What is math!? Topic 0 Mathematical preliminaries · topic 0: mathematical preliminaries midyear 2018 refresher course in mathematical economics (LCabueg) 4 Some terminology Definition

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




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)?




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




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]




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)




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 Greek alphabet

rho (R)sigma (S)tau (T)upsilon (U, Y)phi (PH)chi (CH)psi (PS)omega (Ö)

ρΡσΣτΤυϒϕΦχΧψΨωΩ


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

Documents

What is math!? Topic 0 Mathematical preliminaries · topic 0: mathematical preliminaries midyear 2018 refresher course in mathematical economics (LCabueg) 4 Some terminology Definition