Seeds of a New Renaissance:
THE THREE LEVELS OF MATHEMATICS

by Jonathan Tennenbaum

Conference Speech, Schiller Institute Labor Day Conference, September 5, 1993. Printed in the American Almanac, November 8, 1993


End of Page Paradox Site Map Overview Page

Make your own free website on Tripod.com
Somebody once said that language is the basis of thought, and that no one can think anything, without using words. Whatever truth there may be in such a statement, we should not forget that language changes and develops over time. New conceptions are born somewhere in the depths of the human mind, and later we give them names. And what about languages themselves? Is it possible that an entire language might be born in the mind of a single person?

The earliest development of spoken language has been hidden from us until now. The astronomical observations reported in the Vedic hymns, which were sung and passed on from generation to generation long before they were written down, dates those hymns to a period not less than 6,000 years ago; at that time Sanskrit was already a fully elaborated, complete language. The basic structure of the Indo-European languages, including all the modern European languages and also Russian and other Slavic languages, was fixed thousands of years ago, and has not substantially changed up to the present time. The same is true for Chinese. And similarly for the Semitic languages, Hebrew and the Arabic languages for example, and so on. Despite many sorts of changes that have occurred, the languages human beings speak today are essentially very ancient things.

Not so for mathematics and music.

Mathematics, as the part of language whose development is most immediately correlated with the growth of man's power over nature over the last 2,500 years, has evolved in a series of revolutions, in which its content and form have expanded and changed in a rather dramatic fashion. This newer development of the mathematical--or, in the broadest sense geometrical--component of language, is much more readily accessible to us, than the earlier development of spoken language. From it, we might hope to learn something of great value concerning the human mind.

Looking over the landscape of the last 2,500 years of development of mathematics, two specific events stand out, as revolutions of the relatively most profound sort.

The first of these two events occurred approximately 550 years ago, and is associated with the work of the Nicolaus of Cusa and the launching of the Golden Renaissance in Europe in the 15th and 16th centuries. The second is much more recent, and dates to the development of the concept of the transfinite by Georg Cantor 100 years ago, culminating in Cantor's discovery of what is called the Aleph series. These two events define a division of mathematics into three levels or domains, both from the standpoint of its historical development as well as its present existence. These three levels of mathematics correspond to three different ways of thinking about the universe.

The first level, "A," we can identify historically with the so-called Euclidean geometry of the Greeks, as used for example by Archimedes. The second level, "B," is characterized by the vast development opened up by the introduction of what are called non-algebraic or transcendental functions. That development began essentially with Nicolaus of Cusa, and was transmitted through the geometers of the Renaissance, such as Brunelleschi and Leonardo da Vinci; it came to full blossom beginning in the latter decades of the 17th century, particularly through the work of Huygens, Leibniz, and Bernouilli. One hundred and fifty years later, we find Riemann already pushing vigorously at the limits of this mathematical domain, and not long thereafter Cantor actually breaks through, into the new universe of the transfinite. The third level, "C," the domain characterized by the type of thinking embodied in Cantor's Aleph series, is still in its very beginning stage. It is a very beautiful baby.

Now I propose we have a brief look at these three levels, not chiefly for their own sake, but in order to better grasp what kind of change in the way of thinking carries us from one to the next; that is, how we get from "A" to "B" and then from "B" to "C".

Before jumping into our subject let me caution that I am not aiming here at precision of a formal sort. On the contrary, I am obliged to employ a metaphorical manner of speaking. The reasons why this is necessary will become clear, I hope, by the end.

Secondly, if some of the ideas and terms are unfamiliar to many of you here, don't let that put you off. These matters are basically very simple, although in part quite profound; it is also somewhat difficult to compress them into a short talk. Just go to your nearby Schiller Institute representatives, and pester them until they explain it. Or if they can't, maybe you can work it out together with them. Don't hesitate to challenge them, they are absolutely fearless people.


