Mathematics: created or evolved?

Some ( if not most ) mathematicians think about mathematics as being 'created'. Paul Erdos said that there must be a book, The Book, containing the most important and most beautiful proofs. Who other than God could have written The Book? Einstein once said that 'God does not play dice' and Kronecker said that "God made the integers; all else is the work of man." There is still some creationism left among mathematicians. If there are several ways to attack a problem there is always one best, and that's the one described in The Book.

Where do our mathematical insights come from? Some say they are a form of Divine Intervention. Or is the entirety of mathematics somewhere hidden in our DNA? Evolution Theory versus Intelligent Design. We, the human species, are that part of the physical universe where it becomes self-aware. We ( the physical universe ) describe the behavior of the universe with the help of mathematics. In that sense mathematics evolved over the centuries. In the concept of a created mathematics, mathematics is there, for us humans, to be discovered. Is mathematics finite, created, timeless, deterministic and waiting for intelligent beings to be discovered, understood and applied? Or is mathematics human, a property of our species, a way to handle the complex reality around us. Unlikely as it seems there may be alternative bodies of mathematics possible.

Clearly, compared to the other sciences, mathematics evolves ( is discovered ) at a slow pace. But that's because mathematical truths are eternal ( they say ). The fundamental theorems are, but is the same true for the tools used by engineers and scientists in their daily work? Determinants have lost some of their appeal, quaternions may face the same destiny. Mathematics does change. But the idea that there is one ( created ) mathematics goes deep. You can't patent a mathematical idea for example.

Mathematicians lack a sense of urgency, except for their own careers or immortality maybe. If mathematics in its entirety could be owned by a company, would that ( have ) speed up the development of mathematics? And thus bring the solution of major planetary problems forward? What if there were competing bodies of mathematics, like operating systems, say Windows versus Linux? I think that that would have been possible in some alternate version of history. There -are- different ways of doing things in mathematics. Take geometry for example. In computer graphics you can choose between ( standard ) linear algebra, add quaternions if you like, or a portion of projective geometry so that you can compose translations and linear transformations with matrices, or: you can use geometric algebra which until not so long ago was either abondoned or exclusively used in quantum physics. My point being we may polish the mathematics until its ready for a place in The Book.

Live collisions from CERN on your phone.

Developing Android apps can be challenging and rewarding, especially if graphics and thus mathematics is involved. That applies for the apps I develop myself but even more so for this one that I just found.

If you want to view live 3D displays of collisions direct from CERN on your ( Android ) phone then download this free app called LHSee.

There is also a YouTube video about the development of the app which was done at the University of Oxford.



Did you know that there is even a Lego model of the Atlas collider?

From Intelligent Design to Geometry

Just some thoughts...

When I express my doubts about the success of the Apollo project people smile behind my back but that's about it. Friendships aren't broken and neither are ( employment ) contracts. ( See: AULIS on Apollofor more info ). In the last century science turned into a multi billion dollar industry and that certainly has changed the world of academics. Questioning a general accepted theory can ( and will ) ruin careers. Examples are questioning the cause of Aids, and questioning Darwin's Evolution Theory.

Evolution Theory basically says that we evolved over time by a process called mutation and natural selection. Those who question the Evolution Theory in fact question that very, very complex machines evolved from 'mud'. Darwin didn't answer that question because when Darwin published 'On the Origin of Species on 24 November 1859' he wasn't even remotely aware of the complexity of the cells making up life. DNA wasn't discovered until ten years later in 1869 by Friederich Miescher and it took until 1953 when James Watson and Francis Crick discovered the double helix structure of DNA. Intelligent Design Theory accept Evolution Theory but only up to a certain point. They argue that somewhere in the beginning some information or 'design' had to be injected into the system. Who put it there? I would ask.

To the point.

Anyway, these thoughts entered my mind because I am thinking of building a 'DNA-type-of' geometry building block for a computer program. With the help of a computer these geometries should be able to construct ( divide ) themselves in a scene graph and evolve, multiply and so on. At the moment it's just an idea. I started to look for a way to understand more about DNA by finding popular science books on the subject. I haven't learned much about biology and chemistry and what I have learned seems forgotten. But I am only interested in DNA as a computer, or data structure. Then I found this website 'DNA seen through the eyes of a coder'. Since I am a coder ( computer, Android, programmer ) by profession that was exactly what I was looking for. Take this for example, DNA is not binary, DNA is quaternary. Computer letters normally consist of 8 bits called a byte, so using that system there are 256 possible letters. The equivalent of a byte in DNA is the codon and has three places. So in DNA language there are 64 possible letters. Read more on the site.

More to follow on this geometry project soon, I expect.

Diophantus of Alexandria

Diophantus (+/- 250 AD) of Alexandria has been called 'the father of algebra' and an entire branch of mathematics has been named after him, the study of Diophantine Equations. The most famous problem in this field 'Diophantine Equations' is Fermat's 'Last Theorem'. Fermat was reading Diophantus' comments on the Pythagoran theorem when he conjectured that for an exponent n > 2, the equation \[ x^n + y^n = z^n \] has no integer solutions. This theorem was considered the hardest open problem in mathematics until solved by Andrew Wiles in 1994. Diophantus work was lost to the Western world for thousand years.

Anyway, I thought about Diophantus when I came across this beautiful equation which has an infinite number of integer solutions \[ x^3 + y^3 + z^3 = x^2 +y^2 + z^2.\]

What is mathematical research ?

