The missing page

GrahamH · #1 by **GrahamH** » Oct 17, 2011 6:38 pm

Benarete wrote:Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper — one half inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? One fourth inch thick. And the third page of the book, how thick is this third sheet of paper? One eighth inch thick, etc. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now, slowly lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.

Simon Prosser wrote:In his book Infinity: An Essay in Metaphysics José Benardete described a rather unusual book [1964: 236‐7]. The book has an infinite number of pages, all made from infinitely divisible matter. The first page is half a unit thick; page 2 is a quarter unit thick; page 3 is one eighth of a unit thick, and so on to infinity. The book therefore has a thickness of one unit and although it has a first page it has no last page. For simplicity I shall discuss a version of Benardete’s book that has no cover and the pages are not fastened together; it is nothing more than a pile of pages. Because of the way it is constructed, the back of the book has a topologically open surface (henceforth open surface, i.e. it has no outermost layer of points). The front of the book has a topologically closed surface (henceforth closed surface, i.e. it has an outermost later of points) provided the pages themselves have closed surfaces, which I stipulate to be the case. I shall stipulate, further, that each page is perfectly rigid and non‐porous and that no page is transparent, no matter how thin it is.1 Benardete asks the following question: if the book is placed face‐down with the first page at the bottom and one looks at the book from above, what does one see? Since there is no last page, one does not see a page. Benardete concluded that one sees nothing.

Why is there no (infinitely thin) page on the finite top of the stack?

How can 'topologically closed' pages produce a 'topologically open' stack?

home_ · #2 by **home_** » Oct 18, 2011 8:10 am

It's wrong to pose questions about vision when dealing with such abstract objects. You can imagine such topological spaces (correctly defined etc), but that doesn't mean you can see it as if it were existing in physical world.

andrewk · #3 by **andrewk** » Dec 11, 2011 6:58 am

GrahamH wrote:Why is there no (infinitely thin) page on the finite top of the stack?

For the same reason that there is no largest number in the interval [0,1)

GrahamH · #4 by **GrahamH** » Dec 11, 2011 3:20 pm

andrewk wrote:
GrahamH wrote:Why is there no (infinitely thin) page on the finite top of the stack?

For the same reason that there is no largest number in the interval [0,1)

Yes, but isn't that an issue of countability / numbering? Of course you can't number or place a top page (if you start at the bottom), but surely looking at a finite stack you do see a page on its top face.

If we start with a solid finite block and imagine it infinitely sub-divided into "pages" at what point does the top face, that is surely there at the start, disappear?

andrewk · #5 by **andrewk** » Dec 11, 2011 10:59 pm

There are a number of answers to this, depending on exactly what the question is.

1. If we imagine starting with an undivided block and making the cuts one after another, then the question of what you see when you are finished has no answer, because you will never be finished.

2. What if we have a machine that can make all cuts simultaneously? The answer to this depends whether we are dealing with the universe we know. If not, then answer depends on what the rules of that other universe are. If it's this one then the answer is that the cutting process will not make an infinite number of cuts because matter is not infinitely divisible. Once you are down to a sheet of single particle thickness, you cannot divide further. Additionally, once you get beyond a certain smallness, Heisenberg's Uncertainty principle takes over, and you cannot be sure where any of the cuts are.

3. OK then, what if we are in a hypothetical universe in which quantum mechanics doesn't exist, and matter is infinitely divisible, and we have this magic machine that can simultaneously make an infinite number of cuts to a block of height 1, at intervals 1/2, 3/4, 7/8 etc from the base of the block? Let us also assume that each blade of the cutter is numbered with a natural number and, as it cuts, prints its unique number on the page either side of the blade. The block is solid white, internally and externally, before the cuts are made.
The question then is what do we see if we look along the x axis towards the book? To be precise, let's orient the block in the xyz space, occupying the cubic space defined by
{(x,y,z) : (0<=x<=1) & (0<=y<=1) & (0<=z<=1)}
The cutter makes cuts in planes defined by x=1-2^(-n), for n = 1 to infinity, and prints as described.

What do we see if we look at the last page? By this we mean what do we see when we look at the block from a vantage point of say (x=10,y=0.5, z=0.5)?

The answer is that we see a white face of the block, as we can only see printing that is facing in the positive x direction, and such printing only occurs on a 'face' that is to the left (negative x direction) of a cutter blade. But the rightmost face of the original block is never touched by the left face of a cutter blade, so it receives no imprint.

If the block originally had printing on its rightmost face - say a picture of a donkey - then that's what you'll still see when you look from the vantage point.

