THE PAST AND THE PRESENT

                                             

 Some weeks ago, our lectures Mr. Marsigit give a job. He orders us to search the mathematics concepts, mathematics problems and also mathematics solutions. 

1. The mathematics concepts, mathematics problems and mathematics solutions from the past and is still used by us are:

• Sexagesimal (base-sixty) is a numeral system with sixty is the base. In the first, this system used by Sumerians in the 2000s BC and Babylonians. It is still used for measuring time, angles and geographic coordinates.

The ancient uses:
 The ancient Mesopotamia uses the sexagesimal but it’s not a pure base 60 systems, in the sense that it didn’t use 60 distinct symbols for its digits.
 In the Chinese calendar, a sexagenary cycle is commonly used 

 Base-60 number systems have also been used in some other cultures, for instance the Ekagi of Western New Guinea. 

The modern uses:
 Unlike most other numeral systems, sexagesimal is not used so much in modern times. This numeral system is used in measuring angles, geographic coordinate and time. 
One hour of time is divided into 60 minutes and one minute is divided into 60 seconds. 

The practical unit of angular measure is degree, of which there are 360 in a circle. There are 60 minutes of arc in a degree and 60 seconds of arc in a minute. 

• Zero number concept. This concept, firstly used by Aryabhata from India. He use zero in calculation system and just not empty place. Now, we still use this concept in integers.

• Pythagoras prove a theorem that hypotenuse square of right angled triangle is the total square from two other sides. And now, we still use this theorem which usually named Pythagoras theorem.

• In the Pythagoras period appear a problem which can’t finished by rational number. If a flat line with point 0 and 1, point 0 lie in the left 1 and the negative lie in the right 1. Then q fraction can show with point which divided each unity in the same part of q. The problem is there is point at the line which can’t represent by rational number. So, they must create a new number to show this number, from this problem the irrational number was born. For a several time the root of two is the only one of irrational number. Then the opinions from Plato and Theodorus that the roots of 5,7,8,11,12,13,14,17 are the irrational number too.
  

2. The mathematics concept, mathematics problems and mathematics solutions from the past and not used by us are:

• Rhind papyrus explains that a square area is square from 8/9 of diameter. From this formula, we can get phi= 3,1604…., but now we use phi= 3,14.

• Aryabhata give a pyramid volume is a half of total base times high and sphere volume is phi3/2 .phi3 . Hindu’s people also give several value to phi but they usually use phi=3 and phi=10.
 
3. The mathematics concept, mathematics problems and mathematics in the present which not found in old period are:
• There are many problems in the present that can’t finished with old theorem like closed problem, open-ended problem and open problem. 



Reference: 
• http://wikipedia.com
• The history of mathematics

Speech by Mr. Marsigit

In here, I want to retell what Mr. Marsigit says. Some weeks ago, he tells us about history of mathematics.
History leaving two things:
•Artifacts : something that have form
•Ideas : something that come from mind
There are three streams to explain history:
•Real
•Ideal
•Contemporary
There are some problems to study history:
•The old document
Solution: with internet, we can get new document, which contains the old document.
•Translation
oContext ( geographical, time, society and culture)
oMethodologies
oBearing with other science
There are some manners to study mathematics:
•Empiric: observation, draw a conclusion with induction
Example: Pythagoras develops a theory, then we calls theorem of Pythagoras. The function is c2 = a2 + b2 , then emerge problem, which call irrational number.
•Deduction: this manner develop by Euclid’s, he created book calls “the elements” consist of 13 books containing what referred as definition, postulate or axiom or theorem. Euclid also called as father of mathematics axiom.
Infinite regress is meaning rotation which is no ending.
Aristotle: an elementary assumption preceding a definition. An elementary assumption is a foundation of mathematics.
There are two worlds in mathematics:
•Believe that mathematics have a reason
•Disbelieve that mathematics have a reason
Immanuel Kant said that mathematics must have dynamic foundation. He said that mathematics has foundation calls epistemology but in fact, epistemology is knowledge has not any foundation. Epistemology is a science studying sources of knowledge.
Hill Bert develops a formal mathematics with the logical foundation. Mathematics developed becomes a system having characters deductive and formal.
Hill Bert has a student named Kurt Gödel, he proving complete and incomplete axiom. If mathematics desired consistence, it can be incomplete and if mathematics desired complete, it can be inconsistence.
Plato is an idealist; he said that mathematics stay in kept quiet mind. A way so that we can think is exit from Plato Cave (darkness). Mathematics is given by God.
Aristotle said that mathematics based experience so it is stays out of mind. Mathematics is created by human.
Immanuel Kant: mathematics built with critical mathematics.
Mathematics having character synthetic a priori: we can think it although we never see it.
Synthetic  mathematics as mathematics, experience and ideas.
The opposite of a priori is a posteriori: we see it and then we think it. The law of synthetics is contradiction, in ideas can result a science.
The opposite of synthetics is analytics. The law of analytics is identity.
The constant mathematics foundation is geometry.
The old of mathematics foundation is set.
Mathematics based on intuition.
Common people  intuition is a feeling
Intuition is a place of the nesting fact of mathematics.
Brower is an intuition figure; he is not taking any reason. Intuition is a place or framework: space and time. It means that my mind about mathematics stay in space and time.
The absolute mathematics: mathematics is always true.
The language mathematics: the wrong mathematics is still mathematics.
Many versions about mathematics:
•System
•Structure
•Language
•Body of knowledge