What is mathematical research? And how is it done? Questions that have been on my mind for very long.

We know what researchers do:
- answering questions asked in the literature;
- discovering a new theorem;
- publishing an article in a journal ( i.e. Journal for Number Theory );
- writing a book;
- lecturing about their work and giving talks;
- being part of a research community.

Researchers are either employed by a university or by a corporation. University researchers have a commitment to teach and corporate researchers aren't free to chose their topics. Both are pressed to publish often in the best journals possible.

But -HOW- to they do it? What makes them successful in their field? Questions, I can't answer. Let's go and search for answers elsewhere. Starting with Manning, perhaps.



and this one:



Enjoy.

Conceptual knowledge maps for mathematics

About working and studying, the main topic of this blog: well, it is no longer an option. It is mandatory, society needs people who are committed to their jobs and invest in their lifetime education. Everything changes at an ever faster pace and it is thus our responsibility to stay of value in all of our dynamics.  It is an issue we must align regularly.

Studying in a particular field will gradually align your life to your new interests. For example, I have recently, more or less -landed- in a job where some of my colleagues are mathematicians. In fact, a lot of the work I do involves mathematics. More about that in posts to come.

Everything changes. So do study skills. And a vast amount of tools for computer aided learning is now available. From MIT lectures via YouTube to software for flash cards to sophisticated knowledge databases for personal use. I would like to mention two items I have found recently.

InfoRapid Knowledge Builder is a graphical tool to document conceptual knowledge maps, free for personal use at http://www.buildyourmap.com/. ( Before this I used Mindjet and TheBrain. ) I am working on a map with topics in Analytical Number Theory.

Another item I found is a book. It is called On Course, Strategies for success in College and Life. ( And if you work and study read the title as Strategies for successful combining work and study.) The book is packed with information. It is not a book you should read once but it is a companion you can use, consult, for years. I have the Study Skills Plus edition.

Fixing mathematics education

Conrad Wolfram spoke about math education at the 2010 Wolfram Technology Conference:

Now, a new initiative has been recently announced at computerbasedmath.org to build a completely new math curriculum with computer-based computation at its heart.

Also see Conrad's Wolfram blog here where he announces that Estonia will be the first country that will completely rewrite their mathematics school curriculum.

One day left...

to vote for the most important British innovation of the 20th Century. Alan Turing's Turing Machine is currently on 2nd place with 13% of the votes.

Vote here.

Filling a bottle...

A professor stood before his philosophy class and had some items in front of him. When the class began, he wordlessly picked up a very large and empty mayonnaise jar and proceeded to fill it with golf balls. He then asked the students if the jar was full. They agreed that it was. The professor then picked up a box of pebbles and poured them into the jar. He shook the jar lightly. The pebbles roll ed into the open areas between the golf balls. He then asked the students again if the jar was full. They agreed it was. The professor next picked up a box of sand and poured it into the jar. Of course, the sand filled up everything else. He asked once more if the jar was full.. The students responded with a unanimous ‘yes.’ The professor then produced two Beers from under the table and poured the entire contents into the jar effectively filling the empty space between the sand.The students laughed.. ‘Now,’ said the professor as the laughter subsided, ‘I want you to recognize that this jar represents your life. The golf balls are the important things—-your family, your children, your health, your friends and your favorite passions—-and if everything else was lost and only they remained, your life would still be full. The pebbles are the other things that matter like your job, your house and your car.. The sand is everything else—-the small stuff. ‘If you put the sand into the jar first,’ he continued, ‘there is no room for the pebbles or the golf balls. The same goes for life. If you spend all your time and energy on the small stuff you will never have room for the things that are important to you. Pay attention to the things that are critical to your happiness. Spend time with your children. Spend time with your parents. Visit with grandparents. Take your spouse out to dinner. Play another 18. There will always be time to clean the house and mow the lawn. Take care of the golf balls first—-the things that really matter. Set your priorities. The rest is just sand. One of the students raised her hand and inquired what the Beer represented. The professor smiled and said, ‘I’m glad you asked.’ The Beer just shows you that no matter how full your life may seem, there’s always room for a couple of Beers with a friend. ( Anonymous )

Happy New Year

Thank you for following The Mathematics Blog in 2012. 

December 25th - Sir Isaac Newton ( Day )

Sir Isaac newton (1642-1727), mathematician and physicist and  the 25th of December is Newton's birthday.

Maybe, some day in the future, when all religion is forgotten ( or banned ) it turns out that we kept the good things, like Holidays and presents. In that future the people may very well celebrate Newton Day on December 25th.

Ironically Newton was deeply devoted to religion all his life.

Computers are simple

Computers are simple. They must be. Consider how fast the industry developed and the only conclusion you can draw is that it must be simple. Once you understand how easy it is to implement the basic operations add, move, compare and jump to a machine, and that you can build -any- program from these operations you'll know. Alan Turing started it all.

The smartphones of today are so much more powerful than the mainframes banks had in the sixties and seventies in terms of computing power. Over a billion people on Planet Earth live with a computer. These computers connect us with other people and are turning into our best friends. And that is something what was beyond what Alan Turing foresaw about computers.

Computer Science and Mathematics students need a deep understanding of the Turing Machine. OU M381 did not teach the Turing Machine but a slightly simpler model because the Turing Machine was considered too complex. Not anymore with today's simulators if you ask me.

Turing Machine simulator