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The Fourth Dimension
GA 324a

Questions and Answers XVI

31 March 1920, Dornach

QUESTION: Ordinary mathematics encompasses the forms, surfaces, and lines of force of solids, liquids, and gases. How would you imagine a mathematics of the domains of warmth, chemistry, and life?

First of all, the field of mathematics as such would need to be appropriately expanded if we want to describe higher realms in a way that is analogous—but no more than analogous—to mathematics. As you may know, the need to expand mathematics became evident already in the nineteenth century. [Note 82] Let me just mention a point I have discussed on other occasions—including yesterday, I believe. [Note 83] In the late nineteenth century, it became apparent that a non-Euclidean geometry was needed to supplement Euclidean geometry and to make it possible to carry out calculations involving higher dimensions. Mathematicians of that time were suggesting that mathematics needed to be expanded. [Note 84] In contrast, as long as we are considering ordinary, ponderable matter, there is no appropriate use for dimensions other than the three ordinary Euclidean dimensions.

Mathematicians today, however, are so disinclined to explore appropriate views of the domains of warmth, chemical effects, and the elements of life that extending mathematical thinking into these areas is really very problematic. [Note 85] The views mathematicians propound today certainly do not create a counterbalance to the professed inability of physics to grasp the essential nature of matter. And to be consistent, physicists would have to admit that physics does not deal with the essential nature of light but only with what Goethe calls the image of light. Of course, sensible physicists will refuse to delve into the essential nature of things in the pursuit of their profession. Admittedly, the result is an unfortunate state of affairs: Physicists refuse to deal with the essential nature of things on any level. And those who concoct philosophies from the conventional, material views of physics not only refuse to inquire into the essential nature of things but even claim that it is impossible to do so. As a result, our view of the Earth today is very one-sided, because, in fact, physics is never simply a matter of geology but deals with the sum total of what such a specialized field can yield for general knowledge. Thus, we face the adverse consequences of the mechanistic, non-mathematical worldview that physics has developed over time.

What Goethe meant when he said that we should not talk about the being or nature of light but rather should attempt to become familiar with the facts about it, with its deeds and sufferings—which yield a complete description of the nature of light—is by no means the same as refusing on principle to consider the question of the nature of light. Goethe's statement simply points out that true phenomenology (structured in the way we discussed here yesterday) [Note 86] ultimately does provide an image of the being in question. [Note 87] To the extent that physics is or intends to be real phenomenology, it does provide—at least with respect to mechanics—an image of the essential nature of phenomena.

It can be said therefore that when we are not dealing with merely mechanical aspects of the phenomena of physics—that is, when we are dealing with fields other than mechanics—a mechanistic view hinders our ability to recognize the essential nature of things. To this extent, then, we do need to emphasize the radical difference between Goethe's intended phenomenology, which can be cultivated in Goetheanism, and any system whose principles rule out the possibility of approaching the true nature of things. This has nothing to do with the advantages of mechanistic methods for our urge to control nature. [Note 88] It is quite understandable that the field of technology and mechanics—which has produced the greatest triumphs of the last few centuries—and its mechanistic basis for understanding nature should satisfy our urge to control nature to a certain extent.

But to what extent has this drive to understand and control nature fallen behind in other fields because they refused to press on toward the type of knowledge to which technology aspired? The difference between technology or mechanics and the fields of study beginning with physics and continuing through chemistry to biology is not that these higher fields deal only with qualitative properties or the like. The difference is simply that mechanics and mechanistic physiology are very elementary and easy-to-grasp aspects and have therefore managed to satisfy our desire for control at least to a certain extent.

At this point, however, the question arises, How do we satisfy our urge to control when we move on to higher, less mechanistic fields? In the future, we will have to count on being at least somewhat able to dominate nature in ways that go beyond mere technology. Even in the technological field, we can very easily experience failures to understand and control nature. If someone builds a bridge without adequate knowledge of the laws of mechanics that apply to railways, the bridge eventually will collapse, carrying the train with it.

We react immediately to inadequate control due to faulty information. The proof is not always so easy, however, when control is based on more complicated domains that are derived not from mechanics but from the process of developing a phenomenology. It is fairly safe to say that a bridge that collapses when the third train crosses it must have been built by someone inadequately motivated to understand the mechanics involved. In the case of a physician whose patient dies, it is not so easy to confirm a similar connection between the practitioner's desire to understand and his or her control over nature. It is easier for us to say that an engineer designed a faulty bridge than that a doctor cured the disease but killed the patient. In short, we should be somewhat less hasty to emphasize the importance of our urge to control nature simply because our mechanistic view of nature has proved capable of satisfying this urge only in the domain of mechanistic technology.

Other ways of looking at nature will be able to very differently satisfy our urge to control. Let me point again to something that I believe I mentioned yesterday from a different perspective. We can never bridge the gap between the mechanistic view of the world and the human being except by applying a true phenomenological approach. [Note 89] Goethe's color theory not only presents the physical and physiological phenomena of color but also makes the whole subject humanly relevant by exploring the sensory and moral effects of colors. [Note 90] In our spiritual scientific work, we can move from the effects of colors pointed out by Goethe to the broader subject of understanding the entire human being and then to the still broader subject of understanding all of nature.

