The Axiomatic Method, a lecture by Ricardo Nirenberg. Fall 1996, the University at Albany, Project Renaissance.

I'll start by reminding you: this course, this adventure, this quest upon which we are embarked and in which we will all do our best to shine, is called Project Renaissance. Its main themes are Human Identity and Technology. Let's begin with this fact, our human identity. Each of us is endowed—or maybe we should say stuck—with an identity; this identity goes far beyond a name and a social security number; it is inscribed all over our bodies: our face, our voice, our repertoire of gestures, and, underlying it all, our nervous system, our unique brain, different from the brain of any other individual. Our identity, then, is a fact, a brute fact. But over and beyond that brute fact, the question is, how can we become aware of our identity, conscious of it, how can we come to know ourselves. The answer is very simple: we can't. That is, we cannot hope to know our own self totally; no matter how wise we become, we will always be a surprise to ourselves—this is what makes life interesting. Yet, we can achieve some provisional and imperfect knowledge of our identity, and for that we need the art of memory. Let me say a few words about this art of memory. People tend to think of the Renaissance, of the period, let us say roughly between 1400 and 1600, as a time of the recovery of ancient literature, of great art and of new discoveries in astronomy and the natural sciences, and up to a point this is a true picture, but it is an incomplete picture colored by our present-day concerns, by our 20th century values. The Renaissance intellectual was more often busy with two related disciplines which we tend to forget or to dismiss: magic and the art of memory. We will be talking about these Renaissance disciplines later on; for now let me explain how the art of memory can help us become aware of who we are.

The poet Charles Olson, who was born in Gloucester, Massachussets and died in 1970, said once: "The work of each of us is to find out the true lineaments of ourselves by facing up to the primal features of those founders who lie buried in us." This expresses quite well the importance of memory and history for self-knowledge. Most of us have forgotten most of our first impressions in early childhood—the first time we encountered language, the first time we drank milk: this, too, is a brute fact of nature. But we are able to remember first impressions of later childhood and adolescence. I must clarify: when I say remember I don't mean recapture; we cannot live again any of our past experience in all its richness: what I mean is to keep oneself open to visits by ghosts, just as the old king's ghost appeared to Hamlet at critical moments to remind him to be true to his own self, even if it meant death, and even though Hamlet could not recapture or hold on to his father's ghost, for all ghosts are ungraspable. But not all ghosts are tragic. We will save tragic ghosts for a future lecture on tragedy; today I want to tell you about a personal ghost of mine, the memory of the first time I encountered mathematics.

I was fifteen, a high school student in Buenos Aires, Argentina, and all those algebraic formulas involving factorization and roots of quadratic equations were boring me to death. I flunked not a few math tests; I don't know whether this was because my teachers were themselves bored to death, or my mind was too occupied with more urgent mysteries. Be that as it may, at age fifteen I had to take Euclidean geometry, and for the first time the textbook happened to be one written by a great mathematician who was also a philosopher, an Italian, Federigo Enriques. And there, in this textbook, I had the revelation of mathematics and abstract thought. The revelation had to do with the definition of a circles:

A circle is the collection—or locus—of all points in the plane whose distance from a given point called "center" is a given quantity called "radius."

You have probably encountered the same definition, maybe in different words, when you were in high school. So far, nothing new. But the following theorem followed right after: "The center of a circle is unique. There cannot be two different points both of which are equidistant from all the points on a circle."

In what sense was this a revelation? It seems rather the contrary of a revelation, an anti-revelation, for we already know, or think we know, that the center of a circle is unique. But when I re-read the definition, there was nothing in it that said that the center is unique, no more than "Joe is a man's name" implies that there is only one Joe in the world. Well, but don't we see, don't we perceive that the center of a circle is unique? No, we cannot see this, for the simple reason that we don't see points, we cannot touch them; what we see or touch is always much larger than a point. Okay, we don't see it, but, one is inclined to argue, the fact is "intuitively clear." The problem is, whatever we mean by "intuitively clear," intuition is often a very bad judge. For example, take a yardstick: if you had asked a Babylonian or an Egyptian mathematician 25 centuries ago, can we measure any length with this yardstick, assuming we are allowed to divide it into halves, thirds, or any fraction whatsoever, he would have replied, 'Of course, it is intuitively obvious.' Yet some Greek disciples of Pythagoras discovered, to their dismay, that we cannot measure just any length with a given yardstick: there are lengths which are what they called incommeasurables and we call irrationals, although there is nothing against reason in them—on the contrary, they were discovered by using reason very subtly. So, the only way to show that there is only one center to a circle is to use reason and logic, that is, to prove it. I am not going to do it now, although it is not hard; but this was my revelation, my memorable first encounter with math: we don't rely on our senses, we use intuition, of course, for there's no thought without intuition, but we don't trust it, and we build fact upon fact, using logic as mortar and cement. This is what makes mathematical truths the most enduring: once something has been proved, it is forever true. That was, we may say, my first encounter with eternity, and it was a sweet, intense experience.

