Mathematics, the universal language. Is it really? How stable is mathematics? I'm not questioning its stability or suggesting that anyone should do so. What I am saying is this: always look for the question. Mathematics is the language of the question; it is the formulation of the question. We tend to, and are taught to think of mathematics as the answer. I offer this opposing viewpoint: Mathematics is the question.
What in the known universe is stable...unwavering? The ground we stand on is hurtling through space; an earthquake teaches us that the terra firma isn't firma at all. Stars are exploding, nutrinos are everywhere. Matter and antimatter have existed in harmony for millions of years. Man enters the picture and starts to question. Out of these questions, mathematics arose with the intent of answering the questions. What happened? More questions; better ways to ask the questions that we want answered. Plato, Pythagorus, Euclid, Kepler, Einstein, Newton, the masters and students of great minds all asked questions through mathematics.
Hypothesis, theory, and natural law arise using the scientific method. This is the basic premise of any scientific endeavor including mathematics. Plato defined the platonic solids (cube, tetrahedron, octahedron, icosahedron, and dodecahedron). Kepler spent his life trying to nest the platonic solids to represent the orbits of known celestial bodies, Newton defined gravity, Einstein questioned Newtons' concept of gravity. All these folks came up with some amazing stuff but Kepler didn't succeed in doing what he was driven to accomplish; neither did Einstein. Why? Because great minds never stop. There are always unchartered frontiers for great minds. Great minds know how to suspend disbelief and ask the seemingly unanswerable question.
Nature does things in a logical way according to Phi (not Pi but Phi). Phi is the golden proportion; we see Phi reflected in the human body, growth of plants, and crystal formations, to name a few. Phi is mathematics. Crystal formation, molecular chains, the DNA helix, atomic structure, etc. are more easily understood if one studies the golden proportion and sacred geometry (including the Platonic solids).
Phi (pronounced FEE or FI) is the perfect proportion that occurs in nature everywhere. If you take a line segment and erase the middle third you have two lines. Then you erase the middle third of each of those lines and you have 4 lines, you can keep doing that into infinity and you will always have lines that are Phi proportional. The erased segment of the lines is Phi-proportional to the nonerased segments. Your hand is Phi proportional to your forearm, waist to feet is proportional to body height, joints of fingers are proportionally Phi, plant stems, flowers, buds, cones, and leaves are Phi proportional. Phi is mathematically ~ 1.618; its an irrational number so the .618 is followed by an infinte number of numbers. The mathematical formula for Phi is 1 + sqrt(5)/2 (read as: the quantity 1 plus the square root of 5 divided by 2). We can calculate the Phi proportion for any known length or distance. Nature calculates Phi all the time; when stems grow from a main stem of a plant the stems are distanced according to Phi.
Mathematically, if a whole line segment is called "A", Phi is the position of a point on that line. The larger segment we can call "B" and the smaller segment we call "C". "A" is 1.618 times "B". 'B" is 1.618 times "C". Going the other direction, "C" is 0.618 of "B" and "B" is 0.618 of "A". If you're anything like me you need a visual to understand this concept:
Why is understanding Phi important? Because humans tend to think of balance as equal portions; balance is, rather, Phi proportions. Equality does not equal equality in the natural world. Equality is imbalance in the natural world. I have no surprise that humans exist largely in a state of imbalance; our training and subsequent understanding is incorrect. We try to balance according to what we've learned.
For an in-depth discussion of this subject, please visit Dr. Ron Knott -- The Golden section ratio: Phi.
Bear with me on this as I ramble on please. The pattern of stars in relation to what we see on Earth is shifting and has been doing that for millions of years. This actually has a name: precession. Scientists say it's because the earth rotates on an axis that is tipped 23 degrees off center. Now think of the Dodecahedron overlaid on the Earth made up of Hexagram with shapes of DNA strand-like energy patterns going away from the earth at the Hexagram points. Then think of Dodecahedron shapes surrounding the Earth in space so it's like nested Dodecs. If you drew two parallel lines from one point of a hexagram on earth and connected it to the corresponding point on the next Dodec out in space, since the earth axis is tipped, the the lines connecting the two would twist just like a DNA helix is twisted. At this point you might be wondering, "Hey CJ, how far out there can you actually think?" Well, the answer is " pretty far out there especially since I rarely draw a solid conclusion from my mental meanderings."
If I could build a wire frame model I'd probably drive myself crazy like Kepler did. The inner wire frame shape would represent the earth. The next one out would represent stars that are near the earth and so on until there was a whole bunch of nested Dodecs (I'd probably stop at 2 dodecs though). On the outer dodec, however, I'd line it up so the stars sit at the outer points of the triangles that make up the hexagram shapes that comprise the Dodec. Man, I hope I don't actually get obsessed about this and build this thing! But if I did do this I'm wondering how certain celestial bodies would be placed right on those points I'm talking about? If they were right on those points, this would show something very interesting. More questions? Yes, no doubt.
