Sokolowski on mathematical constitution
HOW THE MATHEMATICAL SCIENCES ARE CONSTITUTED
Modern sciences deal with idealized things: with frictionless surfaces, rays of light, ideal gases, incompressible fluids, perfectly flexible strings, ideally efficient engines, ideal voltage sources, and test particles that do not have any effect on the field in which they move. However, such ideal forms are not fabricated out of thin air. Rather, they are projections that have their roots in the things that we directly experience.
For example, consider how we come to the idea of a geometric surface. We begin with an ordinary surface, such as a tabletop. We sand and polish the surface and make it smoother and smoother. At a certain point, however, we can shift from actual sanding and polishing to an imaginative projection. We imagine sanding the surface until it could not be smoothed out any further; we imagine it as having reached the limit of smoothness. In actual fact we cannot polish the surface to this degree, but we can “take off” from the physical steps of refining it and simply imagine it reaching this unsurpassable limit. This limit is the pure geometric surface, and it is reached from a basis in actual experience. It is a transformation of surfaces that we actually experience.
Another example can be found in optics. We start with a beam of light coming from a flashlight. Then we cover part of the light source and cut the beam, say, in half. Then we cover half of the remaining part. We do this a few times, but then we change gears; we shift from actually blocking part of the light to imagining that we block it, and we go on to imagine that we have cut the light down to a very narrow beam, one so narrow that we could not interrupt any part of it without extinguishing the beam entirely. This tiniest of beams, this uncuttable or atomic beam, becomes a “ray” of light, as it was defined by Newton in his Optics. We could in actual fact never arrive at such a ray of light, but we can imagine or think about it as a limit.
Both the perfectly smooth surface and the ray of light are idealized objects. Such objects could never be experienced in our life world [Lebenswelt]; we establish or constitute them by a special kind of intentionality, one that mixes both perception and imagination. This intentionality starts with something from the life world, but it generates something that seems not to belong to that world any longer. Once we have these idealized objects, however, we may begin to relate them to the concrete objects that we experience. The idealized objects become the perfect versions of what we experience; they seem to be “more real” than the things we perceive because they are more exact. The things we perceive seem to be only imprecise copies of the perfect standard.
Then, if we bring about many such objects, we may think that we have discovered a whole world of things that is far better and more exact than the world of our perception. This is what happens when the kind of science introduced by Galileo, Descartes, and Newton becomes dominant in our culture. People forget that the ideal things referred to in the science have been brought about by a way of thinking; they believe that these things are more real than what we directly experience, and so they credit the sciences that know them with great authority. They take what is the result of a method as being a discovery of a new kind of reality. The scientific experts, the masters in this new domain, are thought to have a much more perfect grasp of the nature of things than the rest of us do, since we deal “merely” with the unscientific world, while they deal with the world as it “truly” is in its perfect exactness. Such idealizations, furthermore, have been projected not only in geometry and physics, but also in the social sciences: in economics, politics, and psychology. Models in game theory, for example, have been used to calculate strategies in warfare and foreign policy.
FURTHER ASPECTS OF SCIENTIFIC OBJECTS
Let us examine in greater detail the procedure by which idealized objects are reached. The object we start with is one in which we can identify a feature in which fluctuations are possible, such as the smoothness of the surface or the size of the beam. There can be variations in these two features: both can be realized in greater or lesser degree, in more or less. The variations are then made smaller and smaller, and the idea arises of a condition in which no further variations are thinkable: they are reduced to zero. The surface becomes perfectly flat, the beam becomes practically a line. We have “geometrized” an object that was once a perceived thing in the world.
It is important to note that when we reach this ideal condition, we retain something of the content or the quality of the thing with which we began. We do not turn everything into pure mathematics. The ideal surface is still a spatial thing, and the ray is still a ray of light. The surface is different from the ray of light, and both are different from, say, the perfectly flexible string or the ideal voltage source, which in their turn are idealizations that began from other worldly objects.
It is the excruciatingly exact identity of the idealized objects that makes them so satisfying intellectually. They are perfect: they are exactly the same wherever they are found, in contrast with the variable surfaces and beams of light that we actually encounter. In earlier chapters of this book, we considered the theme of identity in other contexts; a perceived thing (the cube) was described as an identity in a flow of sides, aspects, and profiles; a mental act was said to be an identity given in the various remembrances we have of it; and even the self was presented as an identity behind our various mental achievements. However, all these identities enclose many variabilities; they are what can be called morphological things or essences. In contrast, the ideal things that mathematical science reaches, the exact essences, do not tolerate any ambiguity or variation. They positively exclude them.
Not all things can be projected toward a limit and constituted as exact essences; a perception or a memory, for example, always retains some vagueness and variability. It would make no sense to try to project things like them toward an ideal limit; they remain “morphological” and not exact kinds of things. Consequently, such things seem, to some people, to be vague and subjective, and attempts are made to introduce an exact science, a kind of mathematical psychology or cognitive science, that will replace such concepts with more exact ones. The attempt to explain human cognition as a form of neuronal computation is an example…
Developments in physics and mathematics in this century have raised questions about the exactness of the natural sciences. Such discoveries as indeterminacy of measurement and observer relatedness in quantum theory, relativity theory, the incompleteness theorem in mathematics, nonlinear systems, chaos theory, and fuzzy logic have cast doubt on the rather tidy understanding of the world that was present in Newtonian physics and the science and mathematics that prevailed during the early years of phenomenology. However, these developments do not affect the problem of the life world and science. All these developments have occurred within the scientific view of the world, which even with them still remains at odds with the world of our spontaneous experience. The newer versions of science may tolerate imprecision, but what they describe is still different from the world in which we live, and the problem of integrating them into that world has not been dissolved. An important contribution to its resolution would lie in the careful analysis of the kinds of intentionalities at work in establishing scientific knowledge.
— Robert Sokolowski
Introduction to Phenomenology (Cambridge, 2000), pp. 148–52.
