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The Fourth Dimension
GA 324a

Questions and Answers XIII

11 March 1920, Stuttgart

FIRST QUESTION: Does my attempt to define the hyper-imaginary through relationships of points on curved surfaces, or manifolds, correspond to reality?

SECOND QUESTION: Is it possible to acquire an enlivened view of the realm of imaginary numbers, and do actual entities underlie this realm?

THIRD QUESTION: Which aspects of modern mathematics, and which formal aspects in particular, need to be developed further along spiritual scientific lines?

Let [Note 54] me begin with your second question. The answer is not easy to formulate because in order to do so, we must leave the realm of visualization to a very great extent. When I answered Dr. Muller's question several days ago, [Note 55] you saw that in order to provide a concrete correlate for a mathematical case, I had to turn to the transition from long bones to head bones, and yet the graphic example was still valid. [Note 56] At least in that case we were still able to visualize the objects and hence the transition from one to another.

When we attempt to look at the domain of imaginary numbers as a spiritual reality, [Note 57] we find that we need to shift from positive to negative, as I recently demonstrated in these lectures on physics. [Note 58] This shift makes our ideas true to reality when we attempt to understand certain relationships between so-called ponderable matter and so-called imponderables. But even when we visualize very ordinary domains, we can see the need to transcend customary ways of illustrating them. Let me mention just one example. On a plane drawing of the ordinary spectrum, we can draw a straight line from red through green to violet. [Note 59] Such a drawing, however, does not symbolize all the relevant aspects, which are encompassed only when we draw a curve, more or less in this plane (reference to a drawing that has not been preserved), to symbolize the red. Then, to depict the violet, we go to the board and behind the board, so that the red, as seen from above, lies in front of the violet. I would have to move out of the plane for the red and back into it for the violet in order to characterize the violet as moving inward toward chemical activity and the red as moving outward toward warmth. [Note 60] Thus, I am forced to expand the straight line here and to see my ordinary drawing as a projection of what I actually ought to draw.

To achieve clarity concerning certain phenomena of higher reality, it is not enough to shift from the positive material aspect to the negative. That is just as unsatisfactory as moving in a straight line from red through green to violet. When we move from the spatial realm to the non-spatial (as symbolized by positive and negative, respectively), we must shift to a higher form of spatial and non-spatial. This process is like moving along a spiral, instead of moving around a circle and returning to our starting point.

Just as elsewhere two different types may be summed up in a union that contains both, we also can imagine the existence of something that is both spatial and non-spatial. We must seek this third element. In the domain of higher reality, if we describe physical reality as positive, we are obliged to describe the etheric realm, where we leave space and begin to enter spirit, as negative. [Note 61] When we take the step into the astral realm, however, space and negative space are no longer enough. We must turn to a third element that relates to positive and negative space in exactly the same way that imaginary numbers relate to positive and negative numbers in formal mathematics. And if we then take the step from the astral realm to the true being of the "I," we need a concept that is hyper-imaginary in relationship to the imaginary. For this reason, I have never been happy with academic antipathy to the concept of hyper-imaginary numbers, because this concept is truly needed when we ascend to the level of the "I" and cannot be omitted unless we want our mathematical formulations to leave the realm of reality. [Note 62] The issue is simply how to use the concept correctly in purely formal mathematics.

Someone I met today discussed the problem of probability, a question that very clearly demonstrates the great difficulty of relating a mathematical procedure to reality. Insurance companies can calculate when a person is likely to die, and their figures are very accurate when applied to groups. It is impossible, however, to conclude from actuarial figures that any individual is going to die exactly in the year that is predicted. Consequently, these calculations lack reality.

The results of calculations are often correct in a formal respect yet do not correspond to reality. We also might have to rectify the formal aspects of mathematics in some instances to accord with such results of hyper-empirical reality. For example, is it correct to state that \(a \times b = 0\) is true only when one of the factors is zero? When either \(a\) or \(b\) is equal to zero, their product certainly is zero. But is it possible for the product to equal zero when neither of the two factors is zero? Indeed, this might be possible if the reality of the situation forced us to turn to hyper-imaginary numbers, which are the correlates of hyper-empirical reality. [Note 63] We must indeed attempt to clarify the relationship of real to imaginary numbers and the relationship of hyper-imaginary numbers to imaginary and real numbers, but we also may have to modify the rules governing calculations. [Note 64]

With regard to your first question, in the human being we can distinguish only what lies above a certain level and below a certain level. I explain this to almost everyone I think will be able to understand it. To anyone who looks at the wooden sculpture in Dornach of Christ in the center as the representative of humanity, with Ahriman and Lucifer on either side, I explain that we truly must imagine the human beings we encounter as existing in a state of balance. On one side is the suprasensible, on the other the subsensible. Each human being always represents only the state of balance between the suprasensible and the subsensible.

