TY - JOUR
AU1 - Gold, Bonnie
AB - This book approaches the issue of the essence of numbers from a very broad perspective: historical, philosophical, and mathematical. The author is a practicing mathematician who has been working in philosophy since the 1990s, with a particular focus on phenomenology, while maintaining his activity in mathematics. As he explains in the preface to the English translation, the original French version began with a series of lectures introducing philosophy students at Nice to the philosophy of science and of mathematics via the philosophy of arithmetic. The initial aim was to develop in the students an understanding of the issues from the origins in classical Greek thought to the point where they would be ready to follow Husserl’s Philosophy of Arithmetic, which Patras considers ‘one of the deepest texts ever written on mathematical thought’. Each chapter is a manageable chunk of about ten pages, presumably covering what was given as a single lecture. Chapter 1 is an introduction, setting up the plan for the book. While our intuitive conception of numbers appears not to have changed much since classical Greece, the development within mathematics of the theory of numbers is quite extensive. The book aims to help resolve this tension by investigating the ‘essence’ of numbers, starting with classical Greece and following the development of the concept through to the modern era with its rigorous definitions. There is then a very brief overview: starting with the Greeks and the ‘problem of the One’ — is one a number? While civilizations before ancient Greece used numbers and calculation, the Greeks were the first to consider its definition. Number, and the study of the arithmetical properties of the continuum, have been a concern of mathematics throughout its history. Developments at the end of the nineteenth century, and especially Frege’s contributions bringing ‘arithmetic back to the pure laws of thought’, were central to the development of the concept of number, ‘a decisive issue for the whole theory of knowledge’. The algebraic nature of numbers and arithmetic led to the possibility of extending beyond the natural numbers. Chapter 2, ‘The lasting influence of Pythagorism’, stopped me in my attempt to read the book every time. It is full of little fragments spanning thousands of years. It quickly mentions Egyptian and Mesopotamian mathematics in the context of the beginning of separating calculation from the specific objects being counted, and briefly mentions Thales, as having given the first definition of number: ‘a collection of units’. It then moves to a mention of the Pythagorean school in the context of the analogical use of numbers — giving numbers qualities: 1 representing the soul, 2 the movement from premises to conclusion, and so on. Patras links this with a range of other mystical attributions of number: Lao Tzu (‘Tao gave birth to the One; One gave birth to Two ...’), the conception of Three in Christianity, until finally the Renaissance gradually banished ‘the most debatable uses of analogy’, though some were still present in Bovelles in 1509. He then moves back to the presocratics, especially Heraclitus’s view of the world in permanent flux, and Parmenides’ response making the One ‘none other than the Being’ (p. 13) leading to the Pythagorean mysticism of numbers. Then on to a distinction in Plato (as discussed by Plotinus) between Ideal/essential numbers and counting numbers — the metaphysical ‘one’ versus the ‘one’ of counting. He then jumps, for five-and-a-half pages, to a modern discussion, in [Connes et al., 2001] Triangle of Thoughts, of different philosophical views of mathematics by modern mathematicians, with Connes’s neo-Platonistic view of mathematics versus Lichnerowicz’s formalist-structuralist-axiomatic view of mathematics. Patras inveighs against the latter as a ‘form of renewed Pythagorism’ (p. 19) that ‘may have contributed to a degeneration of the notion of modelling’ and refers to it as ‘the fetishism of the formal’. He finishes this chapter by returning to Aristotle, who, he writes, refuted the most debatable of Pythagorean or Platonic thought patterns, taking the perspective that mathematical objects are ideal objects that must be anterior to discourse but that it is necessary to distinguish logical anteriority from substantial anteriority — they are not there in the physical world. This is to be the position implicitly adopted in the rest of the book. In contrast with the difficulty of the second chapter, Chapter 3, ‘The One and the multiple’, clarifies a number of notions and terminologies that have been present from the start of the book. After briefly mentioning Plato’s Parmenides in the context of the ‘dialectical opposition of the One and the Multiple’ (p. 23), Patras clarifies his use of the term ‘aporia’ which he has used since page 2, as an apparent internal contradiction, which, on further examination, can be at least partially resolved. He does this via a very clear example of a Parmenidean aporia: Let us assume that the One exists. It cannot be multiple without contradictions. The One cannot therefore be a