It has been argued that the Pythagoreans took their philosophical foundations to be the natural numbers , and if so, the discovery of irrational numbers marks the first crisis in the foundations of mathematics. By the time of Euclid, the theory of natural numbers was grounded instead on supposedly more secure geometric notions, and placed on par with an equally rich theory of rational and irrational magnitudes.
We see these ideas in Euclid’s Elements , possibly one of the most influential books in history , and possibly the most famous textbook in mathematics. Its presentation of axiomatic geometry went largely unquestioned until the 19th century. Then, as mathematicians were becoming more attentive to logical principles, Pasch and Hilbert concluded that the text contained missing assumptions. To be truly rigorous, Pasch had declared that the process of inference in all proofs should be entirely independent of the meaning of any geometrical term , while David Hilbert, anticipating his own “formalist programme”, is famously reported to have said that “mug”, “table”, and “chair” should be substitutes for “point”, “line” and “plane” . The conclusion of this view was Hilbert’s Foundations of Geometry, in which Euclid’s five axioms became nineteen axioms, organised into five groups.
As Poincaré explained in his review of the first edition of the Foundations of Geometry , we can understand this idea of rigour in terms of a purely mechanical symbolic machine. If one feeds the axioms in as input, then all of Hilbert’s theorems should be delivered as output. Thus, we guarantee that intuition does not hide implicit axioms relating the primitive concepts, since a purely mechanical machine has no such intuitions.
Such a mechanical symbolic approach to mathematics was made in earnest by Russell and Whitehead when they wrote their Principia, albeit without the aid of a real logic machine. The task was clearly laborious. Poincaré remarked derisively that “if it requires 27 equations to establish that 1 is a number, how many will it require to demonstrate a real theorem?” . Russell admitted that he never fully recovered from the effort.
Now with modern computers and automated theorem provers, the effort is greatly alleviated. Machines can now check enormous symbolic proofs far more quickly and reliably than any human. Furthermore, the machines assist us with book-keeping and cross-referencing, and relieve us the burden of generating the tedious and simple details that Poincaré presumably considered “puerile”. In fact, with the invention of Herbert Simon’s mechanical Logic Theorist, Russell was forced to reflect on ten years writing proofs by hand as a “wasted” effort . For two excellent introductions to computer assisted formalisation, suitable for non-experts, see Hales  and Harrison .
In keeping with Poincaré’s mechanistic conclusion of logical rigour, we now assure the reader that the work in this paper has been backed up by symbolic definitions and proofs that have been fully formalised and machine-checked using the theorem prover HOL Light . However, for readability, we present our results in the more familiar, and perhaps less “puerile”, mathematical vernacular.
Now as we found when we attempted to formalise Hilbert’s theory, there are some subtle representation issues and choices to be made concerning the underlying logic . In this paper, we are concerned with just one issue: the logical status of the natural numbers.
2. in The Foundations of Geometry
Definition 4 of Book V of the Elements reads:
Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
The way this is used makes it clear that it should have been an axiom, an issue that was corrected in the Foundations of Geometry, where it appears as Axiom V, 1 along with an axiom of completeness.
V, 1 (Axiom of measure or Archimedes’ Axiom). If and are any segments then there exists a number such that segments constructed contiguously from , along the ray from through , will pass beyond the point .
V, 2 (Axiom of line completeness). An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and from V, 1 is impossible.
In this last group of axioms in Hilbert’s system, the assumed logic is substantially stronger than that assumed in the earlier groups. In the first axiom, we are told there exist natural numbers dependent on segments. Natural numbers have not been defined at this point in the text, so here, they are being treated as a logical primitive. This matter seems incongruent with Euclid’s Elements, which treated the natural numbers geometrically.
And in Axiom V, 2, Hilbert is quantifying over sets of points. But the combination of a second-order logic with a theory of natural numbers gives us second-order arithmetic, wherein we can obtain a model of the real numbers. What is the logical priority here? Is the basic logic here as strong as second-order arithmetic, or should we try to recover natural numbers from geometrical notions? To answer this question, we note that Hilbert wanted to avoid the complexity of analysis and produce a strictly synthetic theory , and he went to great length in later chapters to define an arithmetic based on magnitudes. Thus, it would seem we should define the natural numbers geometrically. The question then is where would the definition appear?
3. Group I
The first and largest group contains eight axioms to define properties of incidence. There are three primitive domains, namely points, lines and planes, and two primitive relations, one for point-line incidence, and another for point-plane incidence.
It turns out that the axioms governing these two relations can be minimally realised in four vertices , , , , six lines , , , , , , and the four planes , , and of a tetrahedron (see Figure 1). That is, we can prove that 14 objects in certain relations satisfy the axioms, and prove that the axioms commit us to at least these 14 objects.
