The following text is a written elaboration and extension of a lecture given at the conference “ General Mathematics: Mathematics and Society. Philosophical, Historical and Didactical Perspectives” in Schloss Rauischholzhausen from June 18 to 20, 2015. The invitation to the lecture was based on older epistemological texts, which therefore reappear – partly in spirit, partly verbatim – as set pieces in the present text.
If I look at the world through rose-colored glasses, it appears to me as rose-colored. And accordingly, who looks at the world through mathematical glasses, sees mathematical structures everywhere.[1] Now obviously the color pink is not a property of the world, but one of the glasses. One could add that the world must have pink components so that one can see anything at all through the pink glasses. But nobody would claim that the world consists only of these components, just because all others are blanked out by the glasses. In contrast, Enlightenment reason has managed to confuse reality with the glasses through which it views it, that is, to declare the approach to the world specific to modern science to be a property of reality itself, and thus to declare it to be essentially mathematical.
As a typical and prominent representative of this view, Max Tegmark, physicist at MIT, was quoted in a Spiegel interview on the occasion of the translation of his book “Our Mathematical Universe” (Tegmark 2015) into German:
Spiegel: Professor, if a fairy godmother promised to answer any question about the nature of our world, what would you ask?
Tegmark: Let me think. Hmm, I would probably ask them: What set of formulas provides an accurate description of our world?
Spiegel: And you are convinced that such world formulas exist?
Tegmark: I suspect so. But if the fairy were to shake her head and say, “Sorry, there are no such formulas,” that would also be very exciting to know.
Spiegel, 4.4.2015, 113
In this essay, to stay in the picture, I would like to take the position of the bad fairy and justify why Tegmark’s question is nonsensical, not to say crazy. In his book, Tegmark argues that the “essence of reality” is mathematical and that the universe is pure mathematics, a mathematical structure in which we humans live but whose physical reality is completely independent of us (cf. Tegmark 2015, 370). At least he can be credited for admitting the possibility that the question of the world formula cannot be answered. There are even tougher dogmatists, people who consider themselves particularly enlightened, dismiss religious ideas as “God delusion” (Dawkins 2007), and for their part adhere to the belief that reality follows mathematical laws. But if one knows that the religious forms have sprung from the human head – I am also of this opinion – it should make one think that this is just as valid for mathematics. To locate it so easily in the world, as a property that is independent of us, could therefore be analogously described as “math delusion.”
Unlike mathematics, mathematical – and thus “exact” – natural science and the access to the world associated with it is an invention of modern times. When thinking about the causes and consequences of the mathematization of modern society, for which there was nothing comparable in pre-modern times, one should therefore pay attention to this hinge between mathematics and society. Beyond its original subject area, the mathematical-scientific method has now gained a foothold as a method of “mathematical modeling” in almost all other branches of science and in many non-scientific disciplines. Apparently, the success of this method in physics, chemistry, and recently biology, as well as in the technical subjects related to these natural sciences, leads to its unreflective adaptation, even in those areas where the use of mathematical methods should at least be met with doubts, because they do not fulfill certain prerequisites of the “exact” sciences.
For example, the preface to a standard textbook of economics states:
Economics combines the strengths of political science and natural science. […] By applying scientific methods to political issues, economics seeks to make progress on the fundamental challenges facing all societies.
Mankiw and Taylor (2012, VIII)
Here it is implicitly assumed that scientific methods can be applied to political questions, even if it cannot be claimed that such attempts are crowned with success (cf. Ortlieb 2004), in which they differ from their “exact” models. But even where the idea of making the use of mathematical methods a badge of “scientificity” is not particularly successful, it does help to increase the importance of mathematics for modern society even further, to a certain extent beyond what is necessary.
The thesis put forward here is therefore that mathematics owes its importance in our society on the one hand to the undeniable success of the mathematical natural sciences, but on the other hand also to a false understanding of this success, as expressed for example in the question of the world formula, the belief that reality follows mathematical laws. I would first like to make clear why this belief is unfounded, then venture to explain where it comes from, and finally suggest what harmful consequences it has.
Mathematics as Positivist Magic
The blindness of mathematical-scientific thinking for its own form jumps almost regularly into the eye whenever scientists start to think publicly about the relation of their own science and its mathematical instruments to the real world:
Real science, on the other hand, remains real magic. It is fascinating to see how many physical phenomena adhere to theories and formulas with uncanny accuracy, which has nothing to do with our desires or creative impulses, but with pure reality. It makes one completely speechless when it turns out that phenomena, which were initially only theoretically justified and calculated with formulas, subsequently turn out to be reality. Why should the reality be like this? It is pure magic!
