The missing page

GrahamH · **#21** by **GrahamH** » Dec 13, 2011 10:06 am

andrewk wrote:Just reading over the thread I noticed there's a bit of cross-purposes occurring. The OP considered a book in which the page widths start at 0.5 and form a decreasing geometric sequence, totalling 1. That book has a first page but no last page.

Yes, the original discussion was some time ago and I forgot specifics of decreasing page thickness.

andrewk wrote:However, in post #4 where Graham proposes the cutting machine, he does not specify the widths. I get the impression from your later posts Graham that you have inadvertently switched away from the specification that page widths form a geometric sequence. That explains why you are now imagining that the book is symmetrical and if it has a first page it must have a last one. The book in the OP, and described in the previous paragraph is not symmetrical. Most of the suggestions you make in post #6 are based on an assumption that the book is symmetrical.

To make a symmetrical book you would need a different cutting scheme that delivers an infinite number of cuts within the range of x in (0,1). Possible schemes are to make cuts at:
- all rational values of x
- all algebraic values of x
Under these schemes there would be no pages at all, as none of them could have non-zero width (or put differently, we cannot identify two blades that make cuts between which there exists an undivided page).

If we imagine a universe of perfectly uniform substances, where there are no fundamental particles, such that a "page" can be divided infinitely, we can make an infinite book by successive halving of a block of material. We first get two pages of 1/2 thickness, then four of 1/4 thickness, and so on.

With such a book you can turn arbitrary groups of pages, anywhere in the book, but you can never isolate a single page. No group of pages is finite. You can turn pages to move through the book, but never single pages.

In the original context of an infinite universe this seems more appropriate, with time extending uniformly, the "beginning" and "end" unreachable from any "page" you might happen to be on.

andrewk wrote:A symmetrical cutting scheme that would create an infinite number of pages, each of nonzero width, would be to make cuts at x values of:
* 0.5^(-n) for n = 2 to infinity
* 1 - 0.5^(-n) for n = 2 to infinity
This book is symmetrical. It has two middle pages of width 0.25, encased by pages of width 0.125 etc...
It has no first or last page. But it has right and left edges that are not pages.
If you turn back a bunch of pages at either the front or the back you will have turned an infinite number of pages. However you are able to grab hold of a finite number of pages in the middle of the book.

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The original formulation has the effect of contrasting a finite page thickness at front of the book with infinitely thin pages at the back. It seems more straightforward to have uniform page thickness 1/n n->inf.

Its Calculus vs. Zeno's paradox.

andrewk · **#22** by **andrewk** » Dec 17, 2011 2:25 am

This problem is very entertaining.

Thinking about it some more, the restrictions placed on the types of books that can be created either by the inability to traverse an infinity by successive addition of pages, or by using an infinite-bladed cutting machine to cut into a block, seem unnecessary. What about the following:

We have an infinite-pronged 'seeding machine' that consists of prongs numbered 1 to infinity, arranged in a line, with the distance between the n-th and (n+1)-th prong being 3 * 2^-(n+2). The region z<0 is the ground, made of infinitely divisible fertile 'soil'. The seeding machine is lined up along the positive x axis and inserted into the ground so that the prongs go in and insert seeds at x=1/4, 5/8, 13/16, 29/32 .... Then it is withdrawn so the seeds can grow.
Each seed is different. The n-th seed causes a page-like growth (rectangular-prismal in shape) to arise out of the ground, that has y-dimension = z-dimension = 1 and x-dimension = 2^(-n) and has, on the faces pointing towards the positive and negative x directions, a big black number n on a white background.

This set-up delivers a coverless book the same as was created by the cutting machine previously considered, without the weird far-right edge that is not a page. Nor do we need to traverse an infinity by successive addition, as would be the case where each page is successively halved.

So what do we see when we look at this book from the point (10, 0.5,0.5)? There is nothing between us and the numbered pages, so we must see something, mustn't we?

