On What There is—Infinitesimals and the Nature of Numbers
2015; Taylor & Francis; Volume: 58; Issue: 1 Linguagem: Inglês
10.1080/0020174x.2015.978535
ISSN1502-3923
Autores Tópico(s)Philosophy and History of Science
ResumoAbstractThis essay will be divided into three parts. In the first part, we discuss the case of infintesimals seen as a bridge between the discrete and the continuous. This leads in the second part to a discussion of the nature of numbers. In the last part, we follow up with some observations on the obvious applicability of mathematics. AcknowledgmentsIt is with great pleasure that I thank Johan van Benthem, Sol Feferman, Dagfinn Føllesdal, Mikhail Katz, Øystein Linnebo, Richard Tieszen, Herman Ruge Jervell, Dana Scott, Patrick Suppes, and Ed Zalta for many helpful and critical comments on several earlier drafts of this paper.Notes1 For a detailed study of the history of infinitesimals from Leibniz to Robinson, see Katz and Sherry, 'Leibniz's Infinitesimals'; see also Ehrlich, 'The Rise of Non-Archimedian Matematics and the Roots of a Misconception'; and Feferman, 'Conceptions of the Continuum' for a more extensive discussion of possible conceptions of the continuum.2 'In this test, however, the infinitely small has completely failed'. Fraenkel, Einleitung in die Mengenlehre, 116. The test referred to is how to prove a mean value theorem for arbitrary intervals, including infinitesimal ones; see section 6.1 on the Klein-Fraenkel criterion in Kanovei, Katz and Mormann, 'Tools, Objects, and Chimeras'.3 Skolem, 'Über die Grundlagendiskussionen in der Mathematik'.4 Skolem, 'Über die Nichtcharakterisierbarkeit der Zahlenreihe mittels endlich oder abzählar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen'.5 Robinson, Nonstandard Analysis.6 Keisler, 'The Hyperreal Line'.7 Fenstad, 'Nonstandard Analysis Relevant for the Philosophy of Mathematics?'; and Fenstad, 'The Discrete and Continuous in Mathematics and the Natural Sciences'.8 At this point, the reader is advised to turn to Keisler, 'The Hyperreal Line'. We have made a virtue of the non-uniqueness of the hyperreal line; others would regard this as an argument against the non-standard theory. It is a fact that the existence and the uniqueness of the hyperreals depend on the underlying set theory and how these numbers are constructed relative to the chosen set theoretic basis, see Keisler 'The Hyperreal Line', sections 11 and 12. For many, perhaps most, application in the natural sciences, the important fact is that infinitesimals exit in the geometric continuum. We should also note that there is an alternative approach to an extended arithmetic continuum, the so-called surreal numbers introduced by J. Conway in 1976; see the extensive discussion in Ehrlich, 'Absolute Arithmetic Continuum', where, in particular, the relationship between Robinson's non-standard reals and Conway's surreals is analyzed.9 Robinson, Nonstandard Analysis.10 Robinson, 'Formalism 64'.11 Gödel, 'Remark on Non-Standard Analysis'.12 Albeverio et al., Nonstandard Methods; ch. 7.13 Further information about the theory and the applications of non-standard analysis can be found in Albeverio et al., Nonstandard Methods; Keisler, 'The Hyperreal Line'; and the recent volume edited by van den Berg and Neves, The Strength of Nonstandard Analysis. For our purposes, we recommend the contribution by Keisler, 'The Strength of Nonstandard Analysis'. The debate between Gödel and Robinson is discussed in two 'introductory notes' in Gödel's Collected Works; see Fenstad, 'Introductory Note to Gödel 1974' on Gödel's remarks at the Institute in 1973; and Machover, 'Abraham Robinson: Introductory Note' on the Gödel-Robinson correspondence.14 Skolem, 'Begründung Der Elementären Arithmetik Durch Die Rekurrierende Denkweise Ohne Anwendung Scheinbarer Veränderlichen Mit Unendlichem Ausdehnungsbereich'. The German text is translated as 'Yet, even in providing a foundation for mathematics it is the substance that is important, not the notation' in van Heijenoort, From Frege to Gödel, 333. I am not happy with the translation 'substance' of the word 'Sache'; I would argue that 'Sache' in this context refers to what there is.15 Skolem, 'Untersuchungen über Einige