NP-Complete Challenge
This paper discusses the logic and challenges of NP-complete problems and efforts to solve those problems. I was warned that this topic was "ambitious." That observation is an understatement. I spent many hours researching, learning terminology and concepts, talking to myself and others, and drawing pictures before I understood enough about NP-complete to begin to write about this subject. I discovered that the solution to NP-complete challenges may lie in our ability to approach these problems outside of the learned, three dimensional box around mathematics.
Before proceeding, I must reveal my own world view (bias) of the subject matter. I suspect the interconnectedness of matter at a quantum level. Extracting information at one level of the macrocosm pulls a microcosmic thread that unravels the fabric of logic at the higher levels. Historically, the natural sciences have relied upon mathematics; now the tables are turning. My research revealed that many mathematicians and computer scientists are focusing on physics and the natural sciences when discussing P and NP problems. Perhaps quantum computing is inverting the traditional relationship between math and science. My world view may be a trivial example of the aforementioned inversion relationship; perhaps the inversion is not unlike the operation of a non-deterministic Turing Machine.
Terminology to discuss NP-complete problems is vital to understanding. You may be familiar with the following terms gathered and condensed from several sources including Wikipedia (Wikipedia(1), 2005).
Turing Machine (DTM) -- The algorithm, now called a Turing Machine, was introduced by Alan Turing in 1936 (Bhaskar, n.d.). This algorithm is based upon reading an infinite length of tape with a finite number of symbols written on it. A well-defined procedure can be implemented such as, "If the current state is 'A' and the position contains a '0' then replace the '0' with a '1', move to the right and assume state 'B'. We can see that this algorithm is a loop.
Non-deterministic Turing Machine (NDTM) works like a DTM but branches out rather than following a single computational path.
Polynomial Time is time based upon the length of the input and is considered fast computation. Conversely, super-polynomial time is anything slower than that.
Decision Problems are merely asking the question, "is there a yes/no or true/false procedure that can be implemented to reach a decision."
P Problem (P) is a class of decision problem that is solvable in polynomial time using a DTM.
NP Problem (NP) is a class of decision problems that are solvable in polynomial time using a NDTM. NP Problems can be verified by a DTM in polynomial time.
NP-Complete (NPC) is a class of decision problems that are least likely to fit into the P problem category.
The Clay Mathematics Institute offers a $1,000,000 prize each for solutions to its seven millennium problems (Clay Mathematics Institute, 2000). One of the millennium problems is "P vs. NP." A basic mathematical assumption is that P ? NP although mathematicians have not resolved this issue. This is no small matter. NP-complete problems are the acid test of P = NP.
The essence of the question asked in an NP-complete problem is whether a computer can verify, in polynomial time, that an answer is correct but cannot compute the answer in polynomial time (Wikipedia(3), n.d.). For example, it is a simple task to verify whether a number is a factor of another number, but to extract the factors is more difficult. We know that 6 is a factor of 36 and can verify easily whether 6 is a factor of 21,493, but we cannot easily determine the factors of a 12-digit number. We can write an algorithm to determine the factors of that number, but we cannot guarantee that the algorithm will solve the problem. Why not?
Alan Turing, in his paper On Computable Numbers, with an Application to the Entscheidungsproblem, proved that a universal Turing machine is unable to determine whether any Turing machine will halt after a finite number of steps. In other words, the "halting problem" is undecidable (Bhaskar, n.d.). Once a computer embarks on a calculation there is no way to determine whether it will produce an answer (Yates, 1998). This is known as the halting problem. We see that the halting problem exacerbates the P vs. NP question: is P = NP?
Christopher Langton, a computer scientist at Los Alamos National Laboratory, illustrates the DTM with his simple, two-dimensional concept called Langton's ant (Wikipedia (2), 2005). The defined procedures of Langton's ant allow a 90 degree turn to the right or left, the ability to flip over black and white squares, and a one-unit forward move. This is simple behavior that produces 104 steps of a continuous loop. In other words, the program never halts. The ant continues to move and flip colors indefinitely. If we apply this concept to problem-solving, it is clear that we cannot predict that an answer will ever emerge. This conundrum is the foundation of P vs. NP and one of the reasons why NP problems are considered difficult. Using scientific theory, we must always question a hypothesis and some question whether NP problems are actually hard.
Writers on the Jennifer Dodd blog at the University of Queensland, Australia pose the hypothesis that we cannot solve NP problems until we determine whether the concept of unsolvable difficulty is valid. One writer suggests that we use the history of early physics as a "jumping off point" to begin the arduous task of defining the difficulty in solving NP-complete problems. After all, we must have a place to start (Jennifer Dodd Blog). Quantum physics questions the conservation of matter and energy atrophy; why not question NP-complete from the perspective of the natural world (Jennifer Dodd Blog)?
Paul Rothemund, Nick Papadakis, and Erik Winfree from the California Institute of Technology are exploring this question from that position. They state, "Engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks" (Rothemund et al, 2004). It is important to note that Rothemund et al are approaching this from the perspective that NP decision problems are not only solvable by biological science but are an intrinsic component of biological science.
