DOCSLIB.ORG

Principle of Acquaintance and Axiom of Reducibility

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Principle of Acquaintance and Axiom of Reducibility Principle of acquaintance and axiom of reducibility Brice Halimi Université Paris Nanterre 15 mars 2018 Russell-the-epistemologist is the founding father of the concept of acquaintance. In this talk, I would like to show that Russell’s theory of knowledge is not simply the next step following his logic, but that his logic (especially the system of Principia Mathematica) can also be understood as the formal underpinning of a theory of knowledge. So there is a concept of acquaintance for Russell-the-logician as well. Principia Mathematica’s logical types In Principia, Russell gives the following examples of first-order propositional functions of individuals: φx; (x; y); (y) (x; y);::: Then, introducing φ!zb as a variable first-order propositional function of one individual, he gives examples of second-order propositional functions: f (φ!zb); g(φ!zb; !zb); F(φ!zb; x); (x) F(φ!zb; x); (φ) g(φ!zb; !zb); (φ) F(φ!zb; x);::: Then f !(φb!zb) is introduced as a variable second-order propositional function of one first-order propositional function. And so on. A possible value of f !(φb!zb) is . φ!a. (Example given by Russell.) This has to do with the fact that Principia’s schematic letters are variables: It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. ] (Principia, p. 51) And variables are understood substitutionally (see Kevin Klement, “Russell on Ontological Fundamentality and Existence”, 2017). This explains that the language of Principia does not constitute an autonomous formal language. Principia’s propositional functions are either variables or dummy terms directly standing for “concrete” propositional functions, in contrast with what happens in modern logic à la Tarski. Principia’s propositional functions display all the possible particular functional forms that one can specify without mentioning a concrete propositional function. The type of a propositional function does not only display the number and respective orders of its arguments (real variables), but also the number and respective orders of its apparent variables; and, in case its arguments are functions themselves, it displays the number and respective orders of the arguments and apparent variables of those arguments, and so forth. To that extent, a type codes inductively a quite fine-grained logical structure. It can be formalized as: a a r r ht1 ;:::; tm; t1;:::; tni a where the tj ’s are the respective types of the apparent variables, and r the ti ’s are the respective types of the real variables, ordered according to their occurrences. Examples The type of individuals is o := h−; −i. The type of φ!x is (o) := h−; oi. The type of (x) (x; yb) is ho ; oi. The type of (φ) f !(φ!zb; x) is h(o); oi. The type of (φ) f !(φ!zb; !zb) is h(o); (o)i. The type of (φ) f !(φ!zb; !zb; x) is h(o); (o); (o); oi. The order of a propositional function measures the depth of the presuppositions involved in its expression. It can be determined directly from the type of the function: I the order of an individual is 0 a a r r I the order of a function ' of type ht1 ;:::; tm; t1;:::; tni is a r max(jti j; jtj j)1≤i≤m + 1. 1≤j≤n A propositional function is said to be predicative if its order is the lowest compatible with having its arguments. For instance, (φ) f !(φ!zb; x) is not predicative: it is of order 2 (because of the quantification over φ) whereas its arguments are of order 0. By extension, a variable is predicative when all its possible values are predicative. Russell’s “no loss of generality” claim The Introduction of the (first edition of) Principia suggests that only predicative variables should be adopted in the formal language and that there is “no loss of generality” in doing so: It is important to observe that all possible functions in the above hierarchy [the hierarchy of functions of increasing orders] can be obtained by means of predicative functions and apparent variables. [. ] Thus we need not introduce as variables any functions except predicative functions. (Principia, pp. 53-54) Apart from Church, most commentators acknowledge that all Principia’s variables are predicative. Yet this is puzzling. Firstly, Russell does not provide us with any clear justification of his claim. Secondly, if all schematic letters are variables and all variables are predicative, non-predicative terms are neither variables nor possible values for variables (there are no variables appropriate to them), so what is their purpose? An easy way out consists in claiming that all terms other than individuals are predicative. (This is Gregory Landini’s reading of the formal syntax of Principia.) Two questions The problematic existence of non-predicative terms that could not instantiate any variable prompts two questions: 1. Are all terms predicative? 2. Are all predicative terms (predicative) variables? Landini answers “yes” to both questions. (“Circumflexion was not a term forming operator in Principia.”) According to Landini, Russellian matrices φ!(x), f !