The Foundations of Mathematics

What is the foundations of mathematics?
when i look at the foundations of mathematics books 1st edition (by Thomas Q. Sibley). The first chapter is about laguange, logic and set. The second is about proof.
If the foundations is like that, what's the reason?
I though, foundation is something that make it strong and enough to grew up. The principals are quite similar with the building of a house and all the materials support it to stand up. Language, logic, set, proof is essential part. We can't share mathematics idea without language. Logic is about studying the correct reason, it's true or false. Mathematics investigations is just about true or false, so i think it's relevan, since it was developed a long time ago by Aristotles. Set is the most fundamental concept in mathematics, it's about how to collect a distinct objects. It's also relevan. At the end, i agree with Mr. Sibley.

Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.
From the philosophy it self. There are three school that very influent the mathematicians and philosophers at the first time.

1. Platonism: "Platonist, such as Kurt Godel, hold that numbers are abstrack, necessarily existing objects, independent of the human mind."
2. Formalism : "Formalists, such as David Hilbert, hold that mathematics is no more or less than mathematical language. It is simply a series of games..."
3. Intuitionism : "Intuitionist, such us L. E. J. Brouwer, hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them."
   

Reference :
http://en.wikipedia.org/wiki/Foundations_of_mathematics

Mathematics Discoveries

In this time, there are three points that i want to write in this article:
1. The old mathematics discoveries which is still exist today.
2. The old mathematics discoveries which isn't exist today.
3. The new mathematics discoveries which unrelated to old mathematics.
The explanations is below :

1. The Old Mathematics discoveries Which is Still Exist Today

SEMPOA

Why i take sempoa as one of mathematics discoveries? Since it's calculations tool,i thought this is a great discoveries.

Sempoa is one of old calculations tool which is still exist today. Although there were so many invenstions in calculations tool like calculators and computers with many series, but sempoa still has their existance.

Where sempoa comes from is unknow because sempoa used by many cultures in the world. Some researchers said it was on Babylonia and Tiongkok about 2400 SM and 300 SM. In English, sempoa is well-know as abacus. This word used since 1387, comes from latin "abakos" which is taken from greek "abax" means calculation tables.

Why sempoa still exist today?
I used to watch a live program in one of tv station about someone who expert using sempoa. There were three persons that gave calculation problems with 12 numbers, example :
278673589835 + 675548770098 + 227648652202 = ??

One person using sempoa while others calculate by arithmetic algorithm. Everything goes right but there something strange with the person who use sempoa, he did't use sempoa at all. He just move his thumb and index finger to calculates the problem on the board. He calculate faster than the others and the answer is right. I think it awesome, since he can do it well and defeated others. It's one of the effect of something called "Image Training" i though. Since sempoa in his mind with perfect shape and he can use it anytime. In this case is by using sempoa in a long time and slowly the sempoa is escape phisicaly, thus sempoa is just imaginations.

Perhaps, this is one of the reason why sempoa still exist today. Honestly i love sempoa too when i was a child, but by the time goes on i'm not expert on it again. Hm... I need more practice. What about you? Do you have ever use sempoa? In Indonesia itself thera so many sempoa clubs, one of the famous is ASMA (Adil Sempoa Mandiri) who win Sempoa National Olympics on August 24 2008.

2. The Old Mathematics Discoveries which isn't Exist Today

I determine to take the contradiction of Euclid fifth's Postulate, I hope it's quite relevan to discussed.

Euclids fifth's Postulate / Parallel Postulate

"If two lines are intersect by a transversal in such a way that the sum of the degree measures of the two interior angels on one side of the transversal is less than 180 degrees, then the two lines meet on that side of the transversal."

