Part 5
We have to consider that if we divide or multiply infinity by anything, or if we add anything to it, or subtract anything from it, it still remains infinite. This line of reasoning may seem to militate against anything I have said about the division of infinity leading to the production of finite parts. However, all it really does is establish that infinity cannot be subjected to mathematical processes. Of course, it can be argued that mathematical infinities are quite at home in mathematical processes, but I would contend that this just serves to underline the profound difference between mathematical infinities and physical infinity.
We need to look at the idea of performing any mathematical functions on a physical infinity. If it is truly infinite it is the sum total of all that exists. We cannot add anything to it, because there is nothing to add. We cannot subtract anything from it, because there is nowhere outside it that we could take anything. To talk of multiplying or dividing such an infinity is simply a conceptual exercise that has no real meaning.
Another question presents itself here: Outside of mathematics, does it make sense to talk of an infinity of any specific thing? Elsewhere I have reasoned that it makes no sense to talk of an infinite number of finite objects. What about infinite space, infinite time, infinite matter or infinite energy? It might seem reasonable to argue that it is not possible to have any one without all the others. For example, it would not be possible to have infinite matter without infinite energy (because E=mc^2), then you would need infinite space in which to accommodate it, and infinite time in which it would have to exist, since it must always have been. Perhaps Galileo was right, perhaps our finite minds are poorly equipped to grasp the concepts of the infinite. However, this should not stop us from trying.
Because Hilbert’s Hotel is a firm favourite with those who seek to gain a hold on infinity, or just to perform mathematical “tricks” with infinity, it needs a brief mention. Hilbert’s Hotel is a hypothetical hotel with infinitely many rooms, all of which are occupied; obviously by infinitely many guests. Another guest arrives looking for a room. A typical example of the way in which this guest is accommodated is found at: http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel.
“Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.”
This is little more than a verbal “conjuring trick”, if n = infinity, Hilbert’s trick works because we never have to consider what happens to the nth guest who is moved out of the nth room. Obviously there cannot be an nth room containing an nth guest, because we can never count to infinity, nor can there be a room beyond the nth room, so the whole thing is nonsense. If there were a last room into which the last guest could be moved, the new guest could be put in that room and the infinite shuffling would be totally unnecessary. It might be argued that this is a thought experiment, so it does not need to be practically possible. Even if one accepts Terry Pratchett’s (1989) definition of a thought experiment, that it is: “One that you can’t do, and which won’t work”, it has to bear some relevance to the real world. Hilbert’s Hotel falls far short of that. By what lunacy would anyone suggest that an infinite hotel could fit into a finite universe? What is an infinite number of people? If you already had an infinite number of people in the hotel, where would another one come from? (Forget mathematics; think about a real situation).
(BTW, Pratchett’s explanation, in this same book, of the “Schrödinger's cat” thought experiment is a “must read” for anyone with an interest in quantum physics, and a sense of humour).
The concept of infinity could never be kept permanently out of the realms of mathematics, and rightly so. I often think it is a shame that they didn’t change its name when they turned it into something that could be used in calculations. As we saw when considering renormalisation, anything other than a “doctored” infinity makes nonsense of mathematical calculations.
Two of the eminent mathematicians who, before Cantor, attempted to tame the infinite were Galileo and Leibniz. A good comparison of their respective approaches can be found at:
(
http://steiner.math.nthu.edu.tw/disk5/js/history/infinity.pdf)
so there is no need for extensive repetition here. However, a few relevant extracts may help to clarify certain points. Galileo was quite clear that one could not progress from finite to infinite. “The transition cannot be realized step by step by continued divisions of the divisible, because such successive divisions do not lead to a last division.”
Regarding the nature of this transition Galileo held that: “When a terminated quantity proceeds to the infinite, ‘it meets with an infinite difference, what is more, with a vast alteration and change of character’. And indeed: The corresponding notions result from each other by logical negations: finite becomes infinite, divisible becomes indivisible, quanta become non-quanta”.
Significantly, Leibniz considered “the identification of an infinite number – on condition that there is such a number – with zero, and not, as Galileo had done, with unity”. Identification of infinity with unity would “assume the existence of an actually given infinite number”, which would be absurd.
A major advance arising out of Leibniz’s work was that he facilitated inclusion of indivisibles in mathematics. The way he did that is illustrated in his definition of straight lines as rectangles for the purpose of calculations of space. He lays “….the foundations of the method of indivisibles in the soundest way possible, and it supplies the proof of the method of indivisibles which enables us, ….. to find the areas of spaces by means of ‘sums of lines’. He explained these sums as sums of rectangles with equal breadths of indefinite smallness”. Obviously this is a sleight of hand trick which, while it works perfectly well in its appropriate context, did nothing to change Leibniz’s “lifelong rejection of the actual infinite in mathematics”.
A significant passage, relating to Leibniz’s work, from the article linked above is the following. “The demonstrations of the forty- five theorems that follow are based on this concept of infinitely small and infinite quantities, that is, quantities which are, according to his definition, positive, but smaller than any given quantity or larger than any given quantity. The decisive aspect is that they are quantities, fictive ones certainly, because they were introduced by a fiction, but quantities nevertheless. It does not matter whether they appear in nature or not, because they allow abbreviations for speaking, for thinking, for discovering, and for proving.”
It is not my intention to attempt to diminish the importance of Leibniz’s work. I have neither the mathematical skill, nor the wish to do that. I seek simply to show that infinitely small and infinitely large “quantities” are mathematical tools, and that Leibniz was well aware of that fact. His own assertion that: “We need not take here the infinite in the strict sense of the word” is probably all the testimony we need in that regard.
The way in which circuitous arguments can develop when trying to merge infinity and mathematics is illustrated in this quote from the linked article: “Hence a contradiction with the Euclidean axiom on the part and on the whole seems to emerge, because a part equals the whole. This result, however, was absurd in Leibniz’s opinion; it came about only because the consideration proceeds from a last abscissa which does not exist in reality, or in other words, because the indivisible, the point, is identified with an infinitely small quantity, even though both notions differ basically from each other”.
One thing emerges with certainty; infinity “in the strict sense of the word”, cannot be manipulated mathematically. This leads to a startling realisation. No part of infinity can actually be distinguished from any other. This is tantamount to saying that no part is different from any other. In other words; every part of infinity – if there could be said to be parts – would be the entirety.
The part must be the whole.
1989. Pratchett Terry & Jolliffe Gray. The Unadulterated Cat.
