The Larson Research Center

In discussing the ancient Greek strategy of using the discrete counting numbers to represent continuous magnitudes of line segments, David Hestenes writes:

With admirable consistency, Euclid carefully distinguished between the two concepts. This is borne out by the fact that he proves many theorems twice, once for numbers and once for magnitudes. (See his New Foundations for Classical Mechanics)

The reason for making this distinction between the discrete numbers and the continuous magnitudes of line segments, and keeping them separate in calculations and proofs, is easy to understand, as soon as it’s recognized that measurements involve magnitudes for which there are no counting numbers. The magnitude of the diagonal of the unit square is the simplest example. Hestenes points out that the “Hindus and the Arabs resolved this difficulty directly, by generalizing their notion of number.” However, “Euclid,” he writes, “sidestepped it cleverly by re-expressing problems in arithmetic and algebra, as problems in geometry. Then, he solved for line segments instead of for numbers.”

Euclid’s “sidestepping” approach worked because he labeled products with geometric magnitudes, so that he could represent the product x², as a square with sides of magnitude x, the product xy, as the “rectangle” xy, and the product x³, as a cube with sides of magnitude x. However, Hestenes observes:

But there are no corresponding representations of x⁴ and higher powers of x in Greek geometry, so the Greek correspondence between algebra and geometry broke down. This “breakdown” impeded mathematical progress from antiquity until the seventeenth century, and its import is seldom recognized today.

However, while Hestenes cites this “geometric” approach of the Greeks as a major factor in the “long period of scientific stagnation” that occurred between the “brilliant flowering of science and mathematics in ancient Greece,” and the “explosion of scientific knowledge in the seventeenth century,” he regards it as inevitable, given that no “comprehensive algebraic system” was available until Rene Descartes was finally able to state explicitly what most, by that time, assumed tacitly. As Hestenes puts it:

Descartes gave the Greek notion of magnitude a happy symbolic form by assuming that every line segment can be uniquely represented by a number. He was the first person to label line segments by letters representing their numerical lengths…the aptness of this procedure resides in the fact that the basic arithmetic operations such as addition and subtraction can be supplied with exact analogs in geometrical operations on line segments.

Of course, the objective of Hestenes’ review of these fundamental developments is to show how his recognition of Clifford’s Geometric Algebra (GA), as he calls his version of it, is actually a completion in the understanding of some fundamental aspects of the modern union of numbers and magnitudes that only started with Descartes. He acknowledges the inevitability of Descartes’ explicit articulation of the assumption and notes the independent works of others, such as Fermat, along the same lines, but then he observes:

…Descartes penetrated closer to the heart of the matter. His explicit union of the notion of number, with the Greek geometric notion of magnitude, sparked an intellectual explosion unequalled in all history.

However, the essence of the change that sparked this explosion is subtle. At its heart, is the age-old issue of reconciling the discreteness of quantity with the continuousness of magnitude, which is the very same issue that continues to plague mankind’s quest for fundamental understanding today. Hestenes writes:

The correspondence between numbers and line segments presumed by Descartes can be most simply expressed as the idea that numbers can be put in a one to one correspondence with the points on a geometric line. The Greeks may have believed it at first, but they firmly rejected it when incommensurables were discovered. Yet Descartes and his contemporaries evidently regarded it as obvious…Of course, such a change was possible only because the notion of number underwent a profound evolution [in the intervening centuries].

Certainly, the most important aspect in the profound evolution in the notion of number that Hestenes refers to is the “arithmetization” of the number system, wherein the so-called “real” numbers were defined in terms of “natural numbers and their arithmetic, without appeal to any geometric intuition of ‘the continuum.’” Regarding this crucial aspect of the “profound evolution in the notion of number,” Hestenes asserts:

Some say that this development separated the notion of number from geometry. Rather, the opposite is true. It consummated the union of number and geometry by establishing at last [the assumption] that the real numbers can be put into one to one correspondence with the points on a geometric line. The arithmetical definition of the “real numbers” gave a precise symbolic expression to the intuitive notion of a continuous line (emphasis added).

I have added the emphasis here, because it underscores the importance that the concept of real numbers is universally believed to have “consummated” the union of numbers and the continuum, but if that’s actually the case, one wonders why the issue of uniting these two concepts remains so intractable in the modern theories of physics today. Clearly, however, Hestenes, and almost everyone else in the legacy community, regards this “union” as settling the issue once and for all, at least as far as numbers are concerned.

Nevertheless, reconciling the discrete quantities of natural numbers with the continuous spectrum of real numbers, addresses only part of the deficiencies in the notion of number vis-à-vis the notion of magnitude. As Hestenes summarizes it,

Descartes began the explicit cultivation of algebra as a symbolic system for representing geometric notions. The idea of number has accordingly been generalized to make this possible. But the evolution of the number concept does not end with the invention of the real number system, because there is more to [geometrical magnitudes] than the linear continuum.

Then, Hestenes identifies the two issues remaining to be addressed, and makes a promise:

In particular, the notions of direction and dimension cry out for a proper symbolic expression. The cry has been heard and answered.

