Euclidean or Non-Euclidean: THAT is the Question

A well-known person once chimed that universities everywhere hire entire departments full of mathematicians. He added "They owe their careers not so much to math as to economics. This is because they are hired for the singular reason that it is much cheaper to employ them than to have them institutionalized." Not everyone shares this apprehension of the math department, but let's face it, the discipline can be a little frustrating at times.

Euclid of Alexandria (b. ca. 323 B.C., the same year Alexander of Macedon died after conquering the known world -- Aristotle died the next year -- as times were tough on smart people) is something like the father of basic geometry.He was born during the reign of the next guy who ruled after Alexander, Ptolemy I (called "Soter"), one of Alexander's generals. Many of the smartest people in the ancient world found employment in Alexandria (Egypt), because the rulers favored education to a great extent and spent a good deal of time and money on cultural achievements.

They even built an enormous library there. It was well-known in the ancient world, until someone very controversial burnt it to the ground many centuries later. Libraries were something like fire-magnets in the ancient world. This library experienced several of them over the centuries.The dividing of the Macedonian Greek kingdom into four parts happened, if you recall the book of Daniel, just as God had prophesied by Daniel the prophet, and so the kingdom divided -- one part for each of Alexander's generals.

The Ptolemies -- you say the family name first in the middle and far eastern cultures in most cases -- ruled all the way down to Cleopatra (c.a. 30 B.C.). Then the Romans took over, as the Romans were wont to do. Euclid came up with a set of axioms -- basic rules -- from which he ably derived a system of geometry -- explaining the relationships of all the different parts of familiar shapes like triangles and squares, even parallelograms. He even came up with a way to justify the answers he gave.

Later this "Euclidean geometry" proved very useful for a number of great engineering feats and different kinds of inventions. His original work, Elements, has turned out to be the most influential "mathematical bestseller" in the history of math. Additionally, Euclid's logical rigor, reasoning carefully from a set of first principles by deduction to render his conclusions ended up informing the way most mathematicians still do their work -- even if their math is much more complex today.

A new development, the use of computers and programs to do sophisticated (or simpler) math at high speeds has given rise to other kinds of geometry called "Non-Euclidean," because the categories of Euclids writings don't help much in these newer fields. One such field, which only came about after the invention of computers -- very recently -- is called "Fractal geometry."

Here comes your latin word for the day -- "Fractus" (sounds like cactus) which means "broken" or fractured. This refers to the way the generated images appear, with endless branches, extending now this way and now that, in a somewhat jagged fashion.A man named Benoit Mandelbrot (Ben - NOIT Man-dul-BRAHT) came up with a startling object he created (most say discovered) one day using a very simple formula. But he had his computer run through this formula in a recursive fashion many times and then plot the numbered points (called "coordinates") on a graph.

The picture it yielded had some very fascinating characteristics, one of which was that -- no matter how far you "zomed in" on the image, it kept reproducing smaller copies of itself. It seemed almost infinitely complex, even though it was generated by a very simple formula involving only a few numbers (using Z, then a small z, and a c as variables) and two functions -- addition and multiplication. Filtering a set of numbers through a formula -- over and over -- using the output of the previous run-through as the input for the next one represents a kind of feedback loop which in math they call the process of "iteration."

The Mandelbrot image is an iteration-generated image. The iteration process forms the heart of "fractal" geometry -- it employs the iterative process (or sometimes more than one at the same time) to generate images who characteristics are then studied for comparison with real-life objects -- tree leaves, mountain range tops, cloud formations, oceanic wave characteristics, and anything formed involving processes that seem somewhat random to us. They also study such images to find out how to put them to good use as a tool to develop really cool technology. If you visit wikipedia, you can see the now famous "Mandelbrot image."

It has taken on something of a mythical (maybe even mystical) reputation among some mathematicians. Personally, I thought it had the appearance of a somewhat unimpressive -- but very curious -- mosaic. Reasons for this zeal vary, but the shape resembles in certain ways -- in part usually -- objects with which we are already familiar - from tree leaves to insects. The biggest part of it looks like a heart-shaped something ("cardioid").

Every Christian should know a little something about fractal geometry, especially homeschoolers. There are several really good reasons for this.

1. It has in its very short life span already produced amazing results in several areas of technology and promises to bring about the next technical revolution in science. This post is a heads-up, aiming to put homeschooled Christians in the driver's seat of the next wave of technological innovation.

