explanation within the current paradigm of physics. He does not presuppose or give a verdict of the status of the universe:

Also note that I never assume that there is no real inﬁnity in the physical world. Most physicists today agree that current well-established physics theories accurately describe only physical phenomena above the Planck scale. (p. 19)

A very modest and honourable statement.

19See the FRvsFC-problem and the already mentioned AN-problem, both to be presented in section 1.3.1.

10 1.3 A Logical Explanation of Applicability

1.3.1 The Strategy

To handle the problem of translating inﬁnity of mathematics to their real, physical application, Ye presents the general strategy of developing a SF mathematics which he will elaborate in the next chapters. This is not merely a translation of classical mathematics to a SF version. Ye correctly states that it is a selfstanding project:

Then, to explain the applicability of classical mathematics in an application instance, we can try to show that the application is in principle reducible to an application of strict finitism. Instead of translating the applications of classical mathematics into the applications of strict finitism directly, my strategy is to develop applied classical mathematical theories within strict finitism. (p. 21)

That this project should succeed is – according to Ye – more or less self fulfilling: we need to describe a finite, discrete theory about reality20 with a SF model. Or, formulated in a more negative way: we already have a more or less working (classical) mathematical description of the finite physical theory of reality, hence the strict finitist should show that its use of infinity is not indispensable. As promised in section 1.2.2, we return our attention to the naturalisation of the applicability of mathematics, which is further developed in this section. We highlight two paragraphs:

Applying mathematics in strict ﬁnitism is essentially using a computational device (including a brain) to simulate other physical entities and their properties. We also have realistic premises, mathematical premises and axioms, and bridging postulations here. However, mathematical premises and the axioms of strict ﬁnitism are interpreted as statements about a computational device, and bridging postulations are interpreted as statements about how the computational device simulates other physical entities. These are all realistic statements. Therefore, an application is a series of logical deductions from realistic premises to a realistic conclusion. (p. 21) and

This means that we can in principle reformulate mathematical premises and bridging postulations in those applications as assertions about computational devices and their simulation relations with other physical entities. This should not be very surprising. After all, from the point of view of a naturalistic observer, humans are actually using their brains, assisted by paper-and-pencils or computers, to simulate other physical entities when they apply classical mathematics to those physical entities. The only puzzle for logicians is that when humans use classical mathematical concepts and thoughts that appear committed to infinity, the logic of how those concepts and thoughts simulate finite physical entities is not very clear. Then, the idea here is that the convoluted logic in those abstract mathematical thoughts in classical mathematics can in principle be straightened, to get logically simpler and more transparent (but much lengthier and more tedious) thoughts directly about finite computational devices and their simulation relations with physical entities. (p. 22)

An application is a series of logical deductions from realistic premises to a realistic conclusion, because we can and (and this is important) according to Ye a mathematician should

20A theory about reality, not necessarily reality. We sometimes drop this subtlety further.

11 only make realistic assertions about the neural states or computational devices that make up the mathematical theory or concept. We think this is a gross oversimplification and the cause of many of the philosophical problems we encounter in this fist chapter. First of all, even though we recognise all thoughts are made from, even determined by the neurons that bear them, we don’t have systematic insight in these neural states or how they make up our abstract (mathematical) thoughts/concepts. We link this to the LoD-problem. Secondly, even if we had full insight in all the neural states and their combinations, making up all thoughts (lots of which are undoubtedly mathematical or at least abstract), there is literally no space left for other thoughts! Within this school of thought21, full insight in the neural states of the brain, implies only insight in the neural states of this brain. Thirdly, it doesn’t answer where the current concept of infinity comes from. If his position were correct, we would have difficulty to introduce infinity to mathematics, whereas we now have difficulty showing its non-indispensability. Infinity is currently used, in application of mathematics too, so – oversimplifying a bit – a neural state has to correspond to it. A real neural state, hence a finite one. How to make sense of this? Mind you, we could extend this to god or (other) imaginary creatures: people do have concepts of these, and hence corresponding neural states, but others or the very thinkers themselves don’t acknowledge their actual existence. We will call this problem with the Application of [Ye’s version of] Naturalism the AN- problem. There is another important issue to be addressed, not independent of our previous criticism. In this section we encounter for the first time a problem we will meet throughout the book. It is a confusion in the motivation for SF mathematics. Namely: do we have to develop SF because we intend to apply it to a finite reality or because the calculating devices – be it the mathematician’s brain and the amount of ink stains on a paper it can produce or the limited time and calculating power of a computer – are finite? For further reference, we will call this problem about application to a Finite Reality versus Finite Calculating facilities or calculators the FRvsFC-problem. Note how it is connected to the LoD-problem. FR and FC are independent of each other, in both directions. One is obvious: FC 6⇒ FR, i.e. FC ∧ ¬FR: our finite mathematical calculating devices might still have to describe an infinite universe. The other direction is more subtle. Clearly, strictly speaking we have that FR ⇒ FC. But there is a problem in another sense. Unless a clear answer is provided (and this is not the case up to this point), there is no need of rejecting the idea that even FC might talk about 10100 (in which sense exactly, we will leave aside for the moment), which is more than any estimate of the number of atoms in the universe and hence too much22 for any theory within the realm of FR23, in which case we cannot apply this FC theory directly to the FR. So in a sense we have a situation of FR ∧ ¬FC, i.e. FR 6⇒ FC.