So there is no ambiguity or uncertainty about what you'd see.

OK then, you might say, what if I turn back the last page, and look at the page before that? What do I see then? The answer to that is that you cannot turn the last page, because there is no last page. The surface you see from your vantage point has a right side, but no left side. It is not a page.

This seems very weird, but that's because we have hypothesised a very weird universe, which specifically:
1- allows the infinite divisibility of matter; and
2- allows the creation, within a finite time, of a cutting machine with an infinite number of blades, infinitely close together.

One should not be surprised at anything strange that happens in such a universe.

GrahamH · #6 by **GrahamH** » Dec 12, 2011 12:29 pm

andrewk wrote:
The cutter makes cuts in planes defined by x=1-2^(-n), for n = 1 to infinity, and prints as described.

What do we see if we look at the last page? By this we mean what do we see when we look at the block from a vantage point of say (x=10,y=0.5, z=0.5)?

The answer is that we see a white face of the block, as we can only see printing that is facing in the positive x direction, and such printing only occurs on a 'face' that is to the left (negative x direction) of a cutter blade. But the rightmost face of the original block is never touched by the left face of a cutter blade, so it receives no imprint.

If the block originally had printing on its rightmost face - say a picture of a donkey - then that's what you'll still see when you look from the vantage point.

So there is no ambiguity or uncertainty about what you'd see.

OK then, you might say, what if I turn back the last page, and look at the page before that? What do I see then? The answer to that is that you cannot turn the last page, because there is no last page. The surface you see from your vantage point has a right side, but no left side. It is not a page.

Thanks for your thoughts on this.

What makes the "last page" any different to any other? Given the simultaneous infinite slicing described there would presumably be no first page either. The outer pages are surely interchangeable in every respect except the arbitrary choice of which to number Page 1. If we are interested in the topology rather than the numbering there can be no difference between the pages at ether end, can there?

We could not turn to any page in the book, if the first page is not a page, but could we open the book at any arbitrary page and read it?

Any sub-section of the book is itself an infinite book.

Wherever you might open the book you would have two infinite books, with a last page facing a first page. If these are not pages what then?

Is it necessary to be countable to exist?

VazScep · #7 by **VazScep** » Dec 12, 2011 12:59 pm

The only maths here is the following observation: the union of

{ [(2^n - 1)/2^n, (2^(n+1) - 1)/2^(n+1)] : n in {0,1,2,...}}

is [0,1)

Take it to the physics forum. You'll probably be told that the only physics in the question is that the book as described probably couldn't exist.

ED209 · #8 by **ED209** » Dec 12, 2011 1:09 pm

Why would the outer pages be interchangeable?

You can indeed turn to any page n counting up from page 1 but even opening the book at an arbitrary page does not give two infinite books - the first part from page 1 to the arbitrary split always consists of a finite (and trivially countable) set of pages that does not cause any problems (even in our universe). In terms of counting x=1-2^(-n) over the interval [0,1), there is no problem counting from 0 and going finitely far.

GrahamH · #9 by **GrahamH** » Dec 12, 2011 3:19 pm

ED209 wrote:Why would the outer pages be interchangeable?

You can indeed turn to any page n counting up from page 1 but even opening the book at an arbitrary page does not give two infinite books - the first part from page 1 to the arbitrary split always consists of a finite (and trivially countable) set of pages that does not cause any problems (even in our universe). In terms of counting x=1-2^(-n) over the interval [0,1), there is no problem counting from 0 and going finitely far.

Suppose I open the book about half way (not counting any pages). Before the opening there are approx infinity / 2 pages, and similarly after the opening. These each have infinitely many pages.

Opening at an arbitrary distance through the finite thickness of the book does not entail counting pages. How could it get me to an integer-numbered page?

What do you think makes the outer pages non-interchangeable? The only things that distinguish them seem to be which page is on top / bottom, and whether you decide to start numbering pages from one end or the other, which is surely irrelevant to the topology of the book.

GrahamH · **#10** by **GrahamH** » Dec 12, 2011 3:28 pm

VazScep wrote:The only maths here is the following observation: the union of

{ [(2^n - 1)/2^n, (2^(n+1) - 1)/2^(n+1)] : n in {0,1,2,...}}

is [0,1)

Take it to the physics forum. You'll probably be told that the only physics in the question is that the book as described probably couldn't exist.

Nobody is supposing that such a thing could physically exist, or be made by any sort of process.