(φ!zb; x), . , are free variables in schematic letters’ clothing. Non-predicative wffs (well-formed formulas) can be formed by quantifying some variable but, since only predicative variables are allowed, such wffs are not terms and correspond to terms only by virtue of the axiom of reducibility. The axiom of reducibility (AR) ` ( E ) φx ≡x !x It has much to commend it: I Without (AR), quantification and types would be much more complex: there would be as many second-order variables as there are types of first-order propositional functions. I Defining identity would be impossible, because all first-order propositional functions do not make up a legitimate totality (Principia, p. 49 and p. 57). Thanks to (AR), quantification over all predicative first-order propositional functions amounts to quantification over all propositional functions whatsoever. I The theory of classes is basically equivalent with the adoption of (AR): φx ≡ x z˘(φz) However, the axiom of reducibility is not a justification of Russell’s “no loss of generality” claim, or only retrospectively. (It is an axiom, not a rule of syntax.) A puzzle It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. ] (It should be observed that when φ is assigned [?], we may obtain a function whose values do involve individuals as apparent variables, for example if φ!x is (y) (x; y). But so long as φ is variable [?], φ!x contains no apparent variables.) (Principia, pp. 51-52) If quantification is substitutional, how to conceive of the “assignment” of φ to a propositional function involving apparent variables? Moreover, if all variables are predicative, how could their possible values be non-predicative? A solution may come from the epistemic contextuality of logical analysis. Nicholas Griffin, “Russell on the Nature of Logic (1903-1913)” (1980): [. ] Russell’s type theory is to some extent context sensitive: for example, an item which, in one context, may be taken to be simple may, in another, turn out to be complex; and thus terms like ‘individual’ or ‘first-truth’ are not stable across contexts [. ]. An example occurs in connection with the word ‘Socrates’ which when used by Socrates himself denotes a simple individual of Socrates’ acquaintance; whereas, when used by someone who has never met Socrates, it is a complex hidden description to be analyzed by Russell’s theory of descriptions. Thus Socrates is a possible value of /xb is an individual/ only for Socrates himself, for others, with no acquaintance with Socrates, Socrates is not a possible value for that function [. ]. In general, since different people are acquainted with different items, the range of total variation for functions like /xb is an individual/ will be different for different people. Thus it is intolerable [to Quine and Sommerville] to treat such functions as propositional functions of logic. Russell indeed distinguishes between “Socrates is human” as uttered by Socrates himself, and as uttered by us who only know Socrates by description and thus for whom the sentence contains apparent variables. In the same way, Russell suggests that “Napoleon is a great general” can be analysed either as G(N) or as (φ) f (φ!zb) ⊃ φ!(N) where “f (φ!zb) stands for “φ!zb is a predicate required in a great general.” Epistemic realizations of logical types Let’s consider for instance the proposition expressing that a certain individual a is green. If I am acquainted with both the individual a and the universal Green, I shall say: (1) Ga (elementary truth, first-order proposition). Having only access to a as to “the F:” (2) ( E x) (y) Fy ≡ y = x & Gx (first-order proposition). Having only access to green as to “the colour of grass:” (3) ( E φ) ( ) CoG( ) ≡ = φ & φa (second-order proposition). Combining both descriptions: (4) E ( E x) ( φ) ( ) (CoG( ) ≡ = φ & (y) φy ≡ y = x & φ!x (second-order proposition). In the third case, ‘green’ is nothing but an incomplete symbol. Nevertheless, its occurrence in any context will generate a second-order quantification. In view of this fact, green may be identified with a non-predicative second-order propositional function with one individual argument. The sentences (1)-(4) above express propositions involving propositional functions of different types, even if these propositions correspond, in some way, to the “same” state of affairs. Principia’s ramified type theory allows us to distinguish among between the sentences (1) and (4), even between (3) and (4) —hence the utility of fine-grained types. The logical structure of the situation will vary from subject to subject, and so will, more generally, the realization of the whole theory of logical types.
Load more

Recommended publications
  • Discrete Mathematics
    Discrete Mathematics1 http://lcs.ios.ac.cn/~znj/DM2017 Naijun Zhan March 19, 2017 1Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course. Contents 1. The Foundations: Logic and Proofs 2. Basic Structures: Sets, Functions, Sequences, Sum- s, and Matrices 3. Algorithms 4. Number Theory and Cryptography 5. Induction and Recursion 6. Counting 7. Discrete Probability 8. Advanced Counting Techniques 9. Relations 10. Graphs 11. Trees 12. Boolean Algebra 13. Modeling Computation 1 Chapter 1 The Foundations: Logic and Proofs Logic in Computer Science During the past fifty years there has been extensive, continuous, and growing interaction between logic and computer science. In many respects, logic provides computer science with both a u- nifying foundational framework and a tool for modeling compu- tational systems. In fact, logic has been called the calculus of computer science. The argument is that logic plays a fundamen- tal role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of computer sci- ence as disparate as machine architecture, computer-aided de- sign, programming languages, databases, artificial intelligence, algorithms, and computability and complexity. Moshe Vardi 2 • The origins of logic can be dated back to Aristotle's time. • The birth of mathematical logic: { Leibnitz's idea { Russell paradox { Hilbert's plan { Three schools of modern logic: logicism (Frege, Russell, Whitehead) formalism (Hilbert) intuitionism (Brouwer) • One of the central problem for logicians is that: \why is this proof correct/incorrect?" • Boolean algebra owes to George Boole.