Dewdney (1998, 30)
Why does mathematics, which comes from our own heads, fit so well to nature, which actually has nothing to do with it? Among those who are practically active in positive science, this question, as here with the mathematician Dewdney[2], regularly triggers reverential astonishment, depending upon location either over mathematics, which can accomplish such great things, or over nature, which is so rationally arranged. The only way out of this aporia seems to be to resort to magical notions. However, if even professional science theorists do not get beyond this level, they rightly attract ridicule:
Carnap, one of the most radical positivists, once called it a stroke of luck that the laws of logic and pure mathematics apply to reality. A thinking which has all its pathos in its enlightenment quotes in a central place an irrational – mythical – concept like that of the stroke of luck, only to avoid the insight, which admittedly shakes the positivist position, that the supposed circumstance of luck is none, but product of the nature-dominating … ideal of objectivity. The rationality of reality, registered by Carnap with a sigh of relief, is nothing but the reflection of subjective reason.
Adorno (1969, 30)
Adorno’s criticism covers all the ideas belonging to positivism, that mathematical regularity is a property of external reality, and that science simply consists in grasping the facts and this regularity of things themselves, according to the positivist program of Comte 1844/1994, 17.
In contrast, Adorno insists on the statement – which I will follow here and elaborate on – that mathematics and its laws are not a property of external nature, but part of our instruments of cognition.
An example: Galileo’s laws of falling bodies
The laws of the free fall of heavy bodies are at the beginning of modern physics. They state:
G1 All bodies fall at the same rate.
G2 In a fall from rest, the distances traveled behave like the squares of the times.
With these laws Galileo Galilei (1564-1642) came into contradiction with the Aristotelian science prevailing in his time, whose doctrine said:
Ar Every body has the aspiration to take its rightful place. Light bodies move up, heavy ones fall down. The heavier the body, the faster it falls.
In fact, this is one of the rare cases in which modern physics can be directly confronted with medieval ideas, because it usually deals with questions that people in other or earlier societies did not even ask themselves. It is all the more interesting to see how Galileo’s laws of falling bodies prevailed.
An integral part of the image that modernity in general, and Western science in particular, have of themselves is the idea that they are oriented towards facts, while past cultures followed their myths and other fantasies and have therefore logically and quite rightly passed away. Galileo’s argument with the authority of Aristotelian science and the Catholic Church still serves as a paradigm for this, although mechanics, which goes back to Galileo and Newton, has long since had to give up its claim to general validity. Bertolt Brecht’s play “Life of Galileo,” written around 1945, thrives on the enlightenment pathos of this struggle of the “cold eye of science,” revealing the facts against the “thousand-year-old mother-of-pearl haze of superstition and old words,” through which the reign of “selfish rulers” could continue to be maintained. Brecht’s criticism, which is unavoidable against the background of the dropping of the first atomic bomb, is then also presented exclusively on the moral level, namely that Galileo had allowed himself to be intimidated and had handed over his knowledge to those in power, “to use it, not to use it, to misuse it, just as it served their purposes.” Every well-behaved natural scientist can rightly counter that Galileo’s teachings, despite his retraction, have finally become common knowledge, that truth cannot be stopped, even if this seems to have been of little use to mankind.
Mechanics, associated with the names of Galileo and Newton and today called “classical,” played the role of a leading science from the beginning of modern times until the 19th century. In a certain respect it still is today, even if its results had lost their universal claim by the “modern” physics of the 20th century. For the mathematical-scientific method developed in it and applied with resounding success has gained further importance in the last century and has taken on a model function for Western science of all faculties, at least of their respective mainstreams, so that even the critics of its transfer to the social sciences, for example, still have to deal with it. As correct as their argument is that a method must adapt to its object and that “society” is not the same as “nature,” such discussions often suffer from the fact that positivist empiricism, i.e. the “fact faction,” has won the hegemony of interpretation as to what this method actually does and what kind of results it can achieve. The assertion that these are objective facts, verifiable by everyone, is no longer even questioned.