The issue came up in an argument for god topic as a supposed proof that actual infinities could not, in principle, exit. The problem is not of counting pages, but of a contradiction. A finite-dimensioned book with infinitely many pages that cannot, so it is claimed, have a last page. It can't have a numbered last page, but I don't see why it can't have a page at each end. If it can have a first page then surely it can have a last page. Strangely, having a first page doesn't seem contentious.

Is Physics forum a better place for discussing the conceptualising of infinity than the maths forum?

VazScep · **#11** by **VazScep** » Dec 12, 2011 5:00 pm

GrahamH wrote:The issue came up in an argument for god topic as a supposed proof that actual infinities could not, in principle, exit. The problem is not of counting pages, but of a contradiction. A finite-dimensioned book with infinitely many pages that cannot, so it is claimed, have a last page. It can't have a numbered last page, but I don't see why it can't have a page at each end. If it can have a first page then surely it can have a last page. Strangely, having a first page doesn't seem contentious.

Is Physics forum a better place for discussing the conceptualising of infinity than the maths forum?

The only maths is in the observation I gave. I'm not sure about the physics. But if this has come up in a discussion about god and actual infinities, it belongs on the wibble forum.

GrahamH · **#12** by **GrahamH** » Dec 12, 2011 5:40 pm

VazScep wrote:
GrahamH wrote:The issue came up in an argument for god topic as a supposed proof that actual infinities could not, in principle, exit. The problem is not of counting pages, but of a contradiction. A finite-dimensioned book with infinitely many pages that cannot, so it is claimed, have a last page. It can't have a numbered last page, but I don't see why it can't have a page at each end. If it can have a first page then surely it can have a last page. Strangely, having a first page doesn't seem contentious.

Is Physics forum a better place for discussing the conceptualising of infinity than the maths forum?
The only maths is in the observation I gave. I'm not sure about the physics. But if this has come up in a discussion about god and actual infinities, it belongs on the wibble forum.

The god end of things clearly is wibble.
The mathematical concept I don't understand is related to the term "topologically open". It was claimed that the last page was "topologically open", which I think means something along the lines of ...

Andrewk wrote:OK then, you might say, what if I turn back the last page, and look at the page before that? What do I see then? The answer to that is that you cannot turn the last page, because there is no last page. The surface you see from your vantage point has a right side, but no left side. It is not a page.

Is that wibble?

VazScep · **#13** by **VazScep** » Dec 12, 2011 7:40 pm

GrahamH wrote:The mathematical concept I don't understand is related to the term "topologically open". It was claimed that the last page was "topologically open", which I think means something along the lines of ...
Andrewk wrote:OK then, you might say, what if I turn back the last page, and look at the page before that? What do I see then? The answer to that is that you cannot turn the last page, because there is no last page. The surface you see from your vantage point has a right side, but no left side. It is not a page.

I'm not clear on how the quote in the OP is using "topologically open/closed" either. It can't be saying that the last page is topologically open, because there is no last page. Instead, it talks about the front and back of the book.

As a maths puzzle, the whole thing can be reduced to one-dimension, where we look at the book "edge-on", an ignore the height and width of each page. You can then view each page as the following set of intervals:

[0,1/2], [1/2,3/4], [3/4, 7/8], [7/8, 15/16], ..., [(2^n-1) / 2^n, (2^(n+1) - 1) / 2^(n+1)], ...

What is the "front of the book"? Perhaps it is just any proper prefix of the above sequence, such as (say), the first 10 intervals/pages. What is the "back of the book"? Perhaps it is just any proper suffix of the above sequence, such as (say), everything but the first 10 intervals/pages. Now, if you take the union of any front of the book, you always get a closed interval (yes, topologically closed). And if you take the union of any back of the book, you always get a half-open interval (no, not topologically open).

There is no last page, because the sequence above is infinite. If you want, you could add an infinitely thin last page (the degenerate interval {1}), and call this the ωth page (the infinitieth page). However, you'd have the same problem, only now there is no penultimate page.

Is that wibble?

Yes. It's assuming that the notion of book, the notion of pages of a book, the notion of a front and back of a book, and the idea of how we see books are so conceptually secure that they can be used in physically impossible thought experiments to threaten the conceptual security of the infinite. You only get that kind of confusion among metaphysicians.

Shrunk · **#14** by **Shrunk** » Dec 12, 2011 8:31 pm

andrewk wrote:There are a number of answers to this, depending on exactly what the question is.

1. If we imagine starting with an undivided block and making the cuts one after another, then the question of what you see when you are finished has no answer, because you will never be finished.