21No pun intended. 22Provokingly, one might say “practically inﬁnite”... 23If we take the atom as the smallest unit that makes up the “multitude” in the universe of FR.

12 The witty turn of phrase in note 22 has more severe consequences than one might expect. Indeed, any SF theory that makes – in a sense – sense of too large numbers, cannot automatically be applied to reality. For it takes very little effort to finitely define24 a number like 10100, but that number will be practically infinite when applying it to any FR theory. Note the link with the AN-problem.

1.3.2 The Conjecture of Finitism

Ye formulates the Conjecture of Finitism:

Strict finitism is in principle sufficient for formulating current scientific theories about natural phenomena above the Planck scale and for conducting proofs and calculations in those theories. (p. 23)

The development of his SF theory of analysis, which should be sufficient for the current scientific theories, is indeed his best argument to support this conjecture. Again, that this undertaking should succeed is equivalent to the statement that infinity is not indispensable in these mathematical description of the finite reality. And once again, Ye makes a modest observation, for those not committed to the SF project:

On the other side, even if we end up with a negative answer to the conjecture of finitism, it will still be a valuable thing to know where exactly infinity is strictly logically indispensable for an application to finite things in this physical world and how that can happen. Moreover, recall that I never assume that there is no infinity in the physical world. That question should be left for physicists to answer. The real job here is to explain how infinite mathematics is applicable in current scientific theories about a finite part of the physical world. (p. 25)

We could compare this to the practices of reverse mathematics: start from the mathematics we want/need and investigate what is indispensable for this mathematics, in terms of axioms, techniques, concepts... and in this case: infinity. This would mean a totally different approach to strict finitism, as it assesses the need or redundancy of infinity by analysing the mathematics itself, whereas now one generally develops a system with or without infinity, depending on the philosophical position from which one starts. We think this reverse approach is promising to reveal interesting insights, in either direction, and needs further research. In sections 1.3.3 – 1.3.5, Ye elaborates the Conjecture further and give some more comments and examples on his position. They do not bear new elements and we shall hence not discuss these sections.

24If you wish: substitute “deﬁne” for construct/imagine/write down/explain.

13 Conclusions on Chapter 1

We want to present some conclusions on the first chapter, in which Ye establishes his philosophical motivation for the development of a SF system. Again, we wil not criticise his philosophical position in se, but we will summarise some issues we encountered and develop our own view on the motivation for a SF project. We encountered three interconnected major problems: the LoD-problem, the AN-problem and the FRvsFC-problem. We would deem it a shame if these were to provide counterarguments for Ye’s project of SF. We don’t agree wholeheartedly with Ye’s philosophical motivation for SF mathematics. However, we do endorse the project and indeed hold his contribution in the highest regard, be it by another philosophical approach. We will summarise our position in the following concluding remarks. 1. The fact that every current scientific theory about natural phenomena, considers only a finite universe or multitude, is not sufficient as a motivation for developing a SF mathematics. As Ye correctly states in the opening paragraphs, but tends to forget later: we do not want our mathematics to depend on the current paradigm of physical theories. Moreover, playing devil’s advocate, if we were to endorse the concept of infinity in mathematics (not in physics), we might say that it’s a translation problem, in the sense of the bridging postulations of section 1.2.2. The use of an infinite entity might be unproblematic within mathematics, as long as we can make sense of its application to a realistic physical problem. 2. The finiteness of the calculating devices (human and non-human) where mathematics takes place, is the main motivation for developing a SF mathematics. We recognise that every thought, and hence every mathematical thought, is rooted in a brain and determined by a finite amount of neural states, the exact composition of which, however, cannot be decided. As a consequence, this is not the starting point of our motivation for a SF mathematics. We take a more constructivist position, demanding that the mathematical thought is lifted form this finite brain and constructed. Or at least can be. We will not ask to explicitly construct a physical quantity of 10100 (and not only because this is impossible), however what we do ask for is (1) a finite procedure of explaining a possible way of constructing this number25, or indeed any mathematical concept. By starting from brain states in individuals, we think Ye underexposes the social arena in which mathematics takes place, so we will ask that (2) this construction can be communi- cated, explained in an unambiguous manner. The finite resources, time and extent of the social arena, make that (3) we only recognise finite mathematical objects. As the same constraints apply to manipulations performed by a machine, (4) we can use the same vocabulary and principles for human and non-human actors. As the former constructs and controls the latter, no fundamentally different mathematics is to be considered.26

25E.g. as 100 times the multiplication of 10 by itself, where the 100 can be thought of as a reoccurring action in time, and the 10 as a physical quantity of distinct objects. 26See [11] for the elaboration of these considerations. Another valuable approach is to be found in [10],

14 3. A SF approach does not in itself simplify the application problem. Ye seems to claim that as we can formulate realistic assertions about the ﬁnite brain states, the application problem is naturalised. There is a one-on-one relation between the real world and its (abstract, mathematical) representations in physical brains. And the investigation of how this mathematics works is a purely scientiﬁc endeavour. Were it so easy... As mentioned before, any SF theory could in principle (maybe even should) deal with a number such as 10100. Ye’s SF is no exception, as we will see in the second chapter. But the application of 10100 to any real situation is problematic according to all current theories about physical reality. The problem of why and how 10100 can be applied needs to be addressed and a SF approach in itself does not answer this question. It might even prove to be equivalent to explaining the applicability of ∞...

in which the ﬁnite resources of the mathematician are idealised to a Sheet Mathematician (Shemath). It is good to have this metaphor in mind when looking at our proposal developed in this paragraph.