In some ways it may be beneficial to draw people's attention repeatedly to the fact that a large part of the decadence we experience today in Western culture is related to satisfying our urge to control only from the mechanistic perspective. In this regard, we have done very well. We not only have developed railways, telegraphs, and telephones, and even wireless and multiple telegraphy, but we also have paved over and destroyed large parts of this continent. Thoroughly satisfying our urge to control has led to destruction.

Following the straight line of development that began with our purely technological urge to control has led to destruction. This destructive aspect will be eliminated completely when we replace our pathologically expanding mechanistic view of the phenomena of physics with a view that does not eradicate the specifics of physical phenomena simply by blanketing them in mechanistic ideas. We will move away from the mechanistic view, which admittedly has produced very good physiological explanations, to the specifics of the phenomena of physics. Our new view, which cannot be discussed down to its last consequences in one hour, also will lead to an expansion of mathematics that is based on reality.

We must realize that in the past thirty to fifty years, confused mechanistic ideas have made possible all kinds of opinions about the so-called ether. After much effort, the physicist Planck, whom I mentioned earlier in a different context, arrived at this formulation: If we want to speak about the ether in physics at all, we cannot attribute any material properties to it. [Note 91] We must not imagine it in material terms. Planck forced physics to refrain from attributing material properties to the ether. The errors inherent in ideas and concepts about the ether are not due to having done too little mathematics or anything of that sort. They arose because proponents of the ether hypothesis were completely consumed by the trend that attempted to expand mathematics to cover the specifics of physics. Their mathematics was faulty because they behaved as if they were dealing with ponderable matter when they inserted numbers into formulas in which ether effects played a role. As soon as we realize that when we enter the domain of the ether, we can no longer insert ordinary numbers into mathematical formulas, we also will feel the need to look for a true extension of mathematics itself.

There are only two points that need to be made in this regard. The physicist Planck says that if we want to talk about the ether in physics, we must at least refrain from attributing material properties to it. And Einstein's theory of relativity—or any other theory of relativity, for that matter—forces us to eliminate the ether completely. [Note 92] In reality, we need not eliminate it. I can give only a brief indication here, but the main point is simply that when we shift to the ether, we must insert negative numbers into the formulas of physics—that is, mathematical formulas that are applied to phenomena in physics. These numbers must be negative because when we move from positive matter through zero to the other side, as when we move from positive to negative numbers in formal physics, what we encounter in the ether is neither nothing (as Einstein believes) nor a pure negative (as Planck says) but something that we must imagine as possessing properties that are the opposite of the properties of matter just as negative numbers are the opposite of positive numbers. [Note 93] Although we may debate what negative numbers are, the purely mathematical extension of the number line into negative numbers becomes significant for reality even before we clearly understand the character of negative numbers.

Of course, I am well aware of the significant mathematical debate in the nineteenth century between those who saw qualitative aspects in plus and minus signs and those who saw the minus sign only as a subtrahend lacking a negative minuend. [Note 94] This debate is not especially important, but it is important to note that when physics shifts from ponderable effects to etheric effects, it is forced to take the same route that we take in formal mathematics when we move from positive to negative numbers. We should check the results of the formulas when we decide to handle the numbers in this way. Much good work has been done in formal mathematics to justify the concept of formal imaginary numbers. In physics, too, we are obliged at a certain point to substitute imaginary numbers for positive and negative numbers. At this point, we begin to interact with numbers relevant to nature.

I know that I have sketched all this very briefly and summed it up in only a few words, but I must make you aware of the possibilities. As we move from ponderable matter to the forces of life, we must insert negative numbers into our formulas to signify the inverse of the quantitative aspect of matter. And as soon as we transcend life, we must shift from negative numbers to imaginary numbers, which are not mere formal numbers but numbers with properties derived not from positive or negative matter but from the substantial aspect that is related, qualitatively and intrinsically, to both the etheric aspect or negative matter and the ponderable aspect or positive matter in the same way that the imaginary number line relates to the real number line of positive and negative numbers. Thus, there is indeed a connection between formal mathematics and certain domains of reality.

It would be highly regrettable if attempts to make our ideas approximate reality or to immerse our ideas in reality were to fail because of the trivial notion that the offerings of truly rational, rather than merely mechanistic, physics and physiology would be less effective in satisfying the human urge to control nature. In fact, they would be more effective than applying the mechanistic worldview to the technology that we have glorified to such an extent. This mechanistic technology has certainly produced great results for humanity's cultural development. But people who constantly talk about the glorious progress of the natural sciences as a result of the conventional calculations of physics should keep in mind that other areas may have suffered as a result of turning our attention totally to the technological domain. To escape from the decadence brought on by our merely technical understanding and control of nature, we would do well to turn to a physiology and physics that, unlike our mechanical and mechanistic knowledge, cannot refuse to acknowledge the essential nature of things.

