P is walking. P is thinking. P thinks on the essence of number. P believes the essence of number is individuation. P observes how all things in their being are each one. And in their each being one they contain themselves and themselves only. This must surely be true, or so it seems to P ; there is nothing hidden to him here. Yet P is not naïve he sees how the things do not stay close enough to each be called one for in each’s oneness they suggest the other things around them, they withdraw in to other things. It seems now to him that there here is one thing and there another and then it seems that there is only one thing, a thing which allows the other things to show themselves.
But P has strayed from his thought, which was on the essence of number. What is the essence of number? This is what P thinks. Now P considers that the appearance of individuation is the essence of number. He stops that sense of the things that link together, he puts it clearly out of his mind and thus the separation is clear. Now it seems to P it is very clear; indeed now he wonders to himself how that connecting withdrawing motion ever confused him. He picks up a thing, as if to prove this to himself, and now feels how he is separate from it and it from him. P jumps and down on the earth to indicate this is true in another way. In this way P is convinced of this separation. From here the rest of the argument follows with ease. All things are separate in their appearance. Thus in mans looking around he sees now one thing and now another. Now it seems to P that this now one thing and now another is a state of affairs that must be grasped in some way. This grasping ultimately is the notion that each one of these separate things may be seen in groups, groups of similar things or groups of different things. These groups are quantities.
“Here are several things” P says out loud to himself, to convince himself of his correctness “and these several things are” a quick jotting up of P’s collection occurs “five, yes five things”
P is not so happy with his argument now though, it seemed to him before that once one had seen that things were separate they automatically became numerical. Now P is not so sure. But now P remembers something he thought earlier. The things indicate other things. It seemed that it was impossible to think the things without other things, now it occurs to P that this might be true in a different sense. That is, if we concentrate on the purely numerical side of things, doesn’t one thing imply another in this way too? P likes this idea. In this way every time one has one thing, there is another as another regardless of whether it is there or not.
“Here is one thing” P says to himself holding up an apple “and its oneness tells me there is another” and yet P does not hold another thing “thusly there are two things and three things and so on”. It seems now he is convinced of this without the other things being here.
P’s ruminating on this matter might seem quite pointless but P is in fact laying the ground for something deeper. P felt he needed the essence of number showing to himself before he proceeded. P feels he can now proceed though curiously P has the nagging idea that the essence of number has come no closer than before. Indeed P can no longer really remember what his conclusion has achieved or indeed what his conclusion was. P can remember that it was something satisfactory and this seems to suffice. P will proceed and hope that the conclusion is clear again later; or at least that his later results do not contradict what he says further down the line.
P has an affliction. Now this affliction is not what one would call an ordinary affliction. P is not ill, though P does wonder about this some times. P believes he sees a certain number everywhere. P sees it continuously; it follows him like a friend or a fiend. P cannot decide which, that is it is not particularly auspicious but then neither does its presence denote anything particularly insidious either. P’s numerical friend is 47. This is why it is so important to understand the essence of number, for without it how can Mr 47 be understood? Well surely it can’t, “so its essential” (P pardons his own pun) “that I grasp the essence of number before then moving on to understand how a number can behave in this manner.”
What manner is it that P means? Well in truth he isn’t sure. It seems to him it’s a bit like that other problem, the one about he essence of number. He felt quite sure it was something he had, so he trusts to this sense of having had it. He imagines he’s asked to give an account of it to a body of gentlemen.
“So how is it that this number afflicts you? Tell us accurately and in detail, for as you know we’re a well know body of scientific gentlemen and will settle for nothing less than the essence of the rigorous!”
At this thought P becomes quite flustered. He can see this terrible parliament before him now, seated and hoary, filled with questioning eyes. And my goodness what was it they said they would require? ‘ The essence of the rigorous’, P shudders at the thought. What a terrifying idea, how can his pondering possibly match up to this inquisition? The anxiety brought to bear on P’s mind by ‘the essence of the rigorous’ is too great and he abandons the hideous interrogation.
But now P becomes unsure of his footing, so much so that he fancies he might actually fall over. This notion jars his mind to action. How had that happened? The uncertainty of his case as unsureness was the same as the unsureness of his footing. Indeed they were not separate. This seemed interesting and somehow reminiscent of the problems surrounding the 47, or Mr 47 as perhaps he ought to address him (or her). How had that thought occurred to him? Well it seemed that… that. No, it’s all unclear again. P hasn’t a clue what he’s talking about. And now he shudders to think what that mob of inquisitors would make of this. What if he had thought he had this instance as an example and begged their attention?
