In an earlier article, we asked ‘what is knowledge?’ Here we shall explore the nature of knowledge further.
As previously discussed, knowledge is a symbolic model of something else. To construct such a model, we need symbols. In natural languages, these symbols are words, typically rather loosely defined. In science, the symbols are technical terms which are, ideally, rigorously defined. Either way, the definition of a symbol is a curious business.
One way to define what a symbol represents is to point to it in the real world. If someone were to ask the meaning of the word ‘car’, I could point to a car and state, ‘This is a car.’ Note that to define the word rigorously by pointing, I would have to collect all the cars in existence (including past and future cars) and state that everything in this set is a car and nothing else is a car. Of course that is impractical, and for natural languages, we make do with a few examples and assume that we are all going to extrapolate the full meaning in roughly the same way. Science, on the other hand, demands rigour, so pointing at things is typically not of much use.
Apart from pointing at things, a symbol can be defined in terms of other symbols. This leads to a rather odd characteristic of almost all symbols: a symbol cannot stand alone. In itself, it is devoid of meaning. Its meaning is almost always in reference to other symbols, which in turn are meaningful only in relation to yet more symbols. Let’s consider a few examples.
What is a car? This question can actually be answered in different ways. One way is to refer to a more general symbol: a car is a type of vehicle. It would then be desirable to define what attributes distinguish a car from other vehicles: a car is a vehicle designed to transport a small number of people. Sometimes a definition can be made in the opposite way, by enumerating sub-types, for example a vehicle might be defined as a car or a bus or a lorry or a motorbike…
So far, we’ve considered definitions with regard to sets of objects. There are other sorts of definitions though. One is structural: a car is comprised of a chassis and an engine and some wheels… With such a definition, there also needs to be some definition of how the parts are connected to one another. Another approach is a functional definition: A car is a machine designed to transport people along roads.
There are other flavours of definitions too. One refers to the historic causes leading to the thing: an oak tree is a plant which has grown from an acorn. The opposite of this is to state what is going to become of something: an acorn is a seed which, after germination, grows into an oak tree.
Regardless of the flavour, any definition of a symbol is relational. The meaning of a symbol is determined by how it relates to the meaning of other symbols. This inter-dependence of symbolic meaning has many ramifications.
One consequence of symbolic inter-dependence is that totally abstract languages and models can be constructed, that are formally disconnected from the material world. Pure mathematics is the quintessential example of this. This sort of abstraction is thinking at its best: honest thinking in its own domain, that doesn’t pretend to be a direct reflection of reality.
In an abstract model, truth is determined by whether a statement follows syntactic rules and can be proved to be consistent with other statements known to be true in the model. The proofs themselves have to follow strict logic, which itself forms part of the model. This type of system requires some foundation statements – axioms – which are not in themselves proven. Everything else is derived logically from the axioms. In mathematics, the need for axioms is accepted without any debate; it's obvious that some foundation is needed. In the non-scientific arena, we also need foundation beliefs which are, in themselves, unproven. Oddly, some scientists seem to think that this invalidates all knowledge derived from one’s core beliefs. A more accurate view is simply to accept that the truth of knowledge is dependent on the truth of the core beliefs. A mystic is someone who sees all this and simply accepts that no knowledge can be shown to be truthful in any strictly meaningful sense. Everything is fundamentally unknowable.
A century or so ago, it was assumed that mathematics could be made formally complete. It was believed that a mathematical system could be defined where everything was internally consistent and that any mathematical statement could, in principle at least, be proved to be either true or false. This is the sort of rigour that a scientist is always striving for. Unfortunately for rigour, Gödel proved that any formal system that is powerful enough to be of any use is, fundamentally, incomplete. By incomplete, it is meant that it is always possible to construct a valid statement in a formal language which cannot by proved to be either true or false. Even a formal system has paradox as a characteristic: there are statements which are both true and false at the same time. Or they can be interpreted as neither true nor false. Or it can be said that their truthfulness is undecidable. In any case, we now know that it is not possible to construct a formal system that is logically complete in the way that had previously been assumed. Despite this, everyone (including scientists) continues to use maths because it’s practical and works as long as one steers clear of the paradoxes. This practical approach is reminiscent of the acceptance of scientific theories that have singularities, discussed in a previous article.
I’ll leave it at that for this article, though the mind has that intangible feeling that there is more to be said on this topic…
