CEPA eprint 4103

Remarks of a philosopher of mathematics and science

Friend M. (2017) Remarks of a philosopher of mathematics and science. In: Riegler A., Müller K. H. & Umpleby S. A. (eds.) New horizons for second-order cybernetics. World Scientific, Singapore: 327–332. Available at http://cepa.info/4103
Table of Contents
Structural style
Structural style
1. In the target article, Louis Kauffman discusses the notion of an eigenform. In technical mathematical terms: this is a fixed point for a transformation. In very general and woolly terms, it is a reflexive stability that is reached while something is still changing or moving. The image of a spinning top, before it destabilizes and collapses, can be counted as an image of an eigenform. There is movement (transformation) but also a fixed point (the material top, the axis of rotation and place on the surface of that rotation). The top turns upon itself. It is in this sense reflexive. The rate of spin changes as it dissipates energy. In this sense the “spinning top” (not just the material top at rest) but the top in its spinning state is affected by its own spin. It is, after all, like all material things, subject to the second law of thermodynamics. The energy transmitted to it, by the initial rotation of fingers that set it off, is dissipated in the movement.
2. Eigenforms are relative stabilities, or meta-stabilities, stabilities in motion, but more than that they affect, and draw on, the world around them. We are affected when we watch the spinning top, the physical world is affected since tops make a little noise and they affect small air currents. They draw on the world around them through their design, the aerodynamic, material and balancing properties, these are “put into” the top in its construction. The spin of the fingers that set it off transmits energy to the top, which is then dissipated in the motion.
3. The structural style of Kauffman’s article is itself an eigenform. In the course of the article, Kauffman elegantly spins the ideas about themselves, widening the scope, or going into greater detail, returning to the main theme without repeating himself. The ideas develop in this reflexive circulating manner, and by the end of the article, we are affected by the information, and begin to recognize eigenforms and fixed points around us. We see the world differently. As people read the article, as commentators comment on it, the second-order properties of the eigenform manifest themselves. The recognition of the structure of eigenforms in objects and processes is part of cybernetics. When we reflect upon, or are affected by, such recognition, we move to the meta-level or the second-order level. This reflection iterates – we reflect upon the reflection itself, and it is we who do this, so the reflection is reflexive but transforming at the same time; we become incorporated into the second-order eigenform of the style of the article. We in turn affect other people around us, maybe speaking to them a little differently since our repertoire of concepts has been widened, having been changed ever so slightly by the reading of the article, etc. This wider scope of the eigenform is its third-order transformative effect.
4. I cannot do justice to Kauffman’s structural style by imitating it. I am afraid I shall have to be quite banal in my structural style: more piecemeal, since I want to discuss two other, quite different, themes: objectivity and sustainability.
5. In our daily lives, as we go about our routines, we have a simple account of objects and objectivity. From this perspective, there are physical objects in the world. They are there, and most are there independent of us. We can move some of them, we can alter them, but there are objective limitations to what we can do, since the objects have objective properties that we only rarely alter. The simple conception of objectivity is fine, since it is sufficient for us to navigate the world with some local success. We avoid bumping into hard objects, move out of the way of hard objects that are on a collision course with us, we can pick out groceries, we can estimate how much food to buy, we can put on our clothes fairly well in the morning and take them off at night. Each of these activities involves objects that we can move and alter. If we are interested in the survival of an individual person in the “modern” world, we can safely say that we bump around in the world fairly well. We do this because we can distinguish objects and their properties: hardness, nutritional value, the fit of an item of clothing (we do not wrap a scarf around our foot or try to stuff a sock into our ear). There are objective properties of objects in the world. If we get them wrong, the results can be fatal. Objectivity in this mundane sense is not all that mysterious, and the sense of objectivity is reinforced by our experience in the mundane world.
6. The mundane world, with its medium-sized dry goods and objective properties and relations is left behind in the outer reaches of mathematics and science. There, as we study increasingly complex systems, as we look at the very small, or at the very large, the objects and objectivity of mathematics and science become less tangible and more ethereal. In mathematics and science, the notion of “objectivity” is not as simple as many people suspect.
7. In mathematics we study infinite sets, and various infinite numbers; we study infinitesimals, irrational numbers, lowest upper bounds, self-referring systems, paradoxes and the formal relationship between formal mathematical theories that contradict one another. As we learn about the outer reaches of mathematics, we learn that mathematics, as a discipline, cannot be unified by one mathematical theory. There is no unique and consistent foundation for the whole of mathematics, at least at the present state of play in the discipline.