The first answer is that the pages would become more and more transparent as they get thinner, so we'd see a big grey blur.
We rule that out by saying the pages are completely opaque, no matter how thin.
OK then, so what happens to light that strikes the page surface? It must either be reflected or absorbed, in fact, we should assume that the white parts of the page reflect light and the black parts absorb it. Whether reflected or absorbed, the incident light imparts momentum to the page. For the light to be visible to the observer, it must have a certain minimum nonzero energy, and hence nonzero momentum. Accordingly, once the pages become thin enough, they would be unable to withstand the momentum and would bend or break.
To rule that out, we have to assume that the material out of which these organically-grown pages is constructed is infinitely stiff and infinitely strong.
So then, what do we see from the vantage point?
We see nothing, which means it looks black - an absence of any light. That is because, for any positive n, no light from page n reaches our eyes. Why is that? Because page (n+1) is in the way and, being completely opaque, blocks any light reaching our eyes from page n.

This just brings me back full circle to confirming the statements of Benarete and Prosser. But it's interesting to note just how implausible the scenario has to become to even get to the point of being able to ask the question. In practice, if one were somehow able to create the infinite-paged book, it would have progressively more transparent pages and hence look blurry from the right, or if opaque then the smaller pages would shatter under the impact of even a handful of photons, leaving a rubble of page crumbs on the ground. In the latter case, what we would see would be the number of the smallest page that was able to withstand the photonic onslaught.

GrahamH · **#23** by **GrahamH** » Dec 18, 2011 7:34 pm

andrewk wrote:This problem is very entertaining.

Thinking about it some more, the restrictions placed on the types of books that can be created either by the inability to traverse an infinity by successive addition of pages, or by using an infinite-bladed cutting machine to cut into a block, seem unnecessary. What about the following:

We have an infinite-pronged 'seeding machine' that consists of prongs numbered 1 to infinity, arranged in a line, with the distance between the n-th and (n+1)-th prong being 3 * 2^-(n+2). The region z<0 is the ground, made of infinitely divisible fertile 'soil'. The seeding machine is lined up along the positive x axis and inserted into the ground so that the prongs go in and insert seeds at x=1/4, 5/8, 13/16, 29/32 .... Then it is withdrawn so the seeds can grow.

Haven't you simply substituted prong + seed for the original page? There is no last prong nor a last seed from which a last page could grow.

The ever-diminishing separation is the issue, not the number of "pages". A prong, a seed, or a page, to be a distinct object, must have non-zero separation from other objects, but the series given diminishes the separation to nothing, so no thing is the result.

andrewk wrote:Each seed is different. The n-th seed causes a page-like growth (rectangular-prismal in shape) to arise out of the ground, that has y-dimension = z-dimension = 1 and x-dimension = 2^(-n) and has, on the faces pointing towards the positive and negative x directions, a big black number n on a white background.

Each seed tends to indistinguishability from its neighbour as n -> inf.

This set-up delivers a coverless book the same as was created by the cutting machine previously considered, without the weird far-right edge that is not a page. Nor do we need to traverse an infinity by successive addition, as would be the case where each page is successively halved.[/quote]

I think you have exactly the same situation at the far right edge.

andrewk wrote:So what do we see when we look at this book from the point (10, 0.5,0.5)? There is nothing between us and the numbered pages, so we must see something, mustn't we?

The pages facing you are uncountable, so they cannot be numbered. Whatever you might see it can't be a number.

andrewk wrote:So then, what do we see from the vantage point?
We see nothing, which means it looks black - an absence of any light. That is because, for any positive n, no light from page n reaches our eyes. Why is that? Because page (n+1) is in the way and, being completely opaque, blocks any light reaching our eyes from page n.

But, the light source can be at your observation point, shining towards the finite right edge extent of the set of pages. There are then no pages in the way, and the light cannot penetrate into the "Book". Physics aside, I think there must be a page surface. That page surface cannot have a unique serial number and cannot be turned as a single entity.

andrewk wrote:In the latter case, what we would see would be the number of the smallest page that was able to withstand the photonic onslaught.