Level `A'

The form of mathematics known broadly as Euclidean geometry was codified, in an approximation, to a logical deductive system around 300 B.C. Although the famous 13 books of Euclid's elements provide in some important respects only a distorted and one-sided representation of what Greek geometry really was, they may suffice for the mainly negative purpose I have in mind here, which is to locate a kind of limitation or boundary which Nicolaus of Cusa crossed in launching the revolution leading to the next level of mathematics, level "B."

Euclidean geometry addresses the question of the organization of space and all possible spatial forms, and attempts to reduce these into fundamental elements related to each other is various ways which obey certain rules or laws. The basic element is the point. This is considered as an obvious kind of idea which does not require further investigation. The second element is the straight line, again considered as a self-evident entity, and the third is a plane. And among the basic relationships between these, Euclidean geometry postulates, for example, that whenever two points are given, there is a unique straight line is determined which passes through them.

Euclidean geometry introduces, in addition to point, straight line, and plane, an additional element: the circle, which is connected with a notion of length or distance--measure. The circle, in terms of its circumference, would appear to be made up of all points lying at a certain fixed distance from some fixed point as center. In space, the corresponding form is the sphere. So we have these species of objects--point, straight line, plane, circle, sphere.

And now we try to build up or account for all the forms in space, in terms of those elements. This takes the immediate form of the famous ruler-and-compass constructions.

We make various sorts of triangles, as the simplest forms. Then rectangles and polygons of various types, and the associated solid forms. We examine the relationships of proportion and magnitude, first in terms of the segments defined by pairs of points on a line. We divise ways to duplicate or divide any segment into two, three, or any number of equal segments, to add and subtract lengths, and to multiply and divide them according to the laws of proportionality. All of these operations we can refer to by a shorthand called algebra. Then we go to areas and volumes. We uncover such beautiful relationships as the one attributed to Pythagoras, concerning the areas of the squares formed on the sides of a triangle with one right angle, and which can be demonstrated by dividing up a large square in two alternative ways. And so we go on.

But very soon, already in elementary Euclidean geometry of plane figures, we encounter paradoxes and anomalies, problems which we can formulate very easily, but cannot solve within the world of Euclidean geometry itself. These anomalies point to the (external bounding of Euclidean geometry, cannot account fully for its own existence, there are phenomena which cannot be accounted for).

For example, we encounter the phenomenon of incommensurability among line segments. We can share the astonishment of early geometers in the realization, that in dividing the diagonal and side of a square into arbitrary numbers of equal segments, we never arrive at a common measure, a line segment which fits perfectly into both. This realization leads to the development of various species of numbers, in an attempt to "fill up" the gaps in the line lying between the ordinary fractions or rational numbers.

Another anomaly concerns the division of a circle by inscription of regular, equal-sided polygons. It is a simple matter to construct an equilateral triangle in a circle, also easy to make a square, whose vertices divide the circumference of the circle into four equal arcs. We can also construct a regular five-sided polygon, the pentagon, although the construction is curiously different in a way we can fully understand only from a higher standpoint. The hexagon, or six-sided regular polygon, is again easy. But the next one, the seven-sided heptagon, absolutely defies an exact solution by ruler and compass. Here we seem to encounter a real limit of Euclidean geometry.

All this is closely related to another famous problem: the so-called squaring of the circle. Given a circle, how to construct a square whose enclosed area is precisely equal to that of the circle? An obvious approach to this, which goes back very far into ancient times, is to approximate the circle by a regular polygon with a suitably large number of sides. For those skillful in this sort of geometry, it is a simple matter to divide a regular polygon into right triangles, to reassemble these into a rectangle, and finally to find the side of a square whose area is equal to the rectangle. If we use a polygon inscribed in the circle, the corresponding square will be slightly smaller in area to the circle; the "error" is the small area between the polygonal sides and the subtended arcs of the circle. Similarly, if we circumscribe the circle by a polygon from the outside, we get an area which is slightly too large. The average of the two values would give a good approximation. And for example, we find the Chinese geometer Liu Hui, around 264 A.D. using a polygon of 3072 sides to obtain an approximate value precise to five decimal places.