2. What if we have a machine that can make all cuts simultaneously? The answer to this depends whether we are dealing with the universe we know. If not, then answer depends on what the rules of that other universe are. If it's this one then the answer is that the cutting process will not make an infinite number of cuts because matter is not infinitely divisible. Once you are down to a sheet of single particle thickness, you cannot divide further. Additionally, once you get beyond a certain smallness, Heisenberg's Uncertainty principle takes over, and you cannot be sure where any of the cuts are.

3. OK then, what if we are in a hypothetical universe in which quantum mechanics doesn't exist, and matter is infinitely divisible, and we have this magic machine that can simultaneously make an infinite number of cuts to a block of height 1, at intervals 1/2, 3/4, 7/8 etc from the base of the block? Let us also assume that each blade of the cutter is numbered with a natural number and, as it cuts, prints its unique number on the page either side of the blade. The block is solid white, internally and externally, before the cuts are made.
The question then is what do we see if we look along the x axis towards the book? To be precise, let's orient the block in the xyz space, occupying the cubic space defined by
{(x,y,z) : (0<=x<=1) & (0<=y<=1) & (0<=z<=1)}
The cutter makes cuts in planes defined by x=1-2^(-n), for n = 1 to infinity, and prints as described.

What do we see if we look at the last page? By this we mean what do we see when we look at the block from a vantage point of say (x=10,y=0.5, z=0.5)?

The answer is that we see a white face of the block, as we can only see printing that is facing in the positive x direction, and such printing only occurs on a 'face' that is to the left (negative x direction) of a cutter blade. But the rightmost face of the original block is never touched by the left face of a cutter blade, so it receives no imprint.

If the block originally had printing on its rightmost face - say a picture of a donkey - then that's what you'll still see when you look from the vantage point.

So there is no ambiguity or uncertainty about what you'd see.

OK then, you might say, what if I turn back the last page, and look at the page before that? What do I see then? The answer to that is that you cannot turn the last page, because there is no last page. The surface you see from your vantage point has a right side, but no left side. It is not a page.

This seems very weird, but that's because we have hypothesised a very weird universe, which specifically:
1- allows the infinite divisibility of matter; and
2- allows the creation, within a finite time, of a cutting machine with an infinite number of blades, infinitely close together.

One should not be surprised at anything strange that happens in such a universe.

I actually offered a very similar solution to the problem in another thread here. Not as precisely worded as yours, but it's good to know I'm not alone in thinking on these lines. I also tried to address the whole question of potential vs. actual infinite, though I'm not sure how successfully:

Again, here's how I'm looking at it: I could take my block made out of a miraculous substance that can be cut into ever increasing slices without any limit to its thinness, with my miraculous cutting tool that is able to separate the material into slices without removing any of the material, and produce an infinite number of slices. If I then perfectly reassemble those slices back into the shape of the block with no space between each slice, the block will look like it had never been sliced before. But what happened to the slices? Don't they all still exist? If not, where did they go? If some of them are missing, then the block should no longer be an inch thick. But let me check: No, it's still exactly an inch thick, so all the slices are there.

So if all the slices are there and the block is exactly in the same physical state that is was before it was sliced, then the slices must all already have been there before it was sliced. i.e the physical material of each individual slice is present, and the fact that they are intimately joined to all the other slices does not change the fact of their existence.

Now, it might be objected that this does not represent an "actual infinite" in the sense Craig means, that it simply represents a "potential infinite" in the way that an inch can be potentially divided into an infinite number of fractions, but remains finite itself. However, if that is the case, then the same objection can be made to Benardete's book. Because there is no reason his book could not be produced by the same process I describe: By successive slicing of a solid block of whatever his pages are made of (it cannot be paper, because paper cannot be sliced to infintie thinness) rather than adding pages one by one. So if my argument is invalid because it does not involve an actual infinite, then neither is Benardete's argument valid.

I think your last point is the clincher. The paradoxes that theists use to attempt to prove the impossiblity of an actual infinite (e.g. Hilton's hotel) are based on our experience of time as finite beings. Just because an actual infinite seems weird and incomprehensible to us does not mean it cannot exist. Just the we cannot conceptually grasp it.

VazScep · **#15** by **VazScep** » Dec 12, 2011 8:56 pm

Shrunk wrote:I think your last point is the clincher. The paradoxes that theists use to attempt to prove the impossiblity of an actual infinite (e.g. Hilton's hotel) are based on our experience of time as finite beings.

I made that mistake once, insisting to the receptionist that the hotel could accommodate me even if all rooms were booked out. But I assure you, Hilton Hotels only have finite numbers of rooms.