15 2 Strict Finitism

In this chapter, the model SF is presented. In the subsequent chapters, this system will then be put to work to produce a SF version of the major concepts of mathematical analysis. We will not present a systematic overview of all the technicalities we encounter, but we will address their key features.

2.1 The Formal System SF for Strict Finitism

SF is introduced as

[...] a fragment of quantiﬁer-free primitive recursive arithmetic (PRA) with the accepted functions restricted to elementary recursive functions. Elementary recursive functions are the functions constructed from some base arithmetic functions by composition and bounded primitive recursion. (p. 35)

The bounded PR guarantees that we will be able to make sense of a ﬁnite model. Maybe this looks a bit scanty? But again, he defends his proposal because it is probably suﬃcient for the application in the sciences:

The reason for restricting to elementary recursive functions here is to recognize the fact that in scientiﬁc applications, perhaps elementary recursive functions are all the functions we actually need, since science describes only things above the Planck scale in the universe. (p. 35)

As mentioned before, we do not take the application/applicability as our main motivation for SF mathematics. Interestingly, we want to present an internal-mathematical argument why elementary recursive functions are suﬃcient. We cite the much quoted “Friedman’s Grand Conjecture”:

Every theorem published in the Annals of Mathematics whose statement involves only ﬁnitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA.27

2.1.1 The Language, Axioms and Rules of SF

Language Ye uses λ-calculus, but no expertise is needed. All terms belong to a certain type, with type o intended to be interpreted as numbers. If σ1, ..., σn, σ are types, so is σ1, ..., σn → σ. Higher order types are thus created and are regarded as programs. To deal with such a program as an object in itself, there is the principle of λ-abstraction. However, conversely, the statements about or properties of these higher order types are only determined by the way they manipulate lower order objects, all the way down to the o type. This direction is guaranteed by Ap. Apart from this, new terms are created by using:

• 0 (type o),

27See [7]. EFA is Elementary Function Arithmetic, another name for ERA.

16 • the usual successor function: S (type o → o),

28 • the usual arithmetic operations: +, ·, pow, I< (all of type o, o → o) , ( t1 if t = 0, • definition by cases: J(t, t1, t2), with the intended meaning of: t2 if t > 0, • bounded primitive recursion restricted to numeral types, • finite sum and finite product.

It is important to stress that s = t is a formula if and only if s and t are of type o. There are no equalities between higher order terms.

Intuitively, an equality t = s between two terms of the type (o → o) is implicitly committed to either inﬁnity or abstract entities. (p. 38)

Axioms The axioms that are presented are quite classical, we do note the quantiﬁer-free induction rule: ϕ[0], ϕ[n] → ϕ[Sn] ⇒ ϕ[t]. All this seems to be at least potentially inﬁnite. Where’s the SF spirit of SF?

[...] strict finitism allows abstraction but it is not committed to any idealization. For instance, suppose that the universe is finite and discrete, and therefore there is a limit on how many numerals could really exist. We can still interpret some formulas in strict finitism as assertions about numerals that really exist. If a formula has no such chance of being interpreted as an assertion about concretely existent numerals, then let it be. (p. 38)

If a number is too large for the finite universe, drop it. We refer back to the FRvsFC- problem. We would say: we cannot make statements about numerals for which there is not enough “budget” to construct. SF is formally speaking a potentially infinite system, but intuitively appears to be quite robust when making a finite “cut”, for which it is obviously intended. But if not, referring back to the last remark on section 1.3.2, it is interesting to see that it is compatible for those not committed to the SF project. For those who e.g. were to be interested in a constructivist approach, but do accept a potentially infinite work-space of the mathematician, Ye’s SF is a very valuable contribution as well. 2.1.2 Arithmetic in SF Quite intuitively, the first steps of arithmetic are taken, with the fundamental theorem of arithmetic as a climax. 2.1.3 A Finitistic Interpretation of SF Elementary recursive functions are expressed by terms. The question remains: does the converse hold, might there be terms we cannot “reach” as an ERF? The answer is no (theorems 2.4 – 2.6 in the book). This is essential, for now – as promised – we can indeed interpret every term of SF as a program.

28The <-relation is thus “coded” by assigning a 0/1 truth value to a pair of numbers.