Scott Aaronson, at the Institute for Advanced Study at Princeton, also objects to the hasty conclusion that NP-complete problems are "hard." He hypothesizes that NP problems have not been solved because computer scientists tend to view speculative models as a "diversion" to "serious work" (Aaronson, p. 1). Secondly, Aaronson poses that physicists do not see computational theory as relevant to science (Aaronson, p. 1). Aaronson may be posing a groundbreaking question when he asks, "Can NP-complete problems be solved in polynomial time using the resources of the physical universe?" Aaronson concludes his article by saying, "The NP Hardness Assumption is the belief that such power will be forever beyond our reach" (Aaronson, p. 20). In other words, according to Aaronson, NP problems may be hard only because we think they are hard. Can we break out of that assumption by thinking outside of three dimensional time and space?
The simple algorithm of Langston's ant produces a pattern that appears to emerge out of chaos much like what occurs in fractal design production and molecular self assembly (Rothemund et al, 2004). Langton's ant is two dimensional; the fractal designs that we actually see are three-dimensional but appear two dimensional only because we currently have no technology to represent the results in more than two dimensions. If we can visualize the third dimension, which is relatively simple since we live in three dimensions, we gain an essential, basic understanding of the computational paths of NDTMs at a mentally-controllable level.
Human reliance upon time and location (components of three-dimensional space) prohibits us from visualizing the constructs of a NDTM when the computational paths are more than three. We cannot currently visualize 10-dimensional arrays, but computers use them. Is using a 10-dimensional array harder for a computer than utilizing a two-dimensional array? From a two or three-dimensional perspective, the answer is yes because more memory and processing time are required. However, from a multidimensional perspective, outside of linear time, the complexity of dimensions is not a factor. All matter simply appears wherever it appears without consideration for physical location of that matter. In other words, if we take time and location out of the equation, the number of dimensions is irrelevant. Rothemund et al discovered that molecular self assembly behaves like a NDTM, without regard for halting errors or superpolynomial time. Proteins do not behave according to learned concepts of difficulty. In other words, for proteins, NP decision problems are not "hard."
Our efforts to solve P vs. NP and to solve NP-complete problems may only be hard because we have no association in our experience that allows us to easily determine the solution. Humans focus on working harder if an answer evades us. P vs. NP, the halting problem, and NP-complete verification may all be solved by using physics rather than mathematics. Clearly, the proteins in our bodies solve all of these problems without the benefit of any of our mathematical tools.
References
Aaronson, S. (21 Feb 2005). NP-complete problems and physical reality. Retrieved on October 28, 2005 from http://www.arxiv.org/PS_cache/quant-ph/pdf/0502/0502072.pdf.
Bhaskar, S. K. (n.d.). CMIS102: Introduction to Problem Solving and Algorithm Design. Retrieved on October 9, 2005 from http://tychousa6.umuc.edu/cgi-bin/id/CDI/index.pl?module=1&default=M1-Module_1/S1-Overview.html.
Clay Mathematics Institute. (24 May 2000). Millennium problems. Retrieved on November 3, 2005 from http://www.claymath.org/millennium/.
Jennifer Dodd Blog - University of Queensland. (16 June 2005). Can computational complexity theory motivate physical principles? Retrieved on October 28, 2005 from http://www.physics.uq.edu.au/people/jdodd/?p=13.
Rothemund, P.W.K., Papadakis, N., Winfree, E. (2004). Algorithmic self-assembly of DNA Sierpinski Triangles. PLoS Biol 2(12): e424. Retrieved on November 6, 2005 from http://biology.plosjournals.org/perlserv/?request=cite-builder&doi=10.1371/journal.pbio.0020424.
Wikipedia (1). (24 Septr 2005). Clay Mathematics Institute. Retrieved on October 9, 2005 from http://en.wikipedia.org/wiki/Clay_Mathematics_Institute.
Wikipedia (2). (12 Oct 2005). Langton's ant. Retrieved on November 3, 2005 from http://en.wikipedia.org/wiki/Langton%27s_ant.
Wikipedia (3). (n.d.). NP-complete. Retrieved on November 9, 2005 from http://en.wikipedia.org/wiki/Np-complete.
Yates, M. (May 1998). What computers can't do. Retrieved on November 3, 2005 from http://plus.maths.org/issue5/turing/.
The Fibonacci Sequence was discovered by a guy named Leonardo Fibonacci in Pisa. You can read about him and the sequence on Wikipedia. The mathematics of the sequence is simply to start with a zero and add the numeral 1, which equals 1. Move over one to the right and sum the two numbers, move to the right and sum two numbers. Sound like something Turing might have thought of? You get an infinite sequence of numbers that starts like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584...
0+1=1, 1+1=2, 1+2=3, 2+3+5 and so on.
So what's the big deal? The sheer elegance of this sequence is prevalent everywhere in nature, is based upon Phi and defines the golden mean spiral. If you go to our Links page you'll see a couple of excellent links under Mathematics that explain how Phi, the golden mean spiral, and the Fibonacci sequence relate to one another.
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