A Friendly Introduction to Mathematical Logic Christopher C
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The Mathematical Structure of the Aesthetic
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 June 2017 doi:10.20944/preprints201706.0055.v1 Peer-reviewed version available at Philosophies 2017, 2, 14; doi:10.3390/philosophies2030014 Article A New Kind of Aesthetics – The Mathematical Structure of the Aesthetic Akihiro Kubota 1,*, Hirokazu Hori 2, Makoto Naruse 3 and Fuminori Akiba 4 1 Art and Media Course, Department of Information Design, Tama Art University, 2-1723 Yarimizu, Hachioji, Tokyo 192-0394, Japan; [email protected] 2 Interdisciplinary Graduate School, University of Yamanashi, 4-3-11 Takeda, Kofu,Yamanashi 400-8511, Japan; [email protected] 3 Network System Research Institute, National Institute of Information and Communications Technology, 4-2-1 Nukui-kita, Koganei, Tokyo 184-8795, Japan; [email protected] 4 Philosophy of Information Group, Department of Systems and Social Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan; [email protected] * Correspondence: [email protected]; Tel.: +81-42-679-5634 Abstract: This paper proposes a new approach to investigation into the aesthetics. Specifically, it argues that it is possible to explain the aesthetic and its underlying dynamic relations with axiomatic structure (the octahedral axiom derived category) based on contemporary mathematics – namely, category theory – and through this argument suggests the possibility for discussion about the mathematical structure of the aesthetic. If there was a way to describe the structure of aesthetics with the language of mathematical structures and mathematical axioms – a language completely devoid of arbitrariness – then we would make possible a synthetical argument about the essential human activity of “the aesthetics”, and we would also gain a new method and viewpoint on the philosophy and meaning of the act of creating a work of art and artistic activities. -
“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
A Proof of Cantor's Theorem
Cantor’s Theorem Joe Roussos 1 Preliminary ideas Two sets have the same number of elements (are equinumerous, or have the same cardinality) iff there is a bijection between the two sets. Mappings: A mapping, or function, is a rule that associates elements of one set with elements of another set. We write this f : X ! Y , f is called the function/mapping, the set X is called the domain, and Y is called the codomain. We specify what the rule is by writing f(x) = y or f : x 7! y. e.g. X = f1; 2; 3g;Y = f2; 4; 6g, the map f(x) = 2x associates each element x 2 X with the element in Y that is double it. A bijection is a mapping that is injective and surjective.1 • Injective (one-to-one): A function is injective if it takes each element of the do- main onto at most one element of the codomain. It never maps more than one element in the domain onto the same element in the codomain. Formally, if f is a function between set X and set Y , then f is injective iff 8a; b 2 X; f(a) = f(b) ! a = b • Surjective (onto): A function is surjective if it maps something onto every element of the codomain. It can map more than one thing onto the same element in the codomain, but it needs to hit everything in the codomain. Formally, if f is a function between set X and set Y , then f is surjective iff 8y 2 Y; 9x 2 X; f(x) = y Figure 1: Injective map. -
A Compositional Analysis for Subset Comparatives∗ Helena APARICIO TERRASA–University of Chicago
A Compositional Analysis for Subset Comparatives∗ Helena APARICIO TERRASA–University of Chicago Abstract. Subset comparatives (Grant 2013) are amount comparatives in which there exists a set membership relation between the target and the standard of comparison. This paper argues that subset comparatives should be treated as regular phrasal comparatives with an added presupposi- tional component. More specifically, subset comparatives presuppose that: a) the standard has the property denoted by the target; and b) the standard has the property denoted by the matrix predi- cate. In the account developed below, the presuppositions of subset comparatives result from the compositional principles independently required to interpret those phrasal comparatives in which the standard is syntactically contained inside the target. Presuppositions are usually taken to be li- censed by certain lexical items (presupposition triggers). However, subset comparatives show that presuppositions can also arise as a result of semantic composition. This finding suggests that the grammar possesses more than one way of licensing these inferences. Further research will have to determine how productive this latter strategy is in natural languages. Keywords: Subset comparatives, presuppositions, amount comparatives, degrees. 1. Introduction Amount comparatives are usually discussed with respect to their degree or amount interpretation. This reading is exemplified in the comparative in (1), where the elements being compared are the cardinalities corresponding to the sets of books read by John and Mary respectively: (1) John read more books than Mary. |{x : books(x) ∧ John read x}| ≻ |{y : books(y) ∧ Mary read y}| In this paper, I discuss subset comparatives (Grant (to appear); Grant (2013)), a much less studied type of amount comparative illustrated in the Spanish1 example in (2):2 (2) Juan ha leído más libros que El Quijote. -
