Programming by Demonstration Using Version Space Algebra
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UNDERSTANDING MATHEMATICS to UNDERSTAND PLATO -THEAETEUS (147D-148B Salomon Ofman
UNDERSTANDING MATHEMATICS TO UNDERSTAND PLATO -THEAETEUS (147d-148b Salomon Ofman To cite this version: Salomon Ofman. UNDERSTANDING MATHEMATICS TO UNDERSTAND PLATO -THEAETEUS (147d-148b. Lato Sensu, revue de la Société de philosophie des sciences, Société de philosophie des sciences, 2014, 1 (1). hal-01305361 HAL Id: hal-01305361 https://hal.archives-ouvertes.fr/hal-01305361 Submitted on 20 Apr 2016 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. UNDERSTANDING MATHEMATICS TO UNDERSTAND PLATO - THEAETEUS (147d-148b) COMPRENDRE LES MATHÉMATIQUES POUR COMPRENDRE PLATON - THÉÉTÈTE (147d-148b) Salomon OFMAN Institut mathématique de Jussieu-PRG Histoire des Sciences mathématiques 4 Place Jussieu 75005 Paris [email protected] Abstract. This paper is an updated translation of an article published in French in the Journal Lato Sensu (I, 2014, p. 70-80). We study here the so- called ‘Mathematical part’ of Plato’s Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. -
THE DIMENSION of a VECTOR SPACE 1. Introduction This Handout
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD 1. Introduction This handout is a supplementary discussion leading up to the definition of dimension of a vector space and some of its properties. We start by defining the span of a finite set of vectors and linear independence of a finite set of vectors, which are combined to define the all-important concept of a basis. Definition 1.1. Let V be a vector space over a field F . For any finite subset fv1; : : : ; vng of V , its span is the set of all of its linear combinations: Span(v1; : : : ; vn) = fc1v1 + ··· + cnvn : ci 2 F g: Example 1.2. In F 3, Span((1; 0; 0); (0; 1; 0)) is the xy-plane in F 3. Example 1.3. If v is a single vector in V then Span(v) = fcv : c 2 F g = F v is the set of scalar multiples of v, which for nonzero v should be thought of geometrically as a line (through the origin, since it includes 0 · v = 0). Since sums of linear combinations are linear combinations and the scalar multiple of a linear combination is a linear combination, Span(v1; : : : ; vn) is a subspace of V . It may not be all of V , of course. Definition 1.4. If fv1; : : : ; vng satisfies Span(fv1; : : : ; vng) = V , that is, if every vector in V is a linear combination from fv1; : : : ; vng, then we say this set spans V or it is a spanning set for V . Example 1.5. In F 2, the set f(1; 0); (0; 1); (1; 1)g is a spanning set of F 2. -
Mapping the Meaning of Knowledge in Design Research Kristina Niedderer University of Hertfordshire
V.2:2 April 2007 www.designresearchsociety.org Design Research Society ISSN 1752-8445 Mapping the Meaning of Knowledge in Design Research Kristina Niedderer University of Hertfordshire Knowledge plays a vital role in our life This question has arisen for design in Table of Contents: in that it reflects how we understand the UK, as well as more generally for the world around us and thus deter- creative and practice-led disciplines Articles: 1 Mapping the Meaning of Knowledge mines how we act upon it. In this sense, (CPDs), because research regulations in Design Research knowledge is of particular importance and requirements in the UK remain Kristina Niedderer for designers because they act to shape silent about what knowledge and our world. Conventionally, knowledge understanding mean in the context of 3 Call for Papers: Design Research creation has been assumed by (design) their specifications while implicitly pri- Quarterly research. However developments of oritising propositional knowledge over Case Studies in Research: Knowledge using practice within research have knowledge that cannot be expressed in and Inquiry pointed to knowledge creation within that form (Niedderer 2007). and through practice. This has raised This has led to a number of prob- Listings: the question of the meaning, role and lems concerning the role and format 14 Current Research in Design: format of knowledge in both research of knowledge in research and practice ToCs from Leading Design Journals and practice, and about the compatibil- in the UK. For example, because of the ity between knowledge of research and language-based mode of proposition- 21 Upcoming Events Worldwide practice. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Marko Malink (NYU)
Demonstration by reductio ad impossibile in Posterior Analytics 1.26 Marko Malink (New York University) Contents 1. Aristotle’s thesis in Posterior Analytics 1. 26 ................................................................................. 1 2. Direct negative demonstration vs. demonstration by reductio ................................................. 14 3. Aristotle’s argument from priority in nature .............................................................................. 25 4. Priority in nature for a-propositions ............................................................................................ 38 5. Priority in nature for e-, i-, and o-propositions .......................................................................... 55 6. Accounting for Aristotle’s thesis in Posterior Analytics 1. 26 ................................................... 65 7. Parts and wholes ............................................................................................................................. 77 References ............................................................................................................................................ 89 1. Aristotle’s thesis in Posterior Analytics 1. 26 At the beginning of the Analytics, Aristotle states that the subject of the treatise is demonstration (ἀπόδειξις). A demonstration, for Aristotle, is a kind of deductive argument. It is a deduction through which, when we possess it, we have scientific knowledge (ἐπιστήμη). While the Prior Analytics deals with deduction in -