8. From the perspective of the mundane world that we navigate, this is cause for alarm. We expect to be able to transfer our experience of the everyday to the world of mathematics. We expect objects in mathematics to have properties and assume that these objects and properties are objective. There is an immovable, hard quality to them (“immovable” and “hard” are metaphors transferred from the world of medium-sized dry goods). The object “8” might not be a physical object, but it is objective in the sense that we cannot make up its properties. It does not change its properties, or so we think. The notion of objectivity in mathematics starts to lose its grip when we think of mathematical properties that are true of an object in one theory but false of the “same” object in another theory. Even the innocent number 8 fails to have objective properties, tout court. What it does have are objective properties relative to, i.e., that change with, the theory one is working in. For example, the number 8 has an immediate successor if our domain is the natural numbers, but it does not have one if our domain is the real numbers. The context, or theory, changes the properties of “the” object 8.
9. From the cybernetics perspective, especially the second-order perspective, this is not cause for alarm. The “same” number 8 in the set of natural numbers closely resembles the number 8 in the set of reals. This is because the two 8s share many properties. They are both smaller numbers than 9, for example. Thus, objectivity is a metaphysical feature only found at a higher level of discussion, when we can discuss the object and its context, that is, within a theory. Moreover, we can shift from one theory to another, using a translation, knowing that we lose information in translation. 8 is not an object of mathematics in the sense of one stable well-defined entity such as “the moon.” It is an eigenform in mathematics. “It” changes with mathematical theory development. “It” is defined differently in different mathematical theories, but there is still a recognizable “it” sitting relatively stable through the shifts in theory and perspective. This second-order notion of objectivity is conceptually rich and fruitful. It bears further development and exploration, and I encourage such development. The second-order notion of objectivity can also be found in the “hard” sciences.
10. In the “hard” sciences, the reproducibility of experiments is an indicator of objectivity. As Kauffman writes: “It is the repeatability that makes a successful experiment into an eigenform” (§59). The repeatability tells us that the result is robust or stable within the parameters of what is to count as “similar” experimental conditions. Water boils at 100 degrees centigrade, provided it is sufficiently pure, one is close to zero altitude, the pressure of the air is similar to that which we have on earth, and so on. “Water boils at 100 degrees centigrade” is relatively objective. It is a relatively stable fact. The stability and objectivity depend on, and vary with, the context, what is usually thought of as the experimental conditions. But it is richer than that.
11. The context not only includes the physical experimental conditions, but it also depends on our language and our theory. These in turn, depend on the underlying metaphysics (water is a compound substance), the mathematics we choose for our theory and the underlying logic of that mathematics and language. None of these is untouched, pure, independent of the other or independent of us. They are not simply objective. They feed off each other, inform each other, change each other with new discoveries and knowledge and with new participants in the theory, language, logic, and so on. Together, as a package, they have their second-order, or third-order, eigenforms. The eigenforms give us enough stability that we can do science, but when things become complicated, and we are surprised or puzzled, turning to the second- and third-order questions becomes important. The importance is more immediate in the “softer” sciences, where we are dealing with complex issues that are not as stable.
12. Kauffman’s notion of eigenform as a mark of objectivity in mathematics and science is refreshing, and frees us from many sterile philosophical conundrums about identity conditions on objects, about the role of an observer, about knowing the truth of a theorem. I shall not give examples here. But in all of these areas of philosophical enquiry, the notion of eigenform can give us new insights.
13. “Sustainability” is a word that has come into vogue, but it is poorly defined if defined at all. Here is my suggestion for extending Kauffman’s ideas, this time on the more political and cultural stage. We should look for the eigenforms of “sustainability.” What would these look like? Why do I use the plural?
14. The word “sustainability” contains “sustain.” “Sustain” suggests stability. But it is a stability in motion. It includes internal change, but change within stable parameters, on a first reading. It is also an “ability” that is, a potential, or a predisposition. With “sustainability” we want to preserve the ability of something or the ability to do something.
15. Here are two different conceptions of sustainability. We shall develop a more interesting third conception after exploiting the notion of eigenform. We might want to sustain the current state of affairs, what is commonly referred to as “business as usual.” In this case, our economic, political and infrastructural institutions are sustainable in the very simple sense that we do not have to do anything. By leaving the institutions alone, and letting them change in the ways they have been changing, we sustain business as usual, and we might be tempted to think that this is almost by definition “sustainable.” Here is why: what people usually mean when they favour “business as usual” is that they do not want government interference. They do not want a particular type of institution (such as a government or international agreement) to influence other institutions such as businesses or trade. Thus, to sustain the business-as-usual practice, to ensure that it is sustained, we need to do nothing new. Do nothing, and we sustain business as usual by definition. However, things are not so easy. There is a subtlety. Government institutions change, and this is also “usual.” Thus, to properly sustain the business as usual model, we have to be clear about what counts as usual and what counts as unusual. In fact, we need to ensure against a government, or higher institution, interfering in business at a lower level. The problem with this conception of sustainability will not only come from government and international agreements.
