Proof a historical perspective

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Proof a historical perspective

• 27.03.2007

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References

• 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
• Kleiner, I. Movshovitz-Hadar, N. (1997). Proof
  A many-splendoured thing, The Mathematical
  Intelligencer, 19 (3), 16-26.
• Kleiner, I. (1991). Rigor and Proof in
  Mathematics A Historical Perspective,
  Mathematics Magazine, 64 (5), pp. 291-314.
• History topics
• http//www-groups.dcs.st-and.ac.uk/history/Index
  es/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 (16-17th centuries)
• Vieta, Descartes, Leibnitz (symbolic algebra)
• The calculus of Caushi and Weierstrass (19th
  century) (rigorization of calculus, genetic
  definitions)
• Formalism, logicism, intuisionism (20th 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(16th century)

• Proposition I.32 in Euclids 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.
• Aristotles 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 Aristotles 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 16th 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)(xdx)(ydy)-xyxyxdyydxdxdy-xyxdyydx
  - since the quantity dxdyis 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 nzx, 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 19th century

• Attempts to establish a consistent foundation for
  mathematics one that is free from paradoxes
• In both situations Greeks vs. 16th century and
  16th vs. 19th 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
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Logicism, formalism and intuitionism (Early 20th
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
  mathematicians intuition will guide him in
  avoiding contradictions. Proofs must be
  constructive (Brouwer).

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Israel Kleiner

• 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

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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 2400-593 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)