CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinResources1 Barwise-Etchmendy and Tarski’s World2 Geometry books3 articles in education, philosophy and mathematics mostlyon webCemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinUSA Today – Gallup PollA. Evolution, that is, the idea that human beings developedover millions of years from less advanced forms of life isDefinitely Probably Probably Definitelytrue true false false18 35 16 28B. Creationism, that is, the idea that God created humanbeings pretty much in their present form at one time within thelast 10,000 years isDefinitely Probably Probably Definitelytrue true false false39 27 16 15CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinHow do you tell a sentence is true?Compositional theory of truthThe truth of sentence φ is defined recursively from1 the syntactic rule constructing φ from its components2 the truth value of those components .CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinVertical AnglesGlencoe DefinitionsAn angle is formed by two noncollinear rays that have acommon endpoint. The rays are called the sides of the angle.A linear pair is a pair of adjacent angles whose noncommonsides are opposite rays.Vertical angles are two nonadjacent angles formed by twointersecting lines.CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinVertical AnglesGlencoe DefinitionsAn angle is formed by two noncollinear rays that have acommon endpoint. The rays are called the sides of the angle.A linear pair is a pair of adjacent angles whose noncommonsides are opposite rays.Vertical angles are two nonadjacent angles formed by twointersecting lines.CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinDefinition/theorem Activity1 Consider the definitions on the next slide2 Then using the summary hand-out of Theorems andpostulates from Glencoe, try to prove that ‘vertical anglesare equal’.Glencoe sketches proof of Theorem 2.6 but Theorem 2.3 is anexercise with no hints.CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinWhat is an angleDefinition (Hilbert)Let K be any arbitrary plane and h, k any two distinct half-rayslying in K and emanating from the point O so as to form a partof two different straight lines. We call the system formed bythese two half-rays h, k an angle and represent it by the sym bolh (h, k) or h (k, h).The reason to avoid straight angles is so that each angle willhave a well defined ‘interior’.CemelaSummerSchoolMathematicsas languageFact orMetaphor?John T.BaldwinWrap-up1 What in today’s class was the most useful to you?2 What didn’t work?3 questions or