One of the most beautiful facts emerging from mathematical studies is this very potent relationship between the mathematical process and ordinary language. There seems to be no mathematical idea of any importance or profundity that is not mirrored, with an almost uncanny accuracy, in the common use of words, and this appears especially true when we consider words in their original, and sometimes long forgotten senses.
The fact that the word may have different, but related, meanings at different, but related, levels of concentration does not normally render communication impossible. On the contrary, it is evident that communication of any about the most trivial ideas would be impossible without it.
pg 93: The validity of a proof rests not in our common motivation by a set of instructions, but in our common experience of a state of affairs. This experience usually includes the ability to reason which has been formalized in logic, but is not confined to it. Nearly all proofs, whether about the system containing numbers or not, use the common ability to compute, i.e. to count* in either direction, and ideas stemming from our experience of this ability.
* Although count rests on putare= to prune, correct, (and hence) reckon, the word reason comes from reri = count, calculate,reckon. Thus the reasoning and computing activities of proof were originally considered as one. We may note further that argue is based on arguere = to clarify (literally to make silver). We just find a whole constellation of words to do with the process of getting it right.
pg 100: Notes to Chapter 11: Equations of the second degree
Any evenly subverted equation of the second degree might be called, alternatively, evenly informed. We can see it over a sub-version (turning under) of the surface upon which it is written, or alternatively, as an in-formation (formation within) of what it expresses.
Such an expression is thus informed in the sense of having its own form within it, and at the same time informed in the sense of remembering what has happened to it in the past.
pg 101 ... we have thus already arrived, even at this stage, at a remarkable and striking precursor of the wave properties of material particles.
We may look upon such manifestations as the formal seeds, the existential forerunners, of what must, in a less central stage, under less certain conditions, come about. There is a tendency, especially today, to reguard existence as the source of reality, and thus as a central concept.
But as soon as it is formally examined, existence* ( ex = out, stare = stand.
Thus to exist may be considered as to stand outside, to be exiled) is seen to be highly peripheral and, as such, especially corrupt (in the formal sense) and vulnerable. The concept of truth is more central, although still recognisably peripheral. If the weakness of present-day science is that it centres round existence, the weakness of present-day logic is that it centres round truth.
Throughout the essay, we find no need of the concept of truth, apart from two avoidable appearances (true = open to proof) in the descriptive context. At no point, to say the least, it is a necessary inhabitant of the calculating forms. These forms are thus not only precursors of existence, they are also precursors of truth.
It is, I'm afraid, the intellectual block which most of us come up against at the points where, to experience the world clearly, we must abandon existence to truth, truth to indication, indication to form, and form to void, that has so held up the development of logic and its mathematics.
pg 104: Returning, briefly, to the idea of existential precursors, we see that if we accept their form as endogenous to the less primitive structure identified, in present-day science, with reality, we cannot escape the inference that what is commonly now regarded as real consists, in its very presence, merely of tokens of expressions. And since tokens or expressions are considered to be of some (other) substratum, so the universe itself, as we know it, may be considered to be an expression of a reality other than itself.
Let us then consider, for a moment, the world as described by the physicist. It consists of a number of fundamental particles which, if shot through their own space, appear as waves, and are thus, of the same laminated structure as pearls or onions, and other waveforms called electromagnetic which it is convenient, by Ockhams's razor, to a consider as travelling through space with a standard velocity. All these appear bound by certain natural laws which indicate the form of their relationship.
Now the physicist himself, who describes all this, is, in his own account, himself constructed of it. He is, in short, made of a conglomeration of the very particulars he describes, no more, no less, bound together by and obeying such general laws as he himself has managed to find and to record.
Thus we cannot escape the fact that the world we know is constructed in order (and thus in such a way as to be able) to see itself.
This is indeed amazing.
Not so much in view of what it sees, although this may appear fantastic enough, but in respect of the fact that it can see at all.
But in order to do so, evidently it must first cut itself up into it least one state which sees, and it least one other state which is seen. In this severed and mutilated condition, whatever the sees is only partially itself. We may take it that the world undoubtedly is itself (i.e. is indistinct from itself), but, in any attempt to see itself as an object, it must equally undoubtedly, act* (actor, antagonist. We may note the identity of action with agony.) so as to make itself distinct from, and therefore false to, itself. In this condition it will always partially elude itself.
It seems hard to find an acceptable answer to the question of how or why the world conceives a desire, and discovers an ability, to see itself, and appears to suffer the process. That it does so is sometimes called the original mystery.
Perhaps, in view of the form in which we presently take ourselves to exist, the mystery arises from our insistence on framing a question where there is, in reality, nothing to question. However it may appear, if such desire, ability, and sufferance be granted, the state or condition that it arises as an outcome is, according to the laws here formulated, absolutely unavoidable. In this respect, at least, there is no mystery. We, as universal representatives, can record universal law far enough to say...
...and so on, and so on you will eventually construct the universe, in every detail and potentiality, as you know it now; but then, again, what you will construct will not be all, for by the time you will have reached what now is, the universe will have expanded into a new order to contain what will then be.
In this sense, in respect of its own information, the universe must expand to escape the telescopes through which we, who are it, are trying to capture it, which is us. The snake eats itself, the dog chases its tail.
Thus the world, whenever it appears as a physical universe* (unus = one, vertere = turn. Any given (or captivated) universe is what is seen as the result of a making of one turn, and thus is the appearance of any first distinction, and only a minor aspect of all being, apparent and non-apparent. Its particularity is the price we pay for its visibility.), must always seem to us, its representatives, to be playing a kind of hide and seek with itself.
What is revealed will be concealed, but what is concealed will again be revealed. And since we ourselves represent it, this occultation will be apparent in our life in general, and in our mathematics in particular. What I try to show, in the final chapter, is the fact that we really knew all along that the two axioms by which we set our course were mutually permissive and agreeable. At the certain stage in the argument, we somehow cleverly obscured this knowledge from ourselves, in order that we might then navigate ourselves through a journey of rediscovery, consisting in a series of justifications and proofs with the purpose of again rendering, to ourselves, irrefutable evidence of what we already knew.
Coming across it thus again, in the light of what we had to render it acceptable, we see that our journey was, in its preconception, unnecessary, although its formal course, once we had set out upon it, was inevitable.