“Gentlemen, its like in those cases when one finds connections between a thought quite disparate and seemingly only metaphorically bearing a resemblance to another notion and then they somehow come together.”
The gentlemen looked on interested, P had not foundered yet, what he was saying did seem to be making the sense he wanted it to, yet quite where it came from he had no idea.
“Thus it is with the phenomenon of numbers like the 47, the connections are impossible to delineate and yet clearly apparent”
But now he was stuck, he wanted to end there with ‘and that concludes my presentation on the matter’ but could see how this was a woefully inadequate dealing with of the subject; the gentlemen would never stand for that.
So P abandoned them again and sat down, telling himself that he should not subject himself to the ‘the essence of the rigorous’ as it was clearly getting him nowhere.
Examples, that’s what he needed, for unless he was deluding himself he must surely have examples. The problem was that all P could think of was the fact that it quite often happened that when he looked at him watch it was the hour and 47 minutes or that when he looked at a car it said 47 on the number plate. P did not want to even begin to contemplate the presentation of this as ‘evidenz’ to that panel with their demands that a certain essence be always striven for. Though in this desire the image still emerged:
“And thus we can see that the presence of the 47 on these number plates that I noticed that other day and… well blow me will you look at the time! As if I required a more rigorous proof!”
The gentlemen, having heard his speech about the 4 number plates in a row that he saw the other day and now being asked to look at his watch as it was 14:47, would not be impressed. They would shake there heads and reach for that large stamp with which to stamp the work. ‘WORTHLESS’ the stamp would say on it, and, after it had been duly dunked in the ink sponge it would be brought to bear upon his paper.
Thus would be the pronouncement from those who demanded nothing less than ‘the essence of the rigorous’
This will never do, thought P. Too much toing and froing was getting him nowhere. P needed to relax. P took several deep breaths and thought about why the 47 was important to him. Well there were things that happened, things he couldn’t remember but in a way this was its own proof. That is, why would he be attached to the number at all if it hadn’t done something strange at some point. Yes, there had been things. And there were images too, great and powerful images that grew out of its contemplation. And what was more the things and the images were linked. In this way P’s mind cleared and the phenomenon that he desired to reach seemed to open itself slightly to his mood. Lines of a rhyme drifted into his head:
“In wretched life and sweat wracked slumber,
He seeks the meaning of that number”
And there was more, but P couldn’t remember it, was it something about a boat later on or had he made that part up. No there was definitely something about a boat, but what about the boat? P had no such clue.
P tried to come back to that useful mood of a second ago before the rhyme had disturbed him. What were the things? Well it seemed to P that it was as much to do with shapes as anything. Triangles chiefly, oh and squaring numbers that was important he remembered that. Did that make squares important? Well sort of, but he recalled they only held a kind of red herring role to play.
No shapes were clearly important this much he recollected with clarity, shapes that fed one into the other with endless repetition, angle upon angle, plane upon plane. P seemed quite entranced by the geometrical reverie. But how did this occur? How did that movement come about? How was it that there was an endless oscillation between one and the other? Well, surely, it seemed to P, that the oscillation was metaphorical on the changing flow and ebb of being. But then recalled the unsureness of his footing before and was suddenly doubly unsure. P felt a shudder of vertigo though not from a dizzying height but from…
And P realised the same had happened again, he was going to say the vertigo was metaphorical upon that of the dizzying height, but this would return him only to what he said about the oscillation of the shapes and the ebb and flow of being, which in turn had reminded him of the unsureness of his footing as had occurred a while ago. It seemed to P he wanted to think of one as founded upon the other, but that instance of nearly falling over had somehow jarred this notion. But this was not the time to dwell upon the essence of the metaphorical. So P returned to the matter at hand.
But then again wasn’t this directly related to the matter in hand. Hadn’t P originally tried to discern the essence of number in order to found the understanding of the 47? But no this was different surely, for numbers are not metaphors for the quantity that they designate… are they? P was unsure. For if the senses of the metaphorical in those cases he had just experienced arose together and were not one founded upon the other then why should it not be the case that shape, number, quantity arose together.
