Visualizing the Complex Relationship Between Truth and Falsehood
by Paul Hibbert
Once in a while, in that strange time between sleeping and waking, I have the feeling that I have been endowed with radical insight — an epiphany, if you will. Upon further reflection and scrutinizing in the cold light of day that intuition either evaporates, is seen as trivial or is simply erroneous. I don’t think that my experience is particularly unusual as many people have tried to write down the poems, dreams and so on that they think are insightful, at least on the wee, small hours of the night. However, sometimes I experience something that really has given me a somewhat different way of looking at an issue. I will attempt to describe this to you by way of analogy, but first I must clarify what the analogy will be compared to.
Many readers will already be familiar with the Mandelbrot set, a mathematical construct that has fascinated mathematicians, artists and the general population over the last quarter century or so; but for those who aren’t familiar with it, and for completeness, I’ll give a layman’s description. A depiction of the Mandelbrot set is presented as pixels, as on your computer screen. Each pixel is either black or white — black designating that the pixel is within the Mandelbrot set and white, if not. The method of determining the status is to perform an iterative calculation on a particular non-linear function whose variables include its own result, that is, after an evaluation the result is fed as data to the next evaluation until the result either converges to a constant value, or it doesn’t converge after an arbitrarily large number of iterations. If it does not converge it is within the set (and colored black) or if it does converge it is not in the set (and colored white.) This process of analyzing every pixel is very computing intensive and the Mandelbrot set never could have been discovered without computers. It is also interesting that its beauty and significance would never have been appreciated without depicting it graphically.
The set is shown below. Its properties are that its boundaries are infinitely complex, that is, we could magnify the picture to any degree we wished and as long as we continued to evaluate each new pixel as it emerged, we would continue to see new, complex structures. This is demonstrated in the diagram below. Different levels of magnification have the property of self-similarity, as is evident, but that is not particularly relevant to this discussion. Mandelbrot originally thought that the small, black spots barely visible in the diagram were isolated from all the others but it has been subsequently proven that they are all, in fact, connected by thin tendrils that can be determined by further magnification. This is true at any degree of magnification.
 
The degree of magnification is really a measure of the amount of precision we are computing with. For instance, the pixel represented at coordinate (1.0, 1.0) is actually the value computed
for (1.0000000, 1.0000000), if our computer is capable of eight digits precision, but it represents all the myriad of sub-pixels within it that together might display enormous complexity when magnified.
Suppose we dispense with the Mandelbrot set for the time being. Imagine that we can represent a true statement, for instance, “existence exists,” by a black pixel and a false statement as a white pixel. Knowledge is inexhaustible — we will never be able to know everything about everything, thus there are an infinite number of true statements. As well, the world is infinitely complex, as was succinctly expressed by a programming guru: “Everything is deeply intertwingled.” Thus, all true statements can be linked to one another and they must be mutually supportive. All true statements can be derived from “existence exists.” It is my conjecture that it is possible to devise a method of classifying true statements and relating them one to another, albeit in a very complex manner, and depicting these relationships in a plane. Manifestly true statements such as “existence exists” would reside in the center of a large black mass. That statement is the origin of all other true statements and its coordinates have infinite precision. Other, far less qualified statements and contentious statements would lie near the periphery. All statements would lie in a plane but the level of current knowledge would be at one particular level of magnification and a more basic level would be at a coarser magnification. It is a matter of precision of stipulating the coordinates that correlates with the sophistication of the knowledge.
Looking at successive magnifications under the pixel representing “existence exists” would be very boring, indeed, as all of the pixels would be black and they would all be statements of more and more refinement. If one looked at a pixel at the perimeter of the graphic under coarse magnification, as with the Mandelbrot set, that pixel would represent the evaluation at its “highest precision” coordinates but under greater and greater magnification would exhibit the wonderful complexity comparable to the fringes of the Mandelbrot set. The complexity would demonstrate the intricacies of valid and false argumentation concerning the larger question. Black and white pixels would be interlaced but, without exception, the black pixels are ultimately connected.
This graphically demonstrates Rand’s dictum, “If you discern contradictions, examine your premises.” There is no such thing as a grey area with respect to truth and falsehood — you just haven’t examined the logic at a deep enough level. Grey areas are called grey areas because contradictions exist. Graphically speaking, there are adjacent points or areas in the “grey area” that represent contradictory statements and each of those statements must be examined in greater detail. Of course, the argument will be made that even though you come to a conclusion at one level it is possible to discover that at a deeper level further contradictions arise. New knowledge is always discovered and that is to be acknowledged but we can only act on the knowledge that is accessible to us at any given time. So, when you are confronted by liberals who say, “All you people do is think in terms of black and white — you don’t acknowledge that there are grey areas,” you will know that their mental processes are operating according to a wrong paradigm.
I don’t know if logicians and mathematicians will, even in a thousand years, be able to confront the massive problem of classifying all current knowledge and creating a unique system of interlinking true statements and displaying them in graphical form, but in the meantime, I prefer to visualize the relationship of truth and falsehood as I have described.
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