558 19. LOGIC AND SET THEORYFIGURE 1. The Brouwer fixed-point theorem.Why should you conclude the latter rather than the former? Is this why some
mathematicians have claimed that the practice of mathematics requires faith?19.7. What are the advantages, if any, of building a theory by starting with abstract
definitions, then later proving a structure theorem showing that the abstract objects
so defined are actually familiar objects?
19.8. Brouwer, the leader of the intuitionist school of mathematicians, is also
known for major theorems in topology, including the invariance of geometric di-
mension under homeomorphisms and the Brouwer fixed-point theorem, which as-
serts that for any continuous mapping / of a closed disk into itself there is a point
÷ such that ÷ = /(#)· To prove this theorem, suppose there is a continuous map-
ping / for which f(x) ö ÷ at every point x. Construct a continuous mapping g by
drawing a line from f(x) to ÷ and extending it to the point g(x) at which it meets
the boundary circle (see Fig. 1). Then g(x) maps the disk continuously onto its
boundary circle and leaves each point of the boundary circle fixed. Such a continu-
ous mapping is intuitively impossible (imagine stretching the entire head of a drum
onto the rim without moving any point already on the rim and without tearing the
head) and can be shown rigorously to be impossible (the disk and the circle have
different homotopy groups). How can you explain the fact that the champion of
intuitionism produced theorems that are not intuitionistically valid?
19.9. A naive use of the formula for the sum of the geometric series 1/(1 -f x) =
1 - ÷ + ÷^2 - ÷^3 Ç seems to imply that 1 - 5 + 25 - 125 Ç = 1/(1 + 5) =
1/6. Nineteenth-century analysts rejected this use of infinite series and confined
themselves to series that converge in the ordinary sense. However, Kurt Hensel
(1861-1941) showed in 1905 that it is possible to define a notion of distance (the
p-adic metric) by saying that an integer is close to zero if it is divisible by a large
power of the prime number ñ (in the present case, ñ — 5). Specifically, the distance
from m to 0 is given by d(m, 0) = 5~fc, where 5fc divides m but 5 fc+1 does not
divide m. The distance between 0 and the rational number r = m/n is then by
definition d(m, 0)/d(n,0). Show that d(l,0) = 1. If the distance between two
rational numbers r and s is defined to be d(r — s,0), then in fact the series just
mentioned does converge to | in the sense that d(Sn, g) —> 0, where Sn is the nth
partial sum.
What does this historical experience tell you about the truth or falsity of math-
ematical statements? Is there an "understood context" for every mathematical
statement that can never be fully exhibited, so that certain assertions will be ver-
bally true in some contexts and verbally false in others, depending on the meaning
attached to the terms?
19.10. Are there true but unknowable propositions in everyday life? Suppose
that your class meets on Monday, Wednesday, and Friday. Suppose also that your
instructor announces one Friday afternoon that you will be given a surprise exam at