17 Now, consider how to interpret SF as a realistic theory about concrete computational devices. Terms in SF can be treated as expressions referring to programs in a concrete computer. In particular, a numeral is a program that outputs itself. A closed numerical term in normal form is a composition of numerals, the base functions S, +, ·, pow, I<, and the operators bounded primitive recursion, finite sum and finite product. It is a program producing a concrete numeral output when executed according to the primitive recursive equations defining base functions, bounded primitive recursion, finite sum and finite product. An arbitrary closed numerical term is also a program, since it can be transformed into a normal term. A closed term of an arbitrary type is a program that transforms any sequences of terms s1, ..., sn of appropriate types σ1, ..., σn (as programs) into another term t(s1)...(sn) of the type o (as a program). (p. 50-51)

When we encounter a memory overflow, we reached the maximum number the machine can handle: N. What can we do with this number? Can we divide it by two? Clearly N not, because there is no place left to store 2 or even 2, for if there was place left, SN wouldn’t have caused a memory overflow. We think the N does not need to be introduced or handled explicitly. After all, as Van Bendegem29 points out, with the “largest number” N in itself, we cannot really do much... Another proposal is to “hard-code” a largest number N in a system SFN . We encounter the FRvsFC-problem again: why N? Because it’s the largest number the mathematics/mathematician/machine can handle? Or because it’s the largest number that makes sense for the application, e.g. the maximum number of bacteria in a biological simulation? SF and SFN aren’t entirely equivalent, contrary to Ye’s assertion. SF might as well have an infinite model (in the classical sense). Moreover, SF can carry on calculating until a memory overflow (and hence N) is reached, whereas in SFN , The largest number N is introduced a priori. Inspired by the previous observations, we are inclined to treat “the largest number” as an unspecified entity. Given our position, we know that we could indeed employ all resources and devise clever notation systems to tackle ever growing numbers30, but after a finite time, all these would be exhausted (and we would be very bored indeed). But why would we want to preform that task in the first place? We are satisfied and reassured by the argument that there is a largest number constructable, but that it remains unspecified.31 Another potential problem is this. We are intuitively convinced by the promise that a lot of SF will remain intact if a finite cut needs to be made because we ran out of resources:

However, as long as the numerals involved are not too large, an axiom instance can be interpreted as a literally true assertion about a concrete computer. (p. 51)

On the other hand, there is no way of telling what exactly is still possible and what is lost if we were to encounter a memory overﬂow. Mind you, the same criticism can be applied

29See [13], the section on “the argument of the greatest number itself”. 30E.g. Knuth’s up-arrow notation. 31A nice analogy was suggested to me by professor Van Bendegem. As a quick Google through popular science websites confirms, the number of hairs on the human head is smaller than the population of Brussels. By the pigeon hole principle, there must be two people out there with exactly the same amount of hairs on their head (excluding the trivial case of 0). A fun fact, but the persuasiveness of the argument does not compel me to actually search a pair of those people. I am quite satisfied with their (very real!) existence as an unspecified object.

18 to our proposal... And exactly this non-trivial criticism is quintessential in the debates about strict finitism. In its most general format, we think strict finitism is about the tension between the finite human nature and what it manages to describe. It is about arguing that the finiteness of the descriptive level does have its influence on what can be described on the object level. Unfortunately, the discussion is often narrowed to the conundrum of “a largest number”. This is nonetheless a very important question and a failure to answer it would undermine the SF project. At this stage, we cannot provide the definite answer that will work in all contexts. More research on the various positions and comments needs to be conducted. We will touch upon the subject by referring to [12], an issue dedicated to discussions about strict finitism.32 Don’t specify it: Horsten33 states that

Van Bendegem’s deﬁnition seems to be more analogous to the characterisation of God given in negative theology: God is this creature to which no predicate applies. God is not good, but not bad either; his power is not limited or unlimited...

I believed the central presumption of strict ﬁnitism to be the existence of a largest number. But that appears not to be the case, because one cannot say anything about the largest number (about what?).34

But Horsten thinks this is not essentially a counterargument to strict finitism, although he does accuse this approach of a certain mysticism. Specify it: Heeffer on the other hand, argues that we can without difficulty presuppose a concrete largest number, by reference to the way in which we handle numbers processed by computers:

Every computer uses a ﬁnite – and hence from the axiomatic point of view inconsistent – mathematics. [...] We can learn a lot from the history of the ﬁrst computers. There exists a largest number and that number is determined by a social process of negotiation.35

A case can be made for both positions. However, in the context of Ye’s system SF, we are inclined to treat “the largest number” as an unspeciﬁed notion, as mentioned before.

2.2 Doing Mathematics in Strict Finitism

One could argue that section 2.1 presents a quite classical approach. It is even not immediately clear where the SF spirit is to be found. In this section, however, Ye provides the fruitful insight that we cannot put his SF to work within the setting of a classical logic. Mathematics within strict ﬁnitism does demand an appropriate logic!

32Only available in Dutch. The main article by Van Bendegem however, is the Dutch version of [13]. 33Commenting on Van Bendegem’s proposal to deﬁne the largest number as “the number about which we cannot answer any question”. 34Our translation. 35Our translation.

19 2.2.1 Mathematical claims in Strict Finitism

All terms in SF are to be interpreted as programs of a concrete computing machine. Therefore,

[d]eveloping mathematics in strict finitism means constructing terms (of any types) in SF and proving that those terms satisfy some desired conditions, which means proving some (quantifier-free) formulas containing those terms in SF. Therefore, a claim in strict finitism reports what terms have been constructed and which condition about the terms has been verified. (p. 52)

But in order to complete this task, a delicate yet important principle needs to be introduced, that might as well be at the very core of that funny business we call mathematics: making stuﬀ easier. Be it because it’s more practical, economic or just less tiresome, some simpliﬁcations are to be introduced:

[...] sometimes we do not need to give the constructed terms explicitly. We may be satisﬁed with the recognition that some terms can be (really can be) constructed. Similarly, terms can be constructed hypothetically, that is, assuming that some other terms satisfying some conditions are already available. (p. 53)