Consequently, Hestenes proceeds to demonstrate in his work, that the idea of continuous magnitudes, with their properties of direction and dimension, can be symbolically expressed by further generalizing the concept of number. This is accomplished by exploiting the work of Grassmann and Clifford to introduce a concept of “directed” numbers, together with a concept of a new “geometric product” that consists of the combination of the inner product, analogous to the Greek concept of projection, or what we might call the Euclidean inner product, and Grassmann’s outer product, a key innovation that enables the algebraic expression of magnitude with four properties:

1. Quantity
2. Direction
3. Orientation
4. Dimension

The addition of “orientation” to the three universally accepted properties of magnitude, quantity, direction, and dimension, is a recognition of the importance of polarity, as a property of magnitude, which is similar to our own efforts to distinguish “direction” from direction, in the RSM.

What we have here, then, in Hestenes view, is a notion that the separation of number and magnitude concepts, by the ancient Greeks, the separation of the discrete and the continuous, has been overcome through a centuries long effort to “generalize” the concept of discrete numbers to accommodate the concept of continuous magnitude. So far, this generalization of number has taken the form of two inventions, the invention of the “real numbers,” which consists in the “arithmetization” of the natural numbers, reconciling the continuum of magnitude with the quantum of quantity, and the invention of imaginary numbers, which gives direction to numbers, expressed in terms of rotation.

However, generalizing the concept of number to accommodate the third property of magnitude, dimension, has yet to be accomplished. As pointed out previously, Hestenes regards the “breakdown” of the correspondence between algebra and geometry at x⁴ as a significant development the “import” of which “is seldom recognized today.” In his GA, Hestenes handles the “breakdown” of higher dimensions by showing that no new phenomena are possible beyond x³.

His reasoning follows from the geometric interpretation of GA’s unique geometric product, which is a combination of the inner and outer product. Essentially, the inner product lowers the dimension of a vector, while the outer product raises it, but the effect of the outer product, in raising the dimensionality of a vector, fails when x > 3. Hestenes writes,

No fundamentally new insights into the relations between algebra and geometry are achieved by considering the outer product of four or more vectors. But it should be mentioned that if vectors are used to describe the ordinary space of 3-dimensional geometry, then displacement of the trivector a^b^c [the outer product of three vectors] in a direction specified by d fails to sweep out a four dimensional space segment…This is a simple way of saying that space is 3-dimensional.

Thus, by interpreting magnitudes as scalars and vectors, Hestenes, following Grassmann, understands the Greek’s insistence on the separation of number and magnitude in terms of appropriate algebraic operations: We add scalars and multiply vectors. He writes:

Only in the light of Grassmann’s outer product is it possible to understand that the careful Greek distinction between number and magnitude has real geometric significance. It corresponds roughly to the distinction between scalar and vector. Actually, the Greek magnitudes added like scalars, but multiplied like vectors, so multiplication of Greek magnitudes involves the notions of direction and dimension, and Euclid was quite right in distinguishing it from multiplication of “Greek numbers” (our scalars). Only in the work of Grassmann are the notions of direction, dimension, orientation and scalar magnitude finally disentangled.

Nevertheless, the Chart of Motion (CM), derived from the RST, and its corollary that the space magnitude of distance (vectors) is a measure of the space aspect of motion magnitude, which is the reciprocal of its time aspect, compels us to reconsider the fundamentals underlying the conclusions of Hestenes, and the LST community.

Indeed, it raises the possibility that the Greek separation of number and magnitude and the “consummation” of the union of the two concepts through the modern inventions of numbers, the so-called “real” and “imaginary” numbers, and the latest such invention, the “compactification” of “space” dimensions, in order to accommodate the unphysical numerical abstractions of “extra dimensions,” are simply due to the failure to recognize the true nature of the natural relation between space and time and its exact correspondence with the natural relation between reciprocal numbers.

What the CM tells us is that numbers have direction and “direction” (orientation), just as physical magnitudes do, and that magnitudes have higher dimensions, just as numbers do. To understand this, we simply have to recognize the difference between “space” and “time” magnitudes, and the magnitudes of motion from which they emerge.

To start, let’s go back and take a second look at what started the whole parade, the so-called “incommensurables” of the Pythagoreans. The CM shows us that the magnitudes of 1 are indeed the “mother” of all other magnitudes. It shows us that, while in the numerical sense, 1¹ = 1² = 1³ = 1³⁺ⁿ, in the geometric sense, 1¹ < 1² < 1³ < 1³⁺ⁿ. When we recognize this fact, then the Pythagorean theorem,

1)      a² + b² = c²,

takes on a different meaning when a and b are 1, because we can’t tell if the variables a and b are equal to 1¹, or 1², or 1³. Moreover, Fermat’s last theorem shows that equation 1) only holds for the case of dimension 2, and therefore common sense tells us that we should be careful before we jump to conclusions and generalize the results of equation 1).

Clearly, the sum of two or more sets of n-dimensional magnitudes is straightforward, regardless of the number of units involved. Hence,

2)     1ⁿ + 1ⁿ = 2 * 1ⁿ

and

3)     x(1ⁿ) + y(1ⁿ) = (x+y)(1ⁿ)

for all x, y, and n.