Some mathematicians thought that this is not a postulate, since it can proved by another four postulates. There are some mathematicians who has some effort to prove Euclid's postulate. This work begin when Euclid still alive until 1820. Some of them are:

1. Proclus from Alexandria (410-485)
2. Girolamo Saccheri from Italia (1607-1733)
3. karl Frederich Gauss from Jermany (1777-1855)
4. Wolfgang(Farkas) Boylai from Hungaria (1775-1856) and His son Yanos Boylai (1802-1860)
5. Nicolai Ivanoviteh Lobacevsky (1793-1856)

None of them can prove it, :) Euclid is brilliant. But their effort wasn't useless, because it can find another geometry which called Non-Euclides Geometry.

Since this postulate is difficult to understand, there are some mathematicians who tried to change with another equal postulate. Here are some of them :

• Proclus' Axiom : "If a line intersect one of two parallel lines, both of which are coplanar with the original line, then it must intersect the other also".
• Playfair's Axiom : "Through any point in space, there is at most one straight line parallel to a given straight line".

At the end, what happen with the fifth postulate's Euclid? The discovery of non-Euclidean geometry finally allowed mathematicians to open up their study of this familiar round surface along with other, even stranger, surfaces. Then the expansion of Euclid's axioms led the way to examine what might happen if they were to be eliminated or change.


3. The New Mathematics Discoveries Which Unrelated to Old Mathematics

After a long time, at the end, I couldn't found mathematics discoveries which unrelated with the past discoveries. The new theory is developed from the previous axiom or formulas, for example Jean Christopher Yoccoz who recieved the greatest honour for his work on dynamical systems when he was awarded a Fields Medal, the theory of dynamical systems really began with Poincare who was studying the stability of the solar system. Maria Gaetana Agnesi began working on her most important work, Analytical Institutions, dealing with differential and integral calculus.

During the 20th century, there are so many development on mathematics and at the same time, deep discoveries were made about the limitations to mathematics, it was discovered the truth or falsity of all statement formulated. Some of the most profound discoveries is :

1. In a 1990 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics.
2. In 1976, Wolfgang Haken and Kenneth Apple used a computer to prove the four color theorem.
3. In 1931, Kurt Godel found that this was not the case for natural numbers plus both addition and multiplication: this system, know as Peano arithmetic, was in fact incompletable.

There are also new antry areas of mathematics such as mathematical logic, topology, complexity theory, and game theory.

Reference :
http://id.wikipedia.org/wiki/Sempoa
http://turnbull.mcs.st-and.ac.uk/history/Timelines/TimelineG.html
http://www.mathworld.wolfram.com/PlayfairsAxiom.html
http://www.mathworld.wolfram.com/ProclusAxiom.html
http://en.wikipedia.org/wiki/History_of_mathematicians#20th_century
Moeharti Hw, 1986, Modul Materi Pokok : Sistem-Sistem Geometri, Publisher :karunika Jakarta Open University, Jakarta.

1736 - Leonhard Euler Solves The Problem of The Seven Bridges of Königsberg, In Effect Creating Graph Theory

Fig.1

Based of what i've read, the konigsberg's bridge problem is a unique case. No wonder Euler wouldbe interested on it. Konigsberg located in a strategic position on the river Pregel, that's whay it become a trading center and an importanr medieval city. the river Pregel also divided this area into four land, which is conected with seven bridge : Blacksmith's bridge, connecting's bridge, High bridge, Green bridge, Honey bridge, merchant's bridge and Wooden bridge. As illustrated on figure 1.

Konigsberg is a city with beautiful scenery, the citizens of konigsberg used to spend their sunday afternoons by walking around their city. i don't know why people always make a trouble for themself. The problem was to find a walk trough the city that would cross each bridge once and only once, and if possible, returning to their start point. None can solved it, but Konigsberg is near with St. Petersburg, where the famous mathematicians Euler stay.

