Language Logic and
Mathematics
In the
The Evolution of Mind
Mathematics is precise and
universal. Language is loose and divided.
Held together by their very difference, language and mathematics are closely related yet distinct. Both are abstractions. Language is a loose abstraction from thought. It is loose in that it can be expressed in infinite ways, from a nod of the head (body language), to an hour-long speech. It is divided into a Babylon of tongues, ancient and modern, which evolve and change, so that no one would recognize their own language if carried a thousand years into the past or future.
Mathematics, as an abstraction from individuality, is deep and universal. It is precise in that 2 + 2 = 4 in all lands and climes, ancient and modern and invariant from place to place and time to time. It is the rock around which the winds of language play.
Keeping these aspects of mathematics and language in focus we can now ask what holds the extremes together. Can we turn these relations (precise/ loose, universal/ divided) around? There has to be something precise and universal (we could say math-like), running through language in order to anchor its sense (meaning) within the framework of its free expression. What is this precise element, if, math-like, it is not mathematics itself?
We have in view three categories. The first is language and the third is mathematics, though in another approach, setting out from a physiology of mind we would turn this order around. The second, the in-between, is a mystery item, language's core, which is math-like without being mathematics, and language-like without being language. Described ‘in-itself’, as a complete subject in the not-well-understood Hegelian philosophy, it is Logic.
Hegel was unable to include mathematics in his Logic because the latter's essence is circularity, and the only form of math available at the time was the Euclidean ‘straight to infinity’ result-math taught in the schools. Its genesis in the physiology of the brain and its natural in-mind presence was beyond his horizon. Instead of the extremes he had to conjure his philosophy out of history and his intuition of the coherence of reason in logic itself. Nevertheless his Science of Logic brings the subject before us in its worked detail, strangely (for us) devoid of genesis or any relation to physiology.
This is not surprising given the state of science in his day. The birth-infant-child sequence was known, along with the stages of learning, but microscopes were glass beads held in place by wires and the belief that rain generates earthworms from horsehairs in mud was still current (which Hegel took time to dispute). We can now proceed to put logic in place as the mind-dependent sequence or system that holds reason and intellect in place.
Logic, from the Greek logikos, concerning speech or reasoning is neither mathematics nor language, but partakes of both. As something on its own account it excludes both extremes. Reasoning, as a process within the brain expresses itself as thought. It is the common stem of language and symbol-based mathematics alike. Language, composed of meaning, carries the symbology and intent of mathematics within it. This order is given in the fact that we can speak because we can think, which is something indwelling, and we can learn to calculate according to rule because we have language. The whole is circular and turning.
In this picture, logic holds mathematics and language together as well as apart. To go from the Greek logikos to Aristotle, he defined logic as thought, the ‘highest thing’, which takes the highest thing (namely itself) for its object. This gives us philosophy, the love of wisdom, as thought thinking about thinking, and the science or metaphysics of this relation is logic. Or we can say that logic is the expression of reason. My ‘Mathematics as the Physiology of Mind’ addresses this, but here it is to be presented in a new way.
In Hegel's Science of Logic, which expands and gives body to Aristotle's philosophy, logic is presented as a coherent system devoid of content other than its own nature. This circularity, that of a snake that swallows its own tail, is a turning point in the history of knowledge. The unit in mind and the motor in Hegel's dialectic is paired opposites. The formative threads of reason in a dynamic process then turn each emerging meaning into its opposite. This, his dialectic, welds Logic into the self-sufficient heart (what else can we call it?) of his philosophical science.
Expanding on Spinoza's theory which identifies something by ‘what it is not’, Hegel shows that in absolute abstraction the meaning of being turns into its opposite, nothing. This occurs because being always refers to the being of something, but we want to know what it means bare and naked free from conditional example. Void of such appendage nothing surfaces as being's meaning.
The meaning of nothing rotates in turn into being. The unity of being turning into nothing and nothing turning into being then gives us becoming. Something becoming nothing, and nothing becoming something settles into place as the balance between rest and motion, stability and change, and so on. The derivation continues until every logical relation takes its place within the Hegelian system of logic. This in completion will invert or turn into a fully-fledged theory and physiology of mind.
An initial circle of relations generates another. This generates another, and so on until the whole of logic is unified within an overall dialectical system meaningful in itself. Applying the same principles to mathematics gives us circlemath.