Phenomenal Concepts and the Problem of Acquaintance
Paul Livingston Phenomenal Concepts and the Problem of Acquaintance Abstract: Some contemporary discussion about the explanation of consciousness substantially recapitulates a decisive debate about ref- erence, knowledge, and justification from an earlier stage of the ana- lytic tradition. In particular, I argue that proponents of a recently popular strategy for accounting for an explanatory gap between phys- ical and phenomenal facts — the so-called ‘phenomenal concept strategy’ — face a problem that was originally fiercely debated by Schlick, Carnap, and Neurath. The question that is common to both the older and the contemporary discussion is that of how the presence or presentation of phenomenal experiences can play a role in justify- ing beliefs or judgments about them. This problem is, moreover, the same as what was classically discussed as the problem of acquain- tance. Interestingly, both physicalist and non-physicalist proponents of the phenomenal concept strategy today face this problem. I con- sider briefly some recent attempts to solve it and conclude that, although it is prima facie very plausible that acquaintance exists, we have, as yet, no good account of it. My aim in this paper is to demonstrate that proponents of a recently popular physicalist strategy for explaining consciousness — what has been called the phenomenal concept strategy — share with certain anti-physicalist phenomenal realists a common problem, which was also a problem for Schlick, Carnap, and Neurath in the days of the Vienna Circle. Though it has implications for ontology and metaphys- ics, this problem is essentially one about justification: how can judg- ments, claims, or beliefs about phenomenal experience be justified, at Correspondence: Email: [email protected] Journal of Consciousness Studies, 20, No. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Types of Science Fair Projects: the Good and the Bad Demonstrations, Experiments, Engineering Projects, and Computer Science Projects Science Fair Projects
Types of Science Fair Projects: The Good and the Bad Demonstrations, experiments, engineering projects, and computer science projects Science Fair Projects Experiments Math projects Engineering Computer Science Demonstrations What is a demonstration? • Demonstration projects are not permitted. • A demonstration shows how something works. • An experiment involves an independent and dependent variable. Demo → Experiment • The difference between a demonstration and an experiment is the manipulation of variables. • To change a demonstration to an experiment, modify the project to include an independent and a dependent variable. • Examples: Volcano, Motor Science Fair Projects Experiments Math projects Engineering Computer Science The Process is the Key • Science, engineering, and mathematics each have their own process for coming to new knowledge. • No matter what kind of project you are doing you must follow the process appropriate to your discipline. Image from http://sciencebuddies.org/science-fair-projects/project_scientific_method.shtml Computer Programming Math Projects Engineering Scientific Mathematical Process Method Reasoning/Proof Define a need State your question Define what is known Do background research Do background research Research & define all terminology Establish design criteria Formulate your hypothesis, Make a conjecture/assumption identify variables based on what you know Prepare preliminary designs Design experiment, establish Perform calculations procedure Build & test prototype Test your hypothesis by doing Look for counter examples an experiment Test & redesign as necessary Analyze your results and draw Recalculate and write up steps conclusions to the conclusion Present results Present results Present Results Scientific Method & Engineering Process Comparison used with permission from Science Buddies. Computer Science Projects • Computer science projects are a special type of engineering projects and therefore follow the engineering design process. -
Space-Based Space Surveillance Operational and Demonstration Missions