GrahamH · **#16** by **GrahamH** » Dec 12, 2011 9:10 pm

VazScep wrote:
GrahamH wrote:The mathematical concept I don't understand is related to the term "topologically open". It was claimed that the last page was "topologically open", which I think means something along the lines of ...
Andrewk wrote:OK then, you might say, what if I turn back the last page, and look at the page before that? What do I see then? The answer to that is that you cannot turn the last page, because there is no last page. The surface you see from your vantage point has a right side, but no left side. It is not a page.
I'm not clear on how the quote in the OP is using "topologically open/closed" either. It can't be saying that the last page is topologically open, because there is no last page. Instead, it talks about the front and back of the book.

As a maths puzzle, the whole thing can be reduced to one-dimension, where we look at the book "edge-on", an ignore the height and width of each page. You can then view each page as the following set of intervals:

[0,1/2], [1/2,3/4], [3/4, 7/8], [7/8, 15/16], ..., [(2^n-1) / 2^n, (2^(n+1) - 1) / 2^(n+1)], ...

What is the "front of the book"? Perhaps it is just any proper prefix of the above sequence, such as (say), the first 10 intervals/pages. What is the "back of the book"? Perhaps it is just any proper suffix of the above sequence, such as (say), everything but the first 10 intervals/pages. Now, if you take the union of any front of the book, you always get a closed interval (yes, topologically closed). And if you take the union of any back of the book, you always get a half-open interval (no, not topologically open).

There is no last page, because the sequence above is infinite. If you want, you could add an infinitely thin last page (the degenerate interval {1}), and call this the ωth page (the infinitieth page). However, you'd have the same problem, only now there is no penultimate page.

Is that wibble?
Yes. It's assuming that the notion of book, the notion of pages of a book, the notion of a front and back of a book, and the idea of how we see books are so conceptually secure that they can be used in physically impossible thought experiments to threaten the conceptual security of the infinite. You only get that kind of confusion among metaphysicians.

The book is analogous to the set of rational numbers [0,1] .
The sequence has identifiable end points and the interval, and any sub-interval, is infinitely divisible.
The first page is 0, the last page is 1. A page in the middle of the book is page 1/2.
If the book is split in the middle we get two infinite books - [0,1/2] and [1/2,1]

You can just as easily start counting pages from front or back (start at 0 and count up or start as 1 and count down), so surely first and last page are equivalent.

GrahamH · **#17** by **GrahamH** » Dec 12, 2011 9:20 pm

Just for reference, the original wibble was here:
http://www.rationalskepticism.org/post1 ... l#p1031983

VazScep · **#18** by **VazScep** » Dec 12, 2011 9:42 pm

GrahamH wrote:The book is analogous to the set of rational numbers [0,1] .

If so, they are not even well-ordered. There is no second page. I mean, what's the smallest fraction after 0?

The original thought experiment defines a sequence of pages.

GrahamH · **#19** by **GrahamH** » Dec 12, 2011 9:57 pm

VazScep wrote:
GrahamH wrote:The book is analogous to the set of rational numbers [0,1] .
If so, they are not even well-ordered. There is no second page. I mean, what's the smallest fraction after 0?

The original thought experiment defines a sequence of pages.

I just re-read the original description and see I made a significant error:

Here is a book lying on the table. Open it. Look at the first page. Measure its thickness. It is very thick
indeed for a single sheet of paper – 1/2 inch thick. Now turn to the second page of the book. How thick
is this second sheet of paper? 1/4 inch thick. And the third page of the book, how thick is this third sheet
of paper? 1/8 inch thick, &c. ad infinitum.

This is the crucial difference between the first and last page, and why they get to "there is no last page".

That said, I don't see why the thought experiment could not use equal pages infinitely divided as the set of rational numbers in a finite interval.

The problem as stated is an issue of traversing an infinite set from a finite starting point. That has little relevance to a concept such as a temporally infinite universe. It doesn't seem to establish the impossibility of the existence of actual infinities.

This is off track into wibble, so I'll leave it there.

Many thanks form your input on the less wibbly aspects.

andrewk · **#20** by **andrewk** » Dec 12, 2011 10:15 pm

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. If it is manufactured by the cutting machine Graham suggested, it will have a right 'face' but that face is not the right face of a page. Every page has a unique natural number. If you grab a handful of pages on the left end of the book and turn them back, you will have turned a finite number of pages. Conversely, if you grab a handful of pages on the right end of the book and turn them back, you will have turned an infinite number of pages.

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).

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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