I recently read this wonderful reflection/essay by Eric Shechter on why we study calculus( Direct link HERE )
or, a brief look at some of the history of mathematics |

1/1 | 1/2 | 1/3 | 1/4 | ... |

2/1 | 2/2 | 2/3 | 2/4 | ... |

3/1 | 3/2 | 3/3 | 3/4 | ... |

4/1 | 4/2 | 4/3 | 4/4 | ... |

... | ... | ... | ... | ... |

*list*:

**...**

*countable*-- i.e., it can be arranged into a list; it has the same

*cardinality*as the set of positive integers. Now, run through the list, crossing out any fraction that is a repetition of a previous fraction (e.g., 2/2 is a repetition of 1/1). This leaves a slightly "shorter" (but still infinite) list

**...**

*all*rational numbers is also countable. However, by a different argument (not given here), Cantor showed that the real numbers cannot be put into a list -- thus the real numbers are

*uncountable*. Cantor showed that there are even bigger sets (e.g., the set of all subsets of the reals); in fact, there are infinitely many different infinities.

As proof techniques improved, gradually mathematics became more rigorous, more reliable, more certain. Today our standards of rigor are extremely high, and we perceive mathematics as a collection of "immortal truths," arrived at by pure reason, not even dependent on physical observations. We have developed a mathematical language which permits us to formulate each step in our reasoning with complete certainty; then the conclusion is certain as well. However, it must be admitted that modern mathematics has become detached from the physical world. As Einstein said,

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.For instance, use a pencil to draw a line segment on a piece of paper, perhaps an inch long. Label one end of it "0" and the other end of it "1," and label a few more points in between. The line segment represents the interval [0,1], which (at least, in our minds) has uncountably many members. But in what sense does that uncountable set

*exist*? There are only finitely many graphite molecules marking the paper, and there are only finitely many (or perhaps countably many) atoms in the entire physical universe in which we live. An uncountable set of points is easy to imagine mathematically, but it does not exist anywhere in the physical universe. Is it merely a figment of our imagination?

It may be our imagination, but "merely" is not the right word. Our purely mental number system has proved useful for practical purposes in the real world. It has provided our best explanation so far for numerical quantities. That explanation has made possible radio, television, and many other technological achievements --- even a journey from the earth to the moon and back again. Evidently we are doing something right; mathematics cannot be dismissed as a mere dream.

The "Age of Enlightenment" may have reached its greatest heights in the early 20th century, when Hilbert tried to put all of mathematics on a firm and formal foundation. That age may have ended in the 1930's, when Gödel showed that Hilbert's program cannot be carried out; Gödel discovered that even the language of mathematics has certain inherent limitations. Gödel proved that, in a sense, some things cannot be proved. Even a mathematician must accept some things on faith or learn to live with uncertainty.

Some of the ideas developed in this essay are based on the book *Mathematics: The Loss of Certainty*, by Morris Kline. I enjoyed reading that book very much, but I should mention that I disagreed with its ending. Kline suggests that Gödel's discovery has led to a general disillusionment with mathematics, a disillusionment that has spread through our culture (just as Newton's successes spread earlier). I disagree with Kline's pessimism. Mathematics may have some limitations, but in our human experience we seldom bump into those limitations. Gödel's theorem in no way invalidates Newton, Cantor, or the moon trip. Mathematics remains a miraculous device for seeing the world more clearly.

This web page has been selected as one of the best educational resources on the Web, and has received the coveted "StudyWeb Excellence Award," from StudyWeb. For other web pages about Math History, see StudyWeb's list.