Alain asked:

Does the infinity exist?

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"When we say anything is infinite, we signify only that we are not able to conceive the ends and
bounds of the thing named, having no conception of the thing, but of our own inability."

Thomas Hobbes, Leviathanch.III

If there is one thing that seems to characterise the philosopher's discussions of infinity, it is that the
discussion, like its subject, seems to have gone on and on forever without even the remotest
suggestion of a possible conclusion. Now, I have no intention of re-investigating the weird world of
'transfinite numbers' or debating the merits of 'infinity orders'. What you want to know is whether
infinity is actually a valid concept.

What we do know is that we cannot, by definition, place a limit on infinity, which means that we
struggle to offer up any precise word-image of the term. But that doesn't seem to have stopped even
the most distinguished thinkers from invoking it. Much as I might wish to avoid seeming impious, I
think we have to ask what sort of reasoning led Karl Popper to state that "our ignorance must
necessarily be infinite"? Why was Aristotle so certain that "infinitely various are the incidents in one
man's life"? What made Alfred Whitehead sure that "we are surrounded by possibilities that are
infinite" or led Leibniz to assert that God "selects from an infinite number of possibilities"? And so on.
Has any great thought gone into the invocation of this ultimate pseudo-scientific substitute for 'god'? It
seems not.

But who can blame them? When we're confronted with something which we know carries a range of
possibilities, and we immediately see that the range is beyond ready calculation, it is all too easy to
just throw in the word 'infinite'. What is all too tricky is to take one of those obviously infinite sets and
try to rigorously check whether it does indeed present options without limit. If we're going to
investigate this we'll need to find something, which is self-evidently infinite, which is clearly defined
and which is known and understood by almost everyone. There are many possibilities, but I will
choose to have a look at pictures. It is utterly obvious that the options available in constructing a
picture, or a painting, or a photograph or pattern or piece of graphics is limitless. The number of
possible pictures is infinite indeed. Or is it?

So, how many possible pictures are there? Let us begin with perhaps the simplest form of picture
known to most people today; those austere little icons that litter our computer screens with their
crude, and now very well-recognised images of a document, a printer a magnifying glass or whatever.
It is clear that there can only be a limited number of such icons, because they are constructed within
a bounded system — the abilities of the computer screen. In fact, the system is severely bounded in
that, conventionally, such icons are constructed on a grid just sixteen pixels square. If we further
restrict the possibilities by allowing the use of only two colours, black and white, how many possible
different icons could be devised? What would your guess be? A few hundred? A few thousand?
Perhaps a million or so? In fact the total number of possibilities can be calculated from the number of
options per pixel raised to the power of the total number of pixels, in this case 2 to the power of 256.
This is such an extraordinarily large number that it is probably worth giving in full:

"115,792,089,237,316,195,423,570,985,008,687,907,853,

269,984,665,640,564,039,457,584,007,913,129,639,936"

Is this not remarkable? In just a tiny little grid, with so few options, the total number of possibilities is a
number so vast that it is difficult to find some understandable way to express it. One might point out
that the length of the list of possible sixteen-grid black-and-white images is roughly equivalent in size
to 112 times the age of the universe in seconds multiplied by the volume of the universe in cubic
millimetres, though that barely makes it clearer. Within that list is enumerated, not only every possible
computer icon, but also everything which can possibly be expressed using the same system. So there
is every letter of every alphabet, whether known, or lost, or yet unmade. There is every character
which can be drawn with less than nine vertical or horizontal strokes, including virtually all in every
pictographic script.

But where has this got us? The magnitude of the answer may be surprising, but it is not surprising
that there is a clearly definable answer because we are dealing with a clearly bounded system.
Surely, this cannot be true of pictures in general? No matter what picture is made, what photograph
taken, what pattern formed or what arrangement of lines or colours, no matter how abstract, there is
surely always the possibility that that picture itself could be subtly altered in colour, or viewed from a
different angle, making a new picture. The moustache, it seems, can be drawn onto the Mona Lisa in
an infinite number of ways, because the system within which it is drawn is not bounded by pixels or
by anything else. This is not in fact the case.

