Lectures on the Foundations of Mathematics
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Weyl's Predicative Classical Mathematics As a Logic-Enriched
Weyl’s Predicative Classical Mathematics as a Logic-Enriched Type Theory? Robin Adams and Zhaohui Luo Dept of Computer Science, Royal Holloway, Univ of London {robin,zhaohui}@cs.rhul.ac.uk Abstract. In Das Kontinuum, Weyl showed how a large body of clas- sical mathematics could be developed on a purely predicative founda- tion. We present a logic-enriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several re- sults from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics. Key words: logic-enriched type theory, predicativism, formalisation 1 Introduction Type theories have proven themselves remarkably successful in the formalisation of mathematical proofs. There are several features of type theory that are of particular benefit in such formalisations, including the fact that each object carries a type which gives information about that object, and the fact that the type theory itself has an inbuilt notion of computation. These applications of type theory have proven particularly successful for the formalisation of intuitionistic, or constructive, proofs. The correspondence between terms of a type theory and intuitionistic proofs has been well studied. The degree to which type theory can be used for the formalisation of other notions of proof has been investigated to a much lesser degree. There have been several formalisations of classical proofs by adapting a proof checker intended for intuitionistic mathematics, say by adding the principle of excluded middle as an axiom (such as [Gon05]). -
Revisiting Gödel and the Mechanistic Thesis Alessandro Aldini, Vincenzo Fano, Pierluigi Graziani
Theory of Knowing Machines: Revisiting Gödel and the Mechanistic Thesis Alessandro Aldini, Vincenzo Fano, Pierluigi Graziani To cite this version: Alessandro Aldini, Vincenzo Fano, Pierluigi Graziani. Theory of Knowing Machines: Revisiting Gödel and the Mechanistic Thesis. 3rd International Conference on History and Philosophy of Computing (HaPoC), Oct 2015, Pisa, Italy. pp.57-70, 10.1007/978-3-319-47286-7_4. hal-01615307 HAL Id: hal-01615307 https://hal.inria.fr/hal-01615307 Submitted on 12 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Theory of Knowing Machines: Revisiting G¨odeland the Mechanistic Thesis Alessandro Aldini1, Vincenzo Fano1, and Pierluigi Graziani2 1 University of Urbino \Carlo Bo", Urbino, Italy falessandro.aldini, [email protected] 2 University of Chieti-Pescara \G. D'Annunzio", Chieti, Italy [email protected] Abstract. Church-Turing Thesis, mechanistic project, and G¨odelian Arguments offer different perspectives of informal intuitions behind the relationship existing between the notion of intuitively provable and the definition of decidability by some Turing machine. One of the most for- mal lines of research in this setting is represented by the theory of know- ing machines, based on an extension of Peano Arithmetic, encompassing an epistemic notion of knowledge formalized through a modal operator denoting intuitive provability. -
An Un-Rigorous Introduction to the Incompleteness Theorems∗
An un-rigorous introduction to the incompleteness theorems∗ phil 43904 Jeff Speaks October 8, 2007 1 Soundness and completeness When discussing Russell's logical system and its relation to Peano's axioms of arithmetic, we distinguished between the axioms of Russell's system, and the theorems of that system. Roughly, the axioms of a theory are that theory's basic assumptions, and the theorems are the formulae provable from the axioms using the rules provided by the theory. Suppose we are talking about the theory A of arithmetic. Then, if we express the idea that a certain sentence p is a theorem of A | i.e., provable from A's axioms | as follows: `A p Now we can introduce the notion of the valid sentences of a theory. For our purposes, think of a valid sentence as a sentence in the language of the theory that can't be false. For example, consider a simple logical language which contains some predicates (written as upper-case letters), `not', `&', and some names, each of which is assigned some object or other in each interpretation. Now consider a sentence like F n In most cases, a sentence like this is going to be true on some interpretations, and false in others | it depends on which object is assigned to `n', and whether it is in the set of things assigned to `F '. However, if we consider a sentence like ∗The source for much of what follows is Michael Detlefsen's (much less un-rigorous) “G¨odel'stheorems", available at http://www.rep.routledge.com/article/Y005. -
The Metamathematics of Putnam's Model-Theoretic Arguments
The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. -
Q1: Is Every Tautology a Theorem? Q2: Is Every Theorem a Tautology?