Of course, the human being is a microcosm of sorts and as such is related to the macrocosm. Therefore, we must be able to express the connection between each detail of the human being and a corresponding phenomenon in the macrocosm. Let me illustrate it like this: If this is the plane of balance (reference to a drawing that has not been preserved) and I imagine the subsensible element in the human being as a closed curve and the suprasensible element, or what human beings have in their consciousness, as an open curve, the resulting form is knotted below and opens outward above. This also represents how the human being is incorporated into the macrocosm. This lower, knob-like area pulls us out of the macrocosm, while the open curve of this upper surface incorporates us into the macrocosm. Here is the approximate location of freely willed human decisions. Above the level of free will, human forces are allowed to move out into the macrocosm. Everything below this level encloses macrocosmic forces so that we can assume a specific form.

Within the plane figures formed by this curve, let's note a series of data that I will call \(x\), representing the cosmic thoughts that we can survey. Here we have the cosmic forces that can be surveyed and here the cosmic movements. If I formulate a function involving these numbers up here, the result corresponds to what is down here in the human being. We need a function of factors \(x\), \(y\), and \(z\).

When I attempt to find numbers that express this relationship, however, I cannot find them in the domain of the number system that is available on this plane. In order to connect the suprasensible and the subsensible human being, I must resort to equations containing numbers that belong to systems lying on curved surfaces. These surfaces can be more precisely defined as the surfaces lying on paraboloids of revolution, surfaces that emerge when cones rotate in such a way that each rotating point constantly changes speed. [Note 65] There are also more complicated rotational paraboloids whose points, instead of maintaining fixed relationships among each other, are able to change within the limits of certain laws. Thus, the surfaces that serve my purpose are enlivened rotational paraboloids.

The relationship I am describing is extremely difficult. To date, certain individuals have imagined it, and the need for it has been discovered, but formal calculations will become possible only once esoteric or spiritual science is able to collaborate with mathematics. The path you have outlined for us today constitutes a beginning, a possible initial response to the challenge to discover what corresponds to the association of related functions that refer to number systems on the surfaces of two rotational paraboloids (one that is closed below and one that is open above) whose vertices meet in one point. As I have described, we would simply need to find the numbers lying on these surfaces, which do indeed correspond to a real situation.

With regard to the future development of formal mathematics, I must admit that it seems that much remains to be done and that much is possible. My next comment may do formal mathematics an injustice, since I have been less able to keep up with it in recent years. It has been a long time since I was fully aware of what is going on in this field, and things may have changed. Before the turn of the century, however, I always had the feeling that the papers published in the field of formal mathematics were terribly unconcerned about whether their calculations and operations were actually possible at all, or whether they would need to be modified at a certain point in accordance with some real situation. For example, we can ask what happens when we multiply a one-dimensional manifold by a two-dimensional manifold. Although it is possible to answer such questions, we must nonetheless wonder whether an operation like this corresponds to any reality at all or even to anything we can imagine. In order to get somewhere, it may be necessary to define clearly the concept of "only calculable."

As an example, a long time ago I attempted to prove the Pythagorean theorem in purely numerical terms, without resorting to visual aids. [Note 66] It will be important to formulate the purely arithmetical element so strictly that we do not unwittingly stray into geometry. When we calculate with numbers—as long as we stay with ordinary numbers—they are just numbers, and there is no need to talk about number systems in specific domains of space. When we talk about other numbers, however—imaginary numbers, complex numbers, hypercomplex numbers, hyper-imaginary numbers—we do have to talk about a higher domain of space. You have seen that this is possible, but we have to leave our ordinary space. That is why I feel that before purely formal mathematics sets up numbers that can only be symbolized—and in a certain sense, applying additional corresponding points to specific domains of space is symbolization—we must investigate how such higher numbers can be imagined without the help of geometry, [Note 67] that is, in the sense that I can represent a linear function through a series of numbers.

We would have to answer the question of how to imagine the relationship of positive and negative numbers on a purely elementary level. Although I cannot provide a definitive answer, because I have not concerned myself with the subject and do not know enough about it, Gauss's solution—namely, to assume that the difference between positive and negative is purely conceptual—seems inadequate to me. [Note 68] Dühring's interpretation of negative numbers as nothing more than subtraction without the minuend seems equally inadequate. [Note 69] Dühring accounts for the imaginary number a/-1 in a similar way, but this number is nothing more than an attempt to perform an operation that cannot be carried out in reality, though the notation for it exists. [Note 70] If I have 3 and nothing I can subtract from it, 3 remains. The notation for the operation exists, but nothing changes. In Dühring's view, the differential quotient is only a notated operation that does not correspond to anything else. [Note 71] To me, Dühring's approach also seems one-sided, and the solution probably lies in the middle. We will get nowhere in formal mathematics, however, until these problems are solved.