More formally, Ye has this deﬁnition of a claim in SF:

Suppose that ϕ[x, y, p] is a formula of SF, and suppose that x, y, p are all and diﬀerent free variables in ϕ.A claim in strict ﬁnitism is a symbolic formula

∃x∀y ϕ[x, y, p], (FinC)

which means that we have constructed some terms t of appropriate types and prove that

SF ` ϕ[t, y, p].

t may contain variables in p but not those in y. The variables in p are free variables (as parameters) in the claim. A proof of the claim in strict ﬁnitism consists of the required terms t and a proof of ϕ[t, y, p] in SF. The constructed terms t are witnesses for the claim. (p.53)

It is important to stress that “∃” is not at another level of abstraction, it is only an abbreviation to avoid an explicit construction of the witness, although this witness really can be constructed.

2.2.2 Deﬁned Logical Constants on Claims

What should it mean now to ﬁnitistically assert that “ϕ and ψ” or “there is a z for which ϕ[z] holds”? These would be claims in ﬁnitism as well, hence expressed in the format (FinC). This is why Ye introduces *-connectives that formalise the (FinC)-version of the informal logical connectives (p. 54):

Definition 2.9. Suppose that ϕ ≡ ∃x∀y ϕ1[x, y] and ψ ≡ ∃u∀v ψ1[u, v] are claims in strict finitism, where x, y, u, v are distinct variables. (We suppress the parameters here.) Define

20 ∗ (1)( ϕ ∧ ψ) ≡df ∃xu∀yv (ϕ1 ∧ ψ1);

∗ (2)( ϕ ∨ ψ) ≡df (ϕ1 ∨ ψ1) if x, y, u, v are all empty, otherwise,

∗ ϕ ∨ ψ ≡df ∃nxu∀yv ((n = 0 ∧ ϕ1) ∨ (n 6= 0 ∧ ψ1));

∗ ∗ (3)( ∃ z ϕ) ≡df ∃zx∀y ϕ1 if z does not occur in x, y, otherwise (∃ z ϕ) ≡df ϕ;

∗ ∗ (4)( ∀ z ϕ) ≡df ∃X∀zy ϕ1[X(z), y] if z does not occur in x, y, otherwise (∀ z ϕ) ≡df ϕ;

∗ (5)( ϕ → ψ) ≡df ∃UY ∀xv (ϕ1[x, Y (x, v)] → ψ1[U(x), v]);

∗ ∗ (6)( ¬ ϕ) ≡df (ϕ → S0 = 0) ≡ ∃Y ∀x (¬ϕ1[x, Y (x)]);

∗ ∗ ∗ ∗ (7)( ϕ ↔ ψ) ≡df (ϕ → ψ) ∧ (ψ → ϕ).

The most important idea is that all the *-connectives are reducible to (recursive) constructions of programs, in the sense of the “∃” of section 2.2.1. Ye uses deﬁnitions that afterwards appear to follow most of the principles of standard intuitionistic logic.36 How- ever, for Ye, all claims have to be stated in the format of (FinC), so how should we handle the quantiﬁers in this format? This boils down to the following observation. For Ye37, what would it mean to state that

∀x∃y ϕ[x, y]?

We haven’t encountered set theory at this point and are not committed to inﬁnite entities, so what should the “∀x” mean? It should mean that we have a general procedure, an algorithm, to “process” any x thrown at us, in order to give us a y such that ϕ[x, y] holds, hence: ∃Y ∀x ϕ[x, Y (x)].

We will illustrate this procedure on a few operators. We start with the most extensive case, namely →∗. Ye does this in words, but we think a small FOL calculation is more insightful. “ ” stands for the ﬁnitistic interpretation described above. We also assume, as usual, that x, y do not occur in ψ1 and u, v do not occur in ϕ1.

ϕ → ψ ≡ ∃x∀y ϕ1[x, y] → ∃u∀v ψ1[u, v] (a (FinC)-format of ϕ and ψ)

≡ ... ≡ ∀x∃u∀v∃y (ϕ1[x, y] → ψ1[u, v]) (properties in FOL) ∃UY ∀xv (ϕ1[x, Y (x, v)] → ψ1[U(x), v]) (U depends only on x, Y on x and v, due to the order of quantiﬁers) =: ϕ →∗ ψ.

The second case we will discuss is the deﬁnition38 of ∨∗:

∗ ψ ∨ ψ := ∃nxu∀yv ((n = 0 ∧ ϕ1) ∨ (n 6= 0 ∧ ψ1)) ,

36See [8] for a good introduction. 37And for us and maybe constructivists in general. 38In the non-trivial case, with x, y, u, v not all empty.

21 which at ﬁrst sight, might look like an exclusive “or”. But close inspection reveals that this reﬂects the intuitionist/constructivist spirit very well! What should it mean for Ye that “ϕ or ψ”? It should mean that we have a means to decide (hence the n) which one of the two (“ϕ” or “ψ”) makes “ϕ or ψ” true. And indeed, as mentioned before, it turns out (Theorem 2.11) that ¬∗, ∨∗, ∧∗, →∗, ↔∗, ∃∗, and ∀∗ follow most of the laws of intuitionistic predicate logic. Most of the laws, for we do have this, contrary to standard intuitionistic logic (Lemma 2.12 in the book):

¬∗∃∗x∀∗y ϕ[x, y] ↔∗ ∀∗x∃∗y ¬ϕ[x, y].

