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A Reasonable Line of Enquiry

 
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Peter Blumsom



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Location: Wembley, London, UK

PostPosted: Fri Mar 29, 2019 7:53 am    Post subject: A Reasonable Line of Enquiry Reply with quote

There’s something suspicious here.

I disagree, sir. You are just talking about the way numbers are.

Yes, sergeant, but wouldn’t you say that even what you say is curious in its lack of curiosity?

Why, sir?

Well, what you say is, there’s nothing untoward here; everything appears ship-shape – it’s simply in the nature of numbers.

Quite right. sir. Numbers just being numbers.

Well, it’s curious that you don’t ask yourself, ‘What is the nature of numbers?’

I don’t really understand, inspector.

Well I suppose there are, looking around the scene of the crime, many questions I could ask.

Such as, sir?

You agree that numbers are made up of units, yes?

That is correct sir.

Well, we won’t ask where the unit came from, even though we could. But, instead, consider, if I have a pile of apples, how many apples would I have?

How would I know?

You would have to count them, wouldn’t you?

Of course, sir. That’s obvious.

It’s only obvious because you are a human being. Remember that. It would never occur to a frog to count them.

If you say so, sir.

And counting them, you would find you had five.

If that’s how many there were.

So how would you count?

You’d mark them off in an orderly manner. That is, one, two, three, four, five … and there you’d stop.

But why would you stop?

Because there are none left to count.

You’re getting the idea, sergeant! But what are you counting.

Apples, sir.

In a way you are, but in another way you are counting numbers. For example, suppose I ask you, ‘What is apple number one?’

Whatever apple I start with.

So if you were asked to count again you need not try to remember that same apple you began with … or would you?

Not necessarily. I could start with any of the five, could I not?

You tell me, sergeant. I’m merely asking. All I’m saying is that you wouldn’t have to consult your note book, would you?

No sir, any number it is.

Now listen carefully; is it the same with numbers? Can you begin the count with, say, three?

Ha ha, sir. You are joking.

No, sergeant, I am deadly serious actually. Tell me in your own words, why couldn’t you start with the number three? It’s really a simple sort of question.

Well, to me it’s too silly to talk about, begging your pardon, sir. But if I did, I’d say that that would be in the wrong order. But really, sir, if I may speak off the reord?

Of course, sergeant.

Well, inspector, where is this getting us? We are no nearer to understanding what we’re looking for. I mean, what do we know now that we didn’t know before.

We know that the apples are in no particular order but that the numbers are anything but out of order. They are precisely ordered and they bring their order to everthing that is precisely counted, and that incudes our apples. … How are you sergeant? You look pensive.

I’m fine, sir. But I’d like to get to the bottom of this.

Well, good. Could we say, at a stretch, that number brings the order of a different level of being or knowing to another realm of being that is naturally more disordered?

Er, you could put it that way, at a stretch. But I don’t know where you would be going with that, sir.

Does it matter? It’s enough that you could put it in the way I suggested. And by agreeing that the world that is populated by objects being counted is naturally more disordered than the what is counting them - the numbers themselves, You have agreed also that we have gained some evidence about the nature of numbers. Do you grant your inspector this?

I do.

Really? Without prejudice, without the intervention of assumption?

Yes sir. It is perfectly reasonable, what you said. I can find no flaw in what you say.

Gentle is the path of truth, sergeant, if we take measured paces. So if I were to ask again, what is the nature of number. could you say any more than you said before, which was more or less nothing?

I think I could sir, and perhaps in that pithy way you seem to like. Though I’m feeling strangely light headed I am minded to say number is more ordered than the things it numbers, and that is its nature.

Well, that certainly opens up a line of enquiry. Numbers, then, are not apples, or anything of that sort. But can you say more?

Yes… they are units aren’t they? And a unit means something single or one.

So when you count you say to yourself: one, one, one …

Hardly sir, I mean, respectfully, that’s a bit daft!

But that’s what you said; or perhaps you weren’t being quite so precise as you were before. You make your units to be exactly like apples, except they are more like to each other than apples are or could ever be. In fact they seem identical.

How come?

Supposing this dish with five apples were instead to hold two oranges, two lemons and a banana. How many pieces of fruit would it hold?

Five, obviously.

Obviously, though you haven’t yet demonstrated this. But I’ll accept this for the time being, if you’ll accept what arises from a path of reasoning.

A policemen is compelled to do so sir, as long as he knows it is reasonable.

