Beyond Mathematics and Geometry

[This is one of twenty-one essays in the book Man and the Divine, published in August 2018. The book is available in ePub format from leading retailers of eBooks, such as Barnes & Noble, Blio, Kobo, Itunes, Inktera, Smashwords, etc. Information about Man and the Divine can be found here]

Everything has an inside and an outside. We live in a world in which we mostly react to what is on the outside of things, and readily apparent, rather than what is on the inside, and often invisible to us. What is on the outside is the phenomenon, what is on the inside is the noumenon.

The distinction between what is phenomenal, and what is noumenal, is a major idea in the philosophical outlook of Kant, who pointed out that what things actually are is not only generally unknown to us, but is in many cases actually unknowable.

The argument about this can be quite complex, since not only are there often many simple obstacles to understanding what a thing is, once the observer starts to think in terms of a contrast between a phenomenal representation of what that thing is, and the actual nature of the thing in itself (the noumenon), we become aware of just how little we may know or understand of the relationship between the two. The world is represented to us through the senses, and is interpreted in terms of the categories of understanding which we use to make sense of this information. But how that interpretation relates to what is represented to us in this way is mysterious. Speaking of representation is a good way of reminding ourselves that how something appears, and how it is in itself, is a matter of a conflation of the representation of a thing, and the conventions about what it is, or might be, which exist within the categories of our understanding.

So when we see a tree, we, as a matter of convention, treat the tree and its appearance as what the tree is, because in normal circumstances, there is not likely to be a conflict between our understanding of what is being represented to us by our understanding of the sense data available to us. The appearance fits with the categories of knowledge and understanding which we bring to bear on our experience.

Kant was not the first to observe that there was a distinction between appearance and reality, and between what is understood of a phenomenon by convention, and what the real nature of a thing is, This was a recurring thread in the development of Greek philosophy. Their understanding of the nature of their world was framed within the context of divine powers and agencies, so the idea that reality was hidden from the human understanding was highly developed among the Greeks and other ancient societies.

A question which is sometimes posed to children to illustrate the idea that the representation of something isn’t the same as what it is, is: ‘what is the colour of grass?’ The answer will usually be returned quickly, and be ‘green’.  But of course grass isn’t green. We see grass because the blades reflect particular wavelengths of light more strongly than others, and absorb some wavelengths. So we don’t see what colour grass actually is. In fact we are prompted to ask what we mean by an entity having a property of colour at all. Grass absorbs red transmitted light, and reflects green light, and those are the properties involved in our apprehension of the colour of the grass. We don’t know what the colour of grass in itself is, or even if it is an appropriate question, but we can describe the processes involved in how we apprehend it.

The categories of our understanding serve us from our earliest years, but in a simple form. The development of critical intelligence is the consequence of learning that what is presented to the mind and understanding is often more complex than it appears to be. What we need from our understanding at age six is hopelessly inadequate for us at age twenty. We learn (with the aid of education) to reprogramme the categories of our understanding so that we can process the information in a more sophisticated way than we did at six, and are no longer the prisoners of the illusion that the direct presentation of a thing is the thing itself.

This process of separating ourselves from an interpretation of the world in terms of simple apprehension is driven initially by the practical necessities of our existence. But this process does not need to stop there. Intelligence consists in being able to adjust the categories of our understanding so that we do not mistake one thing for another. It is a mental development which might have no end. This is essentially how Kant understood human intellectual development, which he framed (in his Prolegomena) in terms of a general theory of a priori concepts, not based on empirical sense data, or even a mathematical or geometric understanding of anything in the world. These a priori concepts can have a relationship with sense data and form in the phenomenal world, but they are not derived from these objects and constructs, and can be understood only as concepts entirely stripped of everything which would give them phenomenal or mathematical reality.

What is proposed in the Prolegomena is that real intelligence and understanding is what is shown to the mind by the mind alone, and that these concepts make sense only as purely mental constructs, manipulated and understood by the mind.

So Kant was talking about understanding what is beyond all representation, except in terms of mental abstractions, shorn of scalar and mathematical properties. They are entirely a priori abstractions. The focus of Kant’s metaphysics therefore is the noumenal reality behind all phenomenal appearance. This form of metaphysics he regarded as the basis of a scientific understanding of reality, and that all approaches to understanding the world through how it presents itself to us are faulty, and will not tell us what we may wish to know about what lies behind appearance.