Language comes before logic and math. Children
do not go to school to learn to speak. Competence for speech is inborn and
activates spontaneously. It dwells upon quality, where ‘quality’ implies kind.
Mathematics, the last in the triad dwells upon quantity. Logic sits in between.
Mathematics is also an inborn aptitude but its activation requires cultivation.
Logic, whose elucidation was initiated by Kant in the 18th Century
and consolidated by Hegel in the 19th dwells within the subjectivity
of mind and mediates between language and math.
Logic is a product of the mind, but also forms it by integrating language and mathematics. To scout this in rough, there is a math-language link in mathematics itself, in that the 0 of its 0 1 foundation symbolizes mind. It is our ‘emptiness’ or desire to know. The 1, as the unit for every thing and thought is the same objectified. Mathematics holds to the thread of quantity and lifts from the world of being. Language holds to the thread of quality and lifts from the world given in sense. Logic lifts from and builds back to our understanding of that world in appearance and is the cement in the relation.
If we allow that math is a form of reasoning that abstracts from actuality, we will agree that 1 + 1 + 1 = 3, and that the ‘3’ of “three spoons on the table” abstracts from the nature of the spoons. A directional flow is involved as in a circle, in that given spoons we easily come to a number. A given number does not however suggest spoons. It is less easy to see that language involves abstraction, because as universal we take it for granted. For instance, the meaning in the word spoon is in abstraction from that well-known object, and logic, in abstraction from our understanding holds the two sides together.
We recognize mathematics as a coherent subject in its own right with its own rules and syntax whether we understand it or not. This is also true of language. The test fails when we come to logic, for though we are familiar with the word's application and meaning, our awareness of its corporate identity has yet to be fully established.
This would not have been the case if Hegel's philosophy, initiated by his Phenomenology of Mind and Science of Logic had maintained its momentum in the 19th Century. His Logic, in pawn to the immaturity of science in his day lacked the association of a compatible mathematics, a shortcoming that can now be overcome. Random paragraphs, selected from his Phenomenology or Logic as below, can serve as examples of his style:
“It is sufficient for our purpose here to have indicated the invalidity of the so-called laws of thought from the consideration of the general nature of the case. It falls to speculative philosophy to go more intimately and fully into the matter, and there they show themselves to be what in truth they are, single vanishing moments, whose truth is simply the whole of the thinking process, knowledge itself.” Phen
It is difficult because it is out of context, but the style compounds this. Take the context as well. It is still of little help, for ‘more of the same’ the whole is virtually impenetrable. This is because it is a ‘mathematic of words’. Hegel clarifies this for us in his Prefix:
“In view of the fact that ratiocinative thinking has a content, whether of images or thoughts or a mixture of both, there is another side to its process which makes conceptual comprehension difficult for it.” Phen
“Difficult for us” would have been closer to the mark. A little further on he adds:
“There is a difficulty which might well be avoided. It consists in mixing up the methods of procedure followed by speculation and ratiocination, when what is said of the subject has at one time the significance of its conceptual principle, and at another time the meaning of its predicate or accidental quality. The one mode of thinking invalidates the other; and only that philosophical exposition can manage to become plastic in character which resolutely sets aside and has nothing to do with the ordinary way of relating the parts of a proposition.” Phen
In other words, give the substance junk the explanation. A paragraph from his Logic, any paragraph, shows what this means.
“We have found the syllogism to be the restoration of the Notion in the judgment, and consequently the unity of truth of both. The Notion as such holds its moments sublated in unity: in the judgment this unity is internal, or what is the same thing, external; and the moments, although related, are posited as self-subsistent extremes. In the syllogism the Notion determinations are like the extremes of the judgment, and at the same time their unity is posited.” Logic
It makes sense, but only if we know what we are looking at, and this is missing. The revelation is only a step away. Hegel’s Logic is a ‘mathematic’, not of number but of language and its words. It does not deal, as does mathematics with pure distilled quantity, but pure distilled meaning. It is reason insulated from any content but the nature of its own nature. It assesses the mind as mathematics assesses the world, and as mathematics matures into a physics of the world, it will mature into a physiology of mind.
Stephen W. Taylor MbChB © 2005 September 14