The pages are opaque and uncountable, so you won't see a number of any page inside the book.

andrewk · **#24** by **andrewk** » Dec 19, 2011 3:14 am

GrahamH wrote:
andrewk wrote:We have an infinite-pronged 'seeding machine' that consists of prongs numbered 1 to infinity, arranged in a line, with the distance between the n-th and (n+1)-th prong being 3 * 2^-(n+2). The region z<0 is the ground, made of infinitely divisible fertile 'soil'. The seeding machine is lined up along the positive x axis and inserted into the ground so that the prongs go in and insert seeds at x=1/4, 5/8, 13/16, 29/32 .... Then it is withdrawn so the seeds can grow.
Haven't you simply substituted prong + seed for the original page? There is no last prong nor a last seed from which a last page could grow.

The difference between this scenario and the one cut from a block is that in this scenario there is no matter ("page substance") at any point with x coordinate = 1.

GrahamH wrote:There is no last prong nor a last seed from which a last page could grow.

That is correct.

GrahamH wrote:The ever-diminishing separation is the issue, not the number of "pages". A prong, a seed, or a page, to be a distinct object, must have non-zero separation from other objects, but the series given diminishes the separation to nothing, so no thing is the result.

The formulation above has no separation between consecutive pages, but it can be readily adapted to become one that has separations between pages, by making the width of the n-th page 2^-(n+1) rather than 2^-n. Then there will be a separation of 1/4 between pages 1 and 2, 1/8 between pages 2 and 3 and 2^-(n+1) between pages n and n+1. No page touches any other. This set of pages will look the same from the right as the one that has no page separations.

GrahamH wrote:I think you have exactly the same situation at the far right edge.

Why would you think that? That implies that there is “page substance” at the point (x,y,z)=(1,0.5,0.5), which is the case with the cut block, but not the case with the pages grown from seeds, as there is no seed at x=1.

GrahamH wrote:The pages facing you are uncountable, so they cannot be numbered. Whatever you might see it can't be a number.

They are not uncountable in the sense that mathematicians understand that word, which is that they cannot be put into one-to-one correspondence with the positive integers. These pages can be put into such correspondence by simplying mapping each page to the number of the prong that seeded it, so they are countable. Infinite does not imply uncountable.

GrahamH wrote:
andrewk wrote:So then, what do we see from the vantage point?
We see nothing, which means it looks black - an absence of any light. That is because, for any positive n, no light from page n reaches our eyes. Why is that? Because page (n+1) is in the way and, being completely opaque, blocks any light reaching our eyes from page n.
But, the light source can be at your observation point, shining towards the finite right edge extent of the set of pages. There are then no pages in the way, and the light cannot penetrate into the "Book".

There is no right edge, any more than the set {(x,y,z):x,y,z are all in (0,1)} has a right edge (interpreting ‘right’ to mean the positive x direction). There is no “page substance” at any point with an x-coordinate of 1. Hence there is no way for light emitted from our vantage point at (10,0.5,0.5) to be reflected back to us off any page, as light being reflected in the positive x ("right") direction from any page will be blocked from reaching us by the page to its immediate right.

GrahamH · **#25** by **GrahamH** » Dec 20, 2011 10:59 am

andrewk wrote:
GrahamH wrote:
andrewk wrote:We have an infinite-pronged 'seeding machine' that consists of prongs numbered 1 to infinity, arranged in a line, with the distance between the n-th and (n+1)-th prong being 3 * 2^-(n+2). The region z<0 is the ground, made of infinitely divisible fertile 'soil'. The seeding machine is lined up along the positive x axis and inserted into the ground so that the prongs go in and insert seeds at x=1/4, 5/8, 13/16, 29/32 .... Then it is withdrawn so the seeds can grow.
Haven't you simply substituted prong + seed for the original page? There is no last prong nor a last seed from which a last page could grow.
The difference between this scenario and the one cut from a block is that in this scenario there is no matter ("page substance") at any point with x coordinate = 1.

