From Kant’s mathematics to Schopenhauer’s will…

To give shape to this blog, I’ve been jumping around quite a lot through the fields of mathematics, physics, and the neurological and cognitive sciences.  I decided today to let more of my weight drop into philosophy.

It’s not unusual when reading about 19th century developments in mathematics (the ones that lay the groundwork for mathematics as it is understood today) to see references to the late 18th century work of Immanuel Kant. Often the references describe how the discovery of non-Euclidean geometries contradicted him. This is primarily because Euclidean geometry was the only one possible for him.  But Kant was at work reconciling the conflict between the rationalists, who saw knowledge as a product of the intellect, and the empiricists, who saw it as a product of the senses.   I don’t believe he would have found the development of non-Euclidean geometries inconsistent with his perspective that knowledge emerged from the interplay of sensibility and reason.

Philosophers have influenced developments in physics, but today, the implications of research are mostly discussed by the researchers.  There are interdisciplinary impulses, like the physicists and cosmologists from the Foundational Questions Institute, who organized a multidisciplinary conference on the perception of time.  Yet I don’t think I’ve read any truly philosophical critiques of string theories, for example. The controversies surrounding string theories stem largely from our inability to test the theories, to make an observation that anchors their mathematical meaning to some clear physical meaning as well.  The dispute over the value of this research is not a philosophical one. It’s a pragmatic one.  So today I wondered if the rationalist/empiricist argument is worth remembering in the context of some of the conceptual difficulties in physics.  I pulled this excerpt of Kant from a paper written by David Kaiser .

Without sensibility no object would be given to us, without understanding no object would be thought.  Thoughts without content are empty, intuition without concepts blind.  It is, therefore, just as necessary to make our concepts sensible, that is, to add the object to them in intuition, as to make our intuitions intelligible, that is, to bring them under concepts.  These two powers or capacities cannot exchange their functions.  The understanding can intuit nothing, the senses can think nothing.  Only through their union can knowledge arise.

Kant used mathematics (and in particular Euclidean geometry) as a demonstration of knowledge that doesn’t come from experience.  You can find an outline of his thinking on Philosophy Pages.  About his view of mathematics they say:

Understanding mathematics in this way makes it possible to rise above an old controversy between rationalists and empiricists regarding the very nature of space and time. Leibniz had maintained that space and time are not intrinsic features of the world itself, but merely a product of our minds. Newton, on the other hand, had insisted that space and time are absolute, not merely a set of spatial and temporal relations. Kant now declares that both of them were correct! Space and time are absolute, and they do derive from our minds. As synthetic a priori judgments, the truths of mathematics are both informative and necessary.

I found a website that described itself as being devoted to tackling age-old philosophical questions with the help of cybernetic theories and technologies.  There is an article there (written more than 15 years ago) by Valentin F. Turchin with the title: From Kant to Schopenhauer.

In it Turchin says:

In classical mechanics we use much more of our neuronal world models.

In other words, classical mechanics talks about a world consistent with the one the body organizes with sense data.


There is a three-dimensional space; there is time; there are the concepts of continuity, a material body, of cause and effect, and more.

Mach and Einstein would be, probably, impossible without Kant. They used the Kantian principle of separating elementary facts of sensations and organizing these facts into a conceptual scheme. But the physicists went further. Einstein moved from the intuitive space-time picture given by the classical mechanics down to the level of separate measurements, and reorganized the measurements into a different space, the four-dimensional space-time of the relativity theory. This space-time is now as counterintuitive as it was in 1905, even though we have accustomed to it.

In quantum mechanics, the physicists went even further. They rejected the idea of a material body located in the space-time continuum. The space-time continuum is left as a mathematical construct, and this construct serves the purposes of relating micro and macro-phenomena, where it has the familiar classical interpretation. But material bodies lost their tangible character. … In the relativity theory observations (measurements) at least belonged to the same universe as the basic conceptual scheme: the space-time continuum. In quantum mechanics, on the contrary, there is a gap between what we believe to really exist, i.e. quantum particles and fields, and what we take as the basic observable phenomena, which are all expressed in macroscopical concepts: space, time and causality.

Kant elevated abstract knowledge to the same level of significance as experience or sensation, a perspective that sets the stage for the counter-intuitive conceptual schemes that show us the world we don’t otherwise see.  But quantum mechanics introduces a deeper conceptual difficulty.

Turin goes on to argue that if we want to construct a theory that describes the ultimate reality of physics, one that can, step by step, construct the observables, we need a philosophical basis that goes further than Kant.  He explains:

We must go further down in the hierarchy of neuronal concepts, and take them for a basis. Space and time must not be put in the basis of the theory. They must be constructed and explained in terms of really existing things.

Kant’s metaphysics, Turin explains, needs to be pushed further.  More has to be understood about the relationship between observables and conceptual schemes.   These ‘really existing things’ Turin describes as “the most essential, pervasive, primordial elements of experience.”  And so Turin moved on to Schopenhauer who, finds the essence of reality in action more than substance.

Let us examine the way in which we come to know anything about the world. It starts with sensations. Sensations are not things. They do not have reality as things. Their reality is that of an event, an action. Sensation is an interaction between the subject and the object, a physical phenomenon. Then the signals resulting from that interaction start their long path through the nervous system and the brain. The brain is tremendously complex system, created for a very narrow goal: to survive, to sustain the life of the individual creature, and to reproduce the species. It is for this purpose and from this angle that the brain processes information from sense organs and forms its representation of the world. Experiments with high energy elementary particles were certainly not included into the goals for which the brain was created by evolution. Thus it should be no surprise that our space-time intuition is found to be a very poor conceptual frame for elementary particles.

We must take from our experience only the most fundamental aspects, in an expectation that all further organization of sensations may be radically changed. These most elementary aspects are: the will, the representation, and the action, which links the two: action is a manifestation of the will that changes representation.

Why not see this as an indication that action should have a higher existential status than space, time, matter?”

The Kantian idea that conceptual knowledge is a necessary partner to sensory knowledge lines up with many of the opinions of mathematics expressed in posts I’ve authored.  But Schopenhauer’s view that the universe is apprehensible through introspection approaches the heart of much of how I see things.  I will want to speak to this soon.

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