That is to say, if we combine two or more sets of line magnitudes, or two or more sets of plane magnitudes, or two or more sets of cube magnitudes, the total number of magnitudes in the resultant set of lines, planes, or volumes is the sum of the total number of these magnitudes, in the resulting combined sets. Sometimes, that total number of magnitudes in the combined set is expressible as a numbered squared, and sometimes it is not, depending upon the number. For instance, if a set of 16 2D magnitudes is combined with a set of 9 2D magnitudes, the combined total of 2D magnitudes, 25, can be arranged in a perfect square, as 25 = 5² magnitudes. However, if a set of 4 2D magnitudes is combined with a set of 9 2D magnitudes, the combined total of 13 2D magnitudes cannot be arranged in a perfect square, but this does not mean that all 13 2D magnitudes don’t exist as discrete 2D magnitudes. They do exist; they just can’t be arranged into one perfect square.

However, when we apply equation 1) to the special case of the unit square itself, instead of to multiples of the unit square, we are changing the physical situation. Now, we are combining two, single, 2D magnitudes into one set of two 2D magnitudes and if we try to arrange the resulting set of two, 2D, magnitudes as a perfect square, we find it impossible, in this case, because there is no number times itself that is equal to 2; obviously, the number 2 cannot be divided into 4 whole numbers, the number of sides of a square. Furthermore, the sum of two, 1², magnitudes is clearly 1² + 1² = 2(1²) = 2*1², not 2 * 1¹, so what’s going on here?

Certainly, when we interpret the powers of numbers geometrically, rather than mathematically, the meaning of equation 1) must be interpreted in terms of sets of magnitudes, not single magnitudes. If we don’t do this, then 1² + 1² = 2*(1²) = 2² doesn’t hold, because there is uncertainty in the expression of the dimension of the coefficient, 2, which confuses the meaning of the multiplication operation. If the magnitude of the dimension of the coefficient is taken as a vector, the result of the multiplication operation is 2¹ x 1² = 2³, a volume, but if it is taken as a scalar, then the proper mathematical result is 2⁰ x 1² = 1², but this is problematic too, because 1² + 1² = 2, 2D, magnitudes, not 1, 2D, magnitude, so, again, what’s going on?

When we think about it, what the equation 1) actually says is that the total number of unit 2D magnitudes, in a given set of multiple 2D magnitudes, can be divided into two subsets of multiple 2D magnitudes, which is something that any child knows instinctively. However, the mathematical operation of adding two, individual, 2D magnitudes, to form one set of two, 2D, magnitudes, cannot be expected to follow the same logic. No child would expect that, given only two squares, he could arrange them in some other set of multiple squares. It’s totally illogical to expect such a thing. In this special case, then, the logic doesn’t follow this equation of sets, because a single magnitude isn’t a set of multiple magnitudes, and trying to apply the equation to non-multiple sets leads to confusion.

Thus, it makes no sense to insist that there should be a square root of 2, in this case, and even though the magnitude of the hypotenuse of a triangle is real, and, at the same time, not algebraically expressible in terms of the sum of the squares of the unit magnitude of its sides, it does not follow that an “incommensurability” is found in the nature of discrete numbers, rather we see that it is found in the illogical application of the equation to unit magnitudes.

The CM clarifies this, by demonstrating that the spatial magnitude of 1² can be generated, by any of its three magnitudes of motion. In the case of M₂, or change of position, motion, the 1² magnitude is generated by two separate instances of vectorial motion in orthogonal directions. The fact that the two positions, defining the end points of each instance of orthogonal motion, originating from the same point, define a possible third instance of M₂ motion over the length between them, does not imply anything regarding the necessity to invent irrational numbers, because the magnitude of the third motion (along the “hypotenuse”) is not constrained in any way, as it relates to the original, orthogonal, motion, generating the square magnitude.

In fact, it exists only as a potential magnitude, since the two instances of M₂ motion, generating the square magnitude, do not constitute explicit motion between the two endpoints, representing a length of “space” along the diagonal. Hence, this independence implies that the M₂ “motion,” defining the spatial length between the two endpoints of the orthogonal instances, the hypotenuse of a triangle, may be measured in independent terms of integral units of “space;” that is, the length of the hypotenuse may be taken as a new unit of distance, subdivided into an arbitrary number of subsets, and expressed as a rational number, representing the space aspect of the M₂ motion required to measure its length.

The CM shows that such a change in the metric is permissible, without any inconsistency, due to the fact that the M₂ motion, used to measure the length of the hypotenuse, is a 1D magnitude of length that is completely independent of the 2D magnitude of the area, defined by the original orthogonal motions. What this means, then, is that the brouhaha caused by the so-called incommensurate magnitude of the unit square’s diagonal, was unnecessary, and it therefore implies that Descartes’ assumption that magnitudes of any length may be treated as algebraic variables does not necessarily depend on the “arithmetization” of natural numbers. In other words, if discrete numbers never needed to be distinguished from continuous magnitudes in the first place, then the “consummation” of their union through “arithmetization” of the rational numbers, which Hestenes refers to, is just as imaginary as imaginary numbers.