Math Review1MATH REVIEWLa filosofia è scritta in questo grandissimo libro che continuamente ci sta apertoinnanzi a gli occhi (io dico l'universo), ma non si può intendere se prima non s'impara aintender la lingua, e conoscer i caratteri, ne' quali è scritto. Egli è scritto in linguamatematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i qualimezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsivanamente per un oscuro laberinto.Philosophy is written in this great book that is continually open before our eyes (Imean the universe), but it cannot be understood unless one first learns the language andthe characters in which it is written. The book of Nature is written in the language ofmathematics, and its characters are triangles, circles and other geometric figures; withoutthese tools it is impossible to understand a single word; without them we wander vainlythrough a dark labyrinth.—Galileo Galilei,Il Saggiatore(1623)Galileo Galilei (1564-1642) is rightly regarded as one of the founders, not only of modernphysics, but also of the modern scientific method. He was by no means the first to performexperiments, and he was certainly not the first to theorize; but he did pioneer the idea ofperforming precisequantitativeexperiments under carefully controlled conditions, and thenusing those experiments as the basis for guessing the underlying regularities (“laws ofNature”), which would be expressed inmathematicalform and subjected to furtherexperimental tests.Such a method is by no means guaranteed to work. Why, after all, should the natural worldbe describable in the language ofmathematics(rather than, say, psychology or theology)?The fact that Galileo, Newton, Maxwell, Einstein and their successors were able to produceincreasingly precise descriptions of increasingly wide aspects of the natural world –descriptions that went beyond the experiments originally performed, to correctly predictpreviously unobservedphenomena – says something highly non-trivial about the universe inwhich we live. It points to what twentieth- century physicist Eugene Wigner has called “theunreasonable effectiveness of mathematics in the natural sciences”.11E.P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences,Communications in Pureand Applied Mathematics13(1960), 1‐14.

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Math Review2Mathematics has, of course, moved on since the time of Galileo and Newton. Where theyused complicated geometric arguments to express their reasoning, we can now formulate thesame ideas much more simply usingalgebra.2Indeed, the main tool of this course – besidescareful thinking about concepts – will be high-school algebra (plus a bit of elementarygeometry, notably the Pythagorean Theorem). This first lab is devoted to reviewing whatwe’ll need.

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