compound or have parts. Yet the Being is distributed among all the many beings that exist and, by virtue of Parmenidean thought, the One is never absent from the Being, nor the Being from the One. The One is thus shared in the same way as the Being, and is therefore multiple. He suggests the beginning of a resolution by considering that the number one is indecomposable in positive integers, and so is not multiple, and yet, if considering rational numbers, |$1 = 2 \times {1\over 2}$| and thus is divisible. This tension between what is considered a single unit, and in what ways it can still be composed of other things is what this chapter is devoted to. This includes Plato, then Frege’s criticism of the view that number is gotten from a collection of ‘units’ such as the number 2 being the idea of a lion and a lioness lying next to one another. Then Wittgenstein’s linguistic criticisms that ‘20 apples + 30 apples = 50 apples’ may be a proposition about apples or a proposition about arithmetic depending on its use. Patras focuses here on Frege’s and on Aristotle’s distinctions among the multiple senses of unity, and on Frege’s attempt to separate mathematical propositions from the experimental. His emphasis here is on what is invariant in an object composed of other objects: a book is no longer a book if you shuffle its pages, for example, and yet a pack of cards is still a pack of cards. (Frege’s approach to numbers is considered in much more detail in Chapters 10 and 11.) Chapter 4, ‘Mathematics and reality’, was also informative, with the exception of the last section. The chapter addresses the general problem of ‘participation’ — given that mathematical objects are not objects in the usual sense, not physical (though the idea of number does exist), how can they be useful in an empirical context? The author summarizes three attempts to respond to this problem. First, what he calls the ‘logico-semantic approach’ — that mathematical theories are formal systems devoid of meaning, but models of the theory give real-world interpretations of the theories. Then nominalism, basically that mathematics is just a language and so does not have a separate existence. Third, what he calls ‘schematism of concepts’, due initially to Kant, but he explains that Kant needs some revision: ‘In every mathematical concept is genetically inscribed the possibility of its use, and certain intuitive mechanisms that govern this possibility’ (p. 42). However, the final section of the chapter, on ‘schematism and pragmatism’, I found completely mysterious. Suddenly he moves from talking about the One to talking about ‘unity’. It is not at all clear what he means by this. He says that this concept is used throughout mathematics — but the items he cites, for example ‘topological unity’, are not called this in mathematics and I have no idea what he means by it. Chapter 5, ‘The third man argument’, discusses a problem that arose in Plato’s dialogue Parmenides and traces its development through the modern era. The argument, as it initially appears, concerns not men, but ‘greatness’. Rephrased about men (hence the title), men share common properties whereby they are men, and from that one gets the Idea of man (or ‘man-in-itself’) and then if one puts this idea together with individual man, one gets a ‘third man’, and then continues for an infinite sequence of man-ideas. This infinitude was incoherent, from the classical Greek viewpoint. Aristotle used it to argue that Plato’s claim to demonstrate the existence of ideas was problematic, and discussion continued on this argument. However, starting with Bolzano (beginning with a statement |$A$|⁠, then the statement that ‘|$A$| is true’, call it |$A_{1}$|⁠, one gets an infinite sequence of statements) and continuing with Dedekind, Cantor, and Frege, the argument is used to construct various infinite sets. This led Russell and Whitehead, because of the paradoxes that arose in set theory, to construct type theory. The last part of the chapter is taken up with an explanation of the problems with type theory that made it unwieldy and led to mathematicians largely rejecting it. He concludes the chapter by noting that both the third-man argument and the theory of types produced counter-intuitive, unnatural objects. Chapter 6, ‘Numbers and magnitudes’, finally starts moving from the considerations of classical Greece to modern approaches to mathematics. As such, it is the clearest chapter so far for modern readers. The primary focus is the distinction between arithmetic (discrete, combinatorial) and geometric (continuous, measuring) numbers. It begins with a question, ‘what is the purpose of mathematics?’ (which I see no attempt to respond to in the chapter) and with Aristotle’s division of everything into ten categories (such as substance, quantity, relation, action) and a rejection of that account for numbers. It then returns to the issue of the One, but now makes an Aristotelian observation that for both discrete and continuous quantities, we have to institute a norm, the One, that we can count or measure other objects in terms of. The example he gives is that in a group picture, we choose the notion of individual people as our One. However, for Aristotle, even though one can be used in calculation as if it were a number, because measure implies a plurality, one still cannot be a number. There is also a brief discussion of the difficulties, more obvious, with zero being a number. This is followed by a discussion (in a section, ‘the ontological difference’) of the parallel between Aristotle’s attempt to understand the difference between arithmetic and geometry, and modern mathematicians’ trying to explain how one field of mathematics, such as topology, differs from others such as geometry. This is followed by a discussion of arithmetic and geometry in Euclid: how his Elements was ‘the most important and surprising mathematical text ever written’. This is due to its beginning with an axiomatic approach to geometry, followed by ratios to introduce the arithmetic of magnitudes, and eventually confronting the question of commensurability: whether, given two magnitudes, one can find a unit of measurement with which both can be measured as integral multiples of that unit (leading to irrational numbers). He then jumps to the Cartesian revolution in mathematics, unifying arithmetic and geometry, resulting in the primacy of arithmetic. The chapter concludes with discussion of infinity, which for the Greeks was to be avoided, but for Frege becomes a ‘spatio-temporal phenomenon, based on the intuition of space and time’. Chapter 7, ‘Generalized numbers I’, is the first of two chapters that discuss the extension of what are considered numbers beyond the classical Greek view of numbers as a collection of units (and thus one not being considered a number). He distinguishes this concern, which he classifies as a problem of its ontological status, from the later concerns about their legitimacy. The focus here is on zero and negative numbers. This gives Patras a chance to introduce Husserl and phenomenology, which will be further discussed in Chapter 14. He presents phenomenology as following in ‘the metaphysical and critical tradition’ and ‘concerned with integrating the achievements of the cognitive sciences’ (p. 71). He discusses Husserl’s support for including one among the numbers by analyzing the cognitive processes involved in discussing numbers of items (such as five fruits) and the universality of ‘mental processes at work in everyday life and in scientific thought’ (p. 72). This is used to justify zero as a number, as one can give examples of it being a reasonable answer to ‘how many’ — for example, how many moons a given planet has. He then gives a brief history of negative numbers, starting with Diophantus and later Fibonacci. This is followed by a discussion of assorted ways to justify their use, ending with Kant’s contribution to this discussion. One peculiarity in this chapter (and again later): when discussing developments well in the past, the author sometimes uses the future tense. For example (p. 75) ‘Two attempts at philosophical and methodological clarification ... will take place in France in the eighteenth century ... . ’ Chapter 8, ‘Generalized numbers II’, starts by noting that there is no lack of good books exploring the development of the many number domains such as rational numbers and transcendental numbers; the focus in this chapter will be on the features of these developments most relevant for the philosophy of numbers generally. The chapter concentrates, then, almost exclusively on the complex numbers (though it includes a brief discussion of Cantor’s development of infinite cardinals and ordinals). These began as early as the mid-1500s for the purpose of calculations: assorted ways of representing |$\sqrt{-1}$| to be used in the middle of a calculation to arrive at a solution in real numbers. There was no attempt, initially, to justify complex numbers as more than a calculational maneuver. However, even from their outset there was a concern to guarantee their validity. This was approached from several perspectives. The fundamental theorem of algebra, that a polynomial of degree n has exactly n linear factors over the complex numbers, turns an ugly problem over the real numbers into a beautiful theory. This happens frequently in mathematics but was not entirely satisfactory as a way of justifying the existence of complex numbers. They became more widely accepted with the geometrical representation of a complex number |$x + iy$| as a point |$(x,y)$| in the plane. Frege, in his criticisms of Cantor’s infinite numbers, made similar criticisms of |$\sqrt{-1}$|⁠. Gradually (starting, Patras suggests, with Hankel in the mid-1800s) the ‘principle of permanence of formal laws’ (such as associativity) starts allowing mathematical objects to be accepted as long as there is no contradiction, although this raises concerns about what kind of existence this implies. Also mentioned is the construction of the complex numbers as a quotient ring of real polynomials. Chapter 9, ‘Cantor and set theory’, begins with a brief comparison of the approaches of Cantor, Frege, and Dedekind: Cantor motivated originally by problems in Fourier analysis, Frege by pure logic, and