However, as in intuitionistic logic, the following do not hold in general:

¬∗∀∗∃∗y ϕ[x, y] →∗ ∃∗x∀∗y ¬ϕ[x, y], ¬∗¬∗ϕ →∗ ϕ, ¬∗(ϕ ∧ ψ) → ¬∗ϕ ∨ ¬∗ψ, ¬∗∀∗x ϕ → ∃∗x ¬∗ϕ.

But as most rules are valid and we promise not to use any of the controversial rules in an informal manner, Ye omits the stars on the symbols further on.

2.2.3 Recursive Constructions and Inductions

Recursions can be used to construct sequences of higher types. We do not examine the technicalities of this section, as it’s only important for us to know that one can indeed define a finite sequence of higher type, decoding function giving you the nth element, length function of such a sequence, etcetera. We do, however, cite the following paragraph, which summarises nicely the spirit of strict finitism.

In summary, this is what we will do in developing mathematics within strict finitism. We translate a theorem in classical mathematics into a claim in strict finitism, using logical constants ¬∗, ∨∗, ∧∗, →∗, ↔∗, ∃∗, and ∀∗ to replace classical logical constants. Sometimes we have to modify the classical theorem into a (classically) logically equivalent format before doing the translation, because two classically equivalent statements may have different finitistic content. The claim eventually says that some terms can be constructed to satisfy some condition that is a quantifier-free formula in SF, as in (FinC). We then prove the claim informally, using the axioms and rules of SF, [the *-defined logical operators and derived rules], plus some forms of induction [...] that can be reduced to the quantifier-free induction in SF, plus the techniques in natural deduction, including Deduction Theorem, ∃-Introduction Rule, and so on. (p. 66)

2.3 Sets and Functions

At this point, Ye wants to “translate” classical notions of mathematics. Two important ones are the concept of a set and that of a function.

22 2.3.1 Sets

Unlike a traditional approach, in which a set is a primitive notion from which “all mathematics” is constructed, for Ye a set is a derived concept. It is nothing more than a formal translation of a naive set idea, with a set A as a collection of objects for which there is a certain criterion ϕ[a], stating what it means to be a member of A and a formula ψ[a, b] stating what it means for two “diﬀerent” terms a and b to be indistinguishable for this set A. As a and b are to be seen as programs and not numbers/objects themselves, this is no trivial question. However, there is a price to be paid for this “easy translation” of intuitive notions. Unlike e.g. in ZFC, we cannot treat a set like we treat a term. We cannot quantify over sets or make sets of sets, in particular we cannot construct the notion of a power set P. However, as long as we stay “on the same level”, classical notions can be translated: subsets39, intersection, union, Cartesian product...

2.3.2 Examples: Sets of Numbers

However promising the previous section looked, its application looks a bit artiﬁcial. N is trivial and completely embedded in SF. No real problems here. But then, let us think about what Z should be. We think it natural to start from the quintessential operation “+” in N and construct Z as the closure of N under the inverse of +. Ye however uses the bijection

n 7→ 2n, coding a positive integer, −n 7→ 2n − 1, coding a negative integer, ∼ which is classically used to a posteriori prove that N = Z. The nice intuition of Z as “N plus all the negatives” is lost. The same applies to Q. Ye uses the technical approach, in which rationals are pairs a 2 (quotients) of two integers, up to proportionality. r = b is coded as OP (a, b) := (a + b) + a + 1. This is not a correct coding, because we have e.g. OP (2, −2) = 3 = OP (−7, 10). A correct alternative would be e.g. a standard bijection Z × Z → N × N, after which we apply the coding40 1 J(a, b) = 2 (a + b)(a + b + 1) + a. Even with a correct coding, this approach – in our humble opinion – lacks transparency. We think, intuitively, that Q should be constructed as the closure of Z under inverse of multiplication. Note that our eagerness to construct Z and Q as the closure under the inverse of addition and multiplication respectively, has a link with geometric operations. Starting from a natural n, −n is the reﬂection through 0. Starting from an integer z, seen as a dilation

39No problem here, as long as we do not quantify over all subsets of a set. 40See [2], page 164. The intuitive proof that it is a bijection is rather handsome.

23 1 factor, z is the dilation which gives us back the original unit length. We think that this approach, with numbers seen as geometric operations, is very valuable. But Ye does not aim for the development of a SF geometry, only the 8th chapter will develop semi- Riemannian geometry, with a very formal approach that does not aim to focus on intuitive notions of geometry. We encounter no proposition for applying his number concept – or indeed any of his more basic calculus concepts later in the book – to geometry. We haven’t reached a verdict on what to do with . How satisfying it would be to close √ R the thus constructed set Q under ··· , the inverse of pow. Alas... We then only get the algebraic reals, not the transcendentals, most notably e and π, which would have to be “hard-coded”. Ye uses a traditional approach: code reals as converging sequences (hence of type o → o) of rationals:

1 1 (x ∈ R) ≡df ∀m, n > 0 (x(n) ∈ Q ∧ |x(m) − x(n)| 6 m + n ), 2 (x =R y) ≡df ∀n > 0 (|x(n) − y(n)| 6 n ).

