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Why Does the RSM Constitute "New Mathematics?"

Posted on Saturday, September 9, 2006 at 08:17AM by Registered CommenterDoug | CommentsPost a Comment

In the American Heritage Dictionary, mathematics is simply defined as “the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.” In the beginning, mathematics was used for counting, measuring, bookkeeping, etc, but, since the days of Newton, it has also been used for modeling, using a system of differential equations. In applications of theoretical physics, finding the solutions to these equations is a challenge, but has mostly centered on three equations, or their equivalents: the diffusion equation; the wave equation; and the sine-Gordon equation.

These three equations have been studied extensively, and they form the basis of the LST community’s physical theories, including Maxwell’s equations for electromagnetism, Einstein’s equations for gravity, and Schrodinger equation for quantum mechanics. Not suprisingly, since the equations of these systems involve continuous quantities, their adaptation to the development of theories of quantum physics has worked only to a limited degree, reflected in the fundamental crisis of theoretical physics that we are currently discussing in the The Trouble with Physics blog on this site.

Of course, if energy and matter are quantized, as experiments show they are, this suggests that the use of a discrete system of some kind, as opposed to a continuous system of differential equations, might be a more suitable approach to use, in seeking a solution to the fundamental crisis of physics. In fact, Stephen Wolfram, in his recent book, New Kind of Science, declares that his results show that a discrete approach would greatly simplify the task. He writes:

One of the most obvious differences between my approach to science based on simple programs and the traditional approach based on mathematical equations is that programs tend to involve discrete elements while equations tend to involve continuous quantities.

But how significant is this difference in the end?

One might have thought that perhaps the basic phenomenon of complexity that I have identified could only occur in discrete systems. But from the results of the last few sections, we know that this is not the case.

What is true, however, is that the phenomenon was immensely easier to discover in discrete systems than it would have been in continuous ones. Probably complexity is not in any fundamental sense rarer in continuous systems than in discrete ones. But the point is that discrete systems can typically be investigated in a much more direct way than continuous ones.

Wolfram’s thesis is that it’s not only easier to study nature via a discrete system, but that it’s possible to conceive of an entirely new system of physical theory, which is not based on the traditional system of mathematics, but on a new system of mathematical rules. He writes:

If theoretical science is to be possible at all, then at some level the systems it studies must follow definite rules. Yet in the past throughout the exact sciences it has usually been assumed that these rules must be ones based on traditional mathematics. But the crucial realization that led me to develop the new kind of science in this book is that there is in fact no reason to think that systems like those we see in nature should follow only such traditional mathematical rules.

What has led Wolfram to this conclusion is his discovery that very simple computational algorithms can lead to “behavior that [is] as complex as anything I [have] ever seen.” Though many might have missed the fundamental significance of this discovery, it’s impact on Wolfram was profound:

It took me more than a decade to come to terms with this result, and to realize just how fundamental and far-reaching its consequences are. In retrospect there is no reason the result could not have been found centuries ago, but increasingly I have come to view it as one of the more important single discoveries in the whole history of theoretical science. For in addition to opening up vast new domains of exploration, it implies a radical rethinking of how processes in nature and elsewhere work.

Coming from Wolfram, whose academic and scientific accomplishments are legendary, and who has made a fortune by exploiting the principles of traditional mathematics, and his ability to write software to formulate and solve complex equations from symbolic input, this is not a case of idle speculation. Like Larson, he understands that new ways of thinking are what are needed to successfully advance the scientific knowledge of mankind beyond today’s science. He confesses:

It is not uncommon in the history of science that new ways of thinking are what finally allow longstanding issues to be addressed. But I have been amazed at just how many issues central to the foundations of the existing sciences I have been able to address by using the idea of thinking in terms of simple programs…Indeed, I even have increasing evidence that thinking in terms of simple programs will make it possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known features of our universe will emerge.

However, as might be expected, the LST community is not very receptive to this new approach. Nobel prize winner Steven Weinberg, in reviewing Wolfram’s book, in an article entitled, “Is the Universe a Computer?” writes:

Wolfram himself is a lapsed elementary particle physicist, and I suppose he can’t resist trying to apply his experience with digital computer programs to the laws of nature. This has led him to the view (also considered in a 1981 article by Richard Feynman) that nature is discrete rather than continuous. He suggests that space consists of a network of isolated points, like cells in a cellular automaton, and that even time flows in discrete steps. Following an idea of Edward Fredkin, he concludes that the universe itself would then be an automaton, like a giant computer. It’s possible, but I can’t see any motivation for these speculations, except that this is the sort of system that Wolfram and others have become used to in their work on computers. So might a carpenter, looking at the moon, suppose that it is made of wood.