Klassen Kombinatorischer Probleme. Oslo'.16 This is translated as '… a mathematical definition is a genuine definition if and only if it leads to the goal by means of a finite number of trials' in van Heijenoort, From Frege to Gödel, 333.17 The original text is in Norwegian; see Skolem, Undersøkelser over Potensrester Og over Logisk Karakterisering Av Tallrekken: 'Vil man gå over til en konsekvent formalistisk matematikk, basert på et endelig antall eksakt formulerte aksiomer, så er der intet annet å innvende, enn at der kan være spørsmål om motsigelsesfrihet og hensiktsmessighet o. l. Men innenfor den matematiske praksis, slik den vanligvis drives med kontinua, som slett ikke er gitt i kraft av de og de bestemte oppregnede konstruksjonsprinsipper, er utvalgsaksiomet efter min mening avgjort en uting—en slags videnskabelig svindel'.18 Popper, Objective Knowledge. An Evolutionary Approach, 154.19 Fenstad, 'Is Nonstandard Analysis Relevant?'.20 White, 'The Locus of Mathematical Reality'.21 If 'mind of the species' is too esoteric, substitute the phrase 'cultural heritage', see Grattan-Guinness, 'The Mathematics of the Past'.22 White, 'The Locus of Mathematical Reality'.23 White, 'The Locus of Mathematical Reality'.24 von Neuman, 'The Mathematician'.25 Davies and Hersh, The Mathematical Experience.26 Hersh, What is Mathematics, Really?.27 Mumford, 'Pattern Theory'.28 Atiyah et al., 'The Interface between Mathematics and Physics'.29 Cole, 'Mathematical Domains'; and Cole, 'Toward an Institutional Account of the Objectivity'.30 Kitcher, The Natur of Mathematical Knowledge.31 Leng, 'Introduction', 7–15.32 Linnebo, 'The Nature of Mathematical Objects'; and Linnebo, 'The Individuation of the Natural Numbers'.33 Gödel, 'The Modern Development'. For a general discussion of the relationship between Husserl and Gödel, see Føllesdal, 'Gödel and Husserl'; Føllesdal, 'Introductory Note to Gödel 1961'; and Hauser, 'Gödel's Program Revisited. Part I'.34 Tieszen, 'Mathematics'.35 Tieszen, After Gödel.36 Tieszen, 'Mathematics', 448.37 Fenstad, Grammar, Geometry, and Brain.38 Donald, Origin of the Modern Mind; and Donald, 'Precis and Discussion of Origin of Modern Mind'.39 Fenstad, Grammar, Geometry, and Brain, 39.40 Fenstad, Grammar, Geometry, and Brain.41 Nowak et al., 'The Evolution of Syntactic Communication'.42 Fenstad, Grammar, Geometry, and Brain.43 Dunn et al., 'Evolved Structure of Language'. Recently, we have also seen several examples of an expermentalist approach to the origin and development of language, for an introduction see Normile, 'Experiments Probe Language's Origin and Development'. This is research at an early stage, but it seems to strenghten an evolutionary/cultural point of view. See also the recent analysis in Bergen, Louder than Words.44 Dehaene, The Number Sense.45 Dehaene and Changeux, 'Development of Elementary Numerical Abilities'.46 Dehaene, The Number Sense, 33.47 Amit, Modeling Brain Function.48 Amit, Modeling Brain Function, 252–3.49 Cappelletti and Giardino, 'The Cognitive Basis of Mathematical Knowledge'.50 Dehaene and Brannon, 'Space, Time, and Number'.51 Cappelletti and Giardino, 'The Cognitive Basis of Mathematical Knowledge', 82.52 Joseph, The Crest of the Peacock.53 Gordon, 'Numerical Cognition without Words'; and Beller and Bender, 'The Limits of Counting'. Gordon concludes his study of the Piraha tribe in the Amazon Basin that what is innate is at most the ability to see specific numbers up to three, the rest is culturally determined.54 Suppes, 'Why the Effectiveness of Mathematics in the Natural Sciences is Not Surprising'.55 Ferguson, Engineering and the Mind's Eye.56 Some references here are Laland et al., 'How Culture Shaped the Human Genome'; and Fisher and Ridley, 'Culture, Genes and the Human Revolution'.57 Donald, Origin of the Modern Mind.58 Not everyone will agree to this cultural invariance. It has been a widely shared opinion that the absoluteness of simple arithmetical facts, such as 5 + 7 = 12, is a strong argument against the cultural dependence of mathematics: if mathematics is culture dependent, there could be cultures where the meaning of terms are the same, but at the same time 5 + 7 and 12 are not equal. If this is not possible, the 