You see, this mechanical domain can easily dismiss the essential nature of things precisely because this essential nature is available—spread out in space all around us. It is somewhat more difficult for the entire field of physics to progress in the way that the field of mechanics has progressed. This is the reason for all of this talk of refusing to acknowledge the essential nature of things. When physicists choose to think in purely mechanical terms, they can easily refuse to understand beings. There is no being behind the formulas that are used today to express mechanics in mathematical terms. Beings begin only when we no longer simply apply these formulas but delve into the essential nature of mathematics itself. I hope this addresses the question of how to extend the field of mathematics to cover imponderables.

  1. Questions and answers after Karl Stockmeyer's lecture on "Anthroposophy and Physics" during the conference on "Anthroposophy and the Specialized Sciences" held at the Goetheanum in Dornach from March 21 to April 7, 1920. Ernst August Karl Stockmeyer (1886–1963) was a teacher at the first Waldorf School in Stuttgart. To date, no transcript of his lecture has been discovered. See the brief report on the conference in the journal Dreigliederung des sozialen Organismus ('The Threefolding of the Social Organism"), vol. 1, 1919/1920, no. 45.

  2. See the questions and answers of March 30, 1920, and Steiner's lectures of March 27, 1920 (GA 73a), and January 3, 1920 (GA 320).

  3. Bernhard Riemann (1826–1866), whom Steiner mentions repeatedly, typifies this trend. See also Note 1, Lecture 1 (March 24, 1905) on Bolyai, Gauss, and Riemann.

  4. See the beginning of the question-and-answer session on March 11, 1920 (E. Bliimel's questions) and related notes.

  5. See the question-and-answer session of March 1, 1920.

  6. Goethe says at the very beginning of the Preface to his Zur Farbenlehre ("On Color Theory") [1810]:

    When the subject of color is addressed, the very natural question arises of whether light should be discussed first and foremost. The brief and honest response to this question is that so much has been said about light, and so often, that it seems questionable to repeat or add to what has been said.

    For, in fact, our attempts to express the essential nature of light are in vain. We become aware of the effects of a being, and a complete account of them probably does encompass its essential nature. Our efforts to describe a person's character are all in vain, but if we present all of his actions and deeds, a picture of his character will emerge.

    Colors are the deeds of light, its deeds and sufferings. In this sense, we can expect them to yield conclusions about light. Colors and light are related very precisely, but we must think of both of them as belonging to all of Nature, because through them Nature and Nature alone attempts to reveal itself to the sense of sight.

  7. The editors of the German version, noting that the context requires a meaning of “control" or "understanding," substituted the word Beherrschung (control), here and elsewhere in the lecture for Beharrung) (perseverance), which appeared consistently in the typescript of the stenographic notes.

  8. See also Rudolf Steiner's lecture of March 30, 1920 (GA 312), and the question- and-answer session that took place on the same date.

  9. Goethe, Zur Farbenlehre ("On Color Theory") [1810], section 6, Sinnlich-sittliche Wirkung der Farbe ('The Sensory-Moral Effect of Color"), §758-920.

  10. Max Planck (1858–1947), theoretical physicist in Munich, Kiel, and Berlin. The hypothesis of a quasi-material ether that served as the medium for light processes and electrical phenomena had its roots in the thinking of Isaac Newton (1642–1727) and René Descartes (1596–1650). This qualitative type of ether made it possible to interpret processes whose more precise mechanisms were not yet understood. The chief characteristic of nineteenth-century ether hypotheses was quantifiability, which made it possible to incorporate such processes concretely into mathematical theories on the phenomena of physics. See also the beginning of the question-and-answer session of March 7, 1920, and the corresponding notes.

    The exact wording of Planck's formulation has not been found. Planck [1910] emphasizes, however, "I believe that I will not encounter any serious opposition among physicists when I summarize this position as follows: Presupposing that the simple Maxwell-Hertz differential equations are fully valid for electrodynamic processes in pure ether excludes the possibility of explaining them mechanically" (p. 37). Later he says, "similarly, it is certainly correct to state that the first step in discovering [Einstein's] principle of relativity coincides with the question of what relationships must exist between natural forces if it is impossible to ascribe any material properties to the light ether—that is, if light waves replicate through space without any connection to a material vehicle. In that case, of course, it would be impossible to define—let alone measure—the speed of a moving body with regard to the light ether. I need not emphasize that the mechanical view of nature is virtually incompatible with this view. Thus, anyone who sees this view as a postulate of the thinking of physics will never be comfortable with the theory of relativity. Those who are more flexible in their judgments, however, will first ask where this principle leads us" (p. 39).

  11. See the question-and-answer session of March 7, 1920, and the corresponding notes.

  12. Compare this and the following passages to the question-and-answer sessions of March 11, 1920 (Blümel), and January 15, 1921, and to the corresponding notes.

  13. Comments about the debate surrounding the concept of negative numbers can be found at the end of the question-and-answer session of March 11,1920 (Blümel). See Kowol [1990], chapter IV.B.