The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Chapter 1

Logic

The ideal writing style ascribed to by mathematicians is that in writing mathematics, less is more. If we can convey the exact idea of a concept with 5 words instead of 10, then we will use 5. Thus, we will use the statement Cardinal numbers form a well-ordered collection over the wordier statement The well-ordered property is enjoyed by the collection of cardinal numbers. The second statement is mathematically correct, but it is more than we need to convey the idea.

I have tried to practice this ideal while writing the mathematics in this book. The only exceptions to this ideal are made on the basis of decisions on the educational value of sentence structure, the anecdotal comments, or discussions of this sort that occur between mathematical discourse. Sometimes it is good to sacrifice some mathematical austerity in the interest of getting an important point across to the reader. As the reader will clearly see, this economy of words in mathematical writings is not exercised in the text of a discussion. Discussions and intermediate anecdotes contain examples and illustrations that are the only tools we have to illustrate a concept. Since I have sacrificed a good bit of mathematical rigor in favor of clarity, examples and illustrations are necessary if I am to get some subtle ideas across to the reader. This form of personalized writing style is unavoidable when discussing advanced ideas from mathematics in the popular press.

We have a bit of a mountain to climb ...