'fact' that 5 + 7 = 12 would necessarily be culture and mind independent. Some would see this as an argument in favor of some form of platonism; see Tieszen, 'Mathematical Problem-Solving and Ontology'. I do not agree with this line of reasoning. I shall argue in the next section that structure is the basic entity, syntax is derived and adapted. Thus, if structure is given and a suitable structure-dependent syntax chosen, the arithmetical 'facts' will follow with necessity. Therefore, if I say that 5 + 7 equals 12 and you say no, we have to go back to basic structures. I have argued above for the cultural invariance of numbers and of counting/ addition based on what we currently know about neuroscience and anthropology. This is an observed 'fact', and it follows that I do not expect to be contradicted when I assert that 5 + 7 equals 12. I do not deny that there could be cultures, past or future, where facts and basic structures are otherwise. But for cultures in our invariance class, numbers are true objects in the 'mind of the species', and elementary arithmetical facts are absolute. This is what is needed for the applicability of mathematics.59 Skolem, 'Begründung Der Elementären Arithmetik Durch Die Rekurrierende Denkweise Ohne Anwendung Scheinbarer Veränderlichen Mit Unendlichem Ausdehnungsbereich. Oslo'.60 Suppes, 'Why the Effectiveness of Mathematics in the Natural Sciences is Not Surprising'.61 Mumford, 'The Dawning Age of Stochasticity'.62 Tieszen, 'Mathematics'.63 Wigner, 'The Unreasonable Effectiveness of Mathematics'.64 Suppes, 'Why the Effectiveness of Mathematics in the Natural Sciences is Not Surprising'.65 Ferguson, Engineering and the Mind's Eye.66 Fenstad, 'The Miraculous Left Hand'.67 Capra, The Science of Leonardo.68 Klein, Leonardo's Legacy, 181.69 Eloy, 'Leonardo's Rule'.70 Note that I use the word syntax in an rather extended sense, I could also have spoken of structure and representation, or perhaps of objects and tools. Note further that in the case of Leonardo, the 'syntax' is his left hand, so our representations can in some cases be physical objects. The main thing is that structure is what we 'see' and syntax is what we introduce to make sense of what we 'see'. The reader may at this point also want to consult Mumford, 'The Dawning Age of Stochasticity' on perception and pattern theory. The reader will further see a close relationship of our extended use of structure to the so-called prototype theory, which is to understand structure as a kind of 'fixed point' for what is seen, see Rosch, 'Prototype Classification and Logical Classification'; and Gärdenfors, Conceptual Spaces, the Geometry of Thought. Let me also add one further note of warning, my use of the word 'structure' is similar to the informal use of the same word in contemporery applied mathematics and is related, but does not correspond exactly, to what is known as 'mathematical structuralism' in the philosophy of mathematics; for a careful discussion of the latter, see Nodelman and Zalta, Foundations for Mathematical Structuralism.71 Barwise, 'Model-Theoretic Logics: Background and Aims', 4.72 Fenstad and Wang, 'Thoralf Albert Skolem'.73 Cohen, 'Skolem and Pessimism about Proof in Mathematics', 2417.74 The reader who wishes a deeper insight into the current praxis of mathematical modeling and large-scale scientific computing is strongly adviced to consult the recent survey by Tadmor, 'A Review of Numerical Methods for Nonlinear Partial Differential Equations'.75 Skolem, 'Über Die Mathematische Logik'; see also the discussion on his attitude to Gödel's completeness theorem for first-order logic in Fenstad and Wang, 'Thoralf Albert Skolem', Section 4.3.2.76 Cohen, 'Skolem and Pessimism about Proof in Mathematics', 2409.77 Herken, The Universal Turing Machine.78 Gandy, 'The Confluence of Ideas in 1936'.79 Fenstad and Wang, 'Thoralf Albert Skolem', section 4; and Cohen, 'Skolem and Pessimism about Proof in Mathematics'.80 Skolem, Selected Works in Logic.81 Suppes, Representation and Invariance of Scientific Structures.82 Barrow, 'Simple Really'.83 Consult Fenstad, Grammar, Geometry, and Brain for a discussion of meaning in natural languages.
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