Truth

*A speech given by L. Sue Esch,
in conjunction with receiving the Beachley Distinguished Teaching*

at Juniata College

"Veritas Liberat". Truth sets free. It's more than *just*
a motto; it's what we're all about. Education could easily be defined as "the pursuit
of truth". The problem is "what *is* truth?". A sticky question, to
say the least, but you would think that if *anyone* could get a handle on it, it *might*
be a mathematician. After all, the basic building block of mathematics is logic; the basic
component of logic is the proposition; and the very definition of a proposition is "a
statement which is either true or false". We also have the Law of the Excluded Middle
to guarantee that a statement must be true or false and the Law of Contradiction so that
it can't be both. *Surely*, we must know the difference -- or at least have ways of
determining it. In applied mathematics we seek to determine new *truths* (in the form
of equations) about the relationships among physical objects. In theoretical math, the
"objects" become more abstract, but *truths* about relationships are still
the goal. In both, we must not only determine the relationships but be able to* prove*
that they* are* in fact *truths*. In any such proof, we assume axioms to be *true*
and deduce conclusions or theorems which are then said to be *true*. It certainly
seems that *truth* is an essential ingredient in mathematics -- but you have to watch
out for that word "true" in mathematics. The *truth* is that there are
probably few disciplines more ambivalent about its meaning. As Bertrand Russell said,
"Mathematics is the subject in which we do not know what we are talking about nor
whether what we say is true." Despite the fact that most of you agree *wholeheartedly
*with Russell, and have probably been searching ever since kindergarten for such a
quote to confirm your suspicions, let me backtrack a bit and tell you how mathematics
arrived at such a state and what it all might mean to those of us committed to education
and the pursuit of truth.

In the beginning, truth for mathematics was the same as truth for science; it was
empirical in nature. A statement was true if it accurately described the physical world.
If one needed to know the area of a rectangular plot of land, one just measured the
lengths of the sides and multiplied. Therefore the equation, "the area of a rectangle
equals the length of the base times the length of the height" was deemed a truth for
the simple reason that it worked. Mathematics at this point was totally applied; it was a
science, concerned with relationships among physical objects. Then along came the Greeks,
who insisted that mathematical truth be *more* than empirical truth. It was no longer
sufficient for an equation to merely work; one must be able to *prove* that it works.
One must be able to *logically deduce* it from first principles, called axioms. With
this new emphasis, mathematics entered a new era, an era in which its job was not only to
unearth empirical relationships about the physical world but also to prove them to be
truths.

This view of mathematics continued until the eighteen hundreds with the discovery of
non-Euclidean geometry. As opposed to Euclidean geometry, where parallel lines are
everywhere equidistant, in non-Euclidean geometry, parallel lines either do not exist at
all or if they do, they are "curved" so that the distance between them varies.
This may seem strange to us, but then there are no doubt many things in mathematics that
seem strange. The real *problem* arose in that although one could demonstrate that
non-Euclidean geometry was *logically deducible*, it wasn't *empirically* *consistent*.
Mathematicians could *prove* the logical deducibility of parallel lines that
"curve" and even come arbitrarily close to one another, but everyone *knew*
that *empirically*, in our *physical* world, parallel lines are
"straight" and remain equidistant. After all, the physical world is Euclidean,
or at least in the eighteen hundreds it was. There wasn't any doubt about it. The
scientists assumed it, and the best mathematicians had spent 2000 years devoted to proving
it. Later, with Einstein and relativity, would come "curved space" and the
discovery that the Euclidean model of the world wasn't quite so obvious after all. But,
for the nineteenth century mathematician, the discovery of non-Euclidean geometry was
extremely problematic. *It was "true" because you could logically deduce it,
but could not be "true" because it did not accurately describe the physical
world.* Clearly, mathematics had to redefine its notion of truth. It had to choose
between empirical consistency and logical deducibility. It chose logical deducibility. A
new era. Mathematics ceased to be a science in the physical sense of the word. Its game
became logical deduction and proof. A statement would be true if and only if it was a
theorem, i.e., could be proved.

