Let M be a set with

(a)

(b) and if

then

We have N M.

,1 M

.' Mx

,Mx

Axiom for Mathematical Induction.

Consider the sum

1 + 2 + …. + 2013

= 2013 x 1000 + 7 x 2013= 2013000 + 14091= 2027091.

= (2013 x 2014 ) / 2= 2013 x 1007

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.

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"

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

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 Tmakes 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. Everystep 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 includeall 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:

??

541,18500,13709,12

977265

543

222

222

222

222

cba

How about the sum

?321 2222 n

Here n is a natural number.

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Take a look at n = 100.

.338350100321 2222

??6

)12)(1(

321 2222

nnn

n

.33835067101506

201101100

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 Definition-Speculation-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).