  1. Questions posed by Enist Blümel (1884–1952) after his lecture "Über das Imaginäre und den Begriff des Unendlichen und Unmöglichen" ("On the Domain of the Imaginary and the Concepts of Infinity and Impossibility") on March 11, 1920. Blümel taught mathematics in the school of continuing education at the Goetheanum in Dornach and in the first Waldorf School in Stuttgart. To date, no transcript of his lecture has been found.

  2. Ernst Müller (1884–1954), mathematician, author, and Hebraic and cabalistic scholar, gave a lecture on "Methoden der Mathematik" ('The Methods of Mathematics") in Stuttgart on March 8, 1920. To date, neither a transcript of Müller's lecture nor a record of Steiner's answer to his question has been found.

  3. For further discussion of the metamorphosis of long bones into head bones, see also Steiner's lectures of September 1, 1919 (GA 293),—April 10, 1920 (GA 201); and January 1, 10, 11, 15, and 17, 1921 (GA 323).

  4. On the reality of imaginary numbers, see also Steiner's lectures of March 12, 1920 (GA 321), and January 18, 1921 (GA 323).

  5. ’’Lectures on physics: Rudolf Steiner, Geisteswissenschaftliche Impulse zur Entwickelung der Physik: Zweiter Naturwissenschaftlicher Kurs, Die Wärme auf der Grenze Positiver und Negativer Materialität ("Spiritual Scientific Impulses for the Evolution of Physics: Second Natural Scientific Course. Warmth on the Boundary Between Positive and Negative Matter") (GA 321). See especially the lectures of March 10 and 11, 1920.

  6. Compare the passage that follows with Steiner's lectures of March 12 and 14, 1920 (GA 321). A collection of materials on an experiment in bending the spectrum using a strong magnet can be found in Beiträge zur Rudolf Steiner Gesamtausgabe ("Articles on Rudolf Steiner's Complete Works"), vol. 95/96, 1987.

  7. A variant of the text reads "The red moves outward toward the position/situation/layer," which makes no sense in either English or German.

  8. See Steiner's explanations of the ether and negative space in his lectures of January 8, 15, and 18, 1921 (GA 323); the question-and-answer session of April 7, 1921 (GA 76), the lectures of April 8 and 9, 1922 (GA 82),—and the questions and answers of April 12, 1922 (GA 82).

  9. In a lecture given on May 11, 1917 (GA 174b), Rudolf Steiner tells of a related personal experience during a class at the University of Vienna. According to Steiner's account, Leo Königsberger (1837–1921), a well-known mathematician of the day, rejected the concept of hypercomplex numbers because they would lead to zero factors (see Note 18). Just as complex numbers were slow to gain recognition, hyper-imaginary or hypercomplex numbers were only reluctantly accepted by mathematicians. The difference of opinion between adherents of the calculus of quaternions dating back to William Rowan Hamilton (1805–1865) and advocates of the vector analysis developed by Oliver Heaviside (1850–1925) and Josiah Gibbs (1839–1903) formed the background of the debate Rudolf Steiner alludes to here. Vector analysis initially gained the upper hand in practical applications because of the progress in theoretical physics that accompanied its development. At approximately the same time, however, the development of abstract algebra led to the discovery and classification of different systems of hypercomplex numbers.

    For more information on the above-mentioned debate, see Schouten [1914] (introduction) and Crowe [1967], On the history of the discovery and refinement of hypercomplex number systems, see Van der Waerden [1985],—on the mathematics of hypercomplex numbers, see Ebbinghaus et al. [1988], Part B. These and other generalized number systems have many applications in modern theoretical physics,—see Gschwind [1991] and the bibliography to his book.

  10. In his lecture of May 11, 1917 (GA 174b), Rudolf Steiner reports becoming aware of the mathematical problem of zero factors during a lecture by Leo Königsberger. Zero factors are generalized numbers whose product is zero, though the factors themselves are not equal to zero. Königsberger mentions this problem in the first lecture in his book Vorlesungen über die Theorie der elliptischen Funktionen ("Lectures on the Theory of Elliptical Functions") [1874], pp. 10-12, where he says of the existence of hypercomplex numbers, "Assuming that the validity of common rules of calculation for all arithmetic quantities remains a condition that must be met, if quantities of this sort can be incorporated into pure arithmetic, calculations that involve them and that are carried out according to the rules established for the numbers discussed earlier must lead to results that do not contradict the main propositions/theorems of arithmetic that have been discovered for real and complex imaginary numbers. Thus, according to the rules for multipart expressions, multiplying two numbers of the same type must yield a number of the same type, and the product cannot disappear unless one of the factors becomes zero."