“Oh dear” thought P “But why did I differentiate number and quantity one from the other? This really is becoming very confusing. Let us see now. The ‘shape’, well that’s simply the lines on the paper, I suppose then I must have meant the name for ‘number’ 47 or Mr 47” P smiled to himself “and quantity, well that’s the amount as derived from the appearance of separateness as established earlier.” In this brief moment P found himself getting quite into the ‘essence of the rigorous’; that move of recalling a premise from earlier, it seemed somehow proper and well, the gentlemen would have nodded with approval and ticked the box with the adjacent text: ‘previously established premises later returned to as proof for subsequent arguments’. Yes this would have been very pleasing to them. But P had returned to nonsense again. The gentlemen, quantity, number, shape; all his discussion seemed to belong to nothing other than the essence of gibberish. P needed to begin again.
Well maybe not entirely again, surely out of his ponderings he had made some progress. Though what could possibly count as progress along such a path of thought? No, a new beginning was definitely required even if it did retrace some of the old ground once more. Though now he thought about it again, a new beginning was exactly what he had learnt was not required.
“Just the thing that the doctor expressly forbid…” P said to himself about the beginning-once-more idea, and then added without knowing why “…under pain of death”. In saying this addendum to the clause (CHECK) P pulled a face in which he bared his teeth, his eyes grew wide with anxiety. A passer by noticed and looked fearfully at P before hurrying past.
P decided that he needed to make his way back to that mood in which he had begun to recall things, or rather in which the whole business seemed to open up to him again. He differentiated these two for fear that there were really no such things to recall and that the whole thing was an edifice of number plates and digital times. P found what seemed a reasonable question and asked it to himself.
“What actually made me think of it in terms of an oscillating movement?”
And what a reasonable question it seemed to be. There must be a good answer to this or where would the oscillating motion come from. He could not have thought it completely groundlessly. It seemed to P as he contemplated this ground that it had two sources and neither was even trying to be reliant upon the other and yet both seemed really quite cogent. P suddenly hoped that those gentlemen in their court of science hadn’t heard his usage of the word cogent as he felt that they would hardly approve. Nevertheless, P thought, it was certainly true that one of the ways of oscillation was mathematical and thus deserved the title cogent to some extent even if it did not quite accord with ‘the essence of the rigorous’. P began to think out loud again:
“The first way into the oscillation between the shapes is to think of the shape of the numbers themselves, that it as we commonly draw their symbols, in this way we can observe that one 4 is similar to the upside down version of the other i.e. 7.”
P paused as he became aware of the absurdity of the argument; it seemed that in order to make it ‘cogent’ he should add that the line that hangs vertically down from the top of the 4 should be discarded in order for the similarity to be realised yet he was aware that the discarding of this line was so arbitrary that it compounded the ridiculousness of the argument beyond belief, he decided that the best thing to do was to ignore this slight flaw and carry on. P then realised why the vertical line was not to be dropped. Suddenly he felt that sense returned:
“It will be observed then, that when one fuses these two symbols together in their most obvious synthesis one creates the following symbol:
It thus can be seen that 4 and 7 fuse to make two isosceles triangles joined both at a single point. The single point is the point at which the oscillation takes place. In this way it will be understood that the oscillation between 4 and 7 is the oscillation between two triangles. Does this mean that one triangle is 4 and the other is 7? Well if we posed such a question what could we possibly mean by it? The question mis-leads. But here mis-leading is not a leading-astray in the sense of leading us into errancy. It is rather a question which leads us missingly into places where we can scarcely tread or breath. We cannot say that one triangle is 4 and one is 7 for in truth in delineating this symbol we have merely entered into a chain. We cannot say one triangle is 4 and one is the 7 for we cannot say at which point we have entered the chain and furthermore we do not even know if the chain even means 4 and 7, for in merging these symbols we should understand that the rules that they once were used for have be taken up in the oscillating motion and thus been stripped of their meaning. If each fuses into the other then the identity of both is simultaneously lost, worst than lost, trapped a horrifying movement where it is both and none.”
The gentlemen were not impressed. They did not find the speech cogent or rigorous in its treatment of the matter. He felt he had heard them cough at the point at which he had said ‘their most obvious synthesis’ but he had carried on nevertheless. There might be one of them, so P believed, who had actually been quite interested at one point, but the disdain of his colleagues had been so great as to quell this enthusiasm for the matter. But he had shrunk at the back, deeming it not really worthwhile to quarrel with the other gentlemen. So apart from this one potential gentleman with his occasional pondering of that-which-was-without-rigour the vote was unanimous and out came the stamp to prove it: WORTHLESS was the pronouncement and that was the end of that. The gentlemen had had enough of this nonsense; indeed some others at the back began to look angry with P. Why was he wasting their time? What was all this rubbish about the oscillation of the triangles? Did he want a demonstration on the essence of triangles? If he did they would write one for him: a long and rigorous one.
