The Point of Intersection of Limit and Freedom

Mathematics today can seem an isolated discipline, removed from the questions of life and questions of meaning.  But even a brief look at some of the writing of individuals like Leibniz, Weyl, and Poincare demonstrates substantial interest on the part of the mathematician to reconcile mathematics with common human experience.  I remember one of my teachers in graduate school telling me that he was grateful to be able to do mathematics because it gave him the opportunity to transcend.  I can no longer place this remark in its context, but I have always remembered it because, for me, mathematics does lead beyond appearances.

Today, neuroscience is the place where questions of perception and consciousness are addressed.  And, in the other direction, neuroscientists and cognitive scientists have addressed questions about how mathematics happens.  But we have lost the inclination to look at what mathematics might be telling us about the world of the senses and qualities of thought.  Yet it has been remarked that even in the relatively obscure philosophy of Leibniz, there are the seeds of modern approaches to cognition.

I have found in philosophical writing from the18^th, 19^th, and  20^th centuries (in the works of Leibniz, Herbart, Poincare and Weyl), the distinct impulse to consider how perceived realities are somehow fused from a multiplicity of sensory stuff, and how this may actually be reflected in mathematical thought.  There is a clear tendency to emphasize that the world comes to be in relationship or as Weyl put it, “The world does not exist independently but only for consciousness.”  And this is relevant to the mathematician who is so often standing between the ideal and physical images of the ideal.  Mathematics for many of these mathematicians has been a door to self-understanding and, I believe inevitably, it will be again.

In Science and Hypothesis (1905), Poincare spends a good deal of time making the point that geometric space is not the same as visual space.  He does this by looking into the components of vision.  Vision, he explains begins on a non-homogeneous 2-dimensional retina.  The third dimension is created by “the effort of accommodation made by the convergence of the eyes.” And, since there are two aspects to the perception of distance (the sensation of the muscles and the eyes’ convergence) he argues that visual space could be said to have 4 independent variables and would therefore be 4-dimensional. He takes the time to argue that geometry is not the study of the space we think we see around us.  Its ideal bodies are entirely mental and our experience just enables us to reach the idea.

In three 1932 lectures, collected under the heading The Open World, Hermann Weyl looks at Leibniz’s monad with respect to the problems associated with the infinite divisibility of a line (Zeno’s Paradox for example).   Leibniz says that the conceptual difficulties with the idea of a continuum in mathematics arise because it is not understood that in the ideal or the continuum, the whole precedes the parts, but in substantial things, “the parts are given actually before the whole.”   Ideals divide infinitely, substances only until there is no more.  His perspective, grounded strongly in logic, holds that the ultimate constitutents of the world must be simple, indivisible, and therefore unextended, particles—dimensionless mathematical points.  Extended matter is in reality constructed from simple immaterial substances, monads, or entelechies.   And any monad can be said to represent the world as a whole. While the Leibniz worldview is drawn in a very unfamiliar way, this last detail about every individual monad somehow containing the whole is, I think, a kind of intuition that keeps showing itself, in math, science, and art.

A discussion of Leibniz on philosophy pages tells us this:

What is at work here again is Leibniz’s notion of complete individual substances, each of which mirrors every other. A monad not only contains all of its own past, present, and future features but also, by virtue of a complex web of spatio-temporal references, some representation of every other monad,. . . . In a universe of windowless mirrors, each reflects any other, along with its reflections of every other, and so on ad infinitum.

And this:

But Leibniz held that some monads—namely, the souls of animals and human beings—also have conscious apperception in the sense that they are capable of employing sensory ideas as representations of physical things outside themselves. And a very few monads—namely, spirits such as ourselves and god—possess the even greater capacity of self-consciousness, of which genuine knowledge is the finest example.

Weyl adds the following:

The application of mathematical construction to reality then ultimately rests on the double nature of reality, its subjective and objective aspects: that reality is not a thing in itself, but a thing appearing to a mental ego.  If we assume Plato’s metaphysical doctrine and let the image appearing to consciousness result from the concurrence of a “motion” issuing partly from the ego and partly from the object, then extension, the perceptual form of space and time as the qualitatively undifferentiated field of free possibilities, must be placed on the side of the ego.

And with this observation Weyl concludes very nicely:

Mathematics is not the rigid and uninspiring schematism which the laymen is so apt to see in it; on the contrary, we stand in mathematics precisely at that point of intersection of limitation and freedom which is the essence of man himself.