Shall we take up the point again? The bowl now holds three varieties of fruit, but when you count, the tally goes along the same path: one, two, three … etc.

Naturally.

So, naturally, the numbers didn’t change, when encountering the different kinds of fruit.

That’s obvious.

But if I asked you to count a small number of things in your mind, say, an ant, a skyscraper and laughter, how many things would that be?

Laughter?

'The' laughter.

Fair enough .. three.

And they are completely and utterly different to each other.

As far as could be.

Did this difference make counting them difficult?

Not in the least.

So this confirms something about the nature of units.

What is it?

That they don’t change when the things they count change.

Agreed.

But are you sure? Do you mean that they don’t change in the slightest’ After all, the Empire State Building is very large and an ant, very small, and laughter seems to have no size at all.

But that doesn’t affect the way we count them.

Because … ?

Sorry Sir?

Well, you seem to be about to make a statement about units.

What about, that is is the nature of units that they are … countable.

And even if we change the order of what we count?

It wouldn’t alter a thing.

So these units are identical.

They must be.

So, again, why do we not count one, one one …etc. And this time answer as if you were a little curious.

I’m not sure I follow.

Well we go from one to two don’t we. But when we count to we are only pointing at one – one thing at a time.

It seems such a silly question.

So silly that you seem to be experiencing extraordinary difficulty answering it. Don’t worry, sergeant. The whole subject is extraordinary and difficult, as well as being silly and simple at the same time. But a comprehensive approach to the nature of numbers seems to compel us along this route. Isn’t the answer that we gather units as we go along, and that gathering we call counting?

But here’s another silly question: We have seen that when we count apples we can start with any apple and it doesn’t affect the count, butwe couldn’t begin with any number. We always begin with the first.

Yes I remember that,

But does this then mean we can begin with any unit?

No sir, and I’ve had an idea – doesn’t this prove that the units are identical?

Careful, sergeant, that what you call proof isn’t merely circumstantial evidence.

Well. In a way part of me is of the opinion that we are talking nonsense, yet … if we accept that I believe we can work out a proof.

I’m intrigued … go on.

Therefore in the ridiculous manner of this conversation, let’s say that two was three inches long and three was four inches long and four was five inches long, then 2 x 3 would be a foot long and 2 + 4 would be nine inches long, whereas they both should be equal to one another. And such anomalies would occur in all calculations unless units were exactly identical.

That is as simple proof as one could wish for. And it shows you are developing curiosity regarding the nature of things, and a healthy disregard for the shackling restraints of common sense.

Can I just complete what I was saying sir? If each was one in the way of the above example – and remember we are talking of units, not numbers (and you have shown me there is a difference) - we could call it ‘the case of the unequal unit’ where adding up, dividing and those sorts of things we take for granted could never work. A recipe for chaos is the phrase that comes to mind.

In fact, sergeant, it’s even more chaotic than that. If all the units were unequal it would be an impossibility to even say that two would be equal to three inches, for we assume that the three of these three equal units would not be measurable as rulers would no longer work properly.

A complete and utter disaster, sir. In fact, a case for the Flying Squad.

Well, sergeant, you have taken us over a few fences here, but I’m not sure whether the field are greener or more strewn with weeds.

Thank you sir. What is your opinion?

I think everything is a little greener. It is always brighter when we unburden our souls and begin to look into the nature of things. You don’t suppose the farmer counting his cattle has any thoughts on the nature of numbers, do you?

Well hardly, sir. He counts cows, so that is what his numbers are – cows.

Quite so. Then we have established that things are neither equal nor in any particular order until they are counted, i.e. until they are given numbers. And that these numbers, although perfectly ordered themselves, by counting, can only partially order the things they count.

Sir?

Any counted object receives a number during the count but may receive another number on recount. Remember that any apple could be ‘one’, or any other number of the count. So they are only partially ordered; whereas a number cannot help but be itself. ‘One itself’ cannot become ‘two’ – it’s not like an apple.

I see.

And also established is that these numbers are themselves composed of units, and furthermore, these units are completely identical, as you yourself have proved. Have we missed anything so far?

Well, yes sir, we still haven’t cleared up why units decide to become numbers. What is motive here?

Very true, and this we now begin to address by asking what would happen if units did not decide to become numbers.

I have no idea.

It would certainly seem that another form of chaos would ensue.

In what form, sir?

One even more formidable than the last. We would have two impossible situations, first, a single unit which would be doomed to remain so, and secondly, an unspecified heap of pile of units.

