Laws of Form and Parmenides

This is a notebook about relations between Spencer-Brown book Laws of Form and Plato dialogue Parmenides.

It will be repeatedly updated and maybe, if productive, will transform into a page.

The motivation of starting it comes from Louis Kauffman post    Is mathematics real? .

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1.  one is infinitely many (Parmenides, source used) vs using the empty set to construct all numbers.

Introduction (142b—c) 
b Shall we return to the hypothesis and go over it again from the beginning, to see if some other result may appear? 

By all means. 

Then if unity is, we say, the consequences that follow for it must be agreed to, whatever they happen to be? Not so? 

Yes. 

Then examine from the beginning. If unity is, can it be, but not have a share of being? 

It cannot. 

Then the being of unity would not be the same as unity; otherwise, it would not be the being of it, nor would unity have a share of being; rather, to say that unity is would be like saying that unity is one. But as it is, the hypothesis is not what must follow if unity is 
unity, but what must follow if unity is. Not so? 

Certainly. 

Because "is" signifies something other than "one"? 

Necessarily. 

So when someone says in short that unity is, that would mean that unity has a share of being? 

Of course. 

Then let us again state what will follow, if unity is. Consider: must not this hypothesis signify that unity, if it is of this sort, has parts? 

 How so? 

For the following reason: if being is said of unity, since it is, and if unity is said of being, since it is one, and if being and unity are not the same, but belong to that same thing we have hypothesized, namely, the unity which is, must it not, since it is one, be a whole of 
which its unity and its being become parts? 

Necessarily. 

Then shall we call each of those parts only a part, or must part be called part of whole? 

Part of whole. 

So what is one is a whole and has a part. 

Of course. 

What about each of the parts of the one which is, namely, its unity and its being? Would unity be lacking to the part which is, or being to the part which is one? 

No. 

So once again, each of the parts contains unity and being, and the least part also turns out to consist of two parts, and the same account is ever true: whatever becomes a part ever contains the two parts. For unity ever contains being, and being unity; so that they are ever necessarily becoming two and are never one.

Quite so. 

Then the unity which is would thus be unlimited in multitude? 

It seems so. 

Consider the matter still further. 

In what way? 

We say that unity has a share of being, because it is. 

Yes. 

And for this reason unity, since it is, appeared many. 

True. 

Then what about this: if in the mind we take unity itself, which we say has a share of being, just alone by itself, without that of which we say it has a share, will it appear to be only one, or will that very thing appear many as well? 
One, I should think. 

Let us see. Since unity is not being, but, as one, gets a share of being, the being of it must be one thing, and it must be another. 

Necessarily. 

Now, if its being is one thing and unity is another, unity is not different from its being by virtue of being one, nor is its being other than unity by virtue of being; but they are different from each other by virtue of the different and other. 

Of course. 

So difference is not the same as unity or being. 

No. 
Well then, if we were to pick out, say, being and difference, or being and unity, or unity and difference, would we not in each selection pick out some pair that is rightly called "both"? 

What do you mean? 

This: it is possible to mention being? 

Yes. 

And again to mention unity? 

Yes. 

Then each of two has been mentioned? 

Yes. 

But when I mention being and unity, do I not mention both? 

Yes, certainly. 

Again, if I mention being and difference, or difference and unity, and so generally, I in each case mean both? 

Yes. 

But for whatever is rightly called both, is it possible that they should be both but not two? 

It is not. 

But for whatever is two, is there any device by which each of two is not one? 

No. 

So since together they are pairs, each would also be one? 

It appears so. 

But if each of them is one, then when any one whatever is added to any couple whatever, does not the sum become three? 

Yes. 

Three is odd, and two even? 

Certainly. 

What about this? If there are two things, must there not also be twice, and if three things, thrice, since it pertains to two to be twice 
one, and three, thrice one? 

Necessarily. 

But if there are two things and twice, must there not be twice two, and if three things and thrice, thrice three? 

Of course. 

What about this: if there are three things and twice, and two things and thrice, must there not also be twice three and thrice two? 

Yes, necessarily. 

So there will be even-times even numbers, odd-times odd numbers, even-times odd numbers, and odd-times even numbers. 

True. 

Then if this is so, do you think there is any number left which must not necessarily be? 

None whatever. 

So if unity is, number must also be. 

Necessarily.
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