    [Show full text]
  • “Gödel's Modernism: on Set-Theoretic Incompleteness,” Revisited
    “G¨odel'sModernism: on Set-Theoretic Incompleteness," revisited∗ Mark van Atten and Juliette Kennedy As to problems with the answer Yes or No, the con- viction that they are always decidable remains un- touched by these results. —G¨odel Contents 1 Introduction 1.1 Questions of incompleteness On Friday, November 15, 1940, Kurt G¨odelgave a talk on set theory at Brown University.1 The topic was his recent proof of the consistency of Cantor's Con- tinuum Hypothesis, henceforth CH,2 with the axiomatic system for set theory ZFC.3 His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness:4 (Remarks I planned to make, but did not) Discussion on G¨odel'slecture on the Continuum Hypothesis, November 14,5 1940 There seems to be a difference: between the undecidable propo- sitions of the kind of his example [i.e., 1931] and propositions such as the Axiom of Choice, and the Axiom of the Continuum [CH ]. We used to ask: \When these two have been decided, is then everything decided?" (The Poles, Tarski I think, suspected that this would be the case.) Now we know that (on the basis of the usual finitary rules) there will always remain undecided propositions. ∗An earlier version of this paper appeared as ‘G¨odel'smodernism: on set-theoretic incom- pleteness', Graduate Faculty Philosophy Journal, 25(2), 2004, pp.289{349. Erratum facing page of contents in 26(1), 2005. 1 1. Can we nevertheless still ask an analogous question? I.e.
    [Show full text]
  • Master of Science in Advanced Mathematics and Mathematical Engineering
    Master of Science in Advanced Mathematics and Mathematical Engineering Title: Logic and proof assistants Author: Tomás Martínez Coronado Advisor: Enric Ventura Capell Department: Matemàtiques Academic year: 2015 - 2016 An introduction to Homotopy Type Theory Tom´asMart´ınezCoronado June 26, 2016 Chapter 1 Introduction We give a short introduction first to the philosophical motivation of Intuitionistic Type Theories such as the one presented in this text, and then we give some considerations about this text and the structure we have followed. Philosophy It is hard to overestimate the impact of the problem of Foundations of Mathematics in 20th century philos- ophy. For instance, arguably the most influential thinker in the last century, Ludwig Wittgenstein, began its philosophical career inspired by the works of Frege and Russell on the subject |although, to be fair, he quickly changed his mind and quit working about it. Mathematics have been seen, through History, as the paradigma of absolute knowledge: Locke, in his Essay concerning Human Understanding, had already stated that Mathematics and Theology (sic!) didn't suffer the epistemological problems of empirical sciences: they were, each one in its very own way, provable knowledge. But neither Mathematics nor Theology had a good time trying to explain its own inconsistencies in the second half of the 19th century, the latter due in particular to the works of Charles Darwin. The problem of the Foundations of Mathematics had been the elephant in the room for quite a long time: the necessity of relying on axioms and so-called \evidences" in order to state the invulnerability of the Mathematical system had already been criticized by William Frend in the late 18th century, and the incredible Mathematical works of the 19th century, the arrival of new geometries which were \as consistent as Euclidean geometry", and a bunch of well-known paradoxes derived of the misuse of the concept of infinity had shown the necessity of constructing a logical structure complete enough to sustain the entire Mathematical building.