A picture is, by definition, something, which can be seen by a human eye. It is very probably the case
that dogs, or crayfish, see images of some sort. But 'picture' is the name we give to such images,
and, as far as we are aware, neither dogs, nor crayfish nor any creature other than humans ever
gives names to anything. "A picture", as Wittgenstein put it (admittedly in a slightly different context)
"is a fact". So, let us try to see if the total number of those facts, a purely human thing, has an upper
limit.

But the human eye doesn't resolve an image as definable pixels. Or does it? Visual information is
collected through some 120 million rod and cone cells in each eye. But these are unevenly distributed
and the information from them processed by the brain in such a way that exceptional differences or
exceptional similarities between signals carry more weight. However, no neurology is actually needed
to find out what the resolution of the human eye is — a simple home experiment with a large grid of
coloured dots shows that we can resolve, at very best, about 3,000,000 pixels and about 1,000,000
colours and shades. So the total number of possible different images which could be seen is 1000000
to the power of 3000000. This is a staggeringly large number, but it is most certainly not infinite. In
fact, you could say that it is almost infinitely less than infinity.

That figure of 1M to the power of 3M is the length beyond which the list of all the paintings ever made,
and yet to be made, in every possible variety of combinations and colour varieties, can never go. It is
the upper limit of the length of the list of all possible wallpaper designs, of every possible
representation of every fractal. It has every frame from every movie, with subtitles in every language.
It has every visible page of text, including every page, past or as yet un-thought of, on this website
and the entire text of the as-yet unwritten e-mail demonstrating that this thesis is wrong, and that in
every individual form of handwriting and in every colour and every language and every typeface. It is
the list of all your, and everyone else's, holiday photographs. It is the size of the list of all possible
sights.

But what if one of those pictures had some element in it altered by less than the resolution of the
eye? It would seem that this would constitute a different picture and the validity of the assumption that
there are indeed an infinite number of possible pictures could be restored. However, if the alteration,
say adding a little dot, could not be resolved by a human eye, which is to say it is smaller than any
pixel, then it would not, as far as a human is concerned, be a different picture.

Now, it may well be the case that an infinite number of images could be constructed, though I, nor
any human, cannot know for certain, for that imperceptible alteration to one of them surely cannot be
said in any meaningful way to exist for us? This is the old, old argument of realism versus idealism,
yet with, I think for the first time, some numbers to suggest where the boundary between the two
might lie.

I've only been using pictures as an example, because it seems so obvious that, whatever the picture
is, you could always make a different one, so that the number of possible pictures is infinite. I hope
I've shown you that this isn't true. There is an actual top limit to the number of possible pictures. And
so there is an actual, numerable, top limit to everything — the 'top of tops' being the number of
possible arrangements of all possible entities in the universe. That is so extraordinary a number that I
cannot even begin to think what it might be like, but, as long as you believe that the universe is
physically finite, then there must be an actual top limit of possibilities as well.

So, does the infinite exist? I think there's no objection to the use of 'infinity' as an abstract
mathematical construct. But when it is taken out of the realm of pure numbers and is taken to
represent things in the real universe then the simple answer is NO.

Glyn Hughes

If you think it does, then it does. There is a concept which you might have encountered, 'the space of
all possible x'. This is an infinite space, the same space where your thought resides. But beware of
the error lying in wait for anyone who takes these things literally: this 'space' is not real, but a
conceptual space. There is no possible proof, because the only creatures known to us who are
bothered thinking up such conundrums are humans. We do have this (seemingly ineradicable)
tendency to believe that the only reality there can be is physicalreality. This belief inevitably gets us
into hot water when we then wish to rationally discuss what 'mental' reality is. For one, it is not
physical. Is it not 'real', then? Others, following the same train of argument propose that mental reality
is the 'only' reality. Well, which is what?

You can see the problem, can't you?

Now I'm not sure if with your question you also meant to ask, is the universe infinite? is God truly and
absolutely omnipotent? But if you did, I would answer this in the negative. The whole idea of an
infinite volume is based on our (very limited) understanding of extension. We only have experience of
finite space, of a finite life span, of finite powers, so in such theoretical situations where we suppose
those limitations to be transcended, we also suppose that we are talking about physical reality. Hence
the conflict (as above).

Be aware that others might argue for the opposite.

Jürgen Lawrenz

Sydney