what is math!? topic 0 mathematical preliminaries · topic 0: mathematical preliminaries midyear...
TRANSCRIPT
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 1
Topic 0Mathematical preliminaries Refresher course in
mathematical economics
L Cagandahan Abueg
School of Economics
University of the Philippines Diliman
What is math!?
How do you count?
Problem: One rabbit saw six elephants while going to the river. Every elephant saw two monkeys going toward the river. Each of the monkey has a parrot on the left shoulder.
Question: How many animals went to the river?
Answer: the rabbit, the two
monkeys and the two parrots.
Mathematics defined
Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined (Wolfram Mathematica)
Historically, started with tabulation of quantities, and measurements
Diversification: pure, and applied mathematics
Early mathematicians
Map of the Seven wonders of the ancient world
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 2
MesopotamiaFamous for the true-
value system of
counting (the base ten system), and an algorithm for the existence of √2, and the sexagesimal system
(time and angle measurements).
Hanging Gardens, King Nebuchadnezzar IICity of Babylon, 600BC
Ancient EgyptInitial estimate of πat 3.125, use of fractions 1/n, and the famous cubit
(Egyptian meter), later used by the Israelites. Math was propelled by applied problems in geometry.
Great Pyramid of Khufu , Fourth DynastyGiza, 2550 BC
Ancient GreeceMuch of ancient mathematics problems came from Greece: the doubling of cube, the trisection of an
angle, and the squaring of the
circle.Temple of Zeus with Nike, by Phidias Olympia, 470 BC
A dozen, a gross, and a score
plus three times the square
root of four
divided by seven
plus five times eleven
is nine squared and
not a bit more.
12 144 20+ + 3 4+7
( )5 11+ ⋅ 29 0= +
Why mathematical economics (math econ)?
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 3
Math and econ
Why does modern economic theory
require much mathematical rigour?
It was during the period 1830-1930 when economics was “transformed” from a developmental approach (e.g., Smith, Ricardo, Mill, Marx) to an economics where [almost] everything can be measured quantitatively.
Math and econ
Known in economic history as the period of marginalism or the period of the marginalist
revolution: analysis of different macroeconomic problems (unemployment, business cycles, growth), culminating in the Great Depression of 1929 (Keynes and the birth of macroeconomics)
Math and econ
However, economics literature focused on the optimization by individual consumers and firms:the formal analysis of Smith’s
“invisible hand”, statically
interpreted
[Intense] permeation of calculus in economic theory to solve problems (hence, the birth of mathematical economics)
Math and econ
It is not to be
supposed, however, that because
economy becomes
mathematical in form, it will,
therefore, become
a matter of rigorous
calculation. William Stanley Jevons[1835-1882]
Math and econ
Recognized by the American Mathematical Society, math econ is regarded as one of the specialized areas of applied math.
The American Mathematical Journalpublication of the American Mathematical Society
Some terminology, logic and proofs
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 4
Some terminology
Definition (DEF). A statement of the meaning of a word, word group, sign, or symbol.
Axiom or Postulate (POS). A statement that has found general acceptance, or is thought to be worthy thereof, on the basis of an appeal to intrinsic merit or self-evidence and thus requires no proof of validity.
Some terminology
Lemma (LEM). An auxiliary proposition that has been proved either by the user or elsewhere and that is stated for the expressed purpose of immediate use in the proof of another proposition.
Some terminology
Theorem (THM) or Proposition
(PROP). A formula or statement that is deducted from other proved or accepted formulas or statements and whose validity is hereby proved.
Corollary (COR). Immediately deducible from a proven theorem and that requires little or no additional proof of validity.
Some terminology
Observation (OBS) or Remark
(RMK). A commentary [for some emphasis] related to preceding mathematical statements, which may require proofs, if necessary.
Mathematical proofs
Deduction or deductive
reasoning: from a general state of the world we study specific instances or cases
From Euclidean philosophy in which Euclid himself espoused
Mathematical proofs
Euclid is famous for his book The
Elements, which is known as one of the oldest book on algebra, trigonometry, and geometry.
Euclid of Alexandria[fl.300BC]
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 5
Mathematical proofsThe Elements has thirteen books, which is one of the foundations of deductive logic. From a set of 131 definitions, Euclid proved at least 465 propositions.
The Elements, cover by Sir Henry Billingsley [first English edition] (1570)
Some “faulty” logic
Practice makes perfect.I am not perfect.
So why practice?
[1]
Nobody is perfect.But I am nobody.
So I am perfect!
[2]
Symbols and the Greek alphabet
Elementary symbols
∀∃!∃
for every; for all; for eachthere exists; there isthere exists a unique ...there does not existsupposethereforesincesuch that#
∃/$∴∵
∋#This symbol is not used here because of confusion in set theory (§1).
Elementary symbols
∧∨
if only ifif and only if (also, iff)andorcontradiction (also, C!)is defined asis identical toend of proof (also, Q.E.D.!)
:=
⇐⇔
≡
Note. Q.E.D. stands for quod erat
demonstrandum which means “[the statement] which has
been proven”, and not the phrase quite easily done (or quite elegantly done)
Much attributed to Euclid throughout history, but officially due to Phillipe von Lansberge(1604)
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 6
Quod erat demonstrandum
Triangulorum Geometriæ[1604]
Philippe van Lansberge[1561-1632]
The “struggle” Mathematicians are known for
their usual struggles of “sleepness nights pouring over proofs” and logical answers to mathematical problems
Colloquially known as the Eureka moment (“I have found
it!”) attributed to Archimedes, from the story of the “displacement problem”
Eureka!
Noli turbere,
circulos meos.
Archimedes of Syracuse[287BC-212BC]
Do not obscure
my circles.
The Greek alphabet
alpha (A)beta (B)gamma (G) delta (D)epsilon (E)zeta (Z)eta (E)theta (TH)
αΑβΒγΓδ∆εΕζΖηΗθΘ
The Twelve Great Olympiansfrom Greek Mythology
The Greek alphabet
iota (I)kappa (K)lambda (L)mu (M)nu (N)xi (KS, X)omicron (O)pi (P)
ιΙκΚλΛµΜνΝξΞοΟπΠ
The Twelve Great Olympiansfrom Greek Mythology
topic 0: mathematical preliminaries midyear 2018
refresher course in mathematical
economics (LCabueg) 7
The Greek alphabet
rho (R)sigma (S)tau (T)upsilon (U, Y)phi (PH)chi (CH)psi (PS)omega (Ö)
ρΡσΣτΤυϒϕΦχΧψΨωΩ
The Twelve Great Olympiansfrom Greek Mythology
The Greek alphabetWhy is letter Z the last letter of the English alphabet?When the Romans adapted the Greek alphabet, they dropped the letter Z because they deemed it to be useless.
The Trojan HorseThe Trojan War (Homer)
To end...
Professional mathematicians use the
word “obvious” to indicate that it is
obvious how to give a complete proof.
To use “obvious” to mean “I am sure it’s
true, but I can’t prove it,” is not a
commendable practice.
C. Clark [1982]
“Obvious” is the most dangerous word in
mathematics.
E.T. Bell