However, nothing ever turns out to be quite as simple as we hope. The first problem
that arose should not surprise anyone. It had been there since the Greeks, lurking just
beneath the surface. A statement is true if and only if it can be proved, i.e. logically
deduced. But, logically deduced *from what*? First principles, axioms. But where do *they*
come from? For the Greeks, that was not a problem. Their mathematics had been
fundamentally empirical. Their first principles were merely self-evident statements about
the physical world, statements so basic that everyone accepted their truth. Who could deny
that "two points determine a line" or even more basic, that "equals added
to equals are equal"? But when non-Euclidean geometry forced the nineteenth century
mathematicians to discard empirical consistency in favor of logical deducibility,
mathematics had a problem. Assuming axioms or first principles and deducing theorems, and
then claiming that the truth of the theorems follows from the truth of the axioms, in no
way deals with the truth of the axioms. The classic approach is to *assume* it. But,
on what basis? Well, the primary criterion for a statement to be an axiom in the first
place is that it be self-evident. But, self evident to whom? You can see that this quickly
becomes a philosophical quagmire. In the shift from empirical to logical,
"truth" had lost most, if not all, of its meaning. In fact, if *truth* be
known, mathematics rarely uses the word "true". Rather it uses
"deducible" or "provable". It leaves truth to the philosophers.
"Fools rush in where angels fear to tread."

But, can mathematics totally avoid *any* mention of truth. What about those
propositions and the Law of the Excluded Middle, which demanded that "statements must
either be true or false"? O.K. We can deal with that. If "truth" now means
"provability", we can just translate and demand that "statements must
either be provable or disprovable." But, again, it never turns out to be that simple.
Enter Godel. 1931. What he did was prove that there exist statements which are neither
provable nor disprovable. How he did this is rather interesting. I hope you like mind
games...

Somewhere, you have probably encountered The Liar's Paradox, in one version or another.
At its center lies the statement S which says *about itself* "*this*
statement is false." There's a Catch-22 here though. Is the statement S* itself*
true or false? If S is true, i.e., "this statement is false" is a true
statement, then since "this statement" refers to S itself, what we have is
"S is false" is a true statement, or just "S is false". In other
words, if S is true, it follows that S is false. Oops, that can't be! *So*, since
assuming that S is true leads to a contradiction, the only alternative is that S must be
false. But if S is false, i.e., "this statement is false" is a false statement,
then "this statement" S must be true. Oops, again. In short, S can't be either
true or false, but, by the Law of the Excluded Middle, it *must* be. Catch-22. We
could refuse to consider S, claim that it isn't a legitimate proposition, but we have no
justification for doing that other than avoiding the consequent paradox, which hardly
seems sporting. So, what *do* we do? What else! Let the philosophers worry about it!
After all, it doesn't have anything to do with numbers or equations. For most
mathematicians, it was and is just a cute little oddity -- probably not a paradox at all,
merely an anomaly. For others, however, it raised some serious questions -- first about
the foundations of logic and then, given the close tie between the two, the foundations of
mathematics. And, even though the paradox can be translated into set theory, still for
most working mathematicians, the source of the problem seems to lie more in the realm of
logic and philosophy than mathematics.