Doesn't your seeder have a rightmost seeding limit indistinguishably close to x=1, just as the solid block has a limit at 1? That is not to say it has any discrete object at 1, but that it has a finite extent. As you have it the pages grow from seeds. The position of the page being the position of the seed, and that being the position of the prong. The prongs are equivalent to the pages. I don't see you gain anything with the seeds or growing pages.

Perhaps if the pages grew by division it might seem different.

andrewk wrote:
GrahamH wrote:There is no last prong nor a last seed from which a last page could grow.
That is correct.
GrahamH wrote:The ever-diminishing separation is the issue, not the number of "pages". A prong, a seed, or a page, to be a distinct object, must have non-zero separation from other objects, but the series given diminishes the separation to nothing, so no thing is the result.

The formulation above has no separation between consecutive pages, but it can be readily adapted to become one that has separations between pages, by making the width of the n-th page 2^-(n+1) rather than 2^-n. Then there will be a separation of 1/4 between pages 1 and 2, 1/8 between pages 2 and 3 and 2^-(n+1) between pages n and n+1. No page touches any other. This set of pages will look the same from the right as the one that has no page separations.

True. There will be a surface at the right extent of the finite block or finite row of growing pages, whether or not there are infinitesimal gaps between pages.

andrewk wrote:
GrahamH wrote:I think you have exactly the same situation at the far right edge.
Why would you think that? That implies that there is “page substance” at the point (x,y,z)=(1,0.5,0.5), which is the case with the cut block, but not the case with the pages grown from seeds, as there is no seed at x=1.

There is page substance at a position indistinguishable from x=1, because there was a seed at a position indistinguishable from x=1, because there was a prong at a position indistinguishable from x=1. How is it different to the sliced block?

andrewk wrote:
GrahamH wrote:The pages facing you are uncountable, so they cannot be numbered. Whatever you might see it can't be a number.
They are not uncountable in the sense that mathematicians understand that word, which is that they cannot be put into one-to-one correspondence with the positive integers. These pages can be put into such correspondence by simplying mapping each page to the number of the prong that seeded it, so they are countable. Infinite does not imply uncountable.

Thanks for the correction.
We have an infinte set in ordered sequence and are trying to look at the numbers at the end of the sequence, but the sequence has no end, so you can't see a number.

The tension in the book is the finite extent of the book and the infinite subdivision of that extent into infinitely many pages.

We can access anywhere in the finite extent, just as well from the left or right ends. What we can't do is count pages from the last page. (The same is true of both ends of an infinite book of even page thickness).

andrewk wrote:
GrahamH wrote:
andrewk wrote:So then, what do we see from the vantage point?
We see nothing, which means it looks black - an absence of any light. That is because, for any positive n, no light from page n reaches our eyes. Why is that? Because page (n+1) is in the way and, being completely opaque, blocks any light reaching our eyes from page n.
But, the light source can be at your observation point, shining towards the finite right edge extent of the set of pages. There are then no pages in the way, and the light cannot penetrate into the "Book".
There is no right edge, any more than the set {(x,y,z):x,y,z are all in (0,1)} has a right edge (interpreting ‘right’ to mean the positive x direction). There is no “page substance” at any point with an x-coordinate of 1. Hence there is no way for light emitted from our vantage point at (10,0.5,0.5) to be reflected back to us off any page, as light being reflected in the positive x ("right") direction from any page will be blocked from reaching us by the page to its immediate right.

Again, there is substance at a point indistinguishable from x=1, and no page substance between x=1 and x=10. The right edge may not be resolvable into pages. We may not be able to get into the substance behind the surface, but it doesn't seem there is anything to block light from outside the finite object being reflected back.

To have a black end don't we have to propose some particular property of a page in reflecting light other than the property of page substance? If page substance reflects light we see the surface, if only pages of finite thickness reflect light but all pages block light then the end would appear black, I think.

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