The Myth of Pisa
An example of this phenomenon is the following story, which the historiography of science had to offer as assured knowledge for almost three centuries. It concerns the free fall of heavy bodies, the first part of the Galilean law of fall, and figured as the “blow from which Aristotelian science never recovered”:
At this point we must refer to the famous experiments on the fall of bodies, which are closely connected with the Leaning Tower of Pisa, one of the most curious architectural monuments in Italy. Almost two thousand years earlier, Aristotle had claimed that in the case of two different weights of the same material falling from the same height, the heavier one would reach the ground before the lighter one, according to the ratio of their respective weights. The experiment is certainly not difficult; nevertheless, no one had thought of carrying out such a proof, which is why this assertion was included among the axioms of the science of motion by virtue of Aristotle’s word of power. Galileo, invoking sense perception, now challenged Aristotle’s authority and claimed that the balls fell in the same time, except for an insignificant difference based on the difference in air resistance. The Aristotelians scoffed at this idea and refused to listen to it. Galileo, however, was not intimidated and decided to force his opponents to face the fact like him. Therefore, one morning, in front of the assembled university – professors and students – he climbed the leaning tower, carrying two balls, one ten-pounder and one one-pounder. He placed them on the edge of the tower and dropped them at the same time. And they fell together and hit the ground together.
J.J. Fahie. Galilei, His Life and Work, London 1903, 24 f., quoted after Koyré (1998, 124).
Almost 300 years after Galileo’s death, Alexandre Koyré[3] has finally put an end to the story of his free fall experiments at the Leaning Tower of Pisa, so that no historian of science who wants to be taken seriously can still tell it today. The only truth in the story is that Galileo held a poorly paid, three-year position as professor of mathematics at the University of Pisa around 1590. The legend first appeared 60 years after the incident described and has been embellished by later historians of science. What strikes one without further historical knowledge is its inconsistency: what could possibly have caused the Aristotelian professors, here reproached for their dogmatism, to run together when one of their most insignificant colleagues staged an insane experiment? The story contradicts all customs at universities of that time and probably still of today’s universities. It was never mentioned by Galileo himself, [4]and finally: the experiments would have gone wrong, respectively they were made (1640, 1645, 1650), with big and small iron balls, with clay balls of the same size, one solid, the other hollow, with balls made of different materials, and they all went wrong (in the sense of the legend).[5]
What is really exciting about this modern fairy tale is that for 300 years it belonged to the general educational heritage, to a certain extent to the secured stock of our scientific knowledge. Like all fairy tales, this one also conveys a message, namely that of modern rationality, which lets the facts speak without bias, while the dark Middle Ages only referred to authorities and passed on textbook knowledge. The late proof that this is a myth, the myth of empiricism, does not change its effectiveness. More than 350 years after Galileo, this worldview has become so self-evident that it no longer needs justification. And as a glance at a standard textbook of experimental physics shows, even the fairy tale associated with it is too good to be omitted simply because it is a fairy tale:
First, let us investigate whether the falling motion depends on the type of falling body, e.g. on its size or weight. We make the following experiments: Two balls of the same size, made of aluminum and lead, which therefore have very different weights, are let fall to the ground simultaneously from the same height. We find that they hit the ground at the same time, as Galileo (1590) already found out by drop experiments at the leaning tower of Pisa. If we take three identical spheres made of the same material, they will naturally arrive at the ground at the same time. If we now connect two of these spheres firmly with each other (for example by a passing pin), and if we let this double sphere fall with the third single sphere at the same time, then also these bodies of different size and different weight hit the ground at the same time. However, the following experiment seems to contradict the conclusion that all bodies, independent of shape, type and weight, fall at the same rate: If we let a coin and a piece of paper of the same size fall, we observe that the coin arrives at the bottom much earlier than the piece of paper falling at the same time from the same height; the latter flutters to the ground in irregular motion and needs a longer time to fall through. The contrast, however, is only apparent. In this last experiment, the resistance of the air becomes disturbingly noticeable. The air flowing past the body during the fall inhibits the falling movement, and more strongly the larger the surface of attack of the air on the body in question. If we clench the piece of paper into a small ball, it falls just as quickly as the coin. The disturbing influence of the air resistance on the free fall can be shown by an experiment given by Newton. A glass tube about 2 m long and several centimeters wide, fused at both ends, contains a lead ball, a piece of cork and a down feather. If the three bodies are at the bottom of the tube and the tube is turned rapidly through 180°, the lead ball, then the piece of cork and finally the downy feather are observed to arrive at the bottom. But if we pump the air out of the tube and repeat the experiment, we see that the three bodies hit the bottom of the tube at the same moment. We may therefore state the law of experience: In a vacuum, all bodies fall at the same rate.