I found on some mathematics books about Euler investigated the Konigsberg's bridge by drawing a graph of the city as the figure below :

Fig. 2

But, actually Euler didn't draw the graph. so what did Euler do?
In 1736, Euler wrote up his solution in the Commentarii Academiae Scientriarum Imperialis Petropolitanae under the title 'Solutio Problematis ad Geometrian situs pertinentis'. Euler's diagram of Konigsberg appears in figure 3 :

Fig.3

Altough dated 1736, Euler's paper wasn't actually published until 1741, and was reprented in the new edition on the Commenterii (Novi Acta Commentarii.... ) which is appeared in 1752.
Euler's paper divided into twenty-one numbered paragraph. In his paper, Euler generelized the problem : "Whatever be the arrangement and division of the river into branches, and however many bridges there be, can one find out wheter or possible to cross each bridge exactly once?".

At the first, Euler try to list the possible paths and find the tour which is suitable for Konigsberg's problem. but this method has many weakness, like there so many possible paths, it need a long time and it can't used for more bridges and lands. Hence, Euler decide to look for another method that can find the specific tour with simple step.

Euler use the letter A, B, C, and D to represented the land. If someone walk from A to B right across bridge a or b, the path is AB. A is the initial region and B is the final. In the same way, if someone walk from A to B, B to D and D to C, then the three trips can be represented as ABDC. A is the inital region and C is the final region. Since each land separate by river, this people need to cross 3 bridges. in the same way, if we cross 4 bridges, the path represented with 5 combination letters.

The generalization is, if we cross n bridges the the combination letters is n + 1. The Konigsberg has 7 bridges so the paths can be represented by 8 combination letters.
Euler ignore which bridge is used, as long as the property of initial region and the final region is complete. The Konigsberg's problem can be simplify to look for combination of 8 letters, from A, B, C, and D, and the pair of letters appears with certain frecuency. the region A and B in connected by 2 bridges, so A and B will be used twice, A and C also twice. but, for A and D, B and D, C and D just once.

Before going to the next steps, Euler try to find a formula to indicates wheather possible or not to find the tour. Euler developed the answer from a simpler example on the figure below :

Fig. 4

If A connected with 3 bridges then A will appears twice. If there are five bridges connected to A region, then A will appears 3 times. the generalization is :
if there is an odd number k of bridges, the letter must appear (k + 1 )/2 times.
Since kneiphof is connected by five bridges, the path must contain three As. Similarly. there must be two Bs, two Cs, and two Ds. See the table below:

Summing the final row of frequency gives 9 required letters, but Koingsberg has 7 bridges which exactly once can have only 8 letters. Thus there can be no Konigsberg tour.

Euler continued his analysis :
if there is an even number k of bridges connecting a land mass, then the corresponding letter appears (k / 2) + 1 times if the path begin in that region, and k /2 times otherwise.

Ok, that's all i know about Konigsberg. Thanks for read.

References :
http://www.gss.ucsb.edu/hopkins1.pdf
http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
http://www.math.nmsu.edu/his_projects/Euler-Barnett.pdf

What's the relation between infinity and zero?

I've read some article about The mistery of Zero. It's not the title of a film. Zero is weak but have a big power. How can i say it? one is bigger than zero, but if you try to calculate one divided by zero, you'll get error sign or NaN or infinity, didn't you? zero is netral. Thats way i love it. When zero appear in the first time? Based on what i've read, it come up from india. Infinity is powerful and so infinitely free. It is bigger than any quantity that can be imagined, it's bigger than any finite number. Infinity is one of the fundamental axioms upon which contemporary mathematics is based.

Zero and infinity are perhaps two of the most unique concepts in math. It's like a follower, zero is the reason of the infinity. Infinity it self are uncalculated because it's special. None can solve it, that's why infinity figured with such notation like this :

The sleep eight, some people said eight is perfect or good number. So what's the meaning with this? I guess, eight is a balace. It's hasn't start point and "Futari" or "Berputar-putar" or "Endless point". This is the reason why infinity is figured by "the sleep eight". Perhaps, the true answer is unknown.

inspired by :
http://www.towardsanewera.net/infinity_and_zero.htm