P did not know what to do. P was not entirely sure why the speech was still aimed at this ungrateful audience, who in any reasonable sense, were not actually there. And yet it seemed that the more he tried to tell himself that he did not have to adhere to the criteria of this parliament of logic, the more they came up to listen to his every pondering upon the matter. The cheek of it, he began to think, that they had asked why he was wasting their time? When in fact it was they who were wasting his.
And yet, already he could feel the temptation to try to win them round with the second entry into the oscillation of the 4 and 7. Thusly ignoring the stamped last two papers that now littered the floor, P turned to face the body of gentlemen once more.
“And though he seeks to stay afloat,
The night and bark are slow in boat”
This wasn’t right, this wasn’t right at all. This was the part about the boat from the poem of before. P felt a fool. The gentlemen looked on.
“Boats and night aside though” P continued “let us contemplate the mathematical proof of the oscillation between 4 and 7”
The trick worked. The words ‘mathematical’ and ‘proof’ distracted the gentlemen to the extent that they completely forgot about the boat incident. Now the gentlemen were all ears, all of them! Eager to hear a proof, they murmured excitedly and shuffled in their seats.
“A proof…” began P “…belongs to the essence of the rigorous.” the gentlemen murmured their approval, some even clapped, whilst yet others stamped their feet. “but what, my good gentlemen, belongs to the essence of proof?” an expectant silence gripped the audience “Well, existence in its particular material configurations is ultimately contingent, so in seeking to gain a secure epistemological grip up our matter we must find a body of knowledge which grounds itself in the necessity of its own insights, that is, one formed of propositions which are apodictic and a priori. For only in this way may we proceed. The paradigm model for such a body of knowledge is mathematics, which as the ground of physics is the only secure basis upon which we may ground our knowledge about existence.”
At this the gentlemen roared their approval. The language sang as poetry to them. They stamped their feet, whooped, clapped and threw their hats in the air. Others began to reach for a different stamp in anticipation for the subsequent parts of the paper. The word VALID could be seen clearly (though backwards naturally) on the rubber stamp and already it was dunked in the ink in anticipation of the papers approval.
“Now as the ground of physics we should expect mathematics to ultimately reflect the essence of material reality, thus those propositions which are true of mathematics are necessarily true of being. Furthermore the contingency of a given number system is no proof that the propositions arising out it are in anyway contingent. For as the essence of the mathematical belongs to the individuation of being transcendentally i.e. that it is not merely a matter of my own being that ‘here is one thing and here is another’ –for such individuation may be said to have being regardless of my presence- and thus the essence of mathematics belongs not merely to my manner of apprehending things but rather to being as it is.”
This part of the speech was too, largely taken in by avid listeners. They seemed at least to find it an argument they could tolerate, though clearly now they were waiting for where this was all heading.
“Thus in the following treatment of the those positive integers, 1 through to 9, we may allow that those results acquired be not dismissed on the basis that the system as we commonly use it (1 through to 9) is but arbitrary. As though the system partakes in the individuation as one manifestation of it, it is still equally apodictic in its belonging to the nature of existence.”
At this the gentlemen seemed less sure, it seemed a suspicion began to grow in their minds and yet for the moment they still granted the ears of science to for hearing.
“We begin our discussion with the notion of squaring. Now it is commonly understood, that to square a number we should multiply it by itself. Now the squaring of a number opens the numbers essence further up to us, for we have derived this second number (the square) from nothing other than a simple process and the number itself. However should we not in this most exact science always be striving after that which is simplest and most clear? Now in our number system those numbers which are most clear are those which remain as single integers i.e. they never become more than 9 and thus are not confused to our comprehension by the symbolic addition of an extra number to designate a new ‘higher’ number.”
The discontent within the gentlemen was quite apparent now. Some now muttered amongst themselves, whereas made aeroplanes out of their notes to show their lack of interest whilst others turned their chairs so they faced slightly away from P, nevertheless it was not yet the appalled looks of before so he continued.
“We should then take this kind of scientific notion into consideration when we square numbers for it necessarily happens that the square of a number is greater than the original. The problem of the cumbersome size and inexactitude of numbers can be removed by the simple process of adding the composite digits together. For example in the case of 16 we simply add the 1 and the 6 together. Now we might question the necessity of this addition but it is soon revealed when we consider that if we were to propose a subtraction we would not know which number to subtract from which, hence we seem guided towards this numerical essence by a hidden necessity. This same hidden necessity had previously told us that a square is the opening up of a numbers essence. We must now admit though –no matter how much we may not like it- that the adding together of the composite integers reveals a number which belongs no less to the essence of the original number, indeed is a number which owing to our reduction to simplicity is the most refined part of the essence of the square. Now, having established this we may now show how the oscillation finds its manifestation in the apodictic realm.”