Bergmann-Schaefer (1974, 40)
Why only in a vacuum? After all, it worked in Pisa, too. The conclusion remains as opaque as the reasoning. The reason lies in the fact that statements with completely different methodological status are wildly mixed up here:
- The text contains wrong and right assertions about everyday observations, whereby the right ones are just those which contradict the Galilean law of falling. They are simply interpreted away with reference to the “disturbing” air resistance.
- A thought experiment is carried out (sphere and double sphere), from which the law of falling is logically compelling, but without recourse to any observational results.
- Finally, an experiment is described which requires a high technical effort (pumping the tube empty). Only in the artificial situation thus produced can the claimed law also be observed.
To then call the whole thing an “empirical law” is already strong and indeed presupposes the confusion that first had to be created. Empiricism lives from this confusion.
The text is an example of how little most natural scientists know about the history and method of the science they themselves practice. This was by no means always so, but what can be stated here, rather, has the character of a decay. Galileo himself, at any rate, was quite aware of his approach, unlike most of his epigones – not all of them. It is therefore worth going back to the sources.
What brought the Galilean law of falling into the world, if it could not be the experience, neither the direct observation, because this teaches something else, nor an experiment in the vacuum, which Galilei could not carry out already because he lacked the technical means for it? The simple answer is this: The law of falling bodies results from a logical argument, a mathematical proof or, as one would say today, a thought experiment. The argument had already been published in 1585 by the mathematician Benedetti in Venice and is also contained in the text from the physics textbook quoted above, although there it is completely deprived of its methodical significance.
Proof of the First Law of the Case
Benedetti argued: Two identical bodies fall at the same speed, at least that seems to be undisputed. If they are now connected by a light (ideally massless) rod, their velocity does not change, but it is the same for a body of double mass (cf. Fig. 1). The same can be argued with three, seven or even a hundred thousand bodies, in any case the same velocity results for bodies of arbitrarily different mass.
Galileo (1638/1995, 57/58) used this as a proof of contradiction: If Aristotle’s law of falling bodies Ar were correct, a heavier body would have to move ahead of a lighter one. If both are now connected with a string, the heavier body would have to pull the lighter one behind it, but the lighter one would have to slow down the heavier one (cf. Fig. 2). The result would be a smaller speed than that of the original heavier body, but for an altogether heavier body, a contradiction.
Both proofs of the First Law of Fall G1 abstract from the shape of the bodies, and thus refer only to their mass.[6] So it was shown: If the falling speed does not depend on the shape of the bodies, their mass distribution, then all bodies must fall equally fast. This result, however, is in obvious contradiction to empiricism, because the bodies do not fall at the same speed. If now logic and empiricism would be considered equally, then the conclusion would be to be drawn from it that from the shape of the bodies just may not be abstracted. But Galileo does not draw this conclusion, and exactly here lies the revolutionary novelty of his view of nature: He decides in favor of logic and mathematics and against direct empiricism and thus for a view of nature which antiquity or the Middle Ages could only have regarded as crazy.
The Mathematical-Scientific Method
The connection of the natural laws thus obtained to empiricism lies in the experiment, the second great innovation of modern natural science, whose difference from simple observation cannot be emphasized enough. An experiment is the production of a situation in which the condition of the derived law is fulfilled, in this case: one can abstract from the shape of the bodies, e.g. by vacuum, which Galilei was not yet able to do.
In this respect it can be said that mathematical laws of nature are not based on observation, but are produced. To be more precise: They are instructions for the production of situations (in experiments) in which they are valid.[7] Here lies the reference to the nature-controlling technology of modern times.
The mathematical-scientific method constituted in this way is based on the basic assumption that there are universally valid laws of nature, independent of place and time, which can be described mathematically (the concept of measurement would otherwise be meaningless). For this, a linear flowing, continuous time and a homogeneous space, i.e. not subdivided into different spheres, is presupposed.
The objection that the universal form of laws of nature has long been proven by modern natural science misses the point: The lack of form of laws in any area would never be blamed on nature, but would be justified by the fact that science is not yet ready to recognize them.