Recall that we’re actually dealing with ∀∗ and that (see deﬁnition 2.9) this promised, in the case of the universal quantiﬁer, the existence of a general procedure to handle “all n”. But it is not clear how we should read the formulas above in this SF manner. If we run our of memory, don’t we just end up with a sequence of rationals and no more?

2.3.3 – 2.3.4 Functions & Partial Functions

Functions are introduced without any difficulty and so are derived notions: injective and surjective, composition, inverse... A notion of countable and finite sets is introduced as well. We think this is a very weird and obsolete move. These are probably introduced to “save” these notions if we do consider an infinite model of SF, but doing that corrupts the entire point of SF. In Ye’s jargon: we will run out of calculation resources at some point, and we think that notions of a finite set are therefore incorporated in the very core of SF. To put it negatively: if there were anything else then a finite set for Ye, what would it look like? Partial functions from R to R are introduced, i.e. a function from a subset of R to R. These are more than enough for a theory of integration and hence Ye’s goal of calculus.

24 Conclusions on Chapter 2

Although we had some objections, we think that Ye’s SF is a very valuable contribution to strict finitism. Under 2.1, the general outline and axioms are demonstrated, which appear to be quite innocent. Using λ-calculus is a cunning idea and other operators of the language seem to be free of objection. The axioms used seem appropriate from the SF point of view, even induction. But to actually develop mathematics within this system, Ye makes the sharp observation that we need to address the logic as well. Be it from his nominalistic naturalist approach or our point of view, it seems that a logic along the lines of intuitionist logic41 is the appropriate format. This is tackled under 2.2 and probably Ye’s most original contribution. Especially the way in which e.g. implication and quantifiers are handled, are inspiring. We think SF is compatible with our position stated in the conclusions on chapter 1. The idea of starting from elementary recursive functions, paired with an intuitionist logic, translates the SF spirit quite well. It gives us a mathematics of doing, rather than being, a dynamic tool to describe this world. However, using this tool, Ye makes – in our opinion – strange inferences, as if this new-fledged system needs to revert back to the classical concepts straightaway.

• We think Ye’s concept for a set is very promising: a translation of the intuitive set idea, without the more advanced operations, such as e.g. quantifying over sets, that cause oddities in the classical approach. However, we don’t think Ye’s concrete construction of Z, Q and certainly R does credit to both the nature of these sets of numbers and the SF project...

• Functions are introduced, so far so good. In fact, informally speaking one could say that functions make up the core of SF, as “everything is a function”42, as even the type o elements (the numerical type) are interpreted as a function43 giving me the number, say, n. But with these functions, strange and rather obsolete notions are introduced, such as countability and ﬁnite sets.

We do not claim, however, to provide our own deﬁnitive answer for these questions. Nor do we know how SF should develop a calculus, with or without the (good) parts of Ye’s SF.

41Not entirely, as we have discussed, but with the same constructivistic justiﬁcation. 42Echoing the maxim that “everything is a set”... 43Here function not in the mathematical sense, but in the sense of a computer program or procedure.

25 3 Calculus in SF

Mathematical analysis is merely the extensive study of inequalities.44

The machine is ready. Let it spin. Chapters 3-8 develop a calculus within SF. Supremum, inﬁmum, limits, continuity, sequences, series... all follow rather classically. On the other hand, Ye is inspired by the constructive approach of Bishop and Bridges45, particularly for the introduction of functions exp, sin and cos. E.g. sin is deﬁned as

∞ X x2n+1 sin x := (−1)n . (2n + 1)! n=0 How can we ever link this (rather abstract) sine function to real angles? Or to use the vocabulary of Ye: how to interpret this as a realistic assertion about what really exists? We know it is not within the scope of Ye to link his calculus to geometry, but how can we ever gain an insight in what a function does without making that link? Even the most formal course on calculus will show diagrams, graphs, constructions... Ye does not provide a single figure, so that in fact to gain an insight in e.g. his sine function, one actually has to have the classical sine function in mind already, which undermines his SF as a selfstanding project. exp is easier to interpret in pure calculus terms, as e.g. Bishop and Bridges themselves explain, namely exp as a solution of the differential equation Df = f. We could argue that we have a problem with the very notion of continuity, limits, series and sequences... We think these should be problematic from a SF point of view. However, we think that the fact that these very concepts are definable within SF is because of the preparations in Chapter 2. The harm has been done there, and the controversial concepts of the following chapters are merely a consequence of this. We appear to be bogged down in quite abstract, sometimes infinite sets and functions, while abstract notions and infinity should be problematic for Ye and indeed SF in general. We argued that, in a SF system, defining N, Z and possibly Q might be quite doable. But constructing R, with its standard construction deeply routed within the realm of the infinite and continuous, proves to be particularly fiendish. There appears to be no agreement in the literature or even the vague beginning of a solution and we fear Ye’s proposal does not do the job either. We believe that unless a verdict about the reals is reached, we have no hope for developing an advanced calculus. However, this does not mean Ye’s SF – like proved consequences of the Riemann Hy- potheses – needs to collect dust in the corner until the original question is solved... It should be examined how it can be put to work for solving this very issue!

44Prof. Em. Impens, mathematical analysis professor at Ghent University. 45See [1].

26 Conclusions

In this ﬁnal section, we present our concluding remarks on Ye’s Strict Finitism and the Logic of Mathematical Applications.

