FOM: foundations??

[Harvey Friedman]

As promised in "Golden Age/FOM Plans", here is the e-mail about the meaning
of the phrase "foundations of mathematics."

I want to focus on the following three concepts: integer, rational number,
and curve.

The informal concept of integer, or at least of natural number from 1,
speaks to everyone, and is fundamental to everybody's understanding of the
world. Integers occur in bank accounts, baseball statistics, salaries, date
of birth, etcetera; and this is even with the ordering. E.g., your salary
is higher than mine. Also the idea of zero - e.g., zero salary - is
universally familiar.

DIGRESSION 1.

Now the relationship between natural number and integer, as well as related
concepts such as addition, multiplication, and absolute value, is a vast
subject that can be pursued with great vigor and subtlety. And I am not
even going to get into the psychological aspects of this - e.g., Piaget and
his followers on childhood development.

As a digression, let me show you how subtle foundational projects emerge
even at this most elementary level:

a. should integers be considered a derived concept from natural number, or
rather a concept of independent status and meaning, for which its usual
relationship with natural number is derived? Or are both views equally
valid? How should one formalize the relationship between these two
concepts?

	There does seem to be a case for a geometric understanding of the
integers with <, which is not derived by first thinking of natural numbers.

b. should one distinguish two concepts of natural number: one as in a
series starting with 1, and the other as measuring sizes of (finite) sets?
How should one formalize that these concepts are equivalent?

	There does also seem to be a case for a geometric understanding of
the natural numbers as a whole unit, which might differ from our
understanding of the natural numbers as a never ending unfolding series.

c. is there such a thing as a natural number independent of its place in a
structure? I.e., does it make sense to talk about what 3 really is?

	If we take the view of natural number as measuring sizes of sets,
then we really can talk about what 3 is. However, if we take a geometric
view, or even the unfolding series view, then it seems that we cannot talk
about what 3 really is.

d. more about a. One view of integers is that they measure the path from
one natural number to another, taking into account whether or not you are
going backwards. This also explains zero. Another view is simply: just copy
the natural numbers in reverse order. The first seems more credible, and so
one leans towards the fundamental character of measurement of quantities.

	Under the first view, again one really can talk about what -2
really is. But under the "just copy" view, one cannot talk about what -2
really is.

e. About addition. A very plausible account here is rigid motion. One can
start this discussion with the natural numbers from 1. One simply imagines
a rigid transport of one interval onto another in the well known way. It
appears that the commutative and associative laws of addition may be
derivable from more fundamental geometric principles that do not presuppose
mathematical induction.

There is also the idea of a natural number as a count, and that n+m means
that the successor operation is to be applied n times to m. This seems to
be tied up with induction, and not geometry.

f. About multiplication. A very plausible account is that of measuring the
number of elements in a Cartesian product. Or counting the number of points
in certain geometric figures. There is also the iterative approach - the
number of times you perform addition.

g. what is the exact nature of mathematical induction? is there a purely
geometric account? can induction be proved from something more fundamental?
is there a new kind of independence result that asserts that induction
cannot be proved from a number of surprisingly strong assertions? or is the
opposite true - namely that there is a surprising derivation of induction
(in some form) from some of its most fundamental consequences?

h. how does consideration of globally fundamental physical notions such a
body and moving body and rigidly moving body affect the discussion, as well
as the application of natural number as counting the number of times an
operation or process is performed? and what about naive concepts of time?

i. what are we to make of the fact that there are so many competing
accounts of these notions? what are we to make of the fact that all
approaches turn out to be equivalent? what do we make of the fact that some
approaches support the idea of natural numbers or integers as existing
independently of their place in a given structure, whereas other ideas are
not compatible with that? and when we use an approach that is incompatible
with the idea of natural numbers or integers as existing outside of a given
structure, how do we wish to formalize our reasoning about such? it
certainly seems unsatisfactory to use some obviously less globally
fundamental notions for this such as are used in modern category theory,
etcetera.

j. what are we to make of this entire discussion a) - i)? is it important
and/or interesting? what is to be gained by it? what are we to make of the
fact that it gets so elaborate and subtle and difficult and perhaps
subjective, yet it is directed only at concepts that are supposed to be
assimilated by little children in their first years of school?

END OF DIGRESSION 1.

The concept of rational number is normally derived from that of integer in
mathematics. And for that matter by a process that seems very basic - the
fraction field construction. This fraction field construction seems allied
enough with the general conception of dividing a pie into four pieces, or
allocating proper portions of the food on hand to members of the family,
and mixing proportions of ingredients in cooking, etcetera.

Now for curves. From a completely different point of view than that of
integer, the informal concept of curve is of obvious globally fundamental
importance. Trajectories of moving bodies. Paths of baseballs and
basketballs. Paths of moving fingers, cars, trains, and airplanes.

DIGRESSION 2.

a. how appropriate in the usual formalization in mathematics of curve (in
Euclidean 3-space) as a continuous function from [0,1] into R3? Why are
functions used in as much as they seem to have nothing to do with motion or
time or the basic conception of a curve. In this context, what is a
function anyway? Is it appropriate to have the function concept embedded in
the set theoretic interpretation of mathematics for this purpose? Why
continuity and not some stronger condition? What does all this mean in
finitary terms, since our eyes have finite resolution and maybe the actual
universe is discrete? What then is the meaning of the regularity
conditions? Is analyticity of genuine significance here if we are going to
remain continuous?

b. as an alternative approach, can we take time as a primitive in
mathematics and think of a curve as a single point which changes position
in time? This does not seem to use the function concept in the set
theoretic sense. But what does reasoning look like under this approach? And
if this works out nicely, what do we make of surfaces? Two dimensional time
seems to be a bad idea.

END OF DIGRESSION 2.

We now want to discuss rational points on curves. Almost anybody can
readily understand why a mathematical treatment and study of integers and
curves might be appropriate and interesting and useful. But rational points
on curves??

To anybody but a mathematician, this seems like an acquired taste - and
they are not likely to acquire it. Rationals are used to divide pies,
allocate resources, facilitate transactions, make comparisons, etcetera.
And curves are all over the place, on human bodies, in transportation,
etcetera. Why rational points on curves??? As far as the measurement and
fabrication of curves, there is no sense to infinite precision, so
rationality is in principle irrelevant. And what does dividing pies and
allocating food have to do with curves? Oh, we may want to divide up the
curve so that everybody gets to eat their fair share?

Yes, I am aware that in the history of mathematics rational points on
curves played an important role - a continuing role - and it all is
beautiful and deep and fashionable. Fundamental mathematics? Yes.
Foundations of mathematics? Certainly not. At least not in the way it is
normally communicated. Maybe one day the study of rational points on
polynomial curves with rational coefficients may be used to do something in
Foundaitons of Mathematics - in the normal sense of the word. Even if this
happens, that doesn't make it foundational. And I hope it does happen.

There are only a tiny number of unusual people who could care less about
rational points on curves - and I even include myself as one of them to a
limited extent. But foundational? Not in its present form. Fundamental for
the history of mathematics? Yes. Fundamental for the future development of
mathematics? Perhaps.

Rationals foundational? Yes. Curves foundational? Yes. Rational points on
curves foundational? No. Michael Jordan would agree with me. Foundations
seeks to speak to him (and everybody else).