The procedure then consists first of all in the formulation of ideal conditions, from which conclusions are drawn in the thought experiment in an ultimately mathematical way. The subsequent experiment then consists in the production of these ideal conditions and the verification of the conclusions by measurements. Care must be taken that the measurement procedure, i.e. the physical effort of the experimenter, does not disturb the ideal process. Experiments must be repeatable, and in this respect they also differ from mere observations.[8]
Thus, there can be no question of modern science, in contrast to the Middle Ages, being oriented to “the facts”; rather, the opposite is true. Koyré makes this very clear with the example of the principle of inertia, which as a (mathematical) principle has no direct correspondence in empiricism and nevertheless founded modern physics:
This principle seems to us completely clear, plausible, yes, it is obvious. It seems obvious to us that a body at rest will also remain in it… And once it gets in motion, then it will continue to move in its original direction. And always with the same speed. We don’t really see a reason or cause for why it should happen differently. This appears to us not only as plausible, but also completely natural. Yet it is nothing less than that. The natural, tangible evidence, which these views enjoy, is comparatively recent. We owe it to Galileo and Descartes. In the Greek antiquity as well as in the Middle Ages the same views would have been classified as ‘obviously’ wrong, even absurd.
Koyré (1998, 72)
The question remains to be answered why this misjudgment of the actual mathematical-scientific procedure is so widespread. Koyré explains this by habituation:
We know the basic notions and principles too well, or more correctly, we are too accustomed to them to be able to correctly assess the hurdles that had to be overcome to formulate them. Galileo’s concept of motion (and that of space) seems so ‘natural’ to us that we believe to have derived it ourselves from experience and observation. Although probably none of us has ever encountered a uniformly persisting or moving body – and this simply because such a thing is quite impossible. Equally familiar to us is the application of mathematics to the study of nature, so that we hardly grasp the audacity of the one who claims: ‘The book of nature is written in geometrical symbols.’ We miss Galileo’s audacity in deciding to treat mechanics as a branch of mathematics, that is, to replace the real world of daily experience with a merely imagined reality of geometry and to explain the real from the impossible.
Koyré (1998, 73)
The explanation remains unsatisfactory: The fact that we consider a “manifestly absurd” procedure to be completely “natural” does jump to the eye. Why we do it, however, remains ultimately unexplained here.
Revolution of the Way of Thinking
Immanuel Kant, himself active in the natural sciences for ten years, summarizes the mathematical-scientific method in the preface to the 2nd edition of his Critique of Pure Reason in 1787 in the language peculiar for him:
When Galileo rolled balls of a weight chosen by himself down an inclined plane, or when Torricelli made the air bear a weight which he had previously thought to be equal to that of a known column of water, or when in a later time Stahl changed metals into calx and then changed the latter back into metal by first removing something and then putting it back again, a light then dawned on all natural scientists. They understood that reason has insight only into that which it itself produces according to its design; that it must take the lead with principles for its judgments according to constant laws and compel nature to answer its questions, rather than letting nature guide its movements by keeping reason, as it were, in leading-strings; for otherwise accidental observations, made according to no previously designed plan, can never connect up into a necessary law, which is yet what reason seeks and requires. Reason, in order to be taught by nature, must approach nature with its principles in one hand, according to which alone the agreement among appearances can count as laws, and, in the other hand, the experiments thought out in accordance with these principles – yet in order to be instructed by nature not like a pupil, who has recited to him whatever the teacher wants to say, but like an appointed judge who compels witnesses to answer the questions he puts to them. Thus even physics owes the advantageous revolution in its way of thinking to the inspiration that what reason would not be able to know of itself and has to learn from nature, it has to seek in the latter (though not really ascribe to it) in accordance with what reason itself puts into nature. This is how natural science was first brought into the secure course of a science after groping about for so many centuries.
Kant (1787/1990, B XIII)
On the one hand, it becomes clear here what an important role Kant ascribes to the “principles of reason,” which cannot be derived from empiricism (the Kantian a priori). He thus solves the problem that still troubles modern positivism, namely how objective knowledge is possible.
On the other hand, a typical contradiction of Enlightenment thinking comes through in Kant, which considers “reason” to be a general human quality or ability, but nevertheless claims it exclusively for itself and denies it to other or earlier societies. If this prejudice is brushed aside, it can be said that the mathematical and scientific method did indeed first have to prevail against medieval thinking, and talk of the “revolution of the way of thinking” thus hits the point that this revolution helped a reason to break through which is specific to the bourgeois epoch, against the reason of the Middle Ages, which was completely different, but not therefore unreasonable as such.