    The passage that follows demonstrates concretely that the product of two such hypercomplex numbers can indeed disappear without one of the factors being zero, "which contradicts the basic rule for real numbers that a product of zero results only when one of the factors disappears." Later, Steiner received a copy of Oskar Simony's paper Über zwei universelle Verallgemeinerungen der algebraischen Grundoperationen ("On Two Universal Generalizations of Basic Algebraic Operations") [1885] with a personal dedication by the author. Simony discusses the problem of the existence of zero factors at the very beginning of this article, which is devoted to the concrete construction of two systems of hypercomplex numbers, one of which includes zero factors ([1885], §8). Additional material on this subject can be found in Beiträge zur Rudolf Steiner Gesamtausgabe ("Articles on Rudolf Steiner's Complete Works"), vol. 114/115, Dornach, 1995, p. 5. Schouten's work [1914], also with a personal dedication to Rudolf Steiner, includes an introduction to hypercomplex number systems (which Schouten calls associative systems),—zero factors are mentioned on p. 15.

  11. See Gschwind's investigations [1991] and list of references for further reading.

  12. The typed transcript reads "rotational parallelepopods," a term that does not exist in mathematics and that is probably due to an error in transcription. It seems unlikely from the context that the term "parallelopipeds" was intended. In all the transcripts the archives have received, the term "parallelepopods” is crossed out and replaced by "paraboloids" (in handwriting). Rotational paraboloids are surfaces that result from the rotation of a parabola around its axis of symmetry. This interpretation of the transcript presents the problem of how to relate such a surface to rotating cones. Without going into the problem in greater detail, Gschwind [1991] had good reasons for deciding on this wording and based important and fruitful conclusions on it. Specifically, he demonstrated a relationship between such surfaces and hypercomplex numbers. Exhaustive supplementary material can be found in Beiträge zur Rudolf Steiner Gesamtausgabe ("Articles on Rudolf Steiner's Complete Works"), vol. 114/115, Domach, 1995, pp. 5-7.

  13. Presumably Rudolf Steiner is referring here to the problem in number theory of finding whole numbers that can replace \(a\), \(b\), and \(c\) in the equation \(a^2 + b^2 = c^2\). Such numbers are known as Pythagorean triplets. Algorithms for finding all possible solutions to this equation—that is, all possible Pythagorean triplets have been known since antiquity.

  14. Rudolf Steiners call for establishing the foundations of arithmetic and algebra independent of geometry had already been taken up at the end of the nineteenth century, when the tendency to arithmeticize mathematics sometimes went so far that it threatened to displace geometry. It was one of the most important mathematical accomplishments of the early twentieth century, though initially it remained an internal issue in the field of mathematics. Some time elapsed before this development found its way into textbooks and the teaching of mathematics.

  15. Carl Friedrich Gauss (1777–1855), mathematician in Göttingen who explained negative numbers as simply the opposites of positive numbers. He presented his general views on the subject in his Theoria Residuorum Biquadraticorum [1831], pp. 175ff: "Positive and negative numbers can be applied only where the union of a quantity and its opposite eradicates that quantity. Precisely speaking, this prerequisite does not apply when substances (that is, objects that can be imagined as standing on their own) are involved but only in relationships between objects that are enumerated. It is postulated that these objects are arranged in a series, such as \(A\), \(B\), \(C\), \(D\), . . ., and that the relationship of \(A\) to \(B\) can be considered the same as that of \(B\) to \(C\), and so on. In this case, the concept of opposites means nothing more than reversing the members in a relationship, so that if the relationship between (or transition from) \(A\) to \(B\) is \(+1\), the relationship of \(B\) to \(A\) can be described as \(-1\). Inasmuch as such a series has no limits in either direction, each real whole number represents the relationship between a member that has been selected arbitrarily as the beginning and another specified member of the series." See also the discussion in Kowol [1990], pp. 88ff.

  16. Eugen Dühring (1833–1921), philosopher and author of books on political economy. See especially the book he coauthored with his son Ulrich [1884], which contains harsh criticism of Gauss' definition of negative numbers. According to the Dührings, the contrast or opposition that characterizes negative numbers results from unimplemented subtraction, which they view as the only essential aspect of negative numbers. See [1884], p. 16: "The incisive characteristic of an isolated negative number, however, is that it not only results from a numerical operation in which subtraction cannot be carried out but also points to an operation in which subtraction can be implemented. We must carefully distinguish between these two operations—or, if you will, these two parts of a general operation." For a comparison between Gauss' and Dühring's views on negative numbers, see Kowol [1990], p. 88 ff.

  17. On Dührings view of imaginary numbers, see E. and U. Dühring [1884], Chapters 2-4, and 13. A discussion of Dührings thoughts compared with other attempts to deal with this issue can be found in Kowol [1990], pp. 118ff. and 122ff.

  18. See E. and U. Dühring [1884], Chapters 4, 12, 14, and 15.