Since finite models are possible for the first group of axioms, we know we cannot derive the existence of an infinite collection such as the natural numbers. Thus, we look to the second group of axioms.
4. Group II
The next group of axioms concern order and orientation in terms of a primitive three-place predicate of betweenness, with which we can state that one point lies strictly between two others along a line. The axioms to govern this notion are given in Figure 2.
These axioms have been heavily revised since the first edition as various redundancies were discovered. Originally, Axiom II, 2 had an additional clause asserting that we can always find a point between two others. This turned out to be derivable and is Theorem 3 in the tenth edition. Axiom II, 2 is then effectively a rigorous formulation of the second postulate of the Elements, allowing us to continue a finite straight line (or line segment) along a straight line. Unlike Euclid’s formulation, the use of the betweenness relation makes it clear that this is an axiom of order.
In the first edition, there were further redudancies and these too were later removed111This despite Hilbert’s claim in the metatheory of his first edition that his axioms were clearly independent!. The quantification in Axiom II, 3 was stronger, claiming that, given three points, there is exactly one that lies between the other two. By the tenth edition, the existence of the point was derived as Theorem 4 in a proof attributed to Wald . Finally, a “transitivity” axiom was removed to become Theorem 5, the proof due to Moore . The theorems are given in Figure 3.
After revising, the Group II axioms consist of constraints on the between relation, an axiom to extend a line segment in a given direction, and finally a planar axiom of order. This last axiom, depicted in Figure 4, was identified by Pasch as a crucial missing postulate from Euclid’s Elements. It is used heavily in the proofs of Theorem 4 and Theorem 5, each of which starts with a linear assumption, builds a planar diagram, and then uses Axiom II, 4 repeatedly to produce a linear conclusion.
Hilbert does not give a proof for Theorem 7, but it claims here that the axioms now only have infinite models. Indeed, we could presumably apply Theorem 3 repeatedly to obtain as many points as desired between two others. We start with distinct points and , and then obtain points , , , , , such that the point is between and , the point is between and , the point is between and and the point is between and .
We just need to prove that these points are distinct, and for this, we can use Theorem 5. We know that if is between and and is between and , that must also be between and . And thus, we know that these points are distinct. Repeating this line of reasoning therefore gives us a potentially infinite number of distinct points.
This is not enough to give us the natural numbers unless we can logically “carve out” the results of this process into something which we can quantify over. For us, it suffices that we can exhibit such a process as a function.
6. A Geometric Successor
The function we need is the traditional successor function, and it is possible to witness such a function as a unique object in Group II. We have chosen our witness to have two properties: firstly, it can be understood as a formalisation of Hilbert’s Theorem 7; secondly, the points obtained by the successor function are uniquely defined (unlike the potential infinity of points from the previous section, which we would have to obtain via the axiom of choice). The witness is based on the diagram in Figure 5.222Note that in the diagram, we could have switched the labels, using , , , to denote the moving point .
Formally, each diagram is a set of points , , , , and , satisfying the constraints in Figure 6. The fact that points exist satisfying these constraints is proven in Theorem 3, which uses a diagram exactly as shown in Figure 5(a).
Each diagram is represented by six points. The points , , , and are fixed under the successor function, while the point is moved closer to . Note, however, that at this stage, we only have axioms governing incidence and orientation. In particular, the first two groups of axioms are consistent with a geometry allowing infinitely small distances. So, if the point was infinitely close to , then the points would be infinitely close to . This does not affect the fact that we have a successor function, but is worth bearing in mind nevertheless: when it comes to working in the foundations of geometry, diagrams are often misleading. They represent many more assumptions than are normally in force.
The set of these diagrams is not, of course, provably isomorphic to the natural numbers. In the model of Euclidean geometry, for instance, the set of geometrically similar333Note that we do not have enough axioms at this stage to define this notion. diagrams is isomorphic to an interval of real numbers. To obtain the naturals, we use induction as in set theory or simple type theory , whereby we intersect all sets of diagrams which contain a chosen and are closed under our successor function. It is then sufficient to prove that this successor function is one-one and not onto (something normally taken as axiomatic).
Informally, and for the purposes of explaining our formalisation, we will identify diagrams by the point . In this way, when we talk of the object , we may be referring to the diagram 0, which are the six points shown in Figure 5(a) or to the point 0, which is just one of these six points. Similarly, we shall talk about the diagram that is the successor of 0, as well as the point that is the successor of 0.444In our machine-checked formalisation, these matters are necessarily unambiguous.