Now, as far as we know Archimedes, back around 250 B.C. was the first to consider not one or two, but an entire series of inscribed and circumscribed polygons of increasing numbers of sides. The simplest proceedure is to start with a triangle or square, and progressively double the number of vertices and sides. It is enough to simply bisect the circular arcs or corresponding sides of each successive polygon. So, for example, beginning with an inscribed square, we can construct an octagon, a 16-gon, a 32-gon, a 64-gon and so forth.

Archimedes called his approach the "method of exhaustion", and in fact it is easy to see, that the polygonal area very rapidly "fills up" the circular area when the number of sides becomes large. The remaining area, the "error" as it were, is reduced by much more than a factor of two each time we double the number of sides of the polygon. By going out far enough in the series, we can hypothetically make the error, in terms of magnitude, less than any desired amount. However--and that is a nagging point--strictly speaking we never really ARRIVE precisely to zero error. There is always a slight discrepancy, embodied in the tiny crescent-shaped slivers between any polygon and the circle.

A much more dramatic situation arises, if we take the sphere instead of the circle. In this case we try to approximate the surface of the sphere by surfaces bounded by sections of planes, in other words by polyhedra. Among these, the one closest to the sphere in quality are--in analogy with the regular polygons -- the regular polyhedra, having identical, symmetrical faces. Lo and behold, we find that only five such polyhedra can be constructed -- these are the famous so-called Platonic solids:the tetrahedron, the cube, the octahedron, duodecahedron and icosahedron. Actually, it is really only one, since all the solids can immediately be derived from the duodecahedron. There is no simple doubling proceedure analogous to what we did with the polygons, and certainly none which preserves the property of symmetry. So here the discrepancy between the linear world of planes and straight lines, and the curved world of the sphere seems to take on a very tangible and irreducible form


Cusa's revolution

Now Nicolaus of Cusa came along in the middle of the 15th century, and began to think through what Archimedes had done more than 1,700 years earlier, and to drive it to its conceptual limit. Cusa asks the question, going back to the "exhaustion" approach, whether we can consider that the polygons converge to identity with the circumference of the circle? Or, using a paradoxical manner of speaking, do the polygons become equal to the circle "at infinity"? So that we could say, the circle is a polygon with an infinite number of sides? Nicolaus of Cusa answers this question with an emphatic, no!

Cusa emphasizes that the difference between the circle and the polygons is not merely one of magnitude, but of quality or species. Nicolaus put it this way:

"... since polygons are not magnitudes of the same species as the circle, it is still the case, even though we can always find a polygon which comes closer to the circle than a given polygon, that among things which can be smaller or greater, the absolutely largest cannot be attained in existence or possibility. In fact, the area of the circle is the absolute maximum relative to the areas of the inscribed polygons, which are capable of being more or less and therefore cannot reach the circular area, just as no number can ever attain the encompassing power of the unity nor the composite power of the simple."

Let us explore this further. Note, that at the vertex of any polygon, there is a sudden change in direction of the side, a discontinuity in that sense. Whereas the circle is constantly changing direction, and in that respect all loci on the circle appear to be completely equivalent to each other. So the existence of the discontinuities at the vertices distinguishes a polygon absolute from the circle. And what happens when the number of sides of the polygon increases? The discrepancies represented by the points of discontinuity of direction grow in number and density for example, after doubling the number of sides 50 times starting from the square, we arrive at a situation in which already one degree of arc on the circle contains more than 10 trillion discontinuity-points of the polygon. Evidently, although the areas of the polygons approach that of the circle in mere total amount, the perimeters become qualitatively speaking more and more unlike each other.

We could express this perhaps by saying, that there is a very thin "something" which separates the circle absolutely not only from every individual polygon, but from the combination or sum of all possible polygons. It has no specifiable magnitude in the ordinary sense, but it exists nonetheless. Its width is smaller than any ordinary quantity, but let us refuse to say zero thickness. We can think of it as the region of transition between two worlds, two qualities of being, from polygon-ness to circle-ness. That is what we call a singularity. From the world of polygons we can reach up to just touch that singularity, but we cannot pass through it into the universe of the circle.