ARISTOTLE
Aristotle was a philosopher born in Stagira, Greek in 384 BC. He became Plato’s student at 17 years old. After Plato died, he established an academy named Lyceum. He leaded the school for 12 years. In there, he develop Aristotle’s philosophy included six books discusses about logic. Aristotle’s logic is a deductive reasoning which founded by Euclid. The logical works of Aristotle were compiled into six books:
1. Categories (Latin: Categoriae) consist of substance, quantity, quality, relation, place, time, situation, condition, action, and passion.
2. On Interpretation (Latin: De Interpretatione) introduces Aristotle’s conception of proposition and judgment, discusses the Problem of the futures contingents.
3. Prior Analytics (Latin: Analytica Priora) introduces his syllogistic method.
4. Posterior Analytics (Latin: Analytica Posteriora) discusses about demonstration, definition and scientific knowledge.
5. Topics (Latin: Topica) discusses treats issues in constructing valid arguments.
6. On Sophistical Refutations (Latin: De Sophisticis Elenchis) gives a treatment of logical fallacies.

The different in mathematics between

the periods of the Ancient Egypt and Greek

1. Ancient Egypt

The ancient mathematics born in many areas in east world as practical science to help effort technique and agriculture. It is important to know that in mathematics east world none finds an example from that now we called verification. There is only description a process and not evidence.

For a long time, Egypt become richest area for research ancient history, because the worship that doing by Egypt people to people who have died, besides that area climate also become fascination for history researches. Civilizations in Egypt produce building in the form of temple and mausoleum. That building has important role to save the papyruses. Papyrus is an ancient material to write created by Egypt people and around 650 BC have been introduced in Greek.

There are some omissions from Ancient Egypt:

1. 3100 BC. At Egypt scepter, there are number expressing millions of and hundreds of thousand in Egypt hieroglyph, now this scepter save in Oxford museum.
2. 2900 BC. A big Pyramid in Gizeh builds by 100.000 workers in 30 years.
3. 1850 BC. There is papyrus Moscow contain 25 questions.
4. 1650 BC. There is papyrus Rhind (Ahmes), contains 85 questions in hieratic number by Ahmes. Now, this papyrus saves in British Museum.
5. 1500 BC. Indicative appliance of time with help shadow of the sun, now save in Berlin Museum.
6. 1350 BC. There is papyrus Rollin contains some calculation about bread price, now save in Louver.
7. 1167 BC. There is papyrus Harris, the document is compiled by Rameses IV contain masterworks of his father, Rameses III.

2. Greek

Greek goes forward along with the resignment of Egypt. Modern civilization emerges in small Asia, Greek, Sicilian and Italia. In there, for the first time in mathematics, peoples begin to say “why”. Paul Tannery, T.L. Heath, H.G. Zeuthan, A. Rome, J.L. Heiberg and E. Franle in tracing again history of mathematics in Ancient Greek. They only used sources from report in Arab and Nasrani report. They were reorganizing original report like Euclid, Apollonius, Archimedes, etc. A document about geometry and astronomy, which discusses the period before 355 BC by Eudemus, Aristotle’s student has losses. Then around 450 century, Neo-Platonist Proclus give his work about Commentary on Euclid, it was recognized as Eudemian Summary. This is sources of boldness which most can be trusted in Greek mathematics.