Dedekind the most algebraic, interested in putting analysis on rigorous foundations. (Oddly, twice on page 73 Patras writes of ‘Cauchy series’ when the usual term is ‘Cauchy sequences’ as is made clear by the definition he gives.) The connection to the properties of the natural numbers, which is the focus of the book, is somewhat tenuous: natural numbers can be associated with finite sets by virtue of the fact that two finite sets have a bijective correspondence between them exactly when they have the same number of elements. This is, however, noted as not something new but simply something Cantor’s work emphasizes. The chapter does not describe Cantor’s approach to transfinite numbers in sufficient detail for someone unfamiliar with the work. There is a mention (without giving the proof) of the ‘deeply disturbing’ Cantorian demonstration that the algebraic numbers can be mapped bijectively to the natural numbers, and a discussion of the resulting conclusion that the notion of ‘larger’ is thus problematic for infinite sets. There is a nice description of the diagonal argument’s use to show that the continuum cannot be put in a bijection with the natural numbers. Toward the end of the chapter the emphasis is on Cantor’s discussion of the issue of when sets are well-defined. Chapter 10, ‘Frege’s logicism’, primarily focuses on Frege’s Foundations of Arithmetic, where he attempts to construct the numbers and derive their properties strictly from the rules of pure thought. He criticized mathematicians of his era for accepting definitions as adequate as long as they do not lead to obvious contradictions. Frege sought the meanings of words in the context of the propositions involving them, so that mathematical objects, such as triangles, do not exist in themselves but in the context in which they are used. He insisted on a clear distinction between a term used as an object (say, a horse, in ‘the horse is tied up in the meadow’) and as a concept (‘horses are quadrupeds’). For Frege, the nature of a concept is to be predicative: what is asserted about a term. Patras gives a clear example, ‘the celestial object ... is a star’ (which could be written |$C({\dots}))$| so that when one substitutes ‘sun’ for ‘...’ it is true, but substituting ‘moon’ it is false. I found a discussion on page 106 of the need to distinguish between the way one first obtains the content of a statement, from its method of proof, to be one of the most interesting in the book so far. This chapter also discusses Frege’s relation to Kant’s views, as somewhat following from them but also somewhat different. Patras implies that for Frege, the synthetic and the a posteriori are the same, as are the analytic and the a priori. The chapter ends with Frege’s discussion of the idea of unit: one is not a concept. Rather, concepts are phrases about objects, such as ‘tree in front of me’. If there are four objects falling under that concept, so that the statement ‘There are four trees in front of me’ is true, it is a statement about a concept. It assigns a cardinal number to this multiplicity (here, 4), of which one (if there is only one tree rather than four trees) works as well as four. Chapter 11, ‘Set theory in Frege’, primarily discusses how Frege, in his Foundations of Arithmetic, shows how to produce natural numbers, starting with 0, as objects using the laws of pure thought, and discusses some of the problems with this approach as a way of defining objects in general. Patras calls Frege’s approach to numbers a ‘Copernican revolution’ in arithmetic. He explains that Frege gave an analogy between defining direction in geometry via lines having the same direction if they are parallel and defining number as two sets having the same number if they can be put in one-to-one correspondence. The number 0 is associated to a concept if no objects fall under the concept; in particular, he uses the concept ‘different from itself’. Then 1 is associated to the concept ‘equal to 0’ and |${n} + 1$| is defined in terms of n. Since numbers are based on concepts and their extensions, in general this runs into the assorted set-theory paradoxes such as the set of all sets that are not elements of themselves. There are ways of getting around this by restricting the kinds of sets that can be considered — but this does not fit with Frege’s principal aim, which is to construct the numbers from pure logic. In the end, Frege viewed his efforts in this to have failed. The chapter ends with two sections on Weyl’s criticism of Frege’s approach, that properties should only be affiliated with a definite category of objects, and thus he opposed reducing mathematics to logic. Weyl viewed it as essential to keep in mind the ‘ a priori intuition of iteration and of the sequence of natural numbers’, which Patras notes mixes ontological, logical, and mathematical arguments. But he notes there is some validity to Weyl’s objections, as one understands numbers long before one knows logic and understands the idea of proofs. Chapter 12, ‘Axioms and formalisms’, begins with an interesting discussion of two misunderstandings concerning set theory: first, the belief that