I’m sorry sir but although I can allow you the first option, the ‘second’ would require a number that does not yet exist, needing our standing outside your hypothesis to call it ‘two’ or ‘second’.

Help, hoist on my own petard!

Agreed sir, if I may so remark. But now I note the full chaos of the situation and see little help on the horizon.

What seems to have occurred is that our enquiry into the nature has come to an untimely end before we have even reached the appearance of these entities called numbers because we find there is no single nature to be found that covers the whole range of their existence, and no cause which prompts them to arise. And yet, unless we do we cannot even count from one to that number that comes after one. In fact, we cannot count at all.

I’m afraid that’s true sir. No more times-tables. My young daughter would be very relieved on that count.

On that impossible count, you mean, sergeant. And yet, in spite of this impossibility, Farmer Giles happily counts away until all the cows come home. Let’s see if we can do anything to remedy this situation. Perhaps we can drum up some form of relation between all these impossibilities. Otherwise all our former good work will go for nothing.

I must say, I personally see no hope here sir. I’ve no idea where we are going with this.

I have an idea that may smooth out the path of your thought, sergeant. We can say in general that each is one, and I’m clearly talking of units here.

You probably would be.

I could take it that both are two.

Yes, it’s true that both taken together are two but where does the ‘two’ come from. There is as yet no evidence that it exists.

Very well, let’s talk of something more fundamental. Something that lies in the very nature of the subject we are examining.

I can’t see anything more fundamental than counting, and that has been prohibited. How is it possible to demonstrate this?

A little while ago you baulked when I tried to put before you the chaos that stands between the unit and a pile of units. You would not even allow me to describe the parts of this situation because it meant using the word ‘two’ which as yet we had no way of accounting for. This left us with a chaos that could not even be described, let alone resolved – a kind of phantom situation where we have one aspect and another but have no number which is appropriate to it.

It may have seemed harsh to you sir, but I was only following a valid path line of enquiry.

One which seems to oppose any effort towards a meaningful tally of these units. But now, I believe, I do see a way forward, and this way ensues from the very nature of the units themselves. We are, even within your present restriction, allow to perform a simple motion of dividing this pile, this undifferentiated pile of units, down the middle by matching one with one or any other efficient method which avoids naming a number apart from the unit. This would produce on one side of the division, one pile, and on the other side, another pile, both naturally smaller than the original. Each unit in one pile would face a unit from the opposite pile – you see how carefully I am to remain within your strictures.

I do

Well then, note that one of the resulting piles will always be either equal to the other or in excess or in deficit of the other by a single unit. Is this not so?

Yes, this surely is the case.

And because this is always the case we may say that this occurrence is secured on that solid foundation which is the nature of the one and the many. No numbers are specified here. Is this not so?

Yes.

Have we then established ample evidence for either one or another situation without mentioning that unutterable ‘two’?

Careful sir! You just said it.

But I needn’t have because there are already perfectly respectable terms for these descriptions of ours. When there is no excess or deficit between one division of a pile and the other, the term for the combined pile is ‘even’ – is this not correct?

Yes, and this term would also describe the divisions; each would also be ‘even’. But when there is an excess or deficit of a single unit between the divisions their combination is said to be ‘odd’.

Careful Sergeant, this is not necessarily the case. Each of the two even piles could just as well be odd as even in themselves. When we have command of the numbers we can look into this. But in the case of the odd divisions we can say with confidence that one of the two will be odd and the other even.

Though in this case we couldn’t be sure as to which is which without performing a further process of division upon one of them.

You have it in a nutshell. Does this not pierce the obscurity that enshrouds our anonymous pile of units?

Like a lightning bolt illuminating its hidden nature.

Nicely put, sergeant.

Not so bad yourself, sir.

Then these again are examples of the nature of our subject, which we will defer examination of until some later moment. What we can say, regarding the nature of – shall we give it its time honoured name – the one and the many, that the odd situation and the even are true opposites. A pile cannot be both odd and even.

And I suppose we might add, neither can a number.

So all this is an authentic expression of the nature of number, even though we have not yet given ourselves full permission utter the names of any particular number. What we have discovered is not something imposed externally by thought but rather we have allowed caution to guide us from within.

As placing things side by side is an act of comparison rather than counting we are still innocent of numbers themselves, but the further differentiation of odd and even which seem to be beyond numbers bring us inevitably nearer to the fundamental essence of number itself.