What does it mean to prove a mathematical fact, or theorem? It means to deduce it logically from mathematical facts or theorems (these are synonyms) which are already known. This presents a problem, for the theorems which are already known had to be proved themselves from other theorems already known, and so on. This is what's called an infinite regression, and it won't do. In other words, if we want to prove everything, we end up proving nothing. So the Greeks, mostly the mathematicians of the school of Plato (about whom we will say more in future lectures) devised the axiomatic method, one of the most important contributions of Greek thought, and the method modern mathematics still uses. What they did was to take some facts as axioms, which means that they should not require logical proof. But you cannot just take any bunch of facts as axioms: some requirements must be met. First, the Greeks wanted their axioms to be intuitively clear. If you smell a rat here, you are right: didn't we just say that it isn't clear what is intuitively clear? We will come back to that later on. Second, the axioms should be as few as possible, which means, in particular, that we shouldn't be able to deduce any one of them from the others. Third, they should not lead us into a contradiction, that is, a statement of the form: A is B and A is not B. And fourth, they should be enough to enable us to prove all that we would like to prove about points, lines, circles, triangles, rectangles, etc. It is never easy to come by a good set of axioms: the glory of the Greeks is that they succeeded, although later we will have to qualify what we mean by success.

There are many possible sets of axioms. The one created by the Greeks has come down to us in the most influential and longest lasting textbook ever written, Euclid's Elements. Euclid lived around 300 B.C., learned geometry from the pupils of Plato, and taught in Alexandria, a Greek city in northern Egypt. His way of presenting geometry is not quite the same as that of the textbook I read in high school: for one thing, Euclid did not use the notion of distance. I put in the Web for you the first few pages of Euclid's treatise. He starts by giving 23 definitions, of which I select some examples. "A point is that which has no part," "A line is breadthless length," "A straight line is a line which lies evenly with the points on itself." These definitions don't define much, and they have been thoroughly criticized and ridiculed over the centuries. Modern geometers do not define the most basic notions such as points, because they realize you must start with some undefined objects. Otherwise, we fall again in that horrible trap, infinite regress. This, unfortunately, is not always understood by other professionals. The other day I was browsing in the library and came across the following definition of self-esteem, in a book titled Self-Esteem, published by the American Psych. Association, 1995: "We define self-esteem as a subjective and enduring sense of realistic self-approval..." (Richard L. Bednar and Scott R. Peterson). As you can see, this is not more illuminating than Euclid's definitions of points, lines and straight lines: self-esteem defined as self-approval—great!

Next, Euclid gives us five axioms, or postulates, which I give you in a slightly different form:

  1. Given any two points, there is a straight line containing them.
  2. Given any straight line segment, we can extend it indefinitely to both sides.
  3. There is a circle with any given center and radius.
  4. All right angles are equal. (We need not concern ourselves here with the meaning of equality of angles).
  5. Given any straight line and any point not in it, there is exactly one straight line parallel to the given one and containing that point.

A few remarks: Euclid himself, and many mathematicians after him, felt that the fifth axiom was not as intuitively clear as the other four; people for many centuries, most notably beginning in the Renaissance, unsuccessfully tried to deduce the fifth from the other four. Finally, in the 19th century, it was realized that this is impossible, and indeed, by retaining the first four and changing—actually negating—the fifth, the so-called non-euclidean geometries were created, one of the most fundamental intellectual developments of modern times. I cannot fully explain in this short time why this was so important; one of the reasons is that up to the end of the 18th century the Euclidean axioms, and all the theorems deduced logically from them, were considered absolute truths: this means not only that this is how things are, but that it is inconceivable that things could be otherwise. Let's stop here for a minute. When I say, "This is how things are," I do not mean the things we see, touch, smell. Remember, we cannot see a point ("that which has no part"), not even with the most powerful microscope, even with the most delicate finger we cannot run a finger along a line ("breadthless length"), and the circles we draw are never perfect ones: they can only approximate, imitate, partake as much as they can of, the idea of a circle set forth in the definition. The doctrine that there is a world of ideas, including the mathematical ideas of point, straight line and circle but also many others such as justice, beauty, the perfect city-state, etc., which Plato tried to define as rigorously as circles, is called Platonism. That world we cannot enter with our senses but only with the mind; it is an eternal, timeless, perfect world, of which the things we perceive down here are imperfect imitations we might call it the heaven of ideas. This seemingly mystical doctrine cannot be understood unless we keep in mind its mathematical background; this is why, according to tradition, it was written at the door of Plato's Academy: "No one may enter here who does not know geometry."

Let us look at a picture, School of Athens, by Raphael. Raph2 Plato seems to be the mystic here, and Aristotle the one down-to-earth. And yet, one the most profound ironies of history is that the birth of modern science and technology around 1600 (almost a century after this picture was painted) was a reaction against Aristotle's doctrines and a going back to Platonic ones. The great early scientists, men such as Galileo and Kepler, followed Plato even in their writing style. Our down-to-earth electrical and electronic appliances, our cars and planes, wouldn't have been possible without Plato's seemingly mystic realm of ideas. We will come back to this toward the end of the semester.