set theory gives a basis for the whole mathematical edifice rather than being more like a ‘coding result’, like coding the points in the plane by a complex number. Second, as a result of Gödel’s incompleteness theorems, the Zermelo–Fraenkel axioms (or other axiom systems) for set theory cannot give a complete accounting of sets on which to base all possible mathematical results. In the late 1800s and early 1900s the hope was to use appropriate axioms to describe the natural numbers and sets completely. The discussion here starts with Dedekind’s work constructing the natural numbers by iteration, associating to each integer |$n$| its successor |$n+1$|⁠. This is followed by a brief discussion of Peano’s axioms for the integers. After mentioning the intuitionists, Brouwer and Weyl, Patras notes with approval Weyl’s question, ‘on which fundamental processes should the mathematical edifice be built?’ He then turns to Hilbert’s evolution from his axiomatization of geometry to his attempts to prove that arithmetic is free of contradictions. This latter led to his approach of developing logic and mathematics in parallel based on primitive logical notions of object of thought, object, and combinations. However, Hilbert’s program to prove the consistency of arithmetic by finite processes was doomed to failure, as Gödel’s incompleteness theorems showed. There is a good discussion in a footnote of the coding that Gödel used (now called Gödel numbers) to turn a sequence of symbols into a unique integer. Rather peculiarly, Patras writes (twice) on p. 130 of the impossibility of demonstrating the ‘coherence’ of arithmetic (rather than the standard term ‘consistency’). There is a nice passage in this section, though, on how Gödel’s result — that to show a system is complete or consistent, one has to go to a more powerful system than the one being proven about — is actually quite typical of other results in the history of mathematics. Also he notes that the formal principles of proof lack the expressiveness of those that produced the conceptual structures of mathematics. The chapter ends with a section he calls ‘moderate Platonism’, by which he means one that does not insist on the same reality for all mathematical concepts, and a brief description of Cantor’s transfinite numbers (which seems to me to be out of place). Chapter 13, ‘The brain and cognitive processes’, is a short, interesting chapter which is unnecessarily difficult to read, for reasons I shall discuss later. The chapter begins with mention of Descartes’ distinction between soul and body and the ‘infinitely more abstract’ development of mathematics since the time of Kant. He argues, quite thoroughly, for an ‘inhomogeneity of thought to matter’: that, despite the enormous complexity of the brain, the continuing development of ever more sophisticated mathematical ideas, and the ability of the brain to adapt to them is extremely unlikely to be locatable strictly in physical processes. Patras cites summaries of recent work in neuropsychology, especially Dehaene’s The Number Sense [2011], in the context of the complex process of a child’s learning arithmetic. He also mentions that there is a part of the brain in many animals, not only in humans, that has a primitive number sense. It is noted that, although we view the numbers as going on forever, when we are asked to think of a number, it tends to be small. There is also a quite interesting discussion (likely to be accessible only to mathematicians) at the end of the chapter of the recent evolution of the idea of a point, from its primitive geometric meaning to its connection to maximal ideals. While I have given what I believe is a reasonable summary of what previous chapters contain, and have read Chapter 14, ‘Phenomenology of numbers’, a half-dozen times, trying to figure out what it is about, I have failed. This is material closest to Patras’s interests: he has written numerous articles on the subject, and at places in this chapter, refers the reader to some of them. I believe the chapter is primarily intended as an introduction to phenomenology, using the specific context of numbers (and especially Husserl’s early work, Philosophy of Arithmetic). Patras writes, ‘The detail of the phenomenological analysis can only be understood through examples’ (p. 149). Yet the chapter is almost completely devoid of any examples — the only one I found was in the final section (and it was helpful for that section). The chapter is full of long quotes, mostly from Husserl. On two occasions, there are footnotes (6 and 14) explaining the quotes, which I found very helpful, and if these were added more regularly, it might be possible to learn more than a minimum from this chapter. There are many references to further reading, both by Husserl and by later interpreters. Possibly spending two years reading these would allow a better understanding. As the author notes, ‘we will use the phenomenological vocabulary quite freely’ (p. 145) and further (fn. 5, p. 147) ‘we follow here the French phenomenological tradition’, making it even more distant for those without substantial prior knowledge of phenomenology. Section 