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Sums of infinitely many fractions: 2

A favorite notion held dear by Platonists is the notion of an ‘infinite decimal expansion’. For a fraction, the decimal representation of the fraction is simply a fraction with 10, 100, 1000, 10000, etc as the denominator (the denominator is the number under the line in a fraction), so that we can have, for example 9⁄ 10, 45⁄ 100, 230⁄ 1000, 123⁄ 10000 , etc. But it would be a mistake is to think that there is such a thing as a fraction with a limitlessly large denominator.

Of course, when trying to calculate the value of the square root of 2 (the value that, when multiplied by itself gives the value of 2) we can continue to generate more and more fractions (or decimal places), viz.

1.4

1.41,

1.414,

1.4142,

1.41421,

…

…

1.4142135623730950 488016887242097,

…

This process of generating more and more decimal places can be continued indefinitely. There is no limit to the number of times you can add on an extra digit.

Based on this simple fact a crucial assumption was made, an assumption which is an inherently Platonist assumption. That assumption is that the irrational number is a thing that actually consists of a more than a finite number of decimal digits. For the Platonist, the value of the square root of 2 is actually equal to:

1 plus the ‘fraction’

414213562 … to more than a finite number of digits 1000000000 … to more than a finite number of digits

When confronted with this, the Platonist will reply that that is not what they mean by an ‘infinite decimal expansion’, and that what they really mean is the sum of a limitless quantity of fractions that start off as:

4⁄ 10 + 1⁄ 100 + 4⁄ 1000 + 2⁄ 10000 + 1⁄ 100000 + 3⁄ 1000000 + 5⁄ 10000000 + 6⁄ 100000000 + 2⁄ 1000000000 + …

and that there are limitlessly many such fractions that continue to get smaller and smaller, all added together.

But this doesn’t solve the problem of a limitlessly large denominator; it is simply evading the problem by a rewording rather than a different definition. Each fraction in this ‘sum’ is a definite size and its denominator is 1⁄ 10 of the previous one. But if this is a limitless ‘sum’ that can actually ‘exist’, then there must actually be more than a finite quantity of fractions, each one of a definite size and greater than zero, all added together. From this arises a contradiction:

“Every fraction is related to every other fraction in the ‘sum’, since the value of its denominator must be the value of the denominator of the first fraction in the sum, which is 10, multiplied by 10 a definite number of times. So, if there could be more than a finite number of these fractions, then there would be some fraction that was not related to any of the other fractions by any finite multiple of ten. But that is impossible, since every such fraction has to be related to every other fraction by some finite multiple of 10.”

At this point, the Platonist will resort to virtual hand-waving, and will say something like:

“But you have presumed that every fraction is a finite amount away from the first fraction. But some of the fractions will be an infinite amount away from the first fraction, and so your argument doesn’t apply.”

Unfortunately for the Platonist, this attempt to solve the contradiction doesn’t work. Every one of the fractions has to have a denominator of a definite size, that is, the denominator must be a 1 followed by a definite number of zeros. The alternative is that there are fractions in this infinite sum that do not have a definite value, and that their denominator is 1 followed by an infinite number of zeros – which contradicts the original definition.

Unlike the notion of the sum of an infinite quantity of ever decreasing fractions, a concept that does have a clear meaning is the limit of all the finite sums that can be got from a collection of fractions. For instance, given a collection of fractions that is:

4⁄ 10 , 1⁄ 100 , 4⁄ 1000 , 2⁄ 10000 , 1⁄ 100000 , 3⁄ 1000000 , 5⁄ 10000000 , 6⁄ 100000000 , 2⁄ 1000000000

you can have sums such as:

4⁄ 10 + 1⁄ 100

4⁄ 10 + 1⁄ 100 + 4⁄ 1000

4⁄ 10 + 1⁄ 100 + 4⁄ 1000 + 2⁄ 10000 + 1⁄ 100000

4⁄ 10 + 1⁄ 100 + 4⁄ 1000 + 6⁄ 100000000 + 2⁄ 1000000000

4⁄ 10 + 4⁄ 1000 + 2⁄ 10000 + 1⁄ 100000 + 5⁄ 10000000 + 2⁄ 1000000000

4⁄ 10 + 1⁄ 100 + 4⁄ 1000 + 2⁄ 10000 + 1⁄ 100000 + 3⁄ 1000000 + 5⁄ 10000000

4⁄ 10 + 1⁄ 100 + 4⁄ 1000 + 2⁄ 10000 + 1⁄ 100000 + 3⁄ 1000000 + 5⁄ 10000000 + 6⁄ 100000000

4⁄ 10 + 1⁄ 100 + 4⁄ 1000 + 2⁄ 10000 + 1⁄ 100000 + 3⁄ 1000000 + 5⁄ 10000000 + 6⁄ 100000000 + 2⁄ 1000000000

All of these sums are finite. None of them exceed a certain value. None of them exceed the value 0.414213562. That value is the limiting value for finite sums of fractions taken from the finite collection:

4⁄ 10 , 1⁄ 100 , 4⁄ 1000 , 2⁄ 10000 , 1⁄ 100000 , 3⁄ 1000000 , 5⁄ 10000000 , 6⁄ 100000000 , 2⁄ 1000000000

And if you have more fractions to choose from, you will have many more different finite sums of fractions. The above was a finite collection of fractions from which there can only be a finite number of different sums. But you can also have a concept of a limiting value for all the finite sums that can be got from a limitlessly large collection of fractions. In this case, there are a limitless amount of such sums – but the sums are still all finite. Since all these finite sums are derived from a concept of a collection of fractions that has limitlessly many fractions, then, in order to be able to be able to determine if there is a limit which none of these finite sums can exceed, there must first of all be a way of expressing that concept of a collection of limitlessly many fractions in a finite way, using a finite combination of symbols. Since the concept involves a limitless amount of such fractions, the only way to express that concept is by way of a definition. And similarly, once you have that finite definition, you may be able to use that finite definition to define a limiting value. And that limiting value will be a value that no finite sum of fractions, taken from the overall collection that is given by a definition, can exceed. None of this is anything new; most of the methods of calculating the limits for various series of fractions were known over a hundred and fifty years ago.

Of course, we can have numbers which are finite decimal expansions, such as

3.1416, and

3.14159265359, and

3.14159265358979323846, and

3.141592653589793 23846264338328, and 3.14159265358979323846264 3383279502884197169399,

and so on, which are all numbers that are close to the number Pi, relative to the number 3. And a number has been calculated to billions of digits and this is also closer to the number Pi, relative to the number 3.14159265358979323846. But, in the same way as it is a mistake to think of a ‘large’ number as being close to infinity, it is a mistake to think that the number 3.141592653 is close in some absolute sense to the value of Pi. It isn’t, since there are still infinitely many real numbers between 3.141592653 and Pi. It is only close in relative terms, in that it is closer to Pi than 3.1416.

And more …

For more demonstrations of contradictions arising from the Platonist beliefs in the ‘existence’ of ‘actual’ infinite sets, see The Courant & Robbins contradiction and Sums of infinitely many fractions: 1.

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.

For more on Platonism see Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.