Chapter 1 — Fundamental Concepts¶
1.11 — Number, Axiom and Certainty¶
A proof closes a mathematical question. It does not always close the question that made it necessary.
The hole no rational fills¶
Let us begin with a concrete problem, because big questions almost always begin with a small detail. A set of numbers is bounded above when there is some number, it does not matter which, that is greater than or equal to all its elements: a ceiling. The supremum is the tightest of all those ceilings, the least upper bound, the one that leaves no room to spare. If the set is, for example, that of the rationals less than \(5\), the supremum is exactly \(5\): no smaller number would serve as a ceiling, and any larger number would be an unnecessarily high one.
Now, the set that really matters: the positive rationals whose square is less than \(2\). It is bounded, because no element reaches \(2\): if a rational were greater than or equal to \(2\), its square would be greater than or equal to \(4\). It therefore has rational ceilings to spare. What it lacks is the tightest of them. Any rational proposed as the supremum can be refuted. If its square is less than \(2\), there is another rational a little larger that is still in the set, so it was not a ceiling. If its square is greater than \(2\), there is another rational a little smaller that is still a ceiling, so it was not the tightest. And no rational has square exactly equal to \(2\). The set is left with infinitely many rational ceilings and none that is the best. That is the hole. It has nothing to do with the distance between rationals, which are as close to one another as one likes: it has to do with the fact that, seen from above, the staircase of possible ceilings has no last step.
The real numbers fill that hole by decree. It is postulated that every nonempty set bounded above has a supremum, just as it is postulated that addition is commutative. Within the system of axioms it is derived from nothing: it is one of the starting points. It is a decision about what kind of object the numerical continuum is meant to be. But stopping at the decree leaves a discomfort, and in the nineteenth century two mathematicians tried to replace it with a construction.
Dedekind wanted to construct the number that fills the hole using only rationals. His idea was to split the rationals into two classes: those lying below \(\sqrt{2}\) and those lying above. But "below \(\sqrt{2}\)" presupposes that \(\sqrt{2}\) already exists, which is precisely what cannot be taken for granted. The way out was to describe the partition without ever naming \(\sqrt{2}\). The lower class consists of the negative rationals, zero, and the positive rationals whose square is less than \(2\), the same set as before; the upper class, of the positive rationals whose square is greater than \(2\). Between them they contain all the rationals, because none has square equal to \(2\), and no rational sits on the border: the lower class has no maximum and the upper class has no minimum. That partition, the cut, is declared by definition to be the new number. \(\sqrt{2}\) stops being a quantity that is sought and becomes the very border between "less" and "greater".
Cantor, along with Méray and Heine in the same years, solved the problem by a path that at first sight has nothing to do with it. Instead of splitting the rationals into two classes, he looked at sequences of rationals whose terms come closer and closer to one another, such as \(1\), \(1.4\), \(1.41\), \(1.414\), \(1.4142\), and so on. The property that matters is precise: for any distance one fixes, however small, from some place on any two terms of the sequence are closer than that distance. It is not enough for consecutive terms to come closer; they must all come closer to one another. A sequence with that property is called a Cauchy sequence. The one above comes ever closer to a place where no rational is waiting. Cantor defined the new number from the sequence itself, or more exactly, from the class of all Cauchy sequences that end up coming close to one another, since there are many that approach the same place. \(\sqrt{2}\) stops being a quantity and becomes a process: the infinite promise of an ever better approximation.
A cut that separates two classes; a sequence that approaches without ever arriving. They are objects of completely different nature, built with different tools. Yet when all of arithmetic and order is rebuilt with either of them, one obtains a system that satisfies all the axioms of §1.1, including the supremum axiom, which now stops being a decree and becomes a theorem. And there is more: one can prove that any two systems satisfying those axioms have exactly the same structure, so that one translates into the other without losing anything. It is not that Dedekind and Cantor reached the same place by different paths. It is that there are two genuinely different objects, since a cut is not a sequence, that turn out to be indistinguishable in everything mathematics needs to use them for.
And here the question arises that no later computation resolves:
in completing the rationals, whether by cuts, by sequences, or by the axiom that simply demands it, are numbers discovered that were already there, waiting, or is an object constructed that did not exist before?
If they are discovered, \(\sqrt{2}\) existed before anyone proved it, just as a mountain exists before it is climbed, and the cut and the sequence are merely two different maps of the same territory. If they are constructed, the numerical continuum is an artifact: it depends on a decision about how to fill the gaps, and that decision could, in principle, have taken another form. The supremum axiom takes no side between these two readings; it works the same under either. But choosing to postulate it, instead of deriving it from a construction, is already a stance on what kind of object one wants it to be.
The deductive form and what it hides¶
Almost all the mathematics written today follows the same script: first a definition, then a statement, then its proof. It is worth pausing over that script, because it is not the only possible way of presenting mathematics; it is a choice so old that it has become almost invisible. Euclid fixed it more than two thousand years ago: first one declares which objects one is going to speak about, then certain starting principles are accepted without proof, and only then are the propositions that follow from them built up, step by step.
Let us look closely at what a proof of this kind does. Take the irrationality of \(\sqrt{2}\): one assumes the opposite of what one wants to prove, follows a chain of forced steps and arrives at a contradiction, so that the initial assumption is discarded. The proof is impeccable: each step rests on the previous one, with no room for doubt. But let us note what it does not tell. It does not say how someone first came to suspect that \(\sqrt{2}\) might not be rational. It does not show the failed attempts, the dead ends, the intuition that led to trying precisely parity and not some other property. The proof reconstructs the shortest path for convincing another person, once the answer is already known. That short path almost never coincides with the real path, long and groping, by which someone first reached it.
