Axiom for Mathematical
Induction.
Let M be a set with
(a)
1 M ,
(b) and if
then
xM,
x' M .
We have N  M.
Consider the sum
1 + 2 + …. + 2013
= (2013 x 2014 ) / 2
= 2013 x 1007
= 2013 x 1000 + 7 x 2013
= 2013000 + 14091
= 2027091.
The proof?
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Plato’s idea of Essence
.. > Euclid’s approach to
Geometry
.. > Definition + Theorem +
Proof model of Math.
The Definition-Theorem-Proof
(DTP) Model of Mathematics.
1. Identify the undefined terms &
(unproven) axioms/assumptions.
2. Definitions.
3. Statements/Theorems/
Propositions/Lemmas…
4. Proofs/Arguments/Derivations.
Led by famed mathematicians like
Hilbert, the DTP model of
mathematics is particularly
influential during the early 20th
century.
They sought to put down all
the axioms and undefined terms
of mathematics and logic,
extending the work of Euclid.
At the Paris International Congress of 1900, Hilbert
proposed 23 outstanding problems in mathematics.
These problems have come to be known as Hilbert's
problems, and a number still remain unsolved today.
We must know, We shall know.
Every mathematical question is decidable in a
finite number of steps:
this is the decision problem.
Ironically, the day before Hilbert lectured, the
young Austrian logician Kurt Gödel also lectured
in Königsberg on his incompleteness theorem
After Hilbert was told that a student in his class had
dropped mathematics in order to become a poet, he is
reported to have said "Good--he did not have enough
imagination to become a mathematician"
Liar paradox
• The liar paradox is the sentence
"This sentence is false." An analysis
of that sentence shows that it cannot
be true (for then, as it asserts, it is
false), nor can it be false (for then, it
is true).
• A Gödel sentence G for a theory T
makes a similar assertion to the liar
sentence, but with truth replaced by
provability:
• “This statement has no proof.”
The first incompleteness theorem of Godel
says that, for any rich enough consistent
mathematical theory, there is a statement
that cannot be proved or disproved within
the theory.
Surprise Examination Paradox.
The teacher announces in the class:
“Next week you are going to have a test, but
you will not be able to know on which day
of the week the test is held until that day.”
The second incompleteness theorem of
Godel says that, for any rich enough
consistent mathematical theory, the
consistency of the theory itself cannot be
proved (or disporved) within the theory.
Although there is no decision method
for arithmetic of integers with + and ,
Presburger (1930) gave a decision method
for the part of arithmetic of integers with
+ only.
Tarski (~1939) showed that there are
decision methods for elementary
algebra and geometry.
Tarski at the University of California at Berkeley.
Note that the high standard of
“Proof” in mathematics.
It is not just arguments based on
experience or information. Every
step in a proof must be able to
be traced back to established
or proven statements, which in
turns are based on the axioms.
But the model does not include
all we have in mathematics.
One obvious problem with the
D-T-P model is that it does not
indicate where the statements
for proving are coming from.
In other words, where do
questions coming from?
For students, the answers may
be: from textbooks, tutorials,
teachers, tests, exams….
The problem goes deeper as
asking questions is an
integrated part of doing
mathematics.
The Babylonian tablet records
some interesting relations:
3 4 5
2
2
2
65  72  97
2
2
2
12,709  13,500  18,541
2
2
a  b  c ??
2
2
2
2
How about the sum
1  2  3    n  ?
2
2
2
2
Here n is a natural number.
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1  2  3    n
n(n  1)(2n  1)

??
6
2
2
2
2
Take a look at n = 100.
1  2  3      100  338350.
2
2
2
2
100  101 201
 50 101 67  338350.
6
It is okay!
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The art of making intelligent
guesses, raising questions,
having intuitive and heuristic
`feeling’ on the outcomes, is
grouped under the term
“speculation”
or
“conjectures”.
This leads to the DefinitionSpeculation-Theorem-Proof
(D-S-T-P) model of
Mathematics.
1. Define terms.
2. Raise and have questions,
make guesses and intuition
toward the likely outcomes.
3. Formulate conjectures
(Statements considered to be
true but have not been proven.)
4. Prove or disprove conjectures.
Proven conjectures  theorems.
“No questions are of
such significance
as those that are naive.”
-an idea held by the
Polish poet
Wislawa Szymborska
(1996 Nobel laureate).