    [Show full text]
  • THE 1910 PRINCIPIA's THEORY of FUNCTIONS and CLASSES and the THEORY of DESCRIPTIONS*
    EUJAP VOL. 3 No. 2 2007 ORIGinal SCienTifiC papeR UDK: 165 THE 1910 PRINCIPIA’S THEORY OF FUNCTIONS AND CLASSES AND THE THEORY OF DESCRIPTIONS* WILLIAM DEMOPOULOS** The University of Western Ontario ABSTRACT 1. Introduction It is generally acknowledged that the 1910 Prin- The 19101 Principia’s theory of proposi- cipia does not deny the existence of classes, but tional functions and classes is officially claims only that the theory it advances can be developed so that any apparent commitment to a “no-classes theory of classes,” a theory them is eliminable by the method of contextual according to which classes are eliminable. analysis. The application of contextual analysis But it is clear from Principia’s solution to ontological questions is widely viewed as the to the class paradoxes that although the central philosophical innovation of Russell’s theory of descriptions. Principia’s “no-classes theory it advances holds that classes are theory of classes” is a striking example of such eliminable, it does not deny their exis- an application. The present paper develops a re- tence. Whitehead and Russell argue from construction of Principia’s theory of functions the supposition that classes involve or and classes that is based on Russell’s epistemo- logical applications of the method of contextual presuppose propositional functions to the analysis. Such a reconstruction is not eliminativ- conclusion that the paradoxical classes ist—indeed, it explicitly assumes the existence of are excluded by the nature of such func- classes—and possesses certain advantages over tions. This supposition rests on the repre- the no–classes theory advocated by Whitehead and Russell.
    [Show full text]
  • Sets, Propositional Logic, Predicates, and Quantifiers
    COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete.
    [Show full text]
  • Types, Sets and Categories
    TYPES, SETS AND CATEGORIES John L. Bell This essay is an attempt to sketch the evolution of type theory from its begin- nings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances. 1 THE ORIGINS OF TYPE THEORY The roots of type theory lie in set theory, to be precise, in Bertrand Russell’s efforts to resolve the paradoxes besetting set theory at the end of the 19th century. In analyzing these paradoxes Russell had come to find the set, or class, concept itself philosophically perplexing, and the theory of types can be seen as the outcome of his struggle to resolve these perplexities. But at first he seems to have regarded type theory as little more than a faute de mieux. In June 1901 Russell learned of the set-theoretic paradox, known to Cantor by 1899, that issued from applying the latter’s theorem concerning power sets to the class V of all sets. According to that theorem the class of all subclasses of V would, impossibly, have to possess a cardinality exceeding that of V . This stimulated Russell to formulate the paradox that came to bear his name (although it was independently discovered by Zermelo at about the same time) concerning the class of all classes not containing themselves (or predicates not predicable of themselves).
    [Show full text]
  • The Development of Mathematical Logic from Russell to Tarski: 1900–1935
    The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives .
    [Show full text]
  • Reflexivity and Self-Referentiality 0.8Em in Inverse Monoids and Categories
    Reflexivity and Self-Referentiality In Inverse Monoids and Categories Peter M. Hines York – Mathematics – 2019 www.peterhines.info Reflexivity and Self-Referentiality Some motivation ... This talk is about some inverse category theory closely associated with logic and theoretical computer science. The general topic is models of self-referentiality. We aim to: 1 Describe the historical context & importance. 2 Give concrete axioms & examples. 3 Do all this in the reversible (inverse monoid) setting. www.peterhines.info Reflexivity and Self-Referentiality Historical Context (I) — Foundations & Logic Scenes from the frog-mouse wars www.peterhines.info Reflexivity and Self-Referentiality The historical setting The late 19th and early to mid 20th century saw some- thing of a crisis in the foundations of mathematics. This can be compared to the controversy caused by the introduction of calculus that was resolved by rigorous no- tions of limit & convergence. However, it was more profound, and less easily resolved. Its aftermath is still relevant today. www.peterhines.info Reflexivity and Self-Referentiality The problems of infinity Georg Cantor lit the fuse, and stepped back to a safe distance ... His work was not always appreciated: A ”scientific charlatan”, a ”renegade” and a ”corrupter of youth” — Leopold Kroenecker Mathematics is ”ridden through and through with the per- nicious idioms of set theory”, which is ”utter nonsense” that is ”laughable” and ”wrong” — Ludwig Wittgenstein www.peterhines.info Reflexivity and Self-Referentiality A more balanced approach A very readable contemporaneous account: Mathematical Rigor, past and present – J. Pierpont (1928) The Mengenlehre of Cantor [Set Theory] has brought to light a number of paradoxes which have profoundly disturbed the mathematical community for a quarter of a century.
    [Show full text]
  • Self-Similarity in the Foundations
    Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory .
    [Show full text]
  • What Is Neologicism?∗
    What is Neologicism? 2 by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory. Recently, ‘neologicism’ has emerged, claiming to be a successor to the ∗ What is Neologicism? original project. It was shown to be (relatively) consistent this time and is claimed to be based on logic, or at least logic with analytic truths added. Bernard Linsky Edward N. Zalta However, we argue that there are a variety of positions that might prop- erly be called ‘neologicism’, all of which are in the vicinity of logicism. University of Alberta Stanford University Our project in this paper is to chart this terrain and judge which forms of neologicism succeed and which come closest to the original logicist goals. As we look back at logicism, we shall see that its failure is no longer such a clear-cut matter, nor is it clear-cut that the view which replaced it (that 1. Introduction mathematics is applied set theory) is the proper way to conceive of math- ematics. We shall be arguing for a new version of neologicism, which is Logicism is a thesis about the foundations of mathematics, roughly, that embodied by what we call third-order non-modal object theory. We hope mathematics is derivable from logic alone. It is now widely accepted that to show that this theory offers a version of neologicism that most closely the thesis is false and that the logicist program of the early 20th cen- approximates the main goals of the original logicist program. tury was unsuccessful. Frege’s (1893/1903) system was inconsistent and In the positive view we put forward in what follows, we adopt the dis- the Whitehead and Russell (1910–13) system was not thought to be logic, tinctions drawn in Shapiro 2004, between metaphysical foundations for given its axioms of infinity, reducibility, and choice.
    [Show full text]
  • Constructive Set Theory July, 2008, Mathlogaps Workshop, Manchester
    Constructive Set Theory July, 2008, Mathlogaps workshop, Manchester . Peter Aczel [email protected] Manchester University Constructive Set Theory – p.1/88 Plan of lectures Lecture 1 1: Background to CST 2: The axiom system CZF Lecture 2 3: The number systems in CZF 4: The constructive notion of set Lectures 3,4 ? 5: Inductive definitions 6: Locales and/or 7: Coinductive definitions Constructive Set Theory – p.2/88 1: Background to CST Constructive Set Theory – p.3/88 Some brands of constructive mathematics B1: Intuitionism (Brouwer, Heyting, ..., Veldman) B2: ‘Russian’ constructivism (Markov,...) B3: ‘American’ constructivism (Bishop, Bridges,...) B4: ‘European’ constructivism (Martin-Löf, Sambin,...) B1,B2 contradict classical mathematics; e.g. B1 : All functions R → R are continuous, B2 : All functions N → N are recursive (i.e. CT). B3 is compatible with each of classical maths, B1,B2 and forms their common core. B4 is a more philosophical foundational approach to B3. All B1-B4 accept RDC and so DC and CC. Constructive Set Theory – p.4/88 Some liberal brands of mathematics using intuitionistic logic B5: Topos mathematics (Lawvere, Johnstone,...) B6: Liberal Intuitionism (Mayberry,...) B5 does not use any choice principles. B6 accepts Restricted EM. B7: A minimalist, non-ideological approach: The aim is to do as much mainstream constructive mathematics as possible in a weak framework that is common to all brands, and explore the variety of possible extensions. Constructive Set Theory – p.5/88 Some settings for constructive mathematics type theoretical category theoretical set theoretical Constructive Set Theory – p.6/88 Some contrasts classical logic versus intuitionistic logic impredicative versus predicative some choice versus no choice intensional versus extensional consistent with EM versus inconsistent with EM Constructive Set Theory – p.7/88 Mathematical Taboos A mathematical taboo is a statement that we may not want to assume false, but we definately do not want to be able to prove.
    [Show full text]
  • SEP 0 111993 C
    IMPREDICATIVITY AND TURN OF THE CENTURY FOUNDATIONS OF MATHEMATICS: PRESUPPOSITION IN POINCARE AND RUSSELL by Joseph Romeo William Michael Picard B.A. Honors, Philosophy, University of Calgary, 1986 M.Sc., Philosophy, Massachusetts Institute of Technology, 1990 Submitted to the Department of Linguistics and Philosophy in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN PHILOSOPHY at the Massachusetts Institute of Technology July 1993 0 J.R.W.M. Picard, 1993. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. Signature of Author Department off lnguistics and Philosophy July 8, 1993 Certified by- Professor Richard Caltwright Thesis Supervisor, Department of Linguistics and Philosophy Accepted by Arofessor George Boolos Chairman, Departmental Committee on Graduate Studies MASSACHUSETTS INSTITUTE OFrT•CHNOLOGY 1 SEP 0 111993 -. ·.na uiCO c ABSTRACT The primary purpose of this dissertation is to state a modal account of impredicativity. A (formal or explicit) definition (under a p -ticular interpretation) is impredicative if the object defined )n that interpretation is a value of a bound variable occurring in the detinition. An object may also be called impredicative (with respect to a given language), namely just in case it can be defined in that language but only by means of an impredicative definition. Poincar6 charged in (1906) that impredicative definitions are viciously circular and that impredicative objects dc not exist. Russell agreed with these charges and went on to formulate a logic (ramified type theory) on the basis of a principle which banned imprediwativity (vicious circle principle).
    [Show full text]