14.2 starts by saying that Husserl developed three points of view on arithmetic, ‘psychological, symbolic and algebraic’ which ‘will be further clarified and explained in the following paragraphs’ (p. 145): but I found nothing about the symbolic view until Section 14.7 and nothing about the algebraic at all. Perhaps it is there, but it is not clear to the outside observer! A few concepts are relatively clear, including the need to look for the reasons that concepts such as number have been developed as they were. There is a large amount of discussion of representations: a distinction is made between true representations, which seem to mean presenting an object in a way that its basic intuition is seen, and improper representations, such as referring to something by signs or symbols. To form a true representation of a number from a bunch of objects (say apples), one goes through two stages of abstraction: first, mentally forming the concept of the individuals into a ‘multiplicity’ — I think he means by this seeing them not as unrelated individuals, but as all part of one distinct totality — and then passing to the number that represents how many there are. However, I could literally make no sense of any of the quotes that form essentially the whole of page 150. The last three sections (14.7–14.9) of the chapter are somewhat less opaque. There Patras discusses Husserl’s view of numbers from a symbolic perspective, puzzling over the fact that we get valid results when calculating with number symbols without thinking about their true nature. The final part of this discussion, on ‘horizon structures’, concerns the approach mathematicians take when working with mathematical objects. We carry on some work ( e.g., calculations) via symbols without thinking of what they mean, but then, as needed, reorient our thinking back to the original meanings of the symbols to continue the development of the problem being worked on. This is an interesting description and could use more discussion. Chapter 15, ‘Universal phenomena, algebra, categories’, rather than continuing the previous chapter’s heavy discussion of Husserl, turns to how mathematical developments in the twentieth century have discovered numbers in a range of more general mathematical contexts. The first of these discussed is as invariants. After a brief introduction to the concept of invariants, Patras notes that numbers are invariants associated with equinumerosity classes (he writes this as ‘equinumericity’), that is, classes of sets for which there is a one-to-one correspondence between each pair. He connects this with a range of developments in algebraic topology, though, frankly, this connection seems to me rather forced. Next he considers the set of natural numbers as a universal object in the class of commutative monoids, in that given any commutative monoid |$S$|⁠, and any element |$s$| of |$S$|⁠, there is a unique morphism |$f$| from the set of natural numbers to |$S$| such that |$f(1)=s$|⁠. Third he mentions, in less detail, a construction in category theory involving least upper bounds (which I did not follow). Fourth, there’s an iterative construction due to Dedekind of definition by induction given a function |$g$| from a set to itself, of a function from the natural numbers to the set, basically constructing |$f(n) = g^n(0)$|⁠. The concluding section, ‘Phenomenology of algebra’, gives the final answer from phenomenology to the question of the essence of numbers: roughly, it is not to be found in individual numbers, but in a dynamic conception, the iterative counting process, passing from one element to the next. So one is no longer looking at individual numbers, nor at the completed totality, but at the process of its generation. While this chapter was largely very readable for a mathematician, it is so telegraphic in its descriptions that I do not think a non-mathematician will get any more out of it than I did from the preceding chapter. It is clear on reading this book that the author has an extensive knowledge of the history of philosophy, the history of mathematics, and the history of the philosophy of mathematics. He brings all of this background to bear on what he writes. But his writing is sufficiently condensed in its use of this background that it is difficult for anyone with only some of that background to understand what he is saying. I have a decent mathematical education (Ph.D. in mathematical logic from Cornell) and have done significant reading in relatively recent (especially 1800s on) philosophy of mathematics (though very little in phenomenology, none of it Husserl himself). So some of his telegraphic descriptions of mathematics (for example, p. 159, describing assorted topological concepts) are fairly clear to me, but would be totally mysterious to someone without at least an undergraduate degree in mathematics. On the other hand, I found his discussion in Chapters 2 and 14 extremely difficult. While he explains who the audience initially was, it is not clear who the intended audience for the final book is. One can read it very cursorily and get a rough overall idea of what he is saying. It certainly