There are three important growth lines in the first 300 years in Greek mathematics. The first is the growth from the materials that compiled in unsure started from Pythagoras and his follower, Hippocrates, Eudexus, Theaterus, etc. It included the theorem that proves by Pythagoras until the solution “scandal logic” about irrational number by Eudexus. The second is the growth from ideas about infinitesimal, limit and way of quantifying which has no answer until the finding of calculus in modern-day. Then the last is about the curve in geometry except circle, straight line and area beside spherical object.

History is a key to open any memory in the past. With history, we can study many things, such as to know many famous figures with the finding, take the lesson from many fault in the past and correct it in the future.

In science of mathematics, we also know knowledge that called history of mathematics. We can study many things from a period before century until now. We can know famous figures in the past and the finding so we can use it to support our lesson. We also can more appreciate with the finding from the figures.

A. Activity in class

I never think before in science of mathematics, I will study history of mathematics. At the first meeting, our teacher asked to study history of mathematics we must have many sources, examples from books, CD, internet etc. He also gives us 4-point foundations to lesson history:

1. Dreams
2. Attitude
3. Ability in communication
4. Browsing internet

Those points can support us to study history of mathematics.

Then our teacher shows us containing from “Britannica Encyclopedia”. In there, we can get more information quickly. In the next meeting, our teacher asks many things about history of mathematics. History can shows from two aspects, artifact and idea. In history, we study the fact of the past but there are any problems such as the sources especially documents was very old so we need internet to up date the documents.

There is some manner to study mathematics:

• Empiric: observation, draw a conclusion with induction

Example: Pythagoras develops a theory, then we calls theorem of Pythagoras.

• Deduction: this manner develop by Euclid’s, he created book calls “the elements” consist of 13 books containing what referred as definition, postulate or axiom. Euclid also called as father of mathematics axiom.

Then Aristotle discovers infinite regress: the turning meaning which never ends.

There are two worlds in mathematics:

• Believe that mathematics have a reason
• Disbelieve that mathematics have a reason

A person said that mathematics must have dynamic foundation he is Immanuel Kant. He said that mathematics has foundation calls epistemology but in fact, epistemology is knowledge has not any foundation.

B. Activity outside class

In the second meeting, we get a duty, created the erudite masterpiece. Therefore, we search the sources to answers the questions. There are some questions, examples the different in mathematics between the periods of the Ancient Egypt and Greek, Pythagoras service and role to mathematics growth, Euclid’s finding, etc. Then, I got book entitle “the history of mathematics”, that book I refer to doing the duty. From the book, I get much information that I have never find previously. In there, I know that Pythagoras never find a theorem in right triangle that hypotenuse square is amount of other square two sides, Pythagoras only gives best verification systematically. Babylonian people have recognized that theorem since Hamurabi periods. Therefore, after I read the book I get the fact from the Pythagoras theorem. Besides that, I also read about mathematics in Babylonia and Egypt periods. Babylonian people used burned clay tablet as equipment to write while Egypt people used papyrus. Papyrus is ancient material to write as a paper made from water grass called papu. There are some papyruses in there, likes Moscow Papyrus, Rhind Papyrus, Rollin Papyrus, and Harris Papyrus.

Besides Pythagoras, that book also discuses the finding from many intellectual, likes Thales, Euclid, Eudexus, Plato, etc. Eudexus service in mathematics is proportion theorem that finished “logic scandal” about irrational number. Euclid’s service is created a book calls “The Elements” consist of 13 books discuses geometry, number theorem and elementary or geometry algebra. From the book, we can find intellectual like Paul Tannery, T.L. Heath, H.G. Zeuthan, A. Rome, J.L. Heiberg and E. Franle in tracing again history of mathematics in Ancient Greek. They only used sources from report in Arab and Nasrani report. They were reorganizing original report like Euclid, Apollonius, Archimedes, etc.