Volumes 723that (B,), (B2), and (B3) are valid, a contradiction arises from the assump-
tion that each bounded set in Ea has a volume. Thus there exist bounded
sets in E3 that do not have volumes. We are doing "rigorous mathe-
matics" when we give a definition of volume and prove that a given
spherical shell has a volume. We are still doing "rigorous mathematics"
when we make and use clear statements of provable facts but postpone
or omit the proofs. We are deep in the depths of intellectual degradation
when, without having a definition of volume, we hold aloft a brick or
spherical ball and convey (either explicitly or implicitly) the impression
that the thing "obviously" has a volume. We hit the bottom when
we say that the thing is a volume. Unless we tolerate the idea that
bad mathematics can be acceptable elementary calculus, we must avoid
these degradations. Perhaps we can attain a reasonable view of this
whole matter by recognizing the fact that modern set theory shakes the
foundations of nineteenth-century mathematics as vigorously as modern
atomic theory shakes the foundations of nineteenth-century physics and
chemistry and engineering. Some of us will learn more about these
matters than others, but we can all know that there is much to be learned.