03. Early Greeks & Aristotle

Transcription

1 03. Erly Greeks & Aristotle I. Erly Greeks Topis I. Erly Greeks II. The Method of Exhustion III. Aristotle. Anximnder (. 60 B.C.) to peiron - the unlimited, unounded - fundmentl sustne of relity - underlying sustrtum for hnge - neutrl sustrtum in whih opposites/strife re reoniled Solution to the Prolem of the One nd the Mny: Oservle ojets = omposites of the four elements: erth, ir, fire, wter. Question: How do suh opposing elements omine to form ojets? Answer: Through the medition of to peiron 2. The Pythgorens (Pythgors. 570 B.C.) the physil world = produt of the imposition of pers (limits) on peiron result = order/hrmony sis for this order = nturl numers Pythgors' Theorem Clim: Proof: In ny right tringle, the squre of the hypotenuse is equl to the sum of the squres on the other two sides. Consider two squres, one insried inside the other. re of outer squre = ( + ) 2 = (re of Δ) = (/2) = Or: = = 2

2 Irrtionlity of 2 Clim: The squre root of 2 is not rtionl numer. Proof: Suppose 2 is rtionl numer. Then: There re integers p, q suh tht 2 = p/q. Or: 2 = p 2 /q 2. Suppose: p/q is in lowest terms. Now: 2q 2 = p 2. Hene: p 2 must e even. Hene p must e even. q must e odd. p = 2r, for some integer r. 2q 2 = (2r) 2 = 4r 2. Or: q 2 = 2r 2 But: q 2 is even. This mens q must e even! If q were even, then sine p is even, p/q would not e in lowest terms! 2 Pythgors' Theorem for length of digonl of unit squre: ( ) = 2 3. The Eletis Prmenides of Ele (55 B.C.) Clim: It is meningless to spek of wht is not. Everything is. The One - the metphysilly infinite - indivisile, homogeneous, eternl Further lim: Chnge is n illusion. (Chnge is trnsition from wht is, to wht is not. This is impossile, sine tlk of wht is not is inoherent.) Zeno (490 B.C.) Prdoxes of motion: - Ahilles nd the Tortoise - Prdox of the runner: intended to demonstrte tht motion is not rel A E D C B Clim: Ahilles will never reh the finish-line t B. Proof: () To reh B, must reh C = AB/2. (2) To reh C, must reh D = AC/2, et... (3) Thus there re n infinite numer of finite line segments etween A nd B. (4) So Ahilles would need n infinite mount of time to trverse them ll! Assumptions: () AB is infinitely divisile. () The sum of n infinite numer of finite lengths is infinite. 2

3 4. Plto ( B. C.) peiron - indeterminy/disorder/ hos pers - limits/order 3 ples where notions of the infinite pper in Plto:. Aount of retion of physil world: Result of imposing Forms on indeterminy to produe order 2. Eternl nture of the World of Forms. 3. Infinite diversity in the Physil World. II. The Method of Exhustion (Eudoxus nd Arhimedes) Euliden Geometry: Two notions of the infinite: () infinite divisiility of line segments () infinite extendility of line segments led to prdoxes of infinitely smll nd infinitely ig BUT: Erly Greeks tended to void tlk of the infinite. In geometry, ll ojets re relly finite (like nturl numers: ny one is finite; together re ll infinite). Exmple: Method of Exhustion (Eudoxus BC) s used y Arhimedes to prove re of irle = πr 2. Let C e irle with rdius r. For eturl numer n, let P n e regulr polygon insried in C. Divide P n into n ongruent tringles. n equl sides nd n equl ngles Let n = se of tringle = height of tringle THEN: re of tringle = /2 n AND: re of P n = /2n n P 4 P 8 Now visulize C s P -- polygon with infinitely mny infinitely smll sides. SO: When n = : n n = (irumferene of C) = 2πr = r SO: re of C = re of P = /2n n, when n = = /2(2πr)r = πr 2 The height of eh (infinitely thin!) tringle in P is identified with the rdius of C (nd the se of eh tringle in P is very, very smll... infinitely smll!). Prolems () Wht does it men to multiply y n infinitely smll mount ( n when n = )? (Cn t e sme s multiplying y 0!) (2) Wht is polygon with infinitely mny infinitely smll sides? (3) As n goes to infinity, P n pproximtes C, ut lso C*: C Wht does it men to sy C is wht P n is tending towrds nd not C*? C* 3

4 Arhimedes Solution Proved 2 lims: Clim I: There is regulr polygon s lose in re to C s you re to speify. (i.e., For ny ritrry smll re ε, there is lwys numer n suh tht P n differs in re from C y less thn ε.) Consequene: The re of C is t most πr 2. not true for C* Clim II: The re of C is t lest πr 2. Consequene of I nd II: The re of C is extly πr 2. Signifine of Arhimedes Solution: No mention of infinity! III. Aristotle Empiriist: Pltoni Forms re in the physil world. Relevnt Question: Is nything in nture infinite? Aristotle s Answer: only world there is for Aristotle The infinite exists potentilly nd not tully. untrversile tul infinite: tht whose infinitude exists t some point in time potentil infinite: tht whose infinitude exists over time (not wholly present) Note: For A., this is literlly the distintion: Time is infinite, ut not spe. Aristotle s Response to Zeno s Prdoxes: Ahilles nd the Tortoise The distne etween Ahilles nd the Tortoise is only potentilly infinitely divisile; it is not tully infinitely divisile. And there is no ontrdition in liming tht finite length is potentilly infinitely divisile. or: To trvel potentilly infinitely divisile distne, Ahilles needs potentilly infinite time. And there is nothing wrong in liming he hs suh time ville. potentilly infinitely divisile length : for ny n, it is possile to divide into more thn n prts tully infinitely divisile length : n e divided into infinitely mny prts 4

5 Aristotle s infinite: the untrversile Something is infinite if, tking it quntity y quntity, we n lwys tke something outside. It is not wht hs no prt outside it tht is infinite, ut wht lwys hs some prt outside it. Under Moore s reding, Aristotle rejets the metphysilly infinite nd dopts the mthemtilly infinite. Prolem for Aristotle: Wht out the infinite pst? Alredy trversed? So tully infinite? Wittgenstein s Story: Suppose we ome ross mn sying... 5,, 4, 3., who then proeeds to tell us tht he hs just finished reiting π kwrds for ll pst eternity. Why does this strike us s impossile, wheres someone who just strts reiting π forwrds nd will ontinue for ll future eternity does not (given tht we onede the possiility of living forever). 5