Structure” of Physics: a Case Study∗ (Journal of Philosophy 106 (2009): 57–88)
The “Structure” of Physics: A Case Study∗ (Journal of Philosophy 106 (2009): 57–88) Jill North We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says that the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understand- ing what physics is telling us about the world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects is not easy. But there seems to be no further problem for understanding mathematical structure. ∗For comments and discussion, I am extremely grateful to David Albert, Frank Arntzenius, Gordon Belot, Josh Brown, Adam Elga, Branden Fitelson, Peter Forrest, Hans Halvorson, Oliver Davis Johns, James Ladyman, David Malament, Oliver Pooley, Brad Skow, TedSider, Rich Thomason, Jason Turner, Dmitri Tymoczko, the philosophy faculty at Yale, audience members at The University of Michigan in fall 2006, and in 2007 at the Paci c APA, the Joint Session of the Aristotelian Society and Mind Association, and the Bellingham Summer Philosophy Conference. -
An Update on the Four-Color Theorem Robin Thomas
thomas.qxp 6/11/98 4:10 PM Page 848 An Update on the Four-Color Theorem Robin Thomas very planar map of connected countries the five-color theorem (Theorem 2 below) and can be colored using four colors in such discovered what became known as Kempe chains, a way that countries with a common and Tait found an equivalent formulation of the boundary segment (not just a point) re- Four-Color Theorem in terms of edge 3-coloring, ceive different colors. It is amazing that stated here as Theorem 3. Esuch a simply stated result resisted proof for one The next major contribution came in 1913 from and a quarter centuries, and even today it is not G. D. Birkhoff, whose work allowed Franklin to yet fully understood. In this article I concentrate prove in 1922 that the four-color conjecture is on recent developments: equivalent formulations, true for maps with at most twenty-five regions. The a new proof, and progress on some generalizations. same method was used by other mathematicians to make progress on the four-color problem. Im- Brief History portant here is the work by Heesch, who developed The Four-Color Problem dates back to 1852 when the two main ingredients needed for the ultimate Francis Guthrie, while trying to color the map of proof—“reducibility” and “discharging”. While the the counties of England, noticed that four colors concept of reducibility was studied by other re- sufficed. He asked his brother Frederick if it was searchers as well, the idea of discharging, crucial true that any map can be colored using four col- for the unavoidability part of the proof, is due to ors in such a way that adjacent regions (i.e., those Heesch, and he also conjectured that a suitable de- sharing a common boundary segment, not just a velopment of this method would solve the Four- point) receive different colors. -
Number Theory
“mcs-ftl” — 2010/9/8 — 0:40 — page 81 — #87 4 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . Which one don’t you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in war—which may leave poor Hardy spinning in his grave. It’s also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3. Since we’ll be focusing on properties of the integers, we’ll adopt the default convention in this chapter that variables range over the set of integers, Z. 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak b for some k: D The notation, a b, is an abbreviation for “a divides b.” If a b, then we also j j say that b is a multiple of a. -
UNIVERSALLY BAIRE SETS and GENERIC ABSOLUTENESS TREVOR M. WILSON Introduction Generic Absoluteness Principles Assert That Certai
UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS TREVOR M. WILSON Abstract. We prove several equivalences and relative consistency re- 2 uBλ sults involving notions of generic absoluteness beyond Woodin's (Σ1) generic absoluteness for a limit of Woodin cardinals λ. In particular,e we R 2 uBλ prove that two-step 9 (Π1) generic absoluteness below a measur- able cardinal that is a limite of Woodin cardinals has high consistency 2 uBλ strength, and that it is equivalent with the existence of trees for (Π1) formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many \failures of covering" for the models L(T;Vα) below a measurable car- dinal. Introduction Generic absoluteness principles assert that certain properties of the set- theoretic universe cannot be changed by the method of forcing. Some pro- perties, such as the truth or falsity of the Continuum Hypothesis, can always be changed by forcing. Accordingly, one approach to formulating generic ab- soluteness principles is to consider properties of a limited complexity such 1 1 as those corresponding to pointclasses in descriptive set theory: Σ2, Σ3, projective, and so on. (Another approach is to limit the class ofe allowede forcing notions. For a survey of results in this area, see [1].) Shoenfield’s 1 absoluteness theorem implies that Σ2 statements are always generically ab- solute. Generic absoluteness principlese for larger pointclasses tend to be equiconsistent with strong axioms of infinity, and they may also relate to the extent of the universally Baire sets. 1 For example, one-step Σ3 generic absoluteness is shown in [3] to be equiconsistent with the existencee of a Σ2-reflecting cardinal and to be equiv- 1 alent with the statement that every ∆2 set of reals is universally Baire. -
