A speech given by L. Sue Esch,
in conjunction with receiving the Beachley Distinguished Teaching
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 true but not provable! In other words, what Godel established was that in mathematics, there will always be true statements which are not provable. To put it mildly, this set the mathematical world on its ear. To begin with, it once again demonstrated conclusively that truth cannot be the same thing as provability. But, more fundamentally, in one ingenious stroke, it established that mathematics not only never will but never can prove all true statements, even arithmetical ones. There will always be truths which are not theorems. In short, it put truth out of the reach of mathematics. In the words of Hofstadter in Godel, Escher, Bach, "truth transcends theoremness." Therefore, if anyone can be expected to answer the question "what is truth?", it won't be a mathematician.
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."