The concept of “objective knowledge” thus receives a different meaning than the one that is usual in our linguistic usage, an ahistorical one, independent of the form of society and equally valid for all people, which is why Greiff 1976 also speaks of the “objective form of knowledge.” A representative of another or earlier culture, who does not recognize the basic assumptions of the mathematical-scientific method, the principles of bourgeois reason, would also not be able to be convinced of the truth of scientific knowledge. The only component of natural science which one could demonstrate to him credibly is the experiment: If I carry out this action A, determined to the smallest detail (which probably seems ritualistic or bizarre to the other person), then the effect B regularly occurs. But nothing else follows from this as long as my counterpart does not share my basic assumption of the universal laws of nature that are supposed to be expressed in the experiment.[9]
Fetishism and Gender Dissociation
A fetish is a thing onto which supersensible qualities are projected and which is thus able to exert power over those who fall prey to it. The Enlightenment knows itself to be above such fetishism, as it was attached to West African religions at the beginning of colonialism. Marx, as is well known, saw it differently:
A commodity is therefore a mysterious thing, simply because in it the social character of men’s labor appears to them as an objective character stamped upon the product of that labor; because the relation of the producers to the sum total of their own labor is presented to them as a social relation, existing not between themselves, but between the products of their labor… There it is a definite social relation between men, that assumes, in their eyes, the fantastic form of a relation between things. In order, therefore, to find an analogy, we must have recourse to the mist-enveloped regions of the religious world. In that world the productions of the human brain appear as independent beings endowed with life, and entering into relation both with one another and the human race. So it is in the world of commodities with the products of men’s hands. This I call the Fetishism which attaches itself to the products of labor, so soon as they are produced as commodities, and which is therefore inseparable from the production of commodities.
Marx (1867/1984, 86/87)
The analogy to the positivistic idea of mathematical-scientific knowledge jumps to the eye. It is the attempt to apply products of the human head, specifically numbers and other mathematical forms, to reality, and to form it according to their image, or at least perceive it through them. The end of this story consists in the belief that reality or “nature” itself is law-like, and that the success of natural science is definitive proof of this.
But it is not a mere analogy, and not the coincidental parallelism of two independent fetishisms. Since the late publication of Sohn-Rethel’s approach in 1970, there have been repeated attempts to address the question that was blanked out by the Enlightenment and finally tabooed by positivism, i.e. to illuminate the connection between “commodity form and thought form,” “social form and cognitive form,” “money and mind,” for instance by Greiff 1976, Müller 1977, Bolay/Trieb 1988, Ortlieb 1998. The matter is complex and cannot be clarified in a few pages. The most direct way is taken by Bockelmann 2004, which I briefly sketch here. One of the difficulties on which Sohn-Rethel’s first attempt ultimately failed is to clearly distinguish the modern form of knowledge, as well as that of commodity society, in its specificity from its precursors in antiquity. It is not the mere existence of money or the exchange of the surplus products that set the modern form of thought on its way, rather it was necessary for money to become the determining generality and the actual purpose of production,
when, for the first time historically, it can be said that “all things came to be valued with money, and money the value of all things.” Then money begins – in this sense that is concise for us – to be money, in that it still functions as money alone. The fixed existence, which it had until then only in the material thought to be valuable, then passes over into the fixed generality of the reference of all things to the value of money – and thus into its fixed existence taken for itself. When the acts of buying and selling for provision acquire determining generality, there arises the general necessity to continue the market that must have arisen for it, as the web of these buying acts, simply so that the provision that depends on it does not break off in turn. The necessity to dispose generally of money thus translates into the generality with which the money function continues to be necessary; and thus translates into the solidity of this function as an entity existing for itself.
Bockelmann (2004, 225)
The historically new situation consists of a real abstraction. It demands from the market participants a performance of abstraction which they must carry out without doing so as a conscious mental effort; in Marx’s formulation:
Hence, when we bring the products of our labor into relation with each other as values, it is not because we see in these articles the material receptacles of homogeneous human labor. Quite the contrary: whenever, by an exchange, we equate as values our different products, by that very act, we also equate, as human labor, the different kinds of labor expended upon them. We are not aware of this, nevertheless we do it.
Marx (1867/1984, 88)
It should be pointed out that Bockelmann does not refer to Marx at any point; the concept of (abstract) labor does not appear anywhere in his work. Regarding the question of what commodity production, i.e. production for the sole purpose of acquiring other commodities, mediated by money, brings about in the people subjected to it, however, both explanations are compatible. The subjects of commodities must develop a reflex for the sake of their survivability, which henceforth, as a compulsion of which they are not aware, determines not only their monetary actions, but their access to the world in general:
This is the form in which no one had to think and therefore no one had been able to think, the synthetic performance conditioned by modern times, which people must perform within it: two units related to contents, but themselves not-content-related, in the pure relation of determined against not-determined. This synthesis, thus conditioned, becomes a necessity and a compulsion for thinking… This synthesis has its genuine area in dealing with money, and it is there that people must apply it to everything, regardless of contents, they have to relate the pure unit ‘value’ to any content… The new, functional performance of determining value using non-content-related units lies above the older and likewise synthetic performance of a material way of thinking, which involved thinking value into things and relating them to each other according to this inherently thought value.