Section 3.2: Soundness Theorem; Consistency. Math 350: Logic Class08 Fri 8-Feb-2001 This section is mainly about two questions: Q1: Is every tautology a theorem? Q2: Is every theorem a tautology? What do these questions mean? Review de¯nitions, and discuss. We'll see that the answer to both questions is: Yes! Soundness Theorem 1. (The Soundness Theorem: Special case) Every theorem of Propositional Logic is a tautol- ogy; i.e., for every formula A, if A, then = A. ` j Sketch of proof: Q: Is every axiom a tautology? Yes. Why? Q: For each of the four rules of inference, check: if the antecedent is a tautology, does it follow that the conclusion is a tautology? Q: How does this prove the theorem? A: Suppose we have a proof, i.e., a list of formulas: A1; ; An. A1 is guaranteed to be a tautology. Why? So A2 is guaranteed to be a tautology. Why? S¢o¢ ¢A3 is guaranteed to be a tautology. Why? And so on. Q: What is a more rigorous way to prove this? Ans: Use induction. Review: What does S A mean? What does S = A mean? Theorem 2. (The Soun`dness Theorem: General cjase) Let S be any set of formulas. Then every theorem of S is a tautological consequence of S; i.e., for every formula A, if S A, then S = A. ` j Proof: This uses similar ideas as the proof of the special case. We skip the proof. Adequacy Theorem 3. (The Adequacy Theorem for the formal system of Propositional Logic: Special Case) Every tautology is a theorem of Propositional Logic; i.e., for every formula A, if = A, then A. -
Pure Mathematics Professors Teaching and Leading Research CO-OP OR REGULAR 28
PURE MATHEMATICS Pure Mathematics professors teaching and leading research CO-OP OR REGULAR 28 Mathematician ranked among top 10 TOP 10 jobs from 2011-2017 – Comcast.com of grads are employed Search for a deeper 96.6% within 2 years understanding of mathematics Pure mathematics is at the foundation of all mathematical reasoning. If first-year calculus ALEX teaches you how to drive the car, Pure Mathematics teaches you how to build one. 3B, PURE MATHEMATICS AND Mathematicians know that there could be no general relativity without differential COMBINATORICS AND geometry, and no computer security without advanced number theory. OPTIMIZATION Pure Mathematics at Waterloo is a small, cohesive, and challenging program that will open countless doors for you. Our graduates have used the program as a springboard into careers WHAT DO YOU LOVE ABOUT in information technology, finance, business, science, education, and insurance, often by way PURE MATHEMATICS? The satisfaction from understanding of some of the most prestigious graduate programs in the world. an idea at a deeper level and tying together unrelated branches of ALEX’S FAVOURITE COURSES mathematics or physics for the › PMATH 320 Euclidean Geometry: This course is everything you love about Geometry: first time is the most rewarding Euclid’s axioms, isometries of the Euclidean plane and of Euclidean space, polygons, part of learning and understanding polyhedral, polytopes, and the kissing problem. mathematics. What I really enjoy is › PMATH 351 Real Analysis: It’s a very intuitive and natural approach to real analysis, and the developing a deep understanding of complexity of the course builds very naturally to the end of the semester. -
Proof of the Soundness Theorem for Sentential Logic
The Soundness Theorem for Sentential Logic These notes contain a proof of the Soundness Theorem for Sentential Logic. The Soundness Theorem says that our natural deduction proofs represent a sound (or correct) system of reasoning. Formally, the Soundness Theorem states that: If Γ├ φ then Γ╞ φ. This says that for any set of premises, if you can prove φ from that set, then φ is truth-functionally entailed by that set. In other words, if our rules let you prove that something follows from some premises, then it ‘really does’ follow from those premises. By definition Γ╞ φ means that there is no TVA that makes all of the members of Γ true and also makes φ false. Note that an equivalent definition would be “Γ╞ φ if any TVA that makes all the members of Γ true also makes φ true”. By contraposition, the soundness theorem is equivalent to “If it is not true that Γ╞ φ then it is not true that Γ├ φ.” Unpacking the definitions, this means “If there is a TVA that makes all the members of Γ true but makes φ false, then there is no way to prove φ from Γ.” We might call this “The Negative Criterion.” This is important since the standard way of showing that something is not provable is to give an invalidating TVA. This only works assuming that the soundness theorem is true. Let’s get started on the proof. We are trying to prove a conditional claim, so we assume its antecedent and try to prove its consequent. -
Pure Mathematics
Why Study Mathematics? Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and social systems. The process of "doing" mathematics is far more than just calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results. As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth. The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Mathematics, as a major intellectual tradition, is a subject appreciated as much for its beauty as for its power. The enduring qualities of such abstract concepts as symmetry, proof, and change have been developed through 3,000 years of intellectual effort. Like language, religion, and music, mathematics is a universal part of human culture. -
Starting the Dismantling of Classical Mathematics