Now, so far Cusa's observations seem to be entirely negative. Is he trying to take away what Archimedes gave us? Add one more observation, and you will see that we just gained something very precious. The fact the polygon can never reach the circle, even with a hypothetical "infinite number" of sides, forces us to conceptualize the circle, and all of geometry, in a different way.

If the circle is inaccessible from the world of polygons, where does it come from? Well, the circle, as a form, is just the trace of circular motion which itself is the result of rotational action. And Nicolaus considered rotational action as the unique most direct reflection, in the visible realm of form and motion, of being, that is of the process of creation of the universe itself. That correspondence is demonstrated, as Nicolaus further emphasized, by what is called the isoperimetric or least action characteristic of the rotational action.

This new view of the circle places all of euclidean geometry in a completely new light, while leading beyond it at the same time. We observe that the elementary forms of euclidean geometry are all derived from circular action. For example, if we start with circular action, we generate first the circle. But if we apply the action of rotation once more, to the circle just created, in such a way that the circle is folded upon itself, then we obtain, as the singularity of that folding process, a straight line which is the diameter of the circle. If we fold again, we obtain a second diameter which intersects the first to create a point. Further folding--more degrees of rotation--generates the vertices of the polygons we were using before in the attempt to approximate the circle. Also, an angle is nothing but the result of rotation. So it appears that rotation is everywhere in geometry, as the underlying substance, while at the same time lying just beyond the reach of euclidean geometry itself.

Starting from the purely negative observation of Cusa, pointing to an absolute barrier separating the circle from all polygons, we have broken out of the bounds of euclidean geometry and begun to ascend into a higher realm of mathematics.

Now begins the development of a new type of geometry, in which change, motion is elementary, and not form per se. It is the motion which makes the forms, and a quality of change which determines the characteristics of what we call physical space-time as a whole.


Level `B'

Now if we look at Leonardo da Vinci's work on the design of machines, on anatomy, on hydrodynamics and his wave conception of light and sound, an so forth, it is clear that Leonardo is saturated in the conception of geometry we just began to develop. The formal development of a geometry based on circular action begins much later, in the work of Huygens, Leibniz and Johann Bernoulli. In the words of Leibniz, this work "opens up the fountain and treasure chest of the nonalgebraic functions".

All we do, essentially, is unfold an entire universe of forms or species of motion, starting with nothing but circular action. Huygen's construction of the various cycloids demonstrates the principle very well.

[Note: the next section was demonstrated by David McVey of Baltimore, with a mechanical device which was ingenious in its simplicity and clarity.]

Take one circle and let a second smaller circle role upon it, either on the outside or on the inside. Then, the motion of a point fixed anywhere on the smaller circle describes a curve, called generally a cycloid. What is going on here? The curve is generated by two degrees of circular action acting at the same time. First, the center of the small circle is rotating around the center of the large circle; and second, the small circle is itself rotating. Observe, that when the radii of the two circles are commensurable (relate to each other like two whole numbers), then the cycloids close, and the singular points called cusps, define the positions of the vertices of regular polygons. Perhaps the polygons are to be regarded as mere shadows of these higher entities, the closed cycloids. At the same time, we see that most of elementary arithmetic and number theory is contained in the behavior of these cycloids. An interesting case arises when the radii of the cycloids are not commensurable; then the cusps form an everywhere dense set along the rim of the larger circle.

For the special case of a small circle rolling on a straight line, regarded as the rim of a "very large" circle, the distance between two cusps is equal to the circumference of the small circle. Using this, those skilled in geometry will easily see how to "square the circle."

Now, once we have constructed a cycloid by means of what we might call double rotation, what is to prevent us from carrying the process further? We can roll a third circle on the cycloid, or alternatively, roll a portion of the cycloid on a circle, or on another cycloid. In each case a point fixed on the moving curve describes a new curve. Obviously this process can be continued indefinitely.