What Godel did was to bring it home! Essentially he translated the paradox into
mathematics, replacing "truth" with "provability". First, he
translated each mathematical symbol into an integer, then each axiom (or string of
symbols) into a string of integers which was nothing but a bigger integer. Next, he
translated the rules of logical deduction into arithmetic rules for deriving new integers
(representing theorems) from old ones (representing axioms). The net effect was that,
unlike truth, "provability" became as mechanical as adding and multiplying
integers. If you began with only the integers representing axioms and used only the
arithmetic rules representing logical rules of deduction, then whatever integers could be
produced must represent provable statements, i.e., theorems. The final coup was the
arrival at a statement G which said *about itself* "G is not provable."
Now, as in the Liar's Paradox, ask yourself whether G *itself* is provable or
disprovable. If G is provable, i.e., you can prove the statement "G is not
provable", then G must not be provable. Oops, here we go again. If, on the other
hand, G is disprovable, i.e. you can disprove the statement "G is not provable",
then G must be provable. Oops, again. Our old Catch-22. But what does all this *mean*?
Well, "truth" was *supposed to be* "provability". However, in G
we found a statement which must be true or false by the Law of the Excluded Middle, but,
following our stellar reasoning, can be neither provable nor disprovable. Therefore,
"truth" *can't* be the same thing as "provability". And if that
weren't enough, another *even more* troubling consequence arises if we give this
whole thing a slightly different twist. Let's start again with Godel's statement G that
"G is not provable." This time, appealing directly to the Law of the Excluded
Middle, we know that G must either be true or false. If G is false, i.e., "G is not
provable" is a false statement, then G must be provable, and since if you can prove
something, it follows logically that it must be true, we have again shown that if G is
false, then G is true -- which can't be. And as before, since it can't be false, it *must*
be true. But now we're __ really__ in trouble -- because what we have is a
statement which is

O.K., now that I have shattered whatever faith you may have had in mathematics, what
does all this have to do with education and the pursuit of truth? On a trivial level, it
certainly guarantees that I will never advocate that Juniata students take 120 credits of
mathematics to graduate. On a not so trivial level, it should raise some questions about
the basic tenets of the western intellectual tradition, upon which our whole educational
system is founded. If "truth transcends theoremness", doesn't it seem rather
futile to conduct a search for it *within the confines* of such a rationally based
tradition? The answer is an unequivocal "yes and no". Remember, many truths *do
not* transcend theoremness. Reason and rationality have in fact brought us a long way
in the pursuit of truth. Granted, Godel showed that they are not *sufficient* to go
the distance, but it took some pretty fancy footwork. In mathematics the vast majority of
truths *are* theorems; they *can* be proven. It was only when we made a
statement refer to itself ("*This* statement S is false." "*This*
statement G is not provable.") that we ran into trouble. So, the lesson is certainly
NOT to *abandon *rationality, only to *realize its* *limitations*. Number
one, we must recognize the basic tenet that some truths are and will remain beyond the
reach of rationality and theoremness. Number two, even those "truths" that
appear to be within their grasp are at best *conditional*. They are deducible from
axioms or assumptions, and therefore our acceptance of their truth depends on our
acceptance of the truth of the axioms. And since each discipline, if not each individual,
advances its own axioms and often its own rules for deduction, any kind of *universal*
*acceptance* seems almost unthinkable, even in the most harmonious of communities.
This, at the very least, suggests the need for careful communication. We will undoubtedly
never share each other's truths. We cannot even agree where and how to search for them.
But if we can, through thoughtful communication, somehow come to recognize and respect
alternate avenues of exploration, perhaps acceptance becomes a bit more thinkable. But, as
even we at Juniata have discovered, communication alone is not enough. Plans and reports
do not necessarily lead to understanding and community. We also need TRUST. Trust that
there exist truths beyond theoremness, truths that can neither be determined by nor
irrefutably proven by rigorous rational deduction. Trust that *others*' truths follow
from *their* axioms, even though the passage may seem strange to *us*. And
especially trust that *different and seemingly contradictory axioms systems can be
equally valid*. If this last one seems particularly unattainable, we need only return
to mathematics for inspiration. Euclidean and non-Euclidean geometry would *seem* to
provide prime examples of systems which are not only different but contradictory. Far from
it. Not only are they *not* contradictory and mutually exclusive, but mathematics has
shown that logically, they stand or fall together.

So perhaps mathematics is not as much of a lost cause as we originally thought. It may
not be able to totally capture truth, but it can certainly teach us a thing or two, not
only about theoremness, but also about the importance of communicating our assumptions and
modes of thought, trusting alternative realities, and even recognizing the limits of our
own self-imposed way of looking at the world. Since for the Greeks, mathematics *was*
geometry, perhaps it is more appropriate than traditionally recognized that above the
entrance to Plato's Academy was inscribed the maxim, "Let no one ignorant of geometry
enter this door."

4/30/91