Australasian Journal of Logic Starting the Dismantling of Classical Mathematics Ross T. Brady La Trobe University Melbourne, Australia [email protected] Dedicated to Richard Routley/Sylvan, on the occasion of the 20th anniversary of his untimely death. 1 Introduction Richard Sylvan (n´eRoutley) has been the greatest influence on my career in logic. We met at the University of New England in 1966, when I was a Master's student and he was one of my lecturers in the M.A. course in Formal Logic. He was an inspirational leader, who thought his own thoughts and was not afraid to speak his mind. I hold him in the highest regard. He was very critical of the standard Anglo-American way of doing logic, the so-called classical logic, which can be seen in everything he wrote. One of his many critical comments was: “G¨odel's(First) Theorem would not be provable using a decent logic". This contribution, written to honour him and his works, will examine this point among some others. Hilbert referred to non-constructive set theory based on classical logic as \Cantor's paradise". In this historical setting, the constructive logic and mathematics concerned was that of intuitionism. (The Preface of Mendelson [2010] refers to this.) We wish to start the process of dismantling this classi- cal paradise, and more generally classical mathematics. Our starting point will be the various diagonal-style arguments, where we examine whether the Law of Excluded Middle (LEM) is implicitly used in carrying them out. This will include the proof of G¨odel'sFirst Theorem, and also the proof of the undecidability of Turing's Halting Problem. -
A Quasi-Classical Logic for Classical Mathematics
Theses Honors College 11-2014 A Quasi-Classical Logic for Classical Mathematics Henry Nikogosyan University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/honors_theses Part of the Logic and Foundations Commons Repository Citation Nikogosyan, Henry, "A Quasi-Classical Logic for Classical Mathematics" (2014). Theses. 21. https://digitalscholarship.unlv.edu/honors_theses/21 This Honors Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Honors Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. This Honors Thesis has been accepted for inclusion in Theses by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. A QUASI-CLASSICAL LOGIC FOR CLASSICAL MATHEMATICS By Henry Nikogosyan Honors Thesis submitted in partial fulfillment for the designation of Departmental Honors Department of Philosophy Ian Dove James Woodbridge Marta Meana College of Liberal Arts University of Nevada, Las Vegas November, 2014 ABSTRACT Classical mathematics is a form of mathematics that has a large range of application; however, its application has boundaries. In this paper, I show that Sperber and Wilson’s concept of relevance can demarcate classical mathematics’ range of applicability by demarcating classical logic’s range of applicability. -
Mathematical Languages Shape Our Understanding of Time in Physics Physics Is Formulated in Terms of Timeless, Axiomatic Mathematics
comment Corrected: Publisher Correction Mathematical languages shape our understanding of time in physics Physics is formulated in terms of timeless, axiomatic mathematics. A formulation on the basis of intuitionist mathematics, built on time-evolving processes, would ofer a perspective that is closer to our experience of physical reality. Nicolas Gisin n 1922 Albert Einstein, the physicist, met in Paris Henri Bergson, the philosopher. IThe two giants debated publicly about time and Einstein concluded with his famous statement: “There is no such thing as the time of the philosopher”. Around the same time, and equally dramatically, mathematicians were debating how to describe the continuum (Fig. 1). The famous German mathematician David Hilbert was promoting formalized mathematics, in which every real number with its infinite series of digits is a completed individual object. On the other side the Dutch mathematician, Luitzen Egbertus Jan Brouwer, was defending the view that each point on the line should be represented as a never-ending process that develops in time, a view known as intuitionistic mathematics (Box 1). Although Brouwer was backed-up by a few well-known figures, like Hermann Weyl 1 and Kurt Gödel2, Hilbert and his supporters clearly won that second debate. Hence, time was expulsed from mathematics and mathematical objects Fig. 1 | Debating mathematicians. David Hilbert (left), supporter of axiomatic mathematics. L. E. J. came to be seen as existing in some Brouwer (right), proposer of intuitionist mathematics. Credit: Left: INTERFOTO / Alamy Stock Photo; idealized Platonistic world. right: reprinted with permission from ref. 18, Springer These two debates had a huge impact on physics. -
Church's Thesis As an Empirical Hypothesis
Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 27/02/2019 11:52:41 Annales UMCS Annales UMCS Informatica AI 3 (2005) 119-130 Informatica Lublin-Polonia Sectio AI http://www.annales.umcs.lublin.pl/ Church’s thesis as an empirical hypothesis Adam Olszewski Pontifical Academy of Theology, Kanonicza 25, 31-002 Kraków, Poland The studies of Church’s thesis (further denoted as CT) are slowly gaining their intensity. Since different formulations of the thesis are being proposed, it is necessary to state how it should be understood. The following is the formulation given by Church: [CT] The concept of the effectively calculable function1 is identical with the concept of the recursive function. The conviction that the above formulation is correct, finds its confirmation in Church’s own words:2 We now define the notion, […] of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers […]. Frege thought that the material equivalence of concepts is the approximation of their identity. For any concepts F, G: Concept FUMCS is identical with concept G iff ∀x (Fx ≡ Gx) 3. The symbol Fx should be read as “object x falls under the concept F”. The contemporary understanding of the right side of the above equation was dominated by the first order logic and set theory. 1Unless stated otherwise, the term function in this work always means function defined on the set natural numbers, that is, on natural numbers including zero. I would rather use the term concept than the term notion.