While Weinberg sees no reason to suppose that Wolfram’s ideas represent a promising alternative to the mathematical equations of traditional science, where the structure of the physical universe is described in terms of continuous systems, this opposition is primarily justified by an attempt to show that Wolfram has jumped to conclusions over just how fundamental and far-reachingthe consequences of his simple programs are. However, it’s quite evident that Weinberg is simply protecting his own turf as a particle physicist. He writes:

Wolfram claims to offer a revolution in the nature of science, again and again distancing his work from what he calls traditional science, with remarks like “If traditional science was our only guide, then at this point we would probably be quite stuck.” He stakes his claim in the first few lines of the book: “Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world. My purpose in this book is to initiate another such transformation….

Nevertheless, Weinberg uses the very same word “stuck” to characterize the state of theoretical physics in his own book, Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature, so the issue is not a matter of the efficacy of traditional science. Everyone agrees that modern physicists are “stuck” and that they are finding it very difficult to advance the laws of physics towards a “final theory,” in which the concepts of discrete and continuous phenomena are unified into one physical theory. Hence, while Weinberg may be very skeptical of Wolfram’s approach, the motivations for seeking a discrete system solution to the theoretical crisis should nevertheless be obvious to everyone.  Certainly, the obstacles to considering discrete systems in physics cannot be based on a lack of merit in the notion of discrete systems themselves, since the structure of nature is undeniably discrete in a very real sense.

On the other hand, in recognizing the discrete nature of physical phenomena, there is nothing to restrict us to considering only the rules of discrete forms of simple programs, like cellular automata. It’s important to recognize that natural numbers themselves are discrete and the algebra of their mathematical operations exhibits the same properties as do the interactions of physical objects; that is, when various numbers are combined into greater composite numbers, or existing combinations of numbers are separated into their lesser constituent numbers, the new combinations, or constituents, are totally predictable, given the parameters of the operations acting upon them.

While the vectorial motion of objects is clearly continuous, so that a differential equation gives a relation between the value of some varying quantity and the rate at which that quantity is changing, and perhaps the rate at which that rate is changing, and so on, this means that it is the relationships of the continuously changing numbers represented by the equation that are important, and it is the cause of that change, a force, or a quantity of force, an acceleration, that some of these numbers represent, that is ultimately what we seek to understand in terms of identifying the laws of nature.

For example, if we have a solution to a system of differential equations that can predict the path and energy of a particle through a known environment, given its velocity, mass, charge, spin, etc, using the correct numbers, this means that we can say something definitive about how these properties relate to one another in determining the universal behavior of physical phenomena, and we can classify them accordingly. Indeed, this has been the grand goal of the traditional program of physics, inaugurated by Newton himself.  The description of the structure of the physical universe, in terms of a few fundamental interactions among a few elementary particles, culminates in the finest intellectual achievement of Newton’s program of research, called the standard model of particle physics.

However, when it comes to the physics of discrete entities in the microcosmic realm of the quantum world, ordinary numbers are not sufficient. The solutions to the equations not only have negative roots, which has given rise to the idea of antimatter, but it is impossible to describe the physical phenomena involved, using these equations, without employing complex numbers, which are indispensable, given the concept of uncertainty; that is, we not only need differential equations to deal with discrete phenomena on a continuous basis, but we also need to use imaginary numbers in these equations.

This is a significant point, because the idea of imaginary numbers is an ad hoc invention of the human mind, originally devised to deal with negative numbers (as the square root of -1). In fact, without imaginary numbers, the traditional idea of higher dimensional numbers, such as complex numbers (n1), quaternions (n2), and octonions (n3), would not be possible (except with Hestenes’ GA, where ‘i’ has a geometric significance).  All numbers would be limited to the set of positive, real numbers (n0), and, needless to say, this not only would make the LST community’s formulation of quantum mechanics impossible, it would make all of LST physics impossible, because it would make the concept of vectors impossible.

This is not hard to understand, because it’s the imaginary number that gives numbers direction, that turns these discrete quantities into a something that can be used to define more than simple quantity. With the imaginary number ‘i’, a number can be written that represents a point in any direction from 0; that is, a complex number is not just a point on a number line, that is greater than, or less than, some other number on the line. However, modern mathematicians are quite amazed that the human invention of the imaginary number by mathematicians would turn out to be indispensable to the quantum mechanics of physicists, centuries later.

Yet it’s not improbable that school children of future generations will ask why the mathematicians of our day were so amazed, because to them it will be very clear that direction is a property of motion, and, since physicists study motion using numbers, numbers clearly should have a direction property too.  However, the insight the children will have then, which modern mathematicians and physicists don’t have today, is that numbers do have direction. In fact, they have always had the direction property that was needed, and it had been used routinely in other contexts since the dawn of civilization; there was really no need to invent the “imaginary” number to “give” numbers the direction property.

The truly amazing thing to contemplate is what we claim here at the LRC: Recognizing the inherent property of direction that numbers possess, without invoking the magic of imaginary numbers, revolutionizes the mathematics of physics, transforming it from a continuous system into a discrete system from which the continuous system, and its real numbers, emerge naturally.


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