16. Scientists tell us that this conception of sustainability is impossible in the long run, or that it is unwise, imprudent or immoral. Why? There are physical, biological and environmental limitations that we either have already passed or are about to reach. We live on a finite planet, with a finite amount of fertile land, a finite amount of ocean and a finite amount of biomass that cycles carbon into oxygen. Competing for these resources, aggravated by population growth, becomes increasingly costly (which is why it is an unwise and imprudent conception), and the competition will become ferocious, harming many people; hence the imprudence is immoral.
17. Of course, the situation is more complex than what I have described, thus, it requires more sophisticated conceptual tools to analyse. There are rates of replenishment of resources. A farmer can add more fertilizer, but the crops will still only grow at a certain rate. There are limits to the speed-up of plant growth through irrigation, genetic alteration, adding fertilizer, and so on. Carbon is converted into oxygen at a rate. Aquafer levels are replenished at a rate. Fossil fuels and coal are produced in the earth at a rate. When the rate of consumption of these resources and services (creating oxygen) is greater than the rate of replenishment, we are living unsustainably in a more scientific sense.
18. We have two eigenforms corresponding to the two conceptions of sustainability. One takes as its reflexive domain society and the economy within the context of neo-classical economic thinking. Neo-classical thinking is the fixed point allowing the transformation of business, society and politics. The second conception widens the reflexive domain to include the natural world as we know about it through science. It is us, in the world, shaping and changing the world that together have an eigenform. The relationship between us and the environment, the sustaining of our life by, and within the natural environment is the fixed point. The transformation is the cyclical nature of the seasons and replenishment of resources at, or above, the rate of consumption. That is how we live, or fail to live, sustainably under the second conception.
19. The first conception is first-order. We hardly perceive, or are not aware of, the context. We take neo-classical economics for granted and refuse to think of alternatives. The second conception is second-order. We have more obvious reflexivity, we are interested in a relationship between us and something else, we recognize that we influence the natural environment, and it influences us. Living in harmony with it is what affords stability to the whole system. This is the second eigenform of sustainability. This second form is unacceptable in the “more advanced” cultures.
20. Here is a third conception that is third-order. Today, only quite “primitive,” “indigenous,” “very poor” (living below the poverty line) or a few isolated commune cultures live sustainably under the second conception. Mankind lived in such a state, arguably, up to the industrial revolution. After that, we started to live unsustainably, according to the second conception. Those of us who enjoy the riches of the industrial revolution are often unwilling to give up those riches. The third conception, due to Kozo Mayumi, is that we should decide on a culturally acceptable rate of entropy production per rate of consumption.
21. That is, we accept that we are not living sustainably as per the second conception. We also recognise the science that warns us that we cannot live sustainably as per the first conception. We recognise, with the scientists, that we create entropy. That is, we use up resources at a rate that is greater than the rate of replenishment. Different cultures have different expectations about material and energy consumption. Each culture can make a prima facie decision independently of other cultures at what rate (above the rate of replenishment) they are willing to use up the resources.
22. Of course, we are talking of reflexive domains. The rates decided upon have their own momentum and direction. But cultures are rarely isolated. We influence one another. We exchange information. We try different lifestyles or read and hear about them. Thus, what is an acceptable rate of entropy production per unit of consumption has its own (meta-) rate of change. This is a third-order eigenform. The rate of entropy production per rate of consumption is the fixed point. The culture is the transformation. We have a new conception of sustainability that is quite abstract, and has the structure of an eigenform, and thus can be mathematically represented, and reasoned over rigorously, up to the standards of our best mathematics. As a culture seeking sustainability, we are after this third-order eigenform. We seek to sustain a rate of depletion because it is worthwhile. It is deemed to be worthwhile when we consciously allow ourselves to consume a certain amount of material and energy that will not be replenished. I think that awareness of this third level eigenform will help us understand what is at stake if we want to live “sustainably.”
23. The conception of eigenform developed in an eigenform structural style by Kauffman has rich conceptual possibilities. It is another way of seeing things.
24. When dealing with complex questions, it is usually beneficial to spend time looking at the question from several perspectives. We find several different-looking solutions, and we are then faced with the also complicated task of making sense of the differences. This is all in the nature of enquiry into complex issues.
25. The notion of objectivity in mathematics and science, and the problem of defining and agreeing upon a notion of sustainability are examples of problems that benefit from the perspective suggested by Kauffman. Reading the article already starts us on the path of recognizing eigenforms around us. We then move on to develop our own, to bring the concepts to bear on other areas of enquiry. We form a community of people who share this perspective, and develop it further, creating our own eigenform of eigenforms.
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