2. Because of the extreme success it is likely to have, the pagan world will likely grow in its mystical (or at very least mythical) marvel of the work of their own hands. Mathematicians already refer to the Mandelbrot image as -- I am quoting them -- "THE fingerprint of GOD." The greater the success, the greater the idolatry is sure to become in this regard. This post - like all my posts - is about apologetics (even if accidentally).

Here, Christians can avoid the mathematical idolatry associated with mystical views, while also learning something of a great and powerful application of mathematics, greatly suited to fulfilling the dominion mandate.

3. This field of math has extraordinary potential for use in the medical sciences to alleviate some of the many diseases and conditions whose cure or repair we now regard as impossible or even absurd. This is possible because fractal geometry has already shown it is well able to mimic the biological and crystalline structures that produce the objects around us. For instance, fractal geometry can produce on a computer screen "Growing trees," which show remarkably similar features -- including the diversity of tree features -- to actual trees. You have to see some of the computer-generated images to understand this well.

It can also enable satellites to "see" what the cannot see directly -- filling in image gaps using images generated from the known section of what is viewed, and then by extrapolating the missing section. It can also do incredible things with image and resolution enhancement. This is just the very beginning of the sciences destined to arise from this non-Euclidean geometry.

4. No matter which professional field a Christian goes into, law, medicine, teaching, whatever -- you will need to know something about this field of study because it will likely impact your work in ways you now cannot imagine (just like you could not imagine all the stuff you'd be doing on your cell phone 20 years ago).

5. Mathematicians and scientists working in this area speak just like creationists -- to a man and woman. The extreme complexity they find in simple instruction sets neatly mimics just what DNA does in growing "us" from a very limited supply of four basic proteins. In other words, they have stumbled upon a great analogue (methodological way of mimicking) not only how things replicate -- we know a good deal about how DNA copies itself -- but also how they grow as the replicated instruction sets (originally created by God) -- at least partially responsible for life on this planet -- execute their biological programs.

My inference is this: if you can mimic what the life code does analogically, you could (in principle anyway, applications always introduce unexpected hurdles) teach cells to grow new limbs, cure debilitating disease etc. This means -- from the standpoint of scientists -- this is something of a quantitative and pictorial analogy (i.e. derived, not original) to the biblical concept of creation -- and the more it succeeds, the more silly Neo-Darwinism is going to appear.

Scientists instinctively recognize that they are creating images using iteration -- simple processes repeated many times -- to produce extremely complex and very life-like pictures of real objects. So it appears to them that these real-life objects were ALSO CREATED by a very wise Intelligence using some form of reasoning, at least in some ways analogical to fractal geometry. And this methodology manages what seems an infinitely complex product, and yet remains very simple in terms of the basic elements used. Of course God created originally (ex nihilo), and scientists only create in a secondary fashion.

This is the nature of imagination. And remember, no theory must be true in order to work well. Knowably false ones often yield remarkable results in the history of the sciences. My point is that fractal geometry can imitate real-life quite well, so it has the potential of yielding great real world results or powerful technology. This does not imply that any theoretical notion associated with it is actually true. But most scientists do not realize this. They tend to assume that if it works well, the theory is likely true (the fallacy of affirming the consequent).

Apologetics punchline: Darwin's ideological days are numbered. Given the ordinary assumptions with which scientists operate, Mandelbrot has effectively killed Darwin. Christians need to know this. Every time they say "Darwin," you say "Mandelbrot." This shows that any one field of science -- because the sciences presently are not rooted in the Bible -- may yield results incompatible with the assumptions or conclusions of another science. This also happens with some regularity in the history of the sciences.

6. It is not known what the limits -- every science has them -- of this new science will be. A little history goes a long way. Let me tell you what will happen -- first stage of new scientific revolution -- extreme enthusiasm -- the new science will fix everything.

Second stage: the new science yields extraordinary results in a few key areas with wide-ranging implications, inlcuding economic ones, bringing a higher standard of living for many.

Third stage -- Mandelbrot is an Einstein -- the new superman (this is how the story always goes) -- and he discovered what they call "Gods fingerprint." You can find similar use of overly zealous language in the early 1920's regarding Einsteins theories of general and special relativity (The truth is that humans -- we -- are God's image, bearing his "fingerprint" on our consciences -- so don't buy the baggage but enjoy the results of technological advances).

Stage four -- the science begins more obviously now to encounter its real-world limits and the money poured into it has reached the point of diminishing returns. Still, a few new applications encourage the brethren and the mytho-speak stays alive in certain circles, but the science is now considered more commonplace and the "wow" factor gives way to the "already knew that" factor.

