This file contains an exchange of letters between Wolff and Robert Briggs (Wolff’s step-grandson). In his letter, Mr. Briggs asks the following questions concerning transcendental numbers:
- Where did the term ‘transcendental’ originate and by whom?
- Are transcendental numbers irrational numbers or are they excluded from that group?
- If transcendental numbers are irrational numbers, then how can there be more transcendental numbers than rational numbers?
- What significance do transcendental numbers have, if any?
- Is [the] mathematics of today basically a rediscovery of past knowledge, or is it a creation of expression by the present race?
Wolff responds:
Somehow my first impression upon reading your letter was that you had in mind “Transfinite Numbers” and I soon realized an adequate answer would be almost a book, if the ramifications in Mysticism and Philosophy along with Mathematics was taken into account. However, the error was not so far-fetched when one remembers that Cantor’s proof of the existence of a non-denumerable Transfinite involved the Transcendental Numbers.
Wolff explains that Cantor showed the set of algebraic numbers is denumerable (that is, can be put into one-to-one correspondence with the positive integers) but that the cardinality of the set of transcendental numbers is greater than that of the set of algebraic numbers, and so is non-denumerable (that is, it is an infinite set that cannot be into one-to-one correspondence with the positive integers).
Wolff goes on to note that given that the cardinality of the set of transcendental numbers is greater than the cardinality than the set of all algebraic numbers that “it comes with something of a shock to realize that only two Transcendentals are well known”; these are pi and e. Wolff then makes some tangential remarks about his personal theory of the nature of mathematics (which he designates as the “Gnostic Theory”) before he returns to a discussion of the two transcendental numbers mentioned above. He concludes the letter with some brief answers to Mr. Briggs’ questions above. (7 pages)