It sounds quite imposing when you put is like that, as you said earlier, it all begins to shine like a light. But I have to ask – where does it get us? We still haven’t a single number at our command except for the unit. And that seems like a freeby.

Well, we need to take what we could call, the seminal situation – the root of all singularity and multiplicity, if you like. We can approach this imposing edifice through the following statement:

Each is odd, but together they are even. And we are specifically talking of units here.

This is reasonable, as far as it goes.

But sergeant, it goes quite a long way, maybe further than you suspect. Look at this even; what do you notice about it?

I think I see something. Take you statement: a number cannot be both odd and even, this even which you can call the seminal root of all evens, is divisible into two equal piles, right down to one unit on either side of the division – though I confess its odd to think of single units as a pile.

Nice one, sergeant. But do continue.

Well sir, my thinking is that in such a case each pile is odd but when I place them together oddness disappears and we are left with the even on its own. And this is curious.

Indeed it is sergant, for what disappears is the single unit. You say ‘placed’. How do you ‘place a unit by the side of another.

In thought, I suppose.

So by thought alone the unit is made to disappear. And by the same token our pile, as an unspecified heap, begins to waver. Even more curious.

Bloody remarkable, I’d say. But my thoughts are still telling me that all this is no more than a hill of beans.

However an ever more specific hill of beans; what more do we see?

Delightfully, I think I spy our first number coming over the horizon, apart from the unit which I suppose is a kind of unofficial number.

You mean the number two?

Indeed

We have uttered it into life now we have to justify its utterance by giving it existence..

I reckon we now have what I consider ample evidence of its existence. We could call it by any name but ‘two’ is the name already given to it, just as the single unit we call ‘one’. But I wonder what has happened to single units. Have they really disappeared?

Yes, in a way – but not into ‘nothing’. They are still there but subsumed into the ‘two’. Everybody kind of knows this, but not in the way we are knowing it – along a path of reason.

You mean, I suppose just as a leg of a dog is subsumed into the dog. It’s still there but as a part of a whole.

Something like that. Only the whole in this case is a number. A unity as opposed to a unit. And if we hark back to that process we have been looking at, we could put forward another pair, another two to put beside the first. And although still even, as you will recall, the combination must be even as well. Though no longer two, being too numerous, and we call it four, in England. We also know that by adding or taking away a single unit from that even gathering we would create an odd number.

Yes, either ‘three’ or ‘five’, both odd numbers.

And so on, and we would be seeing that gradually from what was an indistinguishable pile the nature of number begins to emerge in its glory; and that whatever pile, or shall we say, unknown number, not only would we know it to be countable, but also divisible equally by two or with a remainder of one, this holding true for every number into infinity. Always would one or other of these two great forms of the odd and the even hold sway over all numbers. And perhaps more speculatively the individual loses itself in the twoness of the even, where it seems to disappear, only to find itself again in the individuality of oneness accompanying the arising of the odd – as not only the even banishes the odd, but in the opposite case, the odd banishes the even.

Well sir, I can’t quite see it that way. For the numbers we count with, those which start with the unit and end when the full tally has been reached, do they not contain alternate odd and even? So when we count off, say, the number six which according to you has banished the odd, it nevertheless contains three odd numbers, one three and five, as well as the evens.

Yes, that’s true, but the count passes through those numbers on its journey to six. Only there is its completion. Up to that point it is merely a work in progress.

But if you have six apples you also have five apples.

Though do you have five apples and six apples? You either have one or the other you cannot have both. And do you call five a square number because it contains four? A number is a thing in itself not a pile of units. This indeterminateness is banished by the count. Each number being itself rests in its own unity. You still don’t see this because we have more work to be done. It is farther along the path from which we have reached. When we look more deeply into the nature of number we see the numbers themselves not in comparison with other numbers. There are, in such a seeing, no operations to consider, no further calculation needed. Each number is in the place it should be, free of exterior verification, or alteration, sure in its own essence.

Do you think we have enough for a conviction.

Yes I do, sergeant. Fetch the handcuffs!

----------------------
I base this light hearted sketch on some of Socrates strange ‘mathematical’ pronouncements in the latter part of the dialogue PHAEDO by Plato. I’ll probably add more notes and amend it on further reading.
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David Tang
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PostPosted: Tue Apr 02, 2019 9:07 pm    Post subject: Reply with quote

Some short ruminations and evocations on a reading.