Stage five -- a new science arises with all the promise of what this one was thought to carry, and THAT new field or theory becomes the bearer of the ring (also called by scientists "My precious").

7. This one however does have a few unique features -- it actually looks like a great way to imitate (in a derivative fashion --not actually creative but skillfully manipulative of) God's handiwork -- which job it is of all men to think God's thoughts after Him -- and so this may actually produce not only its own great results, but could create a kind of synergy -- enhancing mulitple sciences already in existence -- greatly accelerating the rate of new discoveries in many different -- even apparently unrelated -- arts (i.e. architecture and the graphic arts) and sciences simultaneously -- or nearly simultaneously.

Questions to ask: 1. Who is Euclid of Alexandria?

A. Euclid was the father of modern geometry, whose work, Elements had a profound and lasting impact on the way math is done and the way mathematicians prove their answers are right.

2. What is "fractal geometry?"

Fractal geometry is a field of mathematics which uses the iterative process to make, and also to study, highly complex images useful for new discoveries not only in math, but in many scientific fields, including the biological sciences.

3. Describe the process of "iteration." Give a simple example.

A. The process of iteration begins with a number (or set of numbers) and runs them through a formula which changes the numbers as directed -- adding, mulitiplying, subtracting or dividing (there are other functions too) them -- to produce a new number (or set) to be used as the next field of input on the same formula.

Example. Start with the number "3." Add to it half of its total each time you run the loop. First it yields "4.5" then it adds 4.5 + 2.25 -- yielding 7.75, and so on. The numbers increase only a little at first and then sharply. When these points are plotted on a graph, they produce a slope. if you do this with multiple series at the same time -- you get an image or picture -- made up of overlapping slopes. Don't worry. Start small. It ain't rocket science or brain surgery -- at least not yet.

4. Why was Alexandria important in the ancient world? Alexandria for many centuries both before and after the birth of Jesus became the education hub of the mediterranean world. Hero of Alexandria, for instance invented the first steam engine there in the first century A.D.

5. Who is Benoit Mandelbrot? This names the man who came up with the startling image - computer-generated -- which shows what appear to be either extremely or infinitely complex features -- one of which is self-replication at progressively lower (smaller) levels (or higher levels of magnification). Scientists continue to study this image to determine its nature, structure and attributes for application in several sciences (including image compression in the computer sciences and resolution enhancement in the optical sciences and photograhic arts, to name but a few).

6. Why should Christians know a little something (at the very least) about fractal geometry? This promises to be the mathematical basis for the next scientific revolution and Christians are supposed to be culturally aware, as well as cultural leaders in fields such as math and science. It also provides an excellent analogy to the view of creation Christians hold to be true and seems to refute the most basic concept inherent in all popular forms of evolutionary theory -- that randomness produces high orders of structural and functional complexity.

In fractal geometry, extremely complex structures instead develop over time (seemingly on their own) by following a simple set of instructions from intelligent persons. This provides an analogy for just what Chrstians claim happened "in the beginning."

Moreover, Christians recognize that the Bible itself comes in the form of very simple languages -- a fairly rudimentary form of Hebrew, and Koine Greek -- from ancient cultures -- and yet contains the extremely complex wisdom of the Lord of Wisdom and Knowledge. Simple sets of instructions do not entail a "primitive" message or product -- but may instead prove fascinatingly (or even infinitely) complex. Neither numbers, nor time, appear to have a limit at the upper "end." You can always add a one to the highest number derived.

The same may be true of other aspects of human experience. The Bible says that "God has put eternity [i.e. temporal infinity] in the hearts of men." Presumably this means that the human imagination could invent endlessly. If "of the making of many books there is no end," why not of technological advancements also? Books comprise the source of education and technology is the application of it. If it can be true of math -- a man-made set of axioms developed into a system -- how much more the Word of the Living God.

It is simple in appearance because God condescends to our weaknesses. So it's language is often very plain. But the Bible contains the most poweful message you will ever see, Mandelbrot's "fingerprint" claims notwithstanding.

7. If Euclid and other mathematicians were so smart, and fractal geometry relatively simple, why didn't someone invent it before 1980? Answer: fractal geometry, though based on simpler sets of instructions carried out to an extraordiary extent, could not have been invented until the advent of computers. This shows how one field of technology or science can facilitate others. This is called "intellectual synergy." The team is more than just the sum of its individual parts.

Want to know more? http://www.wikipedia.org/ Learning happens.