Imagine a point on a line. The point has no dimensions, in fact no physical existence. How many such points lie on that line? The answer must be “an infinite number”. The line, however, is finite and is one dimensional even though it needs another two dimensions in order to be seen…but that’s another story. Imagine, now, a square in two dimensions whose sides are made up of four lines exactly the same as the ones above, though arranged so the there is a right angle joining each of them to form the square. How many one-dimensional lines can be placed in the square – say horizontally. The answer, again, must be “an infinite number”. One could argue that the number of lines contained within the square is of the same size as the number of points within the line. But what we can see through this thought experiment is that the number of points in the square must be greater than the number of points in the line – arguably the number will always be the square of that number. Let’s go one dimension further and form similar squares into a cube of three dimensions. How many squares of two dimensions can be placed, say horizontally, into the cube, The answer again must be “an infinite number”. The number of points in the cube must be the cube of the number of points in the line. Some will say that the value of infinity is always the same even though Cantor showed to most people’s satisfaction that it isn’t.

I was prompted to write about this subject by a video I watched recently – with great interest and admiration, by the way – about certain mathematical topics and including a small section on infinity. I will not name the people concerned because they are not the only ones who say the same thing and I am in no way critical of their work on pure mathematics. But when it comes to logic there seems to be a problem to which I will refer below.

What was hypothesised is that an infinite amount of $50 banknotes would have the same value as an infinite amount of $100 banknotes. Whilst that may be true if one regards truth as existing outside of human logic, it makes no sense in reality. The comment I made at the time on the YouTube channel I was watching was that infinity, when applied in this way to physical objects, has no meaning. Let me explain why via thought experimentation. We assume that the bills are of the same size and weight throughout.

- The first problem is to assemble an infinity of banknotes whether they be of $50 or $100 within a finite space in our finite world. If we regard them as already existing, they would have to be stored in every available space and would be so invasive as to render life unliveable. If an infinite number of $50 banknotes were assembled first there could be no space available for the $100s, by definition. So, the hypothesis cannot be true under this scenario. In other words, there is no infinity of physical objects which can exist within a finite space. Moreover, if the space in which we were trying to store an infinite number of physical objects was infinite then the task of storing them could never be finished. It’s worth noticing at this point that we are also introducing the notion of everlasting time – a subject for another piece, perhaps. If both types of banknotes were “stored” at the same time and at the same rate, then there must be the same number of bills of each type at any time – until the end of time.
- Let’s suppose, however, that we were able to divide space into two parts by a force field able to keep separate the two types of banknotes. If there were a machine on either side of the divide producing the notes at the same rate then there would always be twice as much “value” on one side of the divide as on the other, however long we left the machines running. To have different speeds of production on either side renders the hypothesis completely meaningless.
- A third possibility is to create two infinite tubes within our finite space, yet somehow extending beyond, within which we could “load” an infinite number of each banknote. Again, the same principle must apply: if you load at the same rate, then at any moment in time the tube with the $100 banknotes is going to have twice as much value as that holding the $50 banknotes.

Now, what is clear is that it is a purely imaginary world wherein we can create multiple piles of an infinite number of physical objects which only change by the value attributed to them. One might say, if to say so had any value, that the piles might have the same value – but that cannot be applied to the real world because there can be no measurable value of an infinity itself.

In a similar vein, I once heard a well-known personality make the usual mistake of saying that if an infinite number of monkeys armed with an infinite number of typewriters carried on typing for an infinite length of time, they would ultimately produce all of Shakespeare’s works. But there can never be an infinite number of monkeys nor an infinite number of typewriters, nor an infinite supply of paper and ink. More importantly, even if there were, there is nothing to stop one or all of the monkeys typing the same letter three or four times consecutively on multiple pages which would immediately invalidate the hypothesis. Unless, of course, some expert Shakespearian could show me that Shakespeare’s entire works all contain a random number of words with four consecutive letters being the same. Indeed, I would maintain that there is a much greater probability of these monkeys producing complete gibberish over and over again prior to them producing anything at all Shakespearian before their death or the destruction of the typewriters.

I have also heard intelligent people say that parallel lines meet at infinity. Let us place two parallel lines on graph paper – one at y = 1 and one at y = 2. For those two lines to meet we would have to accept that as they approach infinity, they get closer and closer together; that is that 1 starts to equal 2. But that movement towards equality must, by definition, start in finite space which undermines all mathematical logic. Similarly, on our graph paper we can draw two sloping lines: one at y = x and one at y = 2x. The longer those lines are, the more 2x will diverge from x. It is not reasonable to expect us to believe that if they were of infinite length they would converge. In mathematics we accept that it is convenient to treat 2 times infinity as being the same as 1 times infinity. That does not mean that we can give a value to infinity nor that they have the same value in the finite world.

We know from the work of the mathematician Cantor that there are dramatically different sizes of infinity and parts of his work are relatively easy to demonstrate. It is worth noting, however, that Cantor was described as a madman by some of his peers.

I guess the conclusion of all this is quite simple. One should never use the word “infinity” in a casual manner without accepting the consequences of its use, never attribute miraculous qualities to real life objects and never use the word “infinity” as an excuse for omitting to state the assumptions on which any hypothesis is based.

If you have read this far and do not agree with anything that I have said, please be kind enough to add a comment to explain why you disagree.

Danny Barrs, West Sussex, 10 April, 2023