Bockelmann (2004, 229/230)
It is not difficult to see how closely the approach to the world, abstractly described here and enforced by the commodity form, corresponds to that of mathematical natural science, and is still found in the details of its method:
The experiment is the medium for the transformation of nature into function. The modernly changed view on the empirically given is no longer one of observation, but penetrates to find in it what it must presuppose, the lawful behavior.
Bockelmann (2004, 354)
Furthermore, positivistic science’s missing or fetishistic awareness of its method and its object can be easily explained in this way:
World and nature are thought of in a functional way: that is – as long as the genesis of the functional form of thought remains unrecognized – they are thought of as if the functionally conceived form was its real form. After that, the laws of nature must really exist as we think and presuppose them, really in this form of functional non-content.
Bockelmann (2004, 358)
The fact that knowledge of the genesis of its form is necessary for overcoming this consciousness does not mean – and is not claimed by Bockelmann – that it alone will suffice, if it is not accompanied at the same time by the overcoming of the commodity fetish that underlies it.
It should have become clear that any explanation of the connection between the form of society and the form of knowledge must take into account the subject of knowledge, which is always at the same time a citizen of the state and a monad of money, shaped by the society in which knowledge takes place. Even if the aim is only to better understand the phenomenon characterized here as math delusion, independent studies on the constitution of the subject form can certainly be helpful, cf. for example Ulrich 2002, Kurz 2004. One fundamental moment that has not been touched upon so far should be highlighted here, namely the split nature of the modern subject and the (related) gendered connotation of value-shaped socialization, as well as that of the mathematical-scientific form of knowledge.
Objective knowledge, as it takes place for example in the physical experiment, can be described as a process of dissociation, namely the dissociation of those aspects of reality which would disturb the law-like process. One of the “disturbing factors” to be eliminated is the experimenter himself. His physicality and his sensations could disturb the “ideal” course of events and are therefore to be eliminated as far as possible without endangering his observer status, which Greiff (1976) elaborates on the basis of the common, imperatively formulated regulations in textbooks of experimental physics on the execution of experiments. The active intervention in nature carried out in the experiment is thus first and foremost an action of the experimenter on himself, namely his splitting into a mind and a body being. This form of knowledge presupposes a subject that can be split in this way.
Such subjects are by no means to be found in all forms of society, but are rather a specific feature of only one, namely bourgeois society, in which the division into feeling and mind, body and spirit, private and public, together with the corresponding gender connotations, is constitutive. In the public sphere, which is oriented towards abstract calculations, only the “male” parts are in demand, while the “female” parts are to be dissociated. The latter, since they are nevertheless necessary for individual survival and for social reproduction, have not disappeared, but have been delegated to women (“value dissociation,” cf. Scholz 2000, 13 ff. and 107 ff.). Where else, one could object, but these parts become “female” – and the others “male” – only through the corresponding attribution, they are not so by nature. It should also be noted that we are dealing here with a schema that is often broken in individuals; after all, we are not talking about biological determinants, but about social conditions. Thus, not every man is equally “male,” not every woman equally “female,” but the compulsion is great to conform to the gender attributes codified by commodity society, so that still, statistically speaking, the positive correlation between social and biological gender is high.
In this sense, the experimenter, the subject and the bearer of objective knowledge aiming at mathematical laws of nature is “male,” not only structurally but also empirically, and the higher his rank in the scientific hierarchy, the more pronounced this is. It is therefore no coincidence that criticism of the seemingly unassailable mathematical sciences has come almost exclusively from the feminist side in recent decades. Representative of many such critiques are Scheich 1993 and Keller 1995, cf. and also Bareuther 2014. The profound dimension of the problem can, of course, hardly be reached without reference to value dissociation as an equally comprehensive as “in itself broken formal principle of social totality” (Scholz 2004, 19). Those who only consider the institutionalized acquisition of knowledge and its mechanisms for themselves can at best scratch its surface.
Models
If the fetishistic way of thinking about mathematical regularity as a property of things were not so deeply anchored in the social unconscious of modernity, it should have become obsolete at the latest with the emergence of the concept of a model at the end of the 19th century (cf. Ortlieb 2008). Because this term contains – in contrast to Galileo’s idea of the book of nature written in geometric symbols – an ambiguity: Mathematical models do not emerge clearly from the matter, but their development is always subject to arbitrary aspects of expediency (cf.Hertz 1894/1996). The same object of investigation allows different mathematical models that can exist side by side, even if they contradict each other because they cover different aspects. This forbids combining model and reality into one.
