Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com
In this section of the website, the inter-relationship between Platonist beliefs, the misuse of language and the failure to recognize that misuse, and how it applies to the concept of infinity are analyzed. It will be shown that the Platonist notion of an ‘actual’ ‘existing’ infinity leads to contradictions that render such notions logically invalid, and inapplicable to rigorous mathematical analysis.
As long ago as 1638, Galileo warned about ‘how gravely one errs in trying to reason about infinities by using the same attributes that we apply to finites’. He said this in response to the observation that the square numbers (the numbers that result from multiplying a number by itself) is a subset of the set of all natural numbers, but which is also infinite. He concluded that ‘equal, greater and less have no place in the infinite’.
Since the time of Galileo you might expect that the mathematical world, in common with the rest of the modern world, would have moved away from primitive irrational beliefs. But no, Platonists have become even more entrenched in their beliefs in the ‘actual’ ‘existence’ of mathematical things, and the ‘existence’ of an ‘actual’ infinity. (Footnote: For example, Paul Cohen said, ‘For me the essential point is the existence of infinite totalities. The attitude toward infinite sets has traditionally been the great dividing line between mathematicians.’
Paul J. Cohen : Comments on the Foundations of Set Theory, in: Axiomatic Set Theory, ed. D. Scott, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part 1, Amer. Math. Soc., Providence, RI, pp 9-15., p. 10.)
It should hardly be necessary to point out that we cannot simply extend existing concepts that apply to finite sets and apply them to infinite sets without first undertaking a thorough analysis of what one is doing. For example, given any finite set that includes all the positive numbers up to a certain number, there is always one of the elements of the set which is the same as the number of elements in the set. That is, there is a correspondence between a property of that set and one of its elements. For the set of all positive natural numbers, this does not apply. There is no largest element. There is no element of the set that represents the number of elements in that set. There is no correspondence between a property of that set and one of its elements in the same way as there is for finite sets.
It is worth noting that one of the key arguments that Platonists bring to the table in support of their notions is the Indispensability Argument, which argues that since scientific theories use mathematics, then mathematical entities, including infinite entities must actually exist.
So what is infinity? To ask what ‘infinity’ is rather misses the point, because what we actually talk about are things that are infinite, so when we refer to ‘infinite’, we use the term ‘infinite’ as an adjective used to describe something. And while we use the term ‘infinite’ to mean the converse of ‘finite’, i.e., ‘not finite’, we have no word that means the converse of ‘infinity’. So the correct question to ask is: What does ‘infinite’ mean? - and the primary characteristic of something that is infinite is a limitlessness, the lack of a limit of some aspect of whatever we are referring to.
Many people make the error of talking about ‘infinite’ as if it refers to a quantity, whereas in fact it refers to an absence of definite quantity. When someone asserts that one infinity is ‘greater’ than another infinity, they are saying that one absence of definite quantity is greater than another absence of definite quantity.
Platonists believe that such concepts as natural numbers ‘exist’ as ‘actual’ things independently of any human definition of numbers, and similarly that the set of all natural numbers ‘exists’ as an ‘actual’ thing independent of any human definition. Once you have the concept that the set of all natural numbers ‘exists’ as an ‘actual’ thing, then it follows that you also will have the concept that every possible natural number ‘exists’ and is in this set.
Now, numbers are very special in that they can always be set in order; every number can be set into order against all other numbers. And every natural number is defined as being just one greater than another number, and that number in turn is just one greater than another number, and so on. And no matter what natural number we pick, by the definition of a natural number, there will still be only a finite number of these ‘greater by one’ steps back to the number zero.
So, suppose for a moment that there could actually ‘exist’ in some independent ‘actual’ sense, as some sort of ‘reality’, a set of more than a finite quantity of natural numbers. Then since every number has an innate order with respect to every other number, that would mean that there would be at least one number (actually, limitlessly many) for which there are more than a finite number of steps back to the number zero.
But if that was the case then that number would itself be greater than any finite number – which would mean that it was not actually a natural number at all! And that is a contradiction.
This contradiction only arises when there is a Platonist insistence on the ‘existence’ of an ‘actual’ set of natural numbers, independent of human thought. If we simply accept that we have definitions that give rise to limitless series, then we simply say that the set of natural numbers is a set that is given by a certain definition. And with that, there is no problem is stating that there no limit to the size of the set.
For more formal demonstrations of contradictions arising from the Platonist beliefs in the ‘existence’ of ‘actual’ infinite sets, see The Courant & Robbins contradiction, Sums of infinitely many fractions: 1 and Sums of infinitely many fractions: 2.
For demonstrations of how the Platonist belief in infinities results in flawed arguments, see Real numbers and Language, The Diagonal Proof, A list with no Diagonal number, Cantor’s 1874 Proof and The Power Set Proof.
You can also see a formal paper on some of the problems of calculating the measure of some sets that are defined in terms of limitlessness, see On Smith-Volterra-Cantor sets and their measure (PDF).
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Copyright © James R Meyer 2012 - 2018