So long as we remain in phronesis, in practical dealings, immersed in the things, can we speak at all of the region of “units” or, what is the same, “monas”? Both Aristotle and Plato attest that the Egyptian Priests first learned mathimatikos properly speaking, for the reason that they were freed from necessity to ‘leisure’ (that is, school, what we moderns moan about as though it were true ponos [πόνος] itself!; drudgery), to mere “onlooking”, to schola, school. The χωρίς (khōrís), seperation, into the region of thought, thought not in the modern sense, but as a kind of manner of going further into the things, into topos, a really existing region still pursued by Kepler and Leibniz with great intensity and genius. Still reaching into our modern being all forgotten.


There is a special mystery. When we have eidos, I see a green apple, and perhaps am even hungry and consider it with interest, do I yet see the thing “apart”? Which is to say, in one sense, an apple is always there on the table, or on the brilliant green grass of the field, and then, under the late afternoon sun, and a squirrel is nearby with its ears raised and looking shrewdly about in search of food and tense in jealous watch of its safety. But, the eidos, the apple, already in thinking its “unity”, as phronesis thinks it, just in seeing it, abstracts. It thinks only the apple. An apple. Just as though the thing were far away, and not in the midst. The far away is part of our natural and ordinary doings. Yet, so too the earth with its collection of things as mentioned, its alongside and bellow, its higher and lower, its green and not red.
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Peter Blumsom



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Location: Wembley, London, UK

PostPosted: Wed Apr 03, 2019 5:49 pm    Post subject: Reply with quote

You make that apple very appetising. Makes me wonder whether a god, a young Hermes or Krsna could not also be enjoying it in some rustic meadow, in some present moment, as you paint it. To tell the truth I can't comprehend forms in separation, they would lose their joy. The mind can do all sorts of things - the soul has reason and we examine ourselves through the confrontation of assumptions that, as the Bard says, put us beside our parts. Its not unity of Self that is the problem but disunity of selves.
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PostPosted: Wed Apr 03, 2019 7:57 pm    Post subject: Reply with quote

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PostPosted: Wed Apr 03, 2019 8:03 pm    Post subject: Reply with quote

erase

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PostPosted: Wed Apr 03, 2019 8:04 pm    Post subject: Reply with quote

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David Tang
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PostPosted: Wed Apr 03, 2019 8:05 pm    Post subject: Reply with quote

Geometry should be distinguished form arithmetic, in terms of the "separation", since in the latter there is a topos, but, crucially, according to the Greeks, no physical place. There is orientation of the things in the topos perhaps very like orientation of the numbers in higher and lower.

By the abstraction of the unity I mean only, I see a thing amidst the others, distinguishing it, and say, "It's an apple.".

More exactly: the "unity" is something we can distinguish from the mere "look", is it not? The "unity" is a special thinking over of the mere thing seen, the "kind" in the mere seeing of the thing understood to be something.

The intuition, so called, rather than the sense data or the epistemological particular (e.g., the "logical" particular that can appear in syllogistic statements, and so, apart from being there, apart from intuition or seeing the thing and understanding what it is, e.g., "There is the ripe apple! Let us loiter about it in the light of the sun."


What I mean is, the apple, in mere seeing, is already abstracted. Because it is understood, KNOWN, by it "look" or eidos. Stictly speaking, eidos is not "morphe", which is Aristotle's move where he speaks of one of the "whys", the aitoi (αιτίες), the "causes", in the "form", which is not the mere "look" of daily Dasein.


By the way, I understand it this way, in the Socrates of Plato we have very few "rules", so to say, Socrates says, one must remember, and that one must endeavor to understand the same things in the same way, meaning, to use the same words in the same way (in fact, this is the more exact formula, to use the same words in the same way, viz. the Euthydemus, which is no mere comic sortie).

Ergo: when we say, Plato speaks of the eidos, and Aristotle of the morphe or form, we make a distinction with philosophic content. The former is more linked to unthinking, pre-philosophic, or ordinary speech and usage. The use of the word is less stultified by holding it in the scholarly and philosophic cannon, apart from daily usage. Or, in the web of associations of what has come back into daily usage by being overheard by servants and commoners at the dinner table, or by being half understood by inferior minds.

Now, I ask you, is it not that we see a kind of thing, e.g., an apple? And thus, there is an arche? Since, one can think, when no apple is there, of apples, and in imagination, in drawing, in dreams, in clay, on the canvas, in the building unbuilt, the apple may become our theme and multiply. Is it not so. The arche, the origin of apples, is thus, discovered to be there in the kind or eidos, and not as mere "theory" in the modern sense.
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