This leaves an open question:
does the axiomatic-deductive form present mathematical knowledge as it was discovered, or only as it can be justified once discovered?
Neither answer is comfortable. If the deductive form systematically hides the real process, with the mistaken conjecture, the example that refutes it and the correction that follows, then what remains in view is the facade of each result, not the building that had to be raised and torn down several times to reach it. But that same facade, orderly and with no trace of the earlier groping, is what allows anyone, anywhere, to check a result without repeating the whole process of discovery. Perhaps the tension between finding something and justifying that it is true is not a problem to be solved, but a structural feature, uncomfortable but productive, of how mathematics has been written for the last two thousand years.
Translating, not just computing¶
To say that an equation is a circle, that \((x-h)^2+(y-k)^2=r^2\) and the round figure drawn in the plane are simply the same thing, is a gesture so habitual that it no longer draws attention. One does not say "this equation describes a circle" or "this equation corresponds to a circle": one uses the verb to be. It is worth pausing over that verb, because it was not always natural to use it this way.
Before the seventeenth century, geometry and algebra were, in a fairly literal sense, two different mathematics. Greek geometry worked with figures, with ruler and compass, with ratios between magnitudes; the algebra that developed later worked with symbols, equations and unknowns. It was not a lack of ingenuity that kept them apart: what was missing was the bridge. That bridge, the idea of describing each point of the plane by a pair of numbers, was the work of Descartes and of Fermat, almost at the same time and by independent paths, and it turned out so complete that today it is hard to conceive of geometry without it.
But the bridge does not by itself answer a question that remains underneath it, unresolved. When one says that an equation is a circle, is it the same object seen from two angles, like a person seen from the front and in profile, or two genuinely different objects, one algebraic and the other geometric, that a very well built translation makes seem interchangeable? The difference is not a technicality. If they are the same object, algebra adds no information that geometry did not already have; it only makes it easier to handle. If they are two objects connected by a translation, then each time one passes from an equation to its graph, or from a figure to its equation, one is not simply changing notation: one is crossing a border between two different ways of being a mathematical object.
In practice, this chapter made a decision. Distance in the plane and the line were first presented with arguments from Euclidean geometry, Pythagoras and similar triangles, and then algebraic formulas were adopted as definitions. From that moment on, the circle and the line of this book are, by definition, sets of pairs of numbers. That decision allows everything to be proved with algebra, but it does not answer the question: it only chooses which side of the bridge to work on.
What a graph cannot do, then or now¶
There is a distinction that sounds like simple practical advice, that a graph suggests and checks but does not prove, and that, followed calmly, leads much further than any calculator lets one see.
At the end of the nineteenth century, Hilbert proposed a way of understanding axioms that was disturbing at the time. Until then it had been thought that words like "point" or "line" had a fixed, intuitive meaning, and that the axioms described truths about those already known objects. Hilbert reversed the order: the axioms do not describe previously understood objects, but define, through the relations they require among them, anything that may be called a "point" or a "line". According to a well-known anecdote, he said that his geometry should be statable just as well by replacing "point, line, plane" with "table, chair, beer mug", as long as those objects satisfied the same relations required by the axioms. What matters is not what the objects are, but how they relate to one another.
That same move, of axioms that do not describe something already understood but fix the rules of behavior of whatever one decides to call by that name, is what makes the addition and multiplication of real numbers characterized by commutativity, associativity, distributivity and the existence of identities and inverses, without any of those rules saying what a real number is. Certainty, in this picture, stops resting on a prior intuition about the meaning of the symbols and comes to rest entirely on the rules being satisfied.
That idea, that the validity of a piece of reasoning can be checked without appealing to any intuition about the meaning of the symbols, by verifying only that each step respects the rules, is what today allows something that in Hilbert's time was science fiction: an entire proof can be checked, symbol by symbol, by a computer program, without the program understanding, in any reasonable sense of "understand", what it is talking about. Systems such as Lean, with its library of verified results called mathlib, do exactly that: they check that each step of a proof follows from the previous one according to fixed logical rules, with no intuitive shortcuts. The proof that \(\sqrt{2}\) is not rational, the first one in this chapter, has already been formalized this way in more than one of these systems.
With that possibility comes a family of questions that neither Euclid nor Hilbert could have asked in these terms:
is a proof verified by a machine understood by anyone, or simply guaranteed? Are they the same thing? If a formal proof is correct step by step, but no person manages to extract from it why the result is true, if the proof certifies without explaining, in what sense is it still a mathematical proof, and not simply a very long computation that happened to come out true?
The machine can confirm that something is true. Understanding why it is true remains, for now, work done by a person.
What remains open¶
None of the questions raised here has, in this book, an answer. The mathematics built throughout the chapter, with its numbers, powers, expressions, equations, inequalities, analytic geometry and lines, is solid in the sense in which a proof is solid: each step follows necessarily from the axioms and the definitions adopted. But that internal solidity does not say, by itself, whether what was built was waiting to be discovered or was invented. It does not say whether the way it was presented reveals the real path by which those results were reached, or only the path that best justifies them. It does not say whether an equation and the figure it describes are the same object or two translated objects. And it does not say what relation the certainty of a proof written by a person bears to the different certainty that a machine can offer today. All four questions remain open.