Topics Mathematics

Question Irrational numbers and infinity have made me come up with this problem: pi, for example, is an irrational number, which means that it doesn't terminate or repeat. Every new digit found in pi increases the value of the number, no matter the value of the digit (for example, 3.141 is larger than 3.14, and 3.1415 is larger than 3.141). If pi never ends, then that means that there is an infinite amount of digits that will increase the value of pi by a tiny fraction. Therefore, pi should be infinitely large. So, pi = infinity. But there is a problem: pi is between 3.13 < x <3.15. This goes far beyond pi to 1/7 and even rational numbers that don't terminate (ex. 1/3). What is the problem associated with my logic?

Accepted:
June 21, 2016

Comments

This is actually a very good

This is actually a very good question to illustrate how everyday intuitions can lead you astray in thinking about philosophical problems -- or about any problems, really, that require finer and more sophisticated distinctions than needed in typical situations of everyday life.

Everyday intuitions, embedded in the categories of our inherited natural languages, are insensitive to certain fine distinctions made in the more finely-tuned artificial languages we've invented. From an everyday point of view, there is no reason to think that your argument about pi being infinitely large is flawed at all. But in mathematics, we've had to deal with the problem that that argument works in some cases and not in other cases -- some infinite series "converge," as mathematicians say, while others "diverge." The sum of all the reciprocals of the natural numbers (i.e. one half plus one third plus one fourth plus one fifth, and so on) diverges, i.e. it's infinite, as in your intuition. But the sum of all the reciprocals of the powers of two (i.e. one half plus one fourth plus one sixteenth plus one thirty-second, and so on) converges to 1. Why this difference? Well, that's something mathematicians have developed quite an extensive body of theory about over the past couple of hundred years, and if you want to develop educated intuitions about this problem, you have to slog through all those details. Your everyday intuitions are of no help at all in this situation; you have to leave them behind if you want to take this question seriously.