By Kalev Pehme
In one of the great books of the 20th century, Greek Mathematical Thought and the Origin of Algebra, Jacob Klein delineated the difference between ancient and modern mathematics. How we understand mathematics truly speaks to how we believe the entire cosmos or universe is. From the very outset of the book, Klein writes:
The creation of a formal mathematically language was of decisive significance for the constitution of modern mathematical physics. If the mathematical presentation is regarded as a mere device, preferred only because the insights of natural science can be expressed by “symbols” in the simplest and most exact manner possible, the meaning of the symbolism as well as of the special methods of the physical disciplines in general will be misunderstood. True, in the seventeenth and eighteenth century it was still possible to express and communicate discoveries concerning “natural” relations of objects in nonmathematical terms, yet even then—or, rather, particularly then—it was precisely the mathematical form, the mos geometetricus, which secured their dependability and trustworthiness. After three centuries of intensive development, it has finally become impossible to separate the content of mathematical physics from its form.
No matter how a modern physicist explains what the universe is, in the end that understanding cannot be expressed in any way except in a formal mathematical language. One might theorize about something in the universe, but in the end it was to be put in a mathematical language. The physical contents of the universe and their motions cannot be explained in any other way.
Klein’s work points to a very great difference between the ancients and moderns. The ancient Greek mathematicians, including Plato and Aristotle, when thinking about number, always thought of number as a number of things. There is no three, but there are three elephants, three daisies, three gulls, three triangles. You count off things, three objects. A number is always a definite number of things. There is no three that lives in a conventional Platonic heaven. Of course, one can say that the three young girls and three young men and that the three are somehow equivalent, but they are only equivalent insofar as the two sets of three are of three objects, girls and young men.
The implication of this approach is that numbers determine concrete realities. But more importantly it means the unit, the one, is not a number. The first true number as Aristotle tells us is the dyad, two. The unit, the one, is the beginning or the source out of which the not-one, the many, emerge. On a cosmic level that means the whole of all things is an indeterminate, a non-numerical infinite, which, when made determinate becomes “one.” Thus, the non-numerical infinite that is the whole has a dyadic structure, which in turn allows for manyness. It is like the beginning of base-two number system. It begins with zero out of which one comes and then all the rest of the numbers in series of zeros and ones.
When one understands that numbers are the counting off of objects, we understand that individual objects and things are inherently countable, because the cosmos is both an indeterminate unlimited as well as determined units. There is no such thing as an infinite number, because infinity is an indeterminate and every number is a specific determination of how many objects there are. The ancients would not make an equation in which the symbol of infinity would be placed. Moreover, the world is such and objects are such that the unified being of things of each number is possible.
It is very important to understand what a symbol is. All language as well as the mathematical language communicates through arbitrary sensible signs having a meaning imposed on it by convention. A sign can be perceived by the senses. The cause of the symbol is either from nature or from convention. A cloud, which is a sign of rain, has a meaning from nature. The traffic light red has a meaning from convention. Because symbols are by arbitrary agreement, symbols can be both temporary and permanent. Temporary symbols are the signals adopted by a baseball team, the password to a computer operating system, or the colors of a band. Permanent symbols are the soldier’s salute, the nod of yes, hieroglyphics, chemical formulae, and numbers.
When the new science started in Europe in the late 1500s and into the 1600s, a great change occurred in mathematics. It was to put in the simplest form that a number could be separated from objects as if it existed solely in the mind. Three existed as a pure concept. As Hegel said, the ancients begin with things, the moderns begin with concepts. That allowed for a pure general algebra where symbols alone are used. For example, I can write a+b=c and that takes on an objective reality even though we have no idea what the quantities are. Where once the species unity of things enabled the individual numbers of objects to be counted, the moderns made the symbol species an object itself. In other words, abstractions of the mind (like a+b=c) became real objects in and of themselves, even though they have no reality except the one we give it arbitrarily. With that change Descartes was able to create a duality where the symbols made real were part of the mind while the substance of the world, its corporeality, was thought of as extension. The mind of man has only knowledge of mathematics and what man makes, and the world is a mechanical extension whose true physics works through the imagination of the physicist. The scientist imagines and the pure intellect of man, full of symbols and symbolic mathematical procedures, sorts out the various elements of the imagination to give them mathematical form. There is no real reference to the world at all, only to images of the world in the head of the mathematician. What is critical is that what the mathematician does is to create what we call a virtual world or “virtual reality” today. Oddly, in this scheme of things, the virtual world is actually more real than the sensible world. In fact, it is Cartesian geometry that enables us to use our computers for everything from games to word processing, all in a virtual world. What is real is what is abstract in the mind, while the rest of the world is a machine or works mechanically, including living beings. Animals are machines. It is a mechanical, not an electronic view, so to speak.
When it comes to modern physics, the problem is very simply that it accepts the Cartesian duality as inherently true. A theoretical metaphysical change has to be accepted to be true to allow the mathematics to work as it does. Thus, mathematics exists apart from the world in pure form.
The ancient Pythagoreans, for example, did not work that way. Instead, they gave a mathematical interpretation by analyzing the whole of the cosmos and then they used ratios to give it a mathematical ordering. The octave has a ratio of 1:2 and the ordered world is full of the various proportions. These ratios were the logos (speech, reason, ratio) that is inherently a part of the cosmos. But we moderns don’t think that way. The world has no inherent rationality to speak of, in great part, because what constitutes rational today, a form symbolic language called mathematics, exists apart from it.
This duality thus does something else. It helps to distort thinking about the world itself. For example, while infinity is actually not a number logically, in modern mathematics it can be treated as if it were a number. Today, there are physicists who speak of infinite parallel universes. It is a logical nightmare, yet the in the imagination of the physicist it makes perfect sense because his mind and his imagination, he believes, are absolutely apart from all the infinite universes. All he has to do is to find the math to express that, and then see if he can find an experiment to prove it. (It should be noted that there are many problems with experimentation as well, but that is another story.) The Cartesian duality has already framed the scientist’s mind as to what is reality is, and he cannot think of any other reality except the mechanical extension he thinks it is. Inherent in the very conceptualization of number was an inherent understanding of the world. If, however, that duality is not true, then all of modern theoretical physics must be re-evaluated. While some theories are patently true because they are observable, a vast part of mathematical physics may in fact be false or a distorted version of what is true. Did the Big Bang create mathematics or did mathematics create the Big Bang? It is hard to say when the mathematics is understood to be apart in a special realm of the mind. There may not have been a Big Bang at all.
It should be noted that Descartes attempted to locate the mind in the brain itself, i.e., the mind has a material cause. However, he never went to the point of saying that mathematics is in the brain. Inevitably, it doesn’t take much reasoning to see that the Cartesian duality and the existence of mathematics apart from the world is very problematic, but the theoretical physicists don’t see the problem with their view of the world. The solipsistic character of Cartesian thought means that what he regards as knowledge is simply a kind of self-beholding within the mind while the rest of the material world is not truly knowable, because the mind has no place in that material world. It could be that the entire system of physics that we have may be an illusion in its theories of origin, of parallel universes, and so on. Modern theoretical physics may just be in great part a great imaginative picture has very little to do with the way the universe truly is.
The only remedy is that we have to be able to place our knowledge in the world again. But to do that requires that the universe be a cosmos again, a fully integrated whole which itself is reasonable and accounts for itself through reason, logos, and not just through a formal mathematical language of man-made symbols. That is going to be very hard for the moderns to do.