What is the answer to the paradox of the infinitesimal?

Question

I just read this article on npr, which mentioned the following question:

You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're anything greater than zero, then the line would seem to be infinitely long. And if they're zero, well, then no matter how many parts there are, the length of the line would still be zero.

It further mentions that

Today, mathematicians have found ways to answer that question so that modern calculus is rigorous and reliable.

Can anyone elaborate on the modern answers to this question?

Answer 1

The paradox has to do with the additivity with measure; in particular, naively switching from finite additivity to infinite additivity.

We're familiar with the fact that, if we split something up into two parts, the measure of the whole is the sum of the measure of the two parts.

If we repeat this with one of the individual parts, we've now split the original whole into three parts. The measure of the whole is the sum of the individual measures of the three parts.

And so forth; binary additivity of measure does extend to arbitrary, but finite additivity of measure.

Generally speaking, things have to change when you switch from finite to infinite. We can no longer justify additivity of measure when you have infinitely many parts, because you can never get to infinitely many parts by repeatedly splitting the whole into finitely many parts finitely many times.

A priori, there might not even be a reasonable notion of additivity of measure when you have infinitely many parts! However, experience has shown there is a useful extension of additivity to countably many parts, at least when studying a continuum.

i.e. if you split a whole into countably many parts (and in a measurable way), you can expect the measure of the whole to be the sum of the measures of the individual parts.

Note that "sum" must be meant in the sense of an infinite sum from calculus; e.g. as a limit of partial sums. Trying to literally interpret an infinite sum as repeated addition runs into all of the same problems we're trying to work around.

When you split the whole into more than countably many parts -- e.g. you split the number line into its individual points -- you now have too many parts for countable additivity of measure.

Answer 2

Take a line one unit long. You can divide it into n parts each of length 1/n. As n increases the parts get smaller, but at the same time to total length is always 1 unit.

The length is given by n.(1/n) and the modern mathematical concept is that the Limit as n tends to infinity of the function n.(1/n) is 1.

There's a notation for this: $\lim_{n \to \infty} n.(1/n) = 1$

Google and read up on limits of functions and sequences.

Answer 3

It states

...then the line would seem to be infinitely long.

It is infinitely long for a very very small person who walk exactly one real number with each step. For example if she/he is on the point 0.5, then she/he must find the next real point right after (or before) 0.5. Can she/he?

But for us, who use centimeter as a measuring unit, that line would not be infinitely long. So it basically depends on the unit of length that we consider.

Answer 4

If you divide the unit interval $[0,1]$ into $N$ equal parts where $N$ is an infinite number, each of the intervals of the subdivision will be of infinitesimal length. Thus infinitely many subintervals can indeed add to a finite length, but those subintervals can't have appreciable length: they must be infinitesimal. See related discussion at Are infinitesimals dangerous?

To respond the title question, "What is the answer to the paradox of the infinitesimal?", such an answer is given by a construction of a proper extension of the real number field which remains an ordered field (that's the easy part) and moreover is an elementary extension in the sense that "all" (in a suitable sense) properties of the real number field still hold for the proper extension (this requires more work).

Answer 5

Its an idea that relates to how we can indefinitely count in "both directions". In other words we can count in whole steps forever, we can also do the same for the space between the steps.

Consider a ruler, we can count the 12 inches that are clearly marked out and we can count out all the lines in between the inch markings. We can keep dividing and we can imagine the idea of doing this indefinitely. Of course physical limitations will only allow us to divide things so far down.

There really are no paradoxes. These infinitesimal brain teasers rely on how the terms are defined. In other words how the sets of the mathematical system are described. Real Numbers represent the set of all numbers including fractions, decimals, and even whole numbers. This 'set' includes all the other 'sets'. A natural number is the basic 1,2,3,4 etc (or 0,1,2,3 etc). A real number would include pi or 3.3333333 or 1/5 or 1/3 etc.

The idea is that there are more real numbers than natural numbers even though both are 'infinite'. The thing is infinite means boundless. What counting means is we can count forever. The supposed paradox here has to do with confusing numerical symbols and concepts with the quantities they are supposed to represent. These quantities can be real or imagined. These quantities are things like the dimensions of a physical object or its weight, the amount of gas in your car or the amount of debt ( represented by a negative number) you owe.

The sleight of hand of this supposed paradox is the confusing of the numerical symbols and concepts with the quantities they represent. Math simply involves counting of one kind or another and the whole numbers work with the rest of the sets of numbers of the system just fine. The whole system is one numerical system and any divisions are very arbitrary in terms of making metaphysical claims of cosmic mystery paradoxes. (cosmic means order)

We can also simply convert 3.33333 to 333,333 like the metric system allows and we will have only whole numbers and no fractions. We just scale down or up. We can go to nanometers or smaller and we can go to kilometers or bigger. We do not have to resort to fractions. We can extend pi out as far as we like, 3.14159 etc can become 314 or 3142 (rounded) or 314,159 etc. We have to round pi off at some point and all this shows is the limits of math as a human tool. We only need to know what resolution we need and that is our limit. In the real world we can only measure sup to a certain 'resolution' anyway so we are physically limited from truly dividing something down indefinitely.

We can also convert whole numbers to decimal placed numbers by adding as many zero placeholders as we like: 1 becomes 1.000000000 etc.

"Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware."

https://en.wikipedia.org/wiki/Georg_Cantor