sums up the author’s work on the problem, but would probably have to be expanded to three times its current size to be readable without chasing off every page or two to read parts of the books cited, before fully understanding what he is saying. He cites sources for most of his references, but doesn’t give page numbers that would allow one to track down the context without reading the whole work. For example, in Chapter 4 (p. 41) he gives a two short quotes from a book by Heidegger about Kant without page numbers or concrete examples. And at times he just refers to an author, assuming the reader will understand the context ( e.g., p. 36, ‘even after Descartes’). His use of many quotes is so brief that one would have to read through a substantial number of his over one hundred articles and books cited (primarily in German, French, and English) really to get a sense of the context, and thus meaning, of his quotes. Because of this denseness of intricate discussion interweaving many sources, it took me three attempts, with breaks of several months between, to finally finish reading the book. Although the author does not explicitly say so, it appears that he did his own translation. He is highly fluent in English and therefore probably the best person to do this translation. (I speak French, well enough to get by in day-to-day life, but certainly do not claim competence in French to anywhere near Patras’s English proficiency.) However, as I was reading the text, every page or two, there would be a word that was not in my vocabulary, and that often would have been better replaced by a short phrase that would have been clearer. For example, in the first paragraph of Chapter 1, speaking of numbers, he writes ‘However, their theorization within mathematics has progressed surprisingly ...’ [my italics]. It is an English word, but I am pretty sure that he means ‘However, the development of their theory within mathematics has progressed surprisingly ...’ . Chapter 13, unfortunately, contains a high density of such problematic words. These strange words appearing with such frequency interrupts the reader, who then, after puzzling what the author probably means, goes back to start reading the paragraph or section again. In some cases (for example, ‘jouissance’, p. 16), the word is indeed an English word (as well as a common French word), but according to the Oxford English Dictionary (OED), both meanings are obsolete. On a few occasions (‘caudine’, ‘originarity’) the word simply is not an English word — one he uses a lot is ‘reconduce’ (pp. 5, 117, 124, among others). It is not in the OED, and my best guess is it is an anglicization of the French verb reconduire (although the internet says it is Spanish, reconducir). In the first two instances, it seems to mean ‘reduce’, in the third (the sentence reads ‘The key idea of Dedekind ... is to reconduce numbers to a dynamic point of view ...’) a better translation might be ‘to redirect’ or ‘to reconnect’. The discussion in Section 5.1, concerning a distinction between using an individual — say a triangle — as an instance in an argument has some difficulties due to differences between English and French. In this English translation, Patras writes of one saying ‘the man walks’. But one would only say that in English if one were speaking of a particular individual. If one is speaking of men in general, one would say either ‘men walk’ or ‘man walks’ (in the latter case, meaning man, in general, walks). But in French, one always has an article, ‘ le’ or ‘ la’: so ‘ L’homme marche’ for example. Hence he translates it as the awkward not-quite-English sentence ‘The man walks’. All of this is irrelevant to the point of the third-man argument, which works in English or in French, but needs to be written differently in English for it to make sense to an English speaker. This also applies to a discussion of horses in Chapter 10. If one has read almost all the works Patras cites, and has the same background he has in mathematics, philosophy, and history of philosophy, this would be a very worthwhile book, with an interesting view of how numbers should be approached. However, the combination of the terseness of the text, the frequency of the unexplained quotes, and the language issues makes this an unnecessarily difficult book to read. References Connes, A. , Lichnerowicz A., and Schützenberger M.P. [ 2001 ]: Triangle of Thoughts . Gage, J. trans. American Mathematical Society . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Dehaene, Stanislas [ 2011 ]: The Number Sense: How the Mind Creates Mathematics . New and updated ed . Oxford University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Author notes *Orcid.org/0000-0002-8098-8279. © The Authors [2022]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) © The Authors [2022]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
TI - Frédéric Patras.*The Essence of Numbers
JF - Philosophia Mathematica
DO - 10.1093/philmat/nkac028
DA - 2022-11-09
UR - https://www.deepdyve.com/lp/oxford-university-press/fr-d-ric-patras-the-essence-of-numbers-w0GA60uGww
SP - 132
EP - 142
VL - 31
IS - 1
DP - DeepDyve
ER -