210 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSthe modern theory of infinite ordinals. The idea is to progress through the finite
numbers 2,3,4,... until the "first unenumerable" number is reached. This number
corresponds to what is now called ù, the first infinite ordinal number. Then, ex-
actly as in modern set theory, one can consider the unenumerable numbers ù + 1,
ù + 2,..., ù^2 , and so on. We do not have enough specifics to say any more, but
there is a very strong temptation to say that the Jaina classification of enumerable,
unenumerable, infinite corresponds to our modern classification of finite, countably
infinite, and uncountably infinite, but of course it is only a coincidental prefigura-
tion.
Infinite ordinals and cardinals. A fuller account of the creation of the theory of
cardinal and ordinal numbers in connection with set theory is given in Chapter 19.
At this point, we merely note that these theories were created along with set the-
ory in the late nineteenth century through the work of several mathematicians,
most prominently Georg Cantor (1854-1918). The relation between cardinal and
ordinal numbers is an important one that has led to a large amount of research.
Briefly, ordinal numbers arise from continuing the ordinary series of natural num-
bers "past infinity." Cardinal numbers arise from comparing two sets by matching
their elements in a one-to-one manner.
3. Combinatorics
From earliest times mathematicians have been concerned with counting things and
with space, that is, with the arrangement of objects of interest. Counting arrange-
ments of things became a separate area of study within mathematics. We now call
this area combinatorics, and it has ramified to include a number of distinct areas
of interest, such as formulas for summation of powers, graph theory, magic squares,
Latin squares, Room squares, and others. We have seen already that magic squares
were used in divination, and there is a very prominent connection between this area
and some varieties of mystical thinking. It may be coincidental that the elementary
parts of probability theory, the parts that students find most frustrating, involve
these sophisticated methods of counting. Probability is the mathematization of
possible outcomes of events, exactly the matters that are of interest to people who
consult oracles. These hypothetical happenings are usually too many to list, and
some systematic way of counting them is needed.3.1. Summation rules. The earliest example of a summation problem comes
from the Ahmose Papyrus. Problem 79 describes seven houses in which there are
seven cats, each of which had eaten seven mice, each of which had eaten seven seeds,
each of which would have produced seven hekats of grain if sown. The author asks
for the total, that is, for the sum 7 + 7^2 + 7^3 + 7^4 + 7^5 , and gives the answer
correctly as 19,607. The same summation with a different illustration is found in
Fibonacci's Liber abaci of 1202. In this example we encounter the summation of a
finite geometric progression.
A similar example is Problem 34 of Chapter 3 of the Sun Zi Suan Jing, which
tells of 9 hillsides, on each of which 9 trees are growing, with 9 branches on each
tree, 9 bird's nests on each branch, and 9 birds in each nest. Each bird has 9 young,
each young bird has 9 feathers, and each feather has 9 colors. The problem asks
for the total number of each kind of object and gives the answer: 81 trees, 729
branches, 6561 nests, 59,049 birds, 531,441 young birds, 4,782,969 feathers, and