Against Logical Form
Against logical form Zolta´n Gendler Szabo´ Conceptions of logical form are stranded between extremes. On one side are those who think the logical form of a sentence has little to do with logic; on the other, those who think it has little to do with the sentence. Most of us would prefer a conception that strikes a balance: logical form that is an objective feature of a sentence and captures its logical character. I will argue that we cannot get what we want. What are these extreme conceptions? In linguistics, logical form is typically con- ceived of as a level of representation where ambiguities have been resolved. According to one highly developed view—Chomsky’s minimalism—logical form is one of the outputs of the derivation of a sentence. The derivation begins with a set of lexical items and after initial mergers it splits into two: on one branch phonological operations are applied without semantic effect; on the other are semantic operations without phono- logical realization. At the end of the first branch is phonological form, the input to the articulatory–perceptual system; and at the end of the second is logical form, the input to the conceptual–intentional system.1 Thus conceived, logical form encompasses all and only information required for interpretation. But semantic and logical information do not fully overlap. The connectives “and” and “but” are surely not synonyms, but the difference in meaning probably does not concern logic. On the other hand, it is of utmost logical importance whether “finitely many” or “equinumerous” are logical constants even though it is hard to see how this information could be essential for their interpretation. -
Truth-Bearers and Truth Value*
Truth-Bearers and Truth Value* I. Introduction The purpose of this document is to explain the following concepts and the relationships between them: statements, propositions, and truth value. In what follows each of these will be discussed in turn. II. Language and Truth-Bearers A. Statements 1. Introduction For present purposes, we will define the term “statement” as follows. Statement: A meaningful declarative sentence.1 It is useful to make sure that the definition of “statement” is clearly understood. 2. Sentences in General To begin with, a statement is a kind of sentence. Obviously, not every string of words is a sentence. Consider: “John store.” Here we have two nouns with a period after them—there is no verb. Grammatically, this is not a sentence—it is just a collection of words with a dot after them. Consider: “If I went to the store.” This isn’t a sentence either. “I went to the store.” is a sentence. However, using the word “if” transforms this string of words into a mere clause that requires another clause to complete it. For example, the following is a sentence: “If I went to the store, I would buy milk.” This issue is not merely one of conforming to arbitrary rules. Remember, a grammatically correct sentence expresses a complete thought.2 The construction “If I went to the store.” does not do this. One wants to By Dr. Robert Tierney. This document is being used by Dr. Tierney for teaching purposes and is not intended for use or publication in any other manner. 1 More precisely, a statement is a meaningful declarative sentence-type. -
Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
Zero-One Laws
1 THE LOGIC IN COMPUTER SCIENCE COLUMN by 2 Yuri GUREVICH Electrical Engineering and Computer Science UniversityofMichigan, Ann Arb or, MI 48109-2122, USA [email protected] Zero-One Laws Quisani: I heard you talking on nite mo del theory the other day.Itisinteresting indeed that all those famous theorems ab out rst-order logic fail in the case when only nite structures are allowed. I can understand that for those, likeyou, educated in the tradition of mathematical logic it seems very imp ortantto ndoutwhichof the classical theorems can b e rescued. But nite structures are to o imp ortant all by themselves. There's got to b e deep nite mo del theory that has nothing to do with in nite structures. Author: \Nothing to do" sounds a little extremist to me. Sometimes in nite ob jects are go o d approximations of nite ob jects. Q: I do not understand this. Usually, nite ob jects approximate in nite ones. A: It may happ en that the in nite case is cleaner and easier to deal with. For example, a long nite sum may b e replaced with a simpler integral. Returning to your question, there is indeed meaningful, inherently nite, mo del theory. One exciting issue is zero- one laws. Consider, say, undirected graphs and let be a propertyofsuch graphs. For example, may b e connectivity. What fraction of n-vertex graphs have the prop erty ? It turns out that, for many natural prop erties , this fraction converges to 0 or 1asn grows to in nity. If the fraction converges to 1, the prop erty is called almost sure.