The word ‘apodictic’ did its work and the gentlemen still continued to tolerate the speech.
“Let now consider the basic integers each in turn. In taking solitary 1 and squaring him, I mean it, sorry.” P pulled a strange face before continuing “Now in the case of 1 simplicity is always our ally for it will be noted that no amount of squaring can ever force 1 to move beyond itself. In this sense 1 contains its own essence and is complete. The case of the integer 2 is, we should not say more interesting –as what could be more interesting than the self containment of 1- but nevertheless more lively shall we say. The case of 2 runs thusly. The square of 2 is 4, this much is immediately apparent. And yet when we are greeted by that 4 who belongs to the essence of 2 are we not instantly found wanting a further movement, for having arrived at 4 the wisdom of 1 is now apparent to us. For the 1 which clearly demarcates itself as whole and is determined by its own essence is a highly fortunate fellow, I mean number, yes. What I mean is, in arriving at 4, the continuation is already suggested to us, for though 1 repeated itself endlessly, in transforming 2 into 4 we feel we should press on.”
The gentlemen did not look like they felt like pressing on.
“And so, now faced with this 4 who has not yet feel we are on safe ground, we square him also and thusly come to the number 16. Now according to that rule which we laid down earlier as one of the important precepts of mathematics, we should add the 1 and the 6 together. And what has happened now? What is it that that philosophers stone of a principle has achieved? Why nothing other than the transmutation of 4 into 7! And yet are do we still feel secure? We do not. The same method must immediately be brought to bear upon the 7, and in doing so what occurs. Well the square of 7 is 49, but this cumbersome digit will have to be reduced. Now the 49 attempts to resist our attempts to reveal himself by transforming himself into 13; he struggles in our hands. But there is no means of resistance available to the numerical ones that can overcome the inexorable reduction of addition of composite integers. Thusly the 49, reduced to 13 now finds his essence displayed as 4. Yet how can this be? For were we not just moments ago saying how 4 was not stable ground and yet now he is back again as if he were as self determined as the 1. But he is not, for we should not forget the journey that happened, that lead us to 7 and back to 4. Indeed should we begin that journey once more then it is down the same path it will lead us. The 2 does not return, but once transformed to 4, the oscillation between 4 and 7 is eternal and the 2 belongs to this oscillating essence. This is similarly true of 5, who once he has become 25, then becomes 7 and so on. Thus only the numbers 3, 6, 9 and 8 escape this pattern. The group 3, 6, and 9 all belong to the same group insofar as they all come under the sway of 9 whose essence encompasses them all, observe: 3 will become 9, 9 will become 81 and thus back to 9 ad infinitum, 6 will become 36 which will become 9 and so on. This leaves but 8 to unfold. Now the 8 we shall speak on later but for now let it be shown that 8 will become 64, which will return itself to the fold of the great and stable 1. We can now see though how the 4/7 as oscillating phenomenon is inserted with absolute certainty into our being. The numbers 2, 4, 5, and 7 in their particular instances all dissolve into this motion which is neither one number nor the other. The motion of squaring as essence revealing shows the oscillation as inherent in the essence of all things as individuated things. In this way that same chain which we noticed in its symbolic manifestation reasserts itself in the numerical apodictic.”
The gentlemen did not seem to know whether P had finished or not. Though in truth P did not know whether he had finished or not. The whole speech had been a ruse clearly: the dressing up of the number revelation in words that the gentlemen might listen to. But though P now told himself that’s what it was, he once again became unsure. Had he really been so certain of the ruse like nature of it at the time of presentation? Were there not moments at which he had actually believed in that mode of speech that he had carried with him? P had not time for such thoughts. For as the silence endured that board of listeners and watchers finally came to realise the ordeal was over. They ceased playing chess, doodling, staring at the ceiling, tapping their fingers and feet, rattling their chairs and pulling hairs from their noses. All these things they stopped, to acknowledge the end of the diabolical travesty of rigour to which they had been subjected. There were no words to be said. One of them, a particularly ancient looking one, picked up the WORTHLESS stamp and hurled it with a force completely incommensurate with his size. It struck P firmly on the forehead, imprinting its design in the allotted place, P started to splutter an explanation but found the gentlemen had gone and he was back walking on his journey once more.