During the early days of modern science the prestige of Euclid and of the axiomatic method was such that many of the greatest scientists and philosophers followed Euclid's style: in the 1600's Christian Huyghens wrote a book about clocks, and Spinoza wrote a book about God and Ethics imitating Euclid's sequences of definitions, axioms, theorems, corollaries, etc. Once the ideas of modern science took hold and became dominant, though, even if sticking to the axiomatic method, thinkers became very reluctant to entertain the notion that there is a heaven of ideas, a perfect world beyond our senses, for such notions had been much abused by state and church. This pulling away from heaven is called the Enlightment—the age culminating in the American and the French Revolutions—and its deepest philosophic voice was the German Immanuel Kant. No, said Kant, there is no heaven of ideas, or if there is, we cannot know or say anything about it. Still, Kant was very much aware of the peculiar strength of mathematical truths; how could he account for it without a Platonic heaven? Incidentally, I'll let you into a secret: Western notions of truth, from Plato down to our own day, cannot be understood unless viewed against the background of mathematics; philosophers have been either taking math as their ideal of truth or reacting against it. Anyway, here's how Kant solved the problem: the Euclidean axioms and all that follows from them are absolutely true because that's the way we think. This is not merely because our brains are wired that way, but rather because those truths are part and parcel of what we call thinking; there can be no thinking, by us or by E.T.s, where those theorems are not true. Kant gave those absolute truths an impressive name: synthetic a-priori judgments. That was around 1780. Fifty years later, the discovery or invention of non-euclidean geometries, plus subsequent developments in math and physics, showed that not all is peace and quiet in the Platonic heaven, and that, contrary to what Kant had stated,we don't have to think in such a way that all of Euclid's theorems are true. It is no coincidence that the second half of the 19th century saw the beginning of the era of relativism—which means the abandonment of the ideal of absolute truth—and, simultaneously, the era of increasing specialization in fields of knowledge and university disciplines, what we might call the era of parcelled truths. We are living in that era now.

Talking about the era of relativism, I would just like to warn you against a vulgar form of relativism which is often encountered these days in academia: it is the notion that not just some truths but all truths are socially constructed and culture-bound, so that just as we say "tomato" but the British say "tomahto," we may believe that Euclid's fifth axiom is false but the Chinese may believe it's true. This does not agree with the observed facts. Take, for example, another of Euclid's theorems, that there are infinitely many prime numbers: this has been true ever since people have cared to think about prime numbers, and it is a truth firmly held by Chinese, British, Americans and everyone else. If you want to try to show that this is not necessarily so, that there may be, with different cultural assumptions or world-views, only finitely many prime numbers, be my guest and try it but please remember, it won't be enough to point to some "primitive" culture which counts only up to three, for that is not the point: in the same way as the issue of "tomato" versus "tomahto" could arise only after the Spanish conquest of Mexico, when Europeans came in contact with that plant, the question about infinitely many prime numbers could only arise after the notion of infinity was opened to the human mind.

On the other hand, I don't want to leave you with the impression that Euclid's axiomatic method, momentous though it was and still is, produced a monolithic, unchangeable system of truths. Right after the definitions, axioms, etc., when Euclid begins to prove theorems, the very first thing he does is this: on a given straight line segment construct an equilateral triangle. (Do it on the board). Everything flows logically from the axioms, except for a crucial detail: how do we know that the two circles intersect? You may say, "Of course they do, I can see it," but remember, you cannot trust your senses or your intuition (which, in this case, would go wrong). The fact should follow from Euclid's axioms, yet it doesn't. There's a flaw, and not a minor one, in the very first theorem of Euclid's Elements. It took more than two thousand years to patch it up; only at the beginning of this century David Hilbert, a famous German mathematician, produced a system of axioms (19 if I remember right) which made all of Euclid's theorems logically consistent. Is that the last word? Who knows. Math, like all mature sciences, goes through periodic crises which shake its foundations. The intellectual life, just as life in general, is full of marvelous if often upsetting surprises: it is precisely this that makes life worth living.



  1. The best English edition of Euclid's Elements is the translation with commentary by T.L. Heath, Dover, 1956, in 3 vols.
  2. Some general histories of mathematics:Carl B. Boyer, A History of Mathematics, Wiley and Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford.
  3. You may want to take a look at Spinoza's Ethics and see how he arranges his statements as Axioms, Theorems, Corollaries, etc.
  4. On Federigo Enriques, you may consult The Encyclopedia of Philosophy, Paul Edwards, ed.
  5. If you're interested in the Renaissance Art of Memory, consult Frances Yates, The Art of Memory, Chicago U. Press, 1966.
  6. On Kant: his relevant work, Critique of Pure Reason, may be too hard for a first try. You may want to consult The Encyclopedia of Philosophy mentioned above, or E. Cassirer, Kant's Life and Thought, Yale U. Press, 1981. There are some 300 entries on Kant at the University Library.

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