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Proof – a historical perspective 27.03.2007

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References 1.Harel, G., & Sowder, L (in press). Toward a comprehensive perspective on proof, In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, National Council of Teachers of Mathematics. http://www.math.ucsd.edu/~harel/downloadablepapers/TCPOLTOP.pdf http://www.math.ucsd.edu/~harel/downloadablepapers/TCPOLTOP.pdf 2.Kleiner, I. & Movshovitz-Hadar, N. (1997). Proof: A many- splendoured thing, The Mathematical Intelligencer, 19 (3), 16-26. 3.Kleiner, I. (1991). Rigor and Proof in Mathematics: A Historical Perspective, Mathematics Magazine, 64 (5), pp. 291-314. 4.History topics: http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html

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Some phases in development of the concept of proof Pre-Greek math Egyptian mathematics – 4000-3500 BC Babylonian mathematics – 2000-1600 BC (computational procedures, no symbols) Greek math – 600 BC – 200 BC (deductive reasoning) Thales (-600), Pythagoras (-500), Euclid (-300), Aristotle (-300), Apollonius (-200) Syria, Iran, India (200-1100 AC) (preservation and development of the Greek tradition of proof) Omar Khayyam (1100), Adelard and Fibonacci (1100)– brought math knowledge from Islamic countries and Greeks back to Europe Renaissance (16-17 th centuries) Vieta, Descartes, Leibnitz (symbolic algebra) The calculus of Caushi and Weierstrass (19 th century) (rigorization of calculus, “genetic definitions”) Formalism, logicism, intuisionism (20 th century) Hilbert, Zermelo, Fraenkel, Guedel, Computer-based proofs, probabilistic proofs (1970-)

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Pre-Greek vs. Greek math Pre-Greek mathematics concerned actual physical entities and measurements (method: empirical or perceptual reasoning) Geeks: the entities under investigation are idealizations of experiential spatial realities (method: logical deductions, terms without definitions - primary propositions – propositions that should be proved)

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Motives of Greek mathematics The “crisis” caused by Pythagoreans’ proof of incommensurability of the diagonal and side of the square The desire to decide among contradictory results from past civilizations (e.g., debate about Pi, paradox of Zeno) The nature of Greek society The need to teach

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Debates on Greek math (16 th century) Proposition I.32 in Euclid’s Elements: In any triangle, if one of the sides is produces, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangles equals two right angles. Aristotle’s definition of science: “We suppose ourselves to possess unqualified scientific knowledge of a thing,…, unless we think that we know the cause on which the fact depends as the cause of the fact and of no other”. THE IDEA OF CAUSE AND EFFECT Implications: 1) mathematics is not a science in Aristotle’s meaning 2) Proof by contradiction is meaningless and should be removed from mathematics 3) Proofs by exhaustion by Archimedes is unsatisfactory as it is not causal (Rivaltus, 1615)

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Additional motives for re-conceptualization of Greek mathematics in 16 th century Greeks paid no attention to the operations underlying spatial configurations, there were many problems without solutions (e.g., trisection of an angle) Vieta and Leibniz: creation of algebraic notation; proof is a sequence of sentences beginning with identities and proceeding by a finite number of steps of logic and rules of definitional substitution “It is hardly an exaggeration to say that the calculus of Leibniz brings within the range of an ordinary student problems that once required the ingenuity of an Archimedes or a Newton” (Edwards, 1979)

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An example: Leibniz “proof” of the product rule for differentiation d(xy)=(x+dx)(y+dy)-xy=xy+xdy+ydx+dxdy- xy=xdy+ydx - since “the quantity dxdy…is infinitely small in comparison with the rest, and hence can be discarded” (Leibniz, cited in Edwards, 1979)

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An additional example: Euler Now let n be an infinitely large integer and z an infinitely small number The above equation becomes: Let nz=x, which is a finite since n is infinitely large and z infinitely small. Finally,

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Errors were made! Example: Euler “proved” that (counterexample was given by Schwarz) Russel vs. Cantor

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Motives for re-conceptualization of mathematics in 19 th century Attempts to establish a consistent foundation for mathematics – one that is free from paradoxes “In both situations [Greeks vs. 16 th century and 16 th vs. 19 th century], crises had developed which threatened the security of mathematics; and in both cases resorts was taken to explicit axiomatic statement of the foundations upon which one hoped to build without fear of further charges of inconsistency” “Like the Greek mathematicians, the modern mathematicians had only one model in mind, albeit a different model” (Wilder, 1967)

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Greek vs. “modern mathematics” GreeksModern mathematics Ideal existence reference to images Constant reality Selectivity in choosing the entities for investigation Just one model (of geometry) Focus on “operation” (e.g., trisection of an angle) Proof is valid by virtue of its content, not form abstract entities images are not necessary Varying reality Entities for investigation can be of any nature Different models Focus also on “results of operation ” Proof is valid by virtue of it form alone

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Logicism, formalism and intuitionism (Early 20 th century) Logicism: The notion of proof – its scope and limits – a subject of study by mathematicians. Mathematics is a part of logic (Peano). Formalism: Mathematics is a study of axiomatic systems. Mathematics is about symbols to which no meaning is to be attached! (Hilbert) Intuitionism: No formal analysis of axiomatic systems is necessary. Mathematics should not be founded on the system of axioms, the mathematician’s intuition will guide him in avoiding contradictions. Proofs must be constructive (Brouwer).

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1) “Most research mathematicians do not take any course in formal logic!” The logicism is not widespread among mathematicians. 2) The debate among formalists and intuitionists is still unresolved, but “most mathematicians are untroubled, at least, in their daily work, about the debates concerning the various philosophies of mathematics” Israel Kleiner:

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Computer-based and probabilistic proofs Appel and Haken (1976): computer-aided proof of the four-color theorem – verification of 1482 confiducations. Thousands of pages of computer programs that were not published Michael Rabin: 2 400 -593 is a prime number, the statement has a small probability of error “Some have argued that there is no essential difference between probabilistic and deterministic proofs. Both are convincing arguments. Both are to be believed with some probability of error” (Kleiner, 1991)

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ELEMENTARY NUMBER THEORY AND METHODS OF PROOF

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