Thus, the successor of is the diagram obtained by replacing with , the intersection of and . To obtain the next successor, we first find the intersection of and , namely the point . We then replace with the intersection of and .
In general, the successor of a diagram is obtained by finding , the intersection of and , and then finding the intersection of and . Formally, if , , , , and are a diagram, then the successor to this diagram are the points , , , , and that provably unique such that there is a point with:
the points , and collinear;
the points , and collinear;
the points , and collinear;
lying between and ;
lying between and .
We just need to show that these points satisfy the constraints given in Figure 6. The key step is to notice that the diagrams involving the points and satisfy the same constraints. In each case, we apply Axiom II, 4 to and the line to locate a point between and . For our initial diagram, we can apply this argument directly. For the other diagrams, we just need to find the point .
To do this, we apply Axiom II, 4 to and the line , to place the point between and . We can now use the previous argument to locate between and . Finally, since is between and , Theorem 5 shows that the point must also lie between and .
With our function defined, we must show that it satisfies the crucial property of being a successor function: namely, that it is one-one but not onto (and thus that the set of points in its image are Dedekind-infinite). We can show that the function is one-one by reasoning about incidence alone, and it is clearly not onto since the point defines the first diagram, while all images of our successor function use a point which is strictly between and .
We have derived the existence of a set of natural numbers using just the first two groups of Hilbert’s axiom system. Ordinarily, the existence of such a set would be taken as axiomatic, as is the case in ZF set theory555Strictly speaking, the axiom of infinity in ZF set theory merely says there is a set containing zero and closed under successor.. But this is not necessary, and was sometimes considered logically questionable. In the Principia, Russell declared that the existence of an actual infinity of natural numbers was an empirical, not logical matter. Hilbert himself would later argue that the infinite is merely a useful device, and not a logical given .
But in Hilbert’s geometry, we can understand natural numbers as just another geometric concept. They are, in our formalism, a set of diagrams involving a triangle, each generated from its predecessor by applying Axiom II, 4. Thus, when Hilbert mentions natural numbers in his text (such as when formalising Axiom V, 1) we can understand him to be implicitly using a derivative geometric notion, one that we have now explicated based on his first two groups of axioms.
This makes explicit a gap in the Foundations of Geometry, and helps us realise synthetic geometry as a true foundation for number theory, as it was understood by its predecessor, the Elements.
-  John R. Anderson. Obituary: Herbert A. Simon (1916–2001). American Psychologist, 56:516–518, 2001.
-  Carl B. Boyer. A History of Mathematics. John Wiley & Sons, 1991.
-  Alonzo Church. A Formulation of the Simple Theory of Types. Association for Symbolic Logic, 5:56–68, 1940.
-  Thomas Hales. Formal proof. Notices of the American Mathematical Society, 55:1370–1380, 2008.
Technical Report 36, Turku Centre for Computer Science (TUCS),
Lemminkäisenkatu 14 A, FIN-20520 Turku, Finland, 1996.
Available on the Web as
-  John Harrison. HOL Light: a Tutorial Introduction. In Proceedings of the First International Conference on Formal Methods in Computer-Aided Design, volume 1166, pages 265–269. Springer-Verlag, 1996.
-  Thomas L. Heath. Euclid: The Thirteen Books of The Elements. Dover Publications, 1956.
-  Henri Poincaré. Poincaré’s Review of Hilbert’s “Foundations of Geometry”. Bulletin of the American Mathematical Society, 10(1):1–23, 1903.
-  Henri Poincaré. Science and Method. Cosimo Classics, 2007.
-  David Hilbert. Foundations of Geometry. Open Court Classics, 10th edition, 1971.
-  David Hilbert. On the Infinite. In Paul Benacerraf and Hilary Putnam, editors, Philosophy of Mathematics: Selected Readings. Cambridge University Press, 1984.
-  Hubert C. Kennedy. The Origins of Modern Axiomatics: Pasch to Peano. The American Mathematical Monthly, 79(2):133–136, 1972.
-  Wilbur R. Knorr. The Evolution of the Euclidean Elements. Springer, February 1974.
-  Ulrich Majer. The Relation of Logic and Intuition in Kant’s Philosophy of Science, Particularly Geometry. In Emily Carson and Renate Huber, editors, Intuition and the Axiomatic Method. Springer, 2006.
-  Laura I. Meikle and Jacques D. Fleuriot. Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In Theorem Proving in Higher Order Logics, volume 2758, pages 319–334. Springer, 2003.
-  Eliakim H. Moore. On the Projective Axioms of Geometry. Transactions of the American Mathematical Society, 3:142–158, 1902.
-  Hermann Weyl. David Hilbert and his Mathematical Work. Bulletin of the American Mathematical Society, 50:635, 1944.