The same principle, in another form, underlies Huygen's construction of the so-called involute of a curve. Take any circle, attach at some point a flexible piece of string and wrap a portion of the string around the circle. Now slowly unwrap the string while holding it taut. The motion of the string's end describes a new curve, a kind of spiral, called the involute of the original circle

At first glance this proceedure might seem completely different from the generation of a cycloid. But if we examine it closely, we find nothing else than a combination of two degrees of rotational action. For at any moment of the process, the end of the string can be seen to be rotating around the point at which the string touches the circle. That point has for a moment become what the geometers call the center of curvature. At the same time, that center of contact is moving--rotating!--along the original circle. So, we have two degrees of rotation. The additional feature, compared to the cycloid, is that the radius of the second degree of rotation -- which is the distance between the end of the string and the point of contact with the original circle--is constantly increasing. Developing this further, we learn to generate envelops of curves, including the famous "caustic" produced by rays of light reflected in a curved mirror.

Now, what is to prevent us from opening up the "treasure chest" of new curves even further? By applying the operations of involution and rolling one curve on another, to the entirety of the curves already generated at any given point, we produce a new generation of curves. We create a kind of evolutionary process. But observe, that rotational action remains at all times the "hereditary principle" underlying the generation process.

Now, as Leibniz emphasized, this growing family of curves cannot be described by the methods of ordinary algebra. Algebra already fails in the case of the simplest cycloid. So, Leibniz called the new curves "nonalgebraic" or "transcendental". The algebraic shorthand only becomes usable if we introduce entirely new symbols and operations such as the integral and differential of Leibniz. But none of these can be explained or accounted for within the linear world of algebra itself; one must always refer to the generation process outside algebra--which is one reason why Leibniz's calculus caused a great uproar among mathematicians up to this day.

But this is only the beginning of the treasure chest of nonalgebraic functions. Next, take a circle, and rotate it on any diameter; we generate a surface--the sphere!. Fix a smaller circle at right angles to the circumference of a larger one, let the larger circle rotate: we generate a torus. Fix any other curve to the larger circle and rotate it: we create the general type of surface known as a surface of revolution. This new fountain of transcendental functions was tapped especially by Gaspard Monge and his collaborators at the École Polytechnique at the end of the 18th century.

For example, by rotating a line we obtain cones and hyperboloids.

Now take any surface of revolution, and cut it by a series of planes. The intersection of the surface with any plane is a new curve. Those skilled in geometry will know that by cutting a cone, we obtain--depending on the choice of cut--circles of various sizes, ellipses, fat or elongated, parabolas and hyperbolas. By cutting the torus, for example by parallel planes, we get a family of higher-order curves which, as far as I know, has not been given a name. But we could take any of them as the basis of further processes of generation, forming surfaces of revolution from them, or producing their involutes. Thus, a single curve can generate, via its surface of revolution, an entire family of curves and further surfaces; and thus already contain them in the sense of potential--a remarkable thing, if you think about it.

So much for a rapid excursion into the world of non-algebraic functions. Before going further in our journey, let me point out that this generation of nonalgebraic curves of ever higher order is at the same time an exploration of possible designs of machines.

As soon as we have introduced, by the activity of our mind, as Cusa did, the conception of a new quality of change, and the Universe has indicated its agreement in crucial experiments of the sort done by Leonardo da Vinci and later Rmer and Huygens did in the case of light, then we are ready for a technological revolution. This is exactly what occured in the rapid development of machines, anticipated by Leonardo and exploding into life during the industrial revolution. It was for this reason that Leibniz and Huygens refered to the nonalgebraic functions as "mechanical curves"--they are curves described by the parts of machines. There exists thus more than a mere analogy between the generation of higher orders of curves and surfaces, and the progress of technology.