SPACE-BASED SPACE SURVEILLANCE OPERATIONAL AND DEMONSTRATION MISSIONS Diego Escorial Olmos(1), Fernando E. Alemán Roda(2), Kevin Middleton(3), Joris.Naudet(4) (1)GMV, Isaac Newton 11, Tres Cantos 28760, Madrid, Spain, Email: [email protected] (2)GMV, Isaac Newton 11, Tres Cantos 28760, Madrid, Spain, Email: [email protected] (3) RAL Space, STFC Rutherford Appleton Laboratory, Harwell, Oxford Didcot, Oxon, OX11 0QX, United Kingdom, Email: [email protected] (4)QinetiQ Space, Hogenakkerhoekstraat 9, 9150 Kruibeke, Belgium, Email: [email protected] ABSTRACT objects. In order to include a detected object in the catalogue, the accuracy envelope shall be better than GMV is currently leading, under ESA contract, an 2.5km. These requirements shall be achieved by the assessment study to define a demonstration mission for SBSS operational system in conjunction with the space-based space surveillance. The project team ground based segment of the SST (Space Surveillance includes QinetiQ Space as responsible for the platform and Tracking) system that would be also available in the and RAL Space for the payload activities. future. During the first phase of the study a high-level In terms of reactiveness, the system shall be able to definition of a future operational mission has been fulfil a tracking request for any GEO “catalogued” or carried out including the definition of user requirements “pre-catalogued” object before 48 hours after the for a future Space Based Space Surveillance (SBSS) request is issued. This requires that the space segment service. shall be able to be re-planned or re-programmed in less During the second phase of the study a precursor than 24 hours to accommodate the active tracking mission to demonstrate the SBSS operational needs has request. -
ARISTOTLE's FOUNDATIONALISM Breno Andrade Zuppolini Aristotle
ARISTOTLE’S FOUNDATIONALISM Breno Andrade Zuppolini UNICAMP Resumo: Para Aristóteles, conhecimento demonstrativo é o resultado do que ele denomina 'aprendizado intelectual', processo em que o conhecimento da conclusão depende de um conhecimento prévio das premissas. Dado que demonstrações são, em última instância, baseadas em princípios indemonstráveis (cujo conhecimento é denominado 'νοῦς'), Aristóteles é frequentemente retratado como promovendo uma doutrina fundacionista. Sem contestar a nomenclatura, tentarei mostrar que o 'fundacionismo' de Aristóteles não deve ser entendido como uma teoria racionalista da justificação epistêmica, como se os primeiros princípios da ciência pudessem ser conhecidos enquanto tais independentemente de suas conexões explanatórias com proposições demonstráveis. Argumentarei que conhecer os primeiros princípios enquanto tais envolve conhecê-los como explicações de outras proposições científicas. Explicarei, então, de que modo conhecimento noético e conhecimento demonstrativo são, em certo sentido, estados cognitivos interdependentes – ainda que o conceito de νοῦς se mantenha distinto de (e, nas palavras de Aristóteles, mais 'preciso' do que) conhecimento demonstrativo. Palavras-chave: Aristóteles; fundacionismo; ciência; demonstração. Abstract: For Aristotle, demonstrative knowledge is the result of what he calls ‘intellectual learning’, a process in which the knowledge of a conclusion depends on previous knowledge of the premises. Since demonstrations are ultimately based on indemonstrable principles (the knowledge of which is called ‘νοῦς’), Aristotle is often described as advancing a foundationalist doctrine. Without disputing the nomenclature, I shall attempt to show that Aristotle’s ‘foundationalism’ should not be taken as a rationalist theory of epistemic justification, as if the first principles of science could be known as such independently of their explanatory connections to demonstrable propositions. -
Inner Product Spaces
CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn. -
The Effectiveness of Teachers' Use of Demonstrations for Enhancing Students' Understanding of and Attitudes to Learning the Oxidation-Reduction Concept
OPEN ACCESS EURASIA Journal of Mathematics Science and Technology Education ISSN: 1305-8223 (online) 1305-8215 (print) 2017 13(3):555-570 DOI 10.12973/eurasia.2017.00632a The Effectiveness of Teachers' Use of Demonstrations for Enhancing Students' Understanding of and Attitudes to Learning the Oxidation-Reduction Concept Ahmad Basheer The Academic Arab College for Education in Israel - Haifa Muhamad Hugerat The Academic Arab College for Education in Israel - Haifa Naji Kortam The Academic Arab College for Education in Israel - Haifa Avi Hofstein The Academic Arab College for Education in Israel – Haifa & Weizmann Institute of science, Israel Received 02 February 2016 ▪ Revised 14 July 2016 ▪ Accepted 26 July 2016 ABSTRACT In this study we explored whether the use of teachers' demonstrations significantly improves students’ understanding of redox reactions compared with control group counterparts who were not exposed to the demonstrations. The sample consisted of 131 Israeli 8th graders in middle schools (junior high school). Students' attitudes and achievements as well as their understanding of redox and electrolysis were assessed by administering a questionnaire that investigated their attitudes (perceptions) towards a demonstration in chemistry. The findings showed that the experimental group's achievements and understanding of the subject were statistically significantly better than those of their control group counterparts. Keywords: chemistry demonstrations, redox reactions, electrolysis, achievement, attitudes, electrochemical row, chemistry education INTRODUCTION Rationale for the Study The concept of oxidation-reduction is one of the key concepts taught and learned in chemistry in the middle school (junior high schools) in most countries around the world (GCE Ordinary Level). Butts and Smith (1987), for example, reported that high-school students found the concepts of electrolysis relatively difficult to understand.