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Proof – a historical
perspective
27.03.2007
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
2. Kleiner, I. & Movshovitz-Hadar, N. (1997). Proof: A manysplendoured 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
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-)
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)
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
Debates on Greek math
(16th 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)
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)
An example: Leibniz “proof” of the
product rule for differentiation
d(xy)=(x+dx)(y+dy)-xy=xy+xdy+ydx+dxdyxy=xdy+ydx - since “the quantity dxdy…is
infinitely small in comparison with the rest,
and hence can be discarded” (Leibniz,
cited in Edwards, 1979)
An additional example: Euler
(cos z  i sin z )n  cos nz  i sin nz
n(n  1)
n
cos nz  (cos z ) 
(cos z ) n 2 (sin z ) 2 
2!
n(n  1)( n  2)( n  3)

(cos z )n 4 (sin z )4  ...
4!
Now let n be an infinitely large integer and z an infinitely small number
cos z  1, sin z  z, n(n  1)  n 2 , n(n  1)(n  2)(n  3)  n 4
n2 z 2 n4 z 4
cos nz  1 

 ...
The above equation becomes:
2!
4!
Let nz=x, which is a finite since n is infinitely large and z infinitely small. Finally,
x2 x4
cos nx  1    ...
2! 4!
Errors were made!
• Example: Euler “proved” that f xy  f yx
(counterexample was given by Schwarz)
Russel vs. Cantor
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)
Greek vs. “modern mathematics”
Greeks
Modern 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
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
mathematician’s intuition will guide him in
avoiding contradictions. Proofs must be constructive
(Brouwer).
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”
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)