Reflecting upon this, we are bound to ask ourselves: what are the limits of this world of nonalgebraic functions? Observe first the following:

The universe of non-algebraic functions is a universe in which the concept of change is associated with the idea of motion. But, we also have an implied concept of change of quality or species of motion. When we attempt to grasp the physical universe in terms of the concept of motion, we are led to a speciation of qualities of motion.

It is thus characteristic of the nonalgebraic functions, that they develop in species. We already saw a beginning of this in the distinction which Cusa made between between the polygons and the circle. The elaboration of nonalgebraic curves leads to a great multiplicity of new species of functions, and raises a far-reaching question: Is it possible to enumerate those species by some sort of function?

In other words, let us assume for the moment that the concept of multiply-connected circular action, elaborated in the sorts of ways I have indicated, covers all modes of change in physical space-time. Can I now define that space-time itself by means of a function which would include all qualities or species of circular-action-generated motion?

Also is it possible that there is another type of change in our Universe which is not motion but might be the most immediate cause of motion and change in the quality of motion? This touches, by the way, on the problem of the so-called quantum fiel

The attempt to encompass the entire range of possibilities of nonalgebraic functions and to map out their relationships in terms of species, is called function theory. Gauss and Riemann., particularly, succeeded in bringing function theory to the point at which Georg Cantor could come along with an extraordinary insight, carrying us into the world of the transfinite.

A few brief indications will have to suffice for us now. Gauss adopted the method of refering everything back to the sphere, where all displacement is in the form of rotation. This leads among other things to what is called the complex domain. When we redefine the transcendental functions in terms of this spherical or complex domain, we eliminate certain artificial features of the ordinary euclidean space, and the inner organization of these functions reveals itself in a marvelous way. Following the line of Gauss' work, Riemann showed how to characterize a function in the complex domain entirely in terms of the type and configuration of its singularities, without reference to scalar magnitudes. This reduces the determination of the species of transcendental functions to a problem in analysis situs, or topology.

Riemann also tried to characterize the process by which a function can "jump" from a lower to a higher species. The introduction of a new singularity into a function corresponds to the case, where a physical process is driven to and beyond its limits as apparently determined by the finite rates of propagation of action within that process. At that point, we have what the physicists call a phase change. Some new qualities emerge in the process, accompanied--in the happy, negentropic case--by an increase in the number of degrees of freedom. Riemann shows how a functions can be contructed, that pass through a whole series of phase changes, from species or state S1 to state S2, to S3 and so on.

Around the same time, Riemann, Weierstrass and others were pushing the general concept of "function" to its limits. What would be a completely arbitrary, most general sort of mathematical function? Is it possible to encompass the most general kind of curve by some uniform method of representation? The chief form of representation that people looked at was the so-called Fourier series. Essentially, Fourier's method was to explain the changes within a given function in terms of the changing relative phases of a very large number of cyclical processes--a kind of generalized multiple cycloid.


Cantor's revolution

Cantor began his work on the theory of functions by studying the problem of representing an arbitrary function by the so-called Fourier series, and he found that the Fourier representation fails in general: there will generally be a singular domain on which a given function disagrees with its Fourier series. Then, as Cantor indicated, we can order the species of functions according to their degree of nonrepresentability, as reflected in terms of the density of singularities in any interval. This implies that the species of functions have a natural ranking or ordering according to increasing densities of singularities.

We have already seen this sort of thing in our discussion of the circle and the polygons. Each of the polygons represents a species, having 4, 8, 16, and so forth number of singularities. These species are naturally ordered in ascending order of the increasing density of their singularities, the vertices, on the circumference of the circle.

Observe also, that each species contains the preceeding in the following sense: We can immediately obtain the square by joining every other side of an octagon, for example. Note also, that each higher species is obtained by a single action of doubling the number of sides.