That “certain correspondences (must) exist between nature and our mind,” what Hertz (1894/1996, 67) also speaks of, is guaranteed in physics by the fact that nature is adapted to our mind, i.e. to the mathematical ideal conditions, in the experiment and the said correspondence is produced with it first. If, on the other hand, the ideal conditions assumed in the model cannot be produced or can be produced only insufficiently, the laws of nature to be observed remain mathematical fictions in the end, as everybody who has “fitted” models and data at least once knows well. The regularity is only in the mathematical function of the model, while the deviations of the observed data from it are explained by “external disturbances,” which escape the modeling. Fig. 3 gives an arbitrary example for this.
Under the assumption that reality follows mathematical laws, we try to find out the mathematical structure and regularity which best fits the controlled observations. Obviously, this works in many areas, but this is not due to the correctness of the underlying assumption. Conversely, it becomes conclusive that:
By choosing a certain set of instruments – that of the exact sciences – we focus and limit ourselves to the knowledge of those aspects of reality that can be grasped with this set of instruments. And there is nothing to suggest that this is or could become our reality.
This does not define the limits of mathematical knowledge of nature, but at least names them. The unity of nature and mathematics, as Newton and Galileo postulated it, is finally gone, and not least because of the historical development of mathematics and the natural sciences themselves.
As an ideological self-image, however, it is still stuck in many people’s heads. There is no other way to understand why terms such as “artificial intelligence” or “world formula” are used not only for self-promotion and the acquisition of research funds, but also in an emphatic sense, as if they were to be understood literally, as if mathematical machines could really be intelligent and thus possess consciousness, or as if we could have the world “under control” if only we had a formula for it.[10] The mathematical-scientific method is thought to be boundless: there is no question which we would not eventually be able to answer with it, no problem which would be inaccessible to it.
Not being able to see the limits of one’s own instruments – in this case those of the exact sciences, of mathematical modeling – is a sure sign of the unconsciousness with which they are used. In view of the obvious impossibility of being able to solve the great problems of mankind with scientific means alone, a certain modesty would be quite appropriate, as it can only arise – in the sense of the Socratic phrase, “that what I do not know, I do not think I know” (Plato 1994, 18) – from a self-reflective awareness of one’s own thinking and doing.
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[1] For example, David Hilbert, inventor of the “axiomatic method,” in a lecture of 1918 (quoted after Mehrtens 1990, 133): “Indeed, whatever occurrences or phenomena we encounter in nature or in practical life, everywhere the mathematically minded and attuned person will find a mathematical kernel.”
[2] Alexander K. Dewdney is a Canadian mathematician and was responsible for the “Mathematical Recreations” column in Scientific American from 1984 to 1991.
[3] Galileo and the experience of Pisa: About a legend, Annals of the University of Paris 1937, Koyré (1998, 123-134.
[4] In a tract of Galileo from the same year 1590 there is even the opposite hint: If balls of wood and lead are dropped from a high tower, the lead moves far ahead, cf. Fölsing (1996, 85).
[5] see Koyré (1998, 129-132).
[6] Another implicit assumption is that the massless connection of the respective two bodies does not change the velocities.
[7] This is not to say that an experiment can not also show completely unexpected results, namely if the mathematical derivation is based on wrong presuppositions. In the case of Galileo’s law of falling bodies, this would have been the case if the inertial mass was not equal to the heavy mass, i.e. if the doubling of the one would not lead to the doubling of the other. In this respect, an experiment tests whether the assumptions underlying the mathematical considerations are correct.
[8] The law of falling bodies G2, which is not presented here in more detail, is also introduced in the Discorsi according to this scheme: It is proved as a mathematical theorem (Galilei 1638/1995,159), which states that a uniformly accelerated body satisfies the law G2. Galilei comes to the uniform acceleration because of its simplicity, there is no other argument. Only then the experiments follow (Galilei 1638/1995,162). Whether Galileo actually carried them out or only described them is disputed (cf. Koyré 1998,129).
[9] Intentionally manipulated and therefore technically usable effects already existed in antiquity. In contrast, the idea of universal – always and everywhere valid – mathematical laws of nature is modern.
[10] This criticism is not directed against the goal of unifying scientific theories, which is sometimes also subsumed under the term “world formula.”
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