The Larson Research Center

There’s a tendency in our society to understand the history of human thought as a more or less linear progression from primitive to sophisticated. As we think of Western civilization’s technological progress, from horse and buggy, to manned space flight, it’s easy to view our revolutionary capabilities in science and technology as the pinnacle of human achievement, and to suppose that there is no other way forward, but along the way we have traveled.

However, the ancient Hebrews envisioned our days and characterized them, not as the pinnacle of civilization’s progress, but rather as the deterioration of civilization’s worth, inferior to the quality of previous civilizations. The image of the relatively inferior status of modern nations was explained by the Hebrew Daniel, when he saw and interpreted the king of Babylon’s dream, portraying the progressive degradation of the quality of earth’s civilizations, from that time to this.

According to this vision, the ancient Babylonian kingdom was the highest quality civilization in the world, followed by the inferior, but stronger, Persians, who were followed by the still more inferior, but stronger, Greeks, then by the vastly more inferior and stronger Romans, and finally by the remnants of the Romans, mixed in with the conquering Barbarians, the most inferior, totally fragmented, uncivilized nations of all, who were as clay mixed in with the metal of the Romans. The Romans were as iron compared to the more highly prized bronze of the Greeks and to the sliver of the Persians and to the gold of the Babylonians.

Of course, in the end, all of this is irrelevant, as the vision portrayed all of these old kingdoms as being replaced by a new Hebrew kingdom, which would come rolling forth like a stone down a mountain, smashing the image of Western civilization’s heritage, in all these old kingdoms, to dust. Consequently, the dust of the pulverized image simply blows away, like chaff in the wind, and disappears!

But what does this have to do with modern mathematics and science? We don’t know much about the mathematics and science of the Persians and Babylonians, and what we do know comes to us primarily from the Greeks, who learned from the Babylonians, the Persians, and the Egyptians (who, like the Asians, were never a world dominating nation, but nevertheless were sometimes significant players in mathematics and science).

Clearly, however, the strength of the Persians, relative to the Babylonians, and the Greeks, relative to the Persians, and the Romans, relative to the Greeks, and, in general, the modern nations relative to the ancient ones, is based, in part at least, on the progress of technology. Whether it is based on advanced strategic technology, such as provides greater sustenance, infrastructure and internal strength for the nation as a whole, or on advanced tactical technology, providing for improved weapons, communications and mobility to the nation’s armies, navies, and air forces, technology has always played a crucial role in the strength of civilizations.

The interesting aspect of this in the present context is that, while it shows us how understanding the simple fundamentals of mathematics and science makes a profound difference in the power and technological capabilities of nations, it also shows us that there may be nothing particularly enduring about it either. Civilizations come and go, and the particular aspect of their understanding of math and science that made them capable of great feats of organization, engineering and technological exploitation, comes and goes with them. 

From the smallest of means, proceeds that which is great, the ancients said. For example, who could have guessed that the ability of a few Renaissance scientists to deal with the esoteric concepts of irrational and negative numbers would eventually lead to the modern ability to transcend the technology of the ancients so dramatically? But so it is. Without the ability to abstract the square root of 2 and -1, the whole of modern technology would be impossible.

However, knowing this, we are soon lead to ask what other, simple, fundamentals might we be missing? The fundamentals that some future civilization (perhaps the triumphant kingdom of the Hebrews foreseen by Daniel) might discover, might enable them to transcend our technology as much as we have transcended that of the ancients (or even more).

In thinking about this, one might be tempted to revisit the whole notion of irrational and imaginary numbers, the foundation of modern technology, and seek to understand what it is about this whole approach that makes it so powerful. If there is one way to do this, might there be another, maybe even better way to do it?

Of course, readers of these blogs know that here at the LRC we believe there is, and that we are taking our clues on how to proceed from the works of Hamilton, Grassmann, Clifford, Hestenes, and Larson. Hamilton showed us how defining numbers, as traditionally taken for granted, leaves algebra without a suitable scientific basis. Grassmann showed us that there is an underlying connection between geometry and algebra that the Greeks couldn’t make, and Clifford showed us how the two directions of each dimension forms an algebra. Thanks to the work of Hestenes, which brought the works of Grassmann and Clifford to light, we are provided with tremendous insight into the underlying nature of complex and quaternion numbers, and the imaginary numbers that they are built with.

Finally, none of it would have even caught our attention had it not been for the transcendent work of Larson. It is his brilliant recognition and intriguing development of the new and unfamiliar notions of scalar motion that provides us with the motivation for digging into all these ancient mysteries, driving us to uncover the old foundations, in search of new insight into what makes modern math and physics tick.

What we have found astounds us. Could it be that, as Thales and Pythagoras apparently learned from their predecessors, “Everything is number,” after all? The fact that this faith seemed horribly contradicted by the theorem of triangular squares, that squaring the circle could only be approximated, and that the hare could never catch up with the tortoise, and that today, after centuries of effort, we can now name irrational numbers, use them in the calculus to send robots to explore particular parcels of Martian terrain, use computers to calculate π out to a gazillion decimal places, and work with infinite sets, as easily as the Greeks worked with integers, appears to make the whole issue moot.

“Who cares, if the Greeks thought all was number,” one might think. “Our technology, our math and our science reach so far beyond anything ever dreamed of by the Greeks, that it’s patently clear that we have overcome their intellectual obstacles. Let’s just move on.” Ironically, however, that’s just what we can’t do, and the reason that we can’t do it is that the essence of these very same obstacles stands in our way. We now know that nature is both discrete, definitely measured, like numbers, and, at the same time, continuous, infinitely divisible, like distance.

Yet, in spite of the vaunted “work arounds” of our modern mathematics, which have served us so magnificently, irrational, transcendental, and imaginary numbers, finite and infinite sets, etc, we still cannot do as nature does and seamlessly combine the continuous with the discrete. It is frustrating in the extreme. It appears that, if the ancients taught Thales and Pythagoras that all was number, then they were probably just hopelessly naive and the Greeks were simply beguiled by their priestly robes and their high social status. If we can’t do it today, surely the ancient Babylonians and Persians couldn’t do it either.

That may be so, but it doesn’t mean that they didn’t have a valuable insight into numbers and geometry, which has since been lost, one that might prove to be the key to doing what we so desperately want to do. For instance, even though their approximation of π might have been very rough, compared to our very refined approximation, how do we know that it doesn’t matter, in the end? Of course, barring some unexpected archeological find, we are not likely to ever know more about how the ancients thought than the ancient Greeks did, who were in direct contact with them. The point is not, however, that the ancients had the answers we seek. They probably didn’t, but they may have thought about the fundamentals in a way that hasn’t occurred to us, which could prove to be the key for finding the answers.

As it turns out, there are many intriguing clues that the way the ancients thought about numbers, is close to the new way we are thinking about them here at the LRC. In the next post, we will get into some of the details of this.