Now, we already observed that the circle stands, so to speak, above each and the whole series of species of polygons and can generate them all in terms of circular action. So the ordering would apparently be completed if we added, so-to-speak after all the polygonal species, the circular species:

[P(4), P(16), P(32),...], ? ,C

The brackets signify the entire ordered system of species of polygons whose number of sides is a power of two. Cusa insisted that the circle is absolutely separated from each and the sum of all the polygons. That suggests the question, whether there might not be other species located in the "space" between the circle and the polygons. Cantor definitely answers this question with a "yes". Without trying to form any concrete picture for what that might look like, let us simply imagine a hypothetical species S which is the "first species after or higher than all the polygons" and represents only that portion of the power of universal action which exactly suffices to generate all the polygons.

Something of this situation might be captured by a perspective drawing of a railroad going off in the distance as if trying to reach to a star, but converging instead to a vanishing point on the horizon below the star, which is still infinitely far away. The vanishing point is the species S, the star is the species of circular action.

What Cantor then did was to develop what we might call a general theory of types of ordering of species. In that general theory, we leave completely aside the particular nature of the species being ordered, and examine only the way they are ordered in relation to each other. So, we apply Leibniz's method called "analysis situs".

Start with a simple series of species

S(1), S(2),..., S(N)

in which S(2) is the natural successor of S(1), S(3) the natural successor of S(2) and so on. Such series may continue indefinitely, like the polygons, adding degrees of freedom at each step.

Now, the quality of change embodied in the series S(1), S(2), S(3) ..., which is the immediate cause of the progression from each species to the next, constitutes a higher species than any of the individual terms of the series. Call that higher species S'. Following Cantor we place S' immediately above or after the series S(1), S(2), S(3) .... as its natural successor:

[S(1), S(2), ... ], S'

Observe something interesting, which Cusa already pointed out in the case of the circle and polygons: It would appear as if a literally infinite number of steps separate the first species, S(1), from the higher species S'. That kind of numerical infinity is really only a paradoxical reflection of the fact, that S' cannot be reached from any of the lower species S(n) taken in and of itself--that is, separated from the universal creative principle (the absolute) which is the same in all things and stands above everything.

Now, the species S', being not the absolute, is itself capable of development. Now we can begin to see why Cantor used the term "trans-finite" to describe this kind of ordering. So, beginning with S' we have a series S'(1), S'(2), S'(3),... which immediately follows the earlier series in order.

[S(1), S(2), ... ], S', S'(2), S'(3), ...

Once again, there will be a next higher species S'', which embodies the principle of change generating this second series and constitutes the next species that series:

[S(1), S(2), ... ], [S', S'(2), ... ], S''

We could go on like this to S''', S'''' and so forth, but the principle is clear. Reflect on that for a moment: "the principle is clear". Already our mind has formed a concept of a higher series S, S', S'', S'''... which transcends the original first series, and the second, third, and fourth in our ordering, and so on, as well. That higher series embodies itself a higher quality of change, a species we might denote by T, which lies just beyond the reach of each and all of the S-species, and forms their necessary successor in the ordering.

(( [S(1), S(2), ... ], [S', S'(2), ... ], [S'', S''(2) ... ] ...,

[S|, S|(2),... ], ....)), T

But T, of course, begins a new series, and so on. We are exploring the bare beginning of what Cantor called the second number class.

Now, soon a new idea forms in our mind. The whole process by which we started with the first species S(1) and arrived finally at T, could be applied to T also! Of course this would not be literally the same process, but it would be equivalent in the sense of Leibniz's concept of analysis situs. So, formally speaking we can now adjoint, at "T" another copy of the whole ordering, where we just replace S by T everywhere. The next higher species, which embodied to quality of change of the entire T-series, we could call U.

Aha! If we think of the entire process going from S to T as a specific type of transformation, and the same type carries us from T to a still higher level U, then we have the concept of a series S, T, U, V etc and a quality of change embodied in that entire series, which is above all the processes we have discussed to far. We already see the way toward a next higher level of transfinite development.

To continue this exploration, and avoid becoming dizzy and falling off the mountain, we would need a kind of map of the overall kind of ordering which is emerging. Cantor provided this (or at least, an attempt at this) with his theory of the transfinite ordinal types and a special kind of notation, which plays a similar role for the transfinite orderings as the familiar decimal system for ordinary numbers. However, this formal development is not essential for us here.