The case study... We tried to unravel Ye’s philosophical system in which the development of a SF mathematics is embedded. Not wanting to make an assessment of this philosophical position itself, we couldn’t help but pointing out a fundamental problem we found within his nominalistic naturalism, namely the way in which he naturalises mathematical concepts and the application problem of mathematics, by reducing these to the evolving brain-states within the human mind. We called this the LoD- and AN-problem, respectively. More generally, we encountered a problem or confusion we find outside of this particular case study as well, namely with the very reason why SF should be considered. Either with the intent of applying it more directly to this finite reality, or either because of the finiteness of all resources available in the mathematical practices. We called this the FRvsFC-problem and argued why FR can be seen as independent from FC. After this introduction, we discussed Ye’s SF. Both for Ye and our approach to SF, this system looks promising. Schematically, the following steps have been taken: 1. SF is introduced as a format of PRA, cut out for implementing in a finite calculating device. Later we did encounter some controversy, on discussing the non-trivial question of the largest number N. 2. A claim in SF is a claim about what this calculating device can actually produce. A different kind of logic is then introduced, which is needed to construct advanced formulas. The very promising spirit of these two elements is nicely reflected in the paragraph we cited in section 2.2.3. We could endorse Ye’s set concept, however, from its application to the familiar number sets onwards, we encountered some strange steps of reasoning, rendering the status of Ye’s calculus, from the strict finitist point of view, unsure.

... and the open questions it raised To those disappointed by this conclusion, we say:

Oh! pleasant exercise of hope and joy!

For this is only the beginning, a ﬁrst attempt at a fully ﬂedged, concrete SF mathematics. It is apparent that in every section, we encountered open questions, waiting to be tackled. There are the classical debates in the philosophy of mathematics, in which SF has its place. There is the proposition of investigating the need for SF from the same perspective as reverse mathematics. On the technical front, this, and other systems besides SF, are

27 to be developed, in which new and old problems will arise. From the technical question about what on earth to do with R, to the philosophical question of the largest number proper and the very limits of our finite brain. In the meantime, the various approaches towards SF have their own history of at least a few decades, waiting to be systematised, thus eliminating confusion on the subject or possibly revealing new paths to take. All of this will have one thing in common – and indeed in common with Ye’s outline for strict finitism: showing why the infinite is not indispensable. We think the study of mathematical practices might prove to be fundamental in showing this. For mathematics is rooted

Not in Utopia, subterranean ﬁelds, Or some secreted island, Heaven knows where! But in the very world, which is the world Of all of us,—the place where in the end We ﬁnd our happiness, or not at all!

Armed only with a very ﬁnite brain and a slim volume of English poetry, we hope to be able to execute this intriguing exercise of hope and joy in the future. Hopefully without the disappointment of romanticism. Or its opium consumption.

28 References

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[2] G. Boolos and R. Jeﬀrey. Computability and Logic. Cambridge University Press, 1974.

[3] O. Bueno. Nominalism in the Philosophy of Mathematics. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Stanford University, spring 2014 edition, 2014. http://plato.stanford.edu/archives/spr2014/entries/ nominalism-mathematics/.

[4] J.P. Burgess and G.A. Rosen. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Clarendon Press, 1997.

[5] S. Daniel. Physicalism. In Edward N. Zalta, editor, The Stanford Encyclopedia of Phi- losophy. Stanford University, spring 2015 edition, 2015. http://plato.stanford. edu/archives/spr2015/entries/physicalism/.

[6] R. Dedekind. Was sind und was sollen die Zahlen? Vieweg F., 1893.

[7] H. Friedman. FOM: grand conjectures. http://cs.nyu.edu/pipermail/fom/ 1999-April/003014.html, 1999.

[8] J. Moschovakis. Intuitionistic logic. In Edward N. Zalta, editor, The Stanford En- cyclopedia of Philosophy. Stanford University, 2015. http://plato.stanford.edu/ archives/spr2015/entries/logic-intuitionistic/.

[9] A. Paseau. Naturalism in the Philosophy of Mathematics. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Stanford University, summer 2013 edition, 2013. http://plato.stanford.edu/archives/sum2013/entries/ naturalism-mathematics/.

[10] J.P. Van Bendegem. Finite, Empirical Mathematics, Outline of a Model. Rijksuni- versiteit Te Gent, 1987.

[11] J.P. Van Bendegem. Why the largest number imaginable is still a ﬁnite number. Logique et Analyse, 42:107–126, 1999.

[12] J.P. Van Bendegem. Een verdediging van het strikt ﬁnitisme. Algemeen Nederlands tijdschrift voor wijsbegeerte, 102(3):163–211, July 2010.

[13] J.P. Van Bendegem. A defense of strict ﬁnitism. Constructivist Foundations, 7(2):141–149, 2012.

[14] T. Weston. The banach-tarski paradox. http://people.math.umass.edu/~weston/ oldpapers/banach.pdf, year unknown; consulted 2015.

29 [15] R. Whitty. Theorem of the day: Torricelli’s trumpet, year unknown; consulted 2015. http://www.theoremoftheday.org/Analysis/Torricelli/TotDTorricelli.pdf.

[16] F. Ye. Strict Finitism and the Logic of Mathematical Applications, volume 355. Springer, 2011.