FOM: Examples of infinitesimals

[Jon Barwise]

I can't say I am very moved by the concern over not being able to define an
infinitesimal.

1. The hyperreals are an elementary extension of the reals so of course we
can't define any of them.

2. When we expand the reals to the complex numbers we have to add a square
root of -1.  But of course you can't add one without adding two, and there
is simply no way to tell the two apart. (There is an automorphism of the
field of complex numbers taking i to -i.)  Only by taking one of these
roots as "given" can even we define things like the imaginary part of a
complex number.  As a student admit I was bothered by this.  But having
become "mature," I don't find myself bothered any more. I eventually
accepted the idea that i and -i are simply ideal objects added to the reals
to help us model things in a convenient manner.  So too with
infinitesimals.

3. When I taught infinitesimal calculus, here is how I used to motivate the
idea of adjoining infinitesimals as idealized objects to the reals.
Compare the size of a grain of sand to a large heap of sand, a beach, or
all the sand in the universe.  There is some natural number N that is the
number of grains of sand in the whole, and a single grain of sand is 1/N th
of the whole.  Now 1/N is not really infinitesimal, but at an intuitive
level, it behaves a lot like an infinitesimal.  It is unimaginably small.

The sorites paradox (the paradox of the heap) can be taken as showing that
our intuitive ways of thinking about this size is as though it were
infinitesimal compared to the whole.  You recall the argument. Start out
with the big heap of sand.  It cannot have just one grain of sand it in.
Moreover, if you have a big heap of sand and take one grain of sand away,
you still have a big heap of sand.  Now, the argument by induction then is
supposed to show that you can't have a big heap of sand, which contradicts
the obvious that there are lots of big heaps of sand all over the place.
One way to break the paradox, though, it to say that a single grain of sand
is, intuitively, infinitely small compared to a big heap of sand.  That is
why taking one grain away cannot make a significant difference.  In other
words, my claim is, the "paradox" shows that our intuitive understanding of
the ratio of a single grain of sand to a large heap of sand is better
modeled by means of infinitesimals.

This is not the only possible response to the sorites paradox, but it does
seem to me a responsible one and it does make sense to students.  The
feeling that this is a natural way to model huge finite collections of very
small units is reinforced by various applications (by Keilser, Brown, etc.)
of hyperreal analysis to economics, where the economy is modeled not
continuously (as is "standard") but as a *finite set of consumers. Each
individual decision has an infinitesimal impact on the economy, but it all
adds up.  (Ask Bill Gates!)  The people working in hyperreal economics have
proved a number of interesting theorems in economics based on this
intuitive idea.

Jon