You may recognize some similarity to what we have been doing and what Leibniz, Huygens and Gaspard Monge did in opening up the fountain of nonalgebraic functions. It is also the same type of thing we get into when we explore Motivfuhrung and the principle of variations in music. In fact, Cantor's transfinite is nothing but the analysis situs, the underlying ordering principles of all these sorts of development processes.

But I haven't gotten to the essential point yet. All of this is but scaffolding. We started with the question, whether it is possible to enumerate all possible species by means of a single mathematical function, as Riemann and others implicitly posed that problem. The earthshaking answer to that opens the way to Cantor's famous aleph series, and brings our journey to a happy end.

Reflect back briefly on the kind of process we began to unfold. At any point in that process, the entire ordering of species comprehended up to that point can implicitly be enumerated; we have merely to retrace the steps that brought us so far. But what about the principle which keeps leading us to extend the process further? We drive the process forward by forming, in our mind, the concept of a new quality of progression, which we name in terms of a mathematical function. In unfolding the process based on that concept, we become aware of its limits or relative finiteness, and at the same a new concept forms. But as soon as we have conceptualized a range of development in terms of a function--that is, an ordered progression of the sort we have indicated--at exactly that point, a new phase of development beyond that function begins.

Aha! That means, that the potential--the power of this process of concept formation transcends the concept of function itself--at least in the sense of function we have inherited from level "B" of mathematics. Following Cantor, we call "Aleph zero" the power of that ordinary notion of function, enumeration or ordered succession of values. Then call "Aleph One" the power of the concept of a generating principle of the transfinite process we just described, by a process of conceptualizing ever higher ordering functions, a process which by its very nature transcends any possible function.

Now, observe, that Aleph one is the necessary successor of Aleph zero. even though they are separated by what would seem to be an unbridgeable, an almost indescribably vast gap.

Observe, also, that Aleph one is absolutely unreachable by any form of language. We cannot communicate it directly, linearly, but we can only cause it to be evoked, called forth, in the creative mental processes of another person's mind. Aleph one is in this sense, only communicable by metaphor.

Yet--and this is crucial--Aleph one is a very precisely defined thought object. It is not creative action or creative concept-formation in general, in some very vague sense; it is certainly not God. But it is that thought object immediately above any ordinary mathematical notion of function, and which immediatly generates that notion. We could say, Aleph one is the type of that level of physical reality which immediately bounds ordinary space-time from the outside.

By making this immediate??? boundary layer now an object of conciousness, we push Physics into a new domain, a scientific renaissance.

In a sense, the concept of function inherited from level "B" of mathematics, now plays the role which the polygons played for Nicolaus of Cusa. That tells us also, that the usual space-time of present-day mathematical physics is externally bounded by something else, which we are beginning to conceptualize right now.

We've come not to the stretto.

Need I tell you, that there are higher alephs, and in fact a whole transfinite series of them?

Think about the step from "Aleph Zero" to "Aleph One". That characterizes, I think pretty well, Cantor's revolutionary step from level "B" of mathematics, the domain of function theory, to the new level "C," the fountain and treasure chest of the transfinite.

Now compare that step, to the step Nicolaus of Cusa made, from level "A" to level "B." Isn't there a kind of higher equivalence between these two revolutions? Now go back and look at what Lyndon LaRouche has been writing about the higher hypothesis and the hypothesizing the higher hypothesis.

Does that not suggest, that we are at the threshhold of a new renaissance? The first steps have already been taken.

Now somebody says: What? A new renaissance? Are you crazy? I haven't seen that on television! Well, you won't, at least not in the beginning. But don't forget: before the flowers come the seeds. And we are planting them right now.


Top of Page Paradox Site Map Overview Page


The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.


Publications and Subscriptions for sale. See: Publications and Subscriptions

Readings from the American Almanac. Contact us at: american_almanac@yahoo.com.