Manuscripts on the Foundations of Geometry
In this section we present a collection of texts written by Leibniz on several topics in elementary geometry. Leibniz worked at the foundations of geometry throughout his life, and left many hundreds of pages and manuscripts on this subject. Almost none of them were published in his lifetime, and only a few are currently available in print. Leibniz’ studies on the foundations of geometry deal with a vast array of topics: history of mathematics, commentaries on Euclid’s Elements, philosophical discussions on definitions of basic geometric figures, new developments of axiomatics, formalization of geometry through different symbolic systems (a characteristica geometrica), new proofs of well-known theorems, discussions on diagrams and their role in mathematics, foundations of higher mathematics in elementary geometry, and much more. Sometimes, Leibniz labeled the whole of these researches with the general name of analysis situs, and this new discipline stays at the crossroads of mathematics, logic, epistemology and metaphysics of space. Leibniz’ analysis situs is likely to be the highest point of foundational research in mathematics in the 17th century.
Ars Representatoria
Overview of Ars Representatoria (1691)
The Ars Representatoria is a general and programmatic paper interested in problems of characteristics. Leibniz reaffirms the starting point and purpose of his analysis situs, meant to avoid the use of extrinsic algebraic proceedings in geometry, without however having to give up the advantages of the characteristica speciosa and go back to the elegant, but complicated and unproductive, synthetic geometry. It is a matter of reforming algebraic formalism and finding another one that may be proper to geometry. Although it can be blindly manipulated, such formalism must have this advantage on ordinary algebra: that, when interpreted, it immediately offers the geometrical object to our imagination. It is therefore a cogitatio caeca that promises a vast view. The discussion in the Ars Representatoria in many respects anticipates Leibniz’s late essays on the theory of knowledge and interlocks with the theory of the imaginatio distincta that a little earlier Leibniz began to envision as the solution to several epistemological questions in mathematics.
Specimen Analyseos Figuratae in Elementis Geometriae
Overview of Specimen Analyseos Figuratae in Elementis Geometriae (1683)
The Specimen analyses figuratae offers a detailed analysis of the first proposition of Euclid’s Elements, teaching how to construct an equilateral triangle. This proposition had been the object of foundational analysis innumerable times before Leibniz, and since antiquity several issues were found in its demonstration. In particular, the early modern discussion on this proposition had concentrated on the implicit assumption of Euclid that the two circles drawn in the construction of the problem should have a point in common. This assumption seemed to require some discussion on the continuity of the circles themselves, as the ground of the existence of their point of intersection. Already in 1532, Oronce Fine had added an intersection axiom to Euclid’s original principles in order to bridge this demonstrative gap, and since then several 17th-century mathematicians recognized the need of an axiomatization of continuity. Leibniz will bring this kind of reflections to a further level of refinement. The demonstration of Elements I, 1, moreover, had been at the center of the modern logical debate on the possibility of reducing Euclidean demonstrations to syllogistic chains. This was an important epistemological problem in the Renaissance, that touched the methodology of science and the possibility of reconciling logic and mathematics. Semi-formal proofs of Elements I, 1, through four syllogisms, had been given by Piccolomini, Herlinus and Clavius in the 16th century (drawing on Greek materials). In the 17th century, Jungius had extended syllogistic logic with a few non-syllogistic inferences, and had produced a remarkable analysis of the proof of this proposition, reshaping it into a chain of twenty-one elementary inferences. Building on these researches, Leibniz offers here a truly remarkable example of a detailed proof composed by forty inferential steps grounded on explicit logical and geometrical principles.
Scheda
Overview of Scheda (mid–1690s)
This "Scheda" from the mid-1690s, which Leibniz himself characterized as "good" amidst many other writings of the same period, provides a good overview of the foundations of analysis situs in this period. In it Leibniz presents his own thinking in deductive form (which is uncommon, but not altogether rare in writings on geometry), thus showing which propositions he considers more elementary and which are derivable from previous ones.
In particular, it is relevant that Leibniz assumes here as the initial term of the whole essay the notion of an extensum, which is defined as a coexistent continuum (a definition that had already appeared in the first writings on Characteristica geometrica). The notion of a point, however, which in the 1680s had been coordinated as a second primitive to that of an extensum is seen here as derived from it.
Some fundamental ideas appear in the text that often recur in the writings of the time. Principal among these is probably that all points have situational relations to each other, and therefore they all belong to the same all-encompassing extensum, which Leibniz here calls "Universal Space." Here Leibniz also develops some fundamental ideas for continuity, and mainly that in any continuous extensum it is always possible to take another continuous extensum as its own part (today this is an important axiom of the open sets of a topological space). Finally, the distinction between homogeneity and congeneity recurs in the text, which is a topic Leibniz particularly studied in these years. Congeneity is a more general relation than classical homogeneity (which has various definitions, but generally denotes geometric objects of the same dimension), and Leibniz calls congeneous those figures that can be transformed into each other through continuous motion (a point and a line, for example). With such a notion he wants to give a characterization of what is properly spatial (an instant is not congeneous to a geometric line, whereas points, surfaces and solid bodies are), and at the same time to have a notion useful for discussing some classical foundational difficulties (a curvilinear angle is not congeneous to a rectilinear angle). The concept recurs with the name "homogonum" or with the Greek συγγενές in other essays of the period.
De curvis similibus et similiter positis et parallelis
Overview of De curvis similibus et similiter positis et parallelis
The text discusses the notion of parallelism in relation to curves. This is the concluding text in a series of studies by Leibniz on parallelism, in which Leibniz had defined parallelism as equidistance. The problem Leibniz saw with this notion is that in general a curve equidistant to a given curve is not similar (in the sense of Euclidean similarity) to the original curve. This had been noted by Leibniz about conic sections, and had to do with the transformations by evolution that Leibniz (following Huygens) was studying in the 1680s. The present essay offers a full discussion of this topic.
The fundamental problem underlying these Leibnizian investigations, however, was the provability of the famous Parallel Postulate. Many attempts to prove this postulate, in fact, were based on a definition of parallelism through equidistance, and then in the assumption that the line equidistant to a straight line is also a straight line. This last assumption is false in non-Euclidean geometry, and if one assumes it, it actually becomes possible to prove the Parallel Postulate. Giovanni Alfonso Borelli, in his Euclides Restitutus (1658) had noted that therefore several demonstrations of the Parallel Postulate (such as that of Clavius, 1589) were in fact petitiones principii.
Leibniz treats the argument from a more general point of view, and showing that in curved lines the equidistant curve is not similar to the original curve, he exposes an argument to doubt that the equidistant from a straight line is also a straight line—thus concurring with Borelli on the need to reject as incomplete those demonstrations of the Parallel Postulate.
Leibniz’ Notes to Arnauld’s Nouveaux elemens de geometrie
Overview of Leibniz’ Notes to Arnauld’s Nouveaux elemens de geometrie
The Nouveaux Éléments de Géométrie were published by Arnauld in 1667, and in a second edition in 1683. It is a work that made school in the study of the foundations of geometry, and had a vast influence on French and European geometry in general. Arnauld's approach to the foundations of geometry was, however, very different from Leibniz', and was based on the assumption of a large number of principles and axioms that Arnauld considered "self-evident" and therefore impossible to prove. Leibniz discussed this work many times during his lifetime, and frequently noted that Arnauld had assumed propositions that in his opinion needed to be proven in order to establish the foundations of geometry. These reading notes testify to the attention with which Leibniz read this fundamental text, and they are the origin of a large number of essays and subsequent observations on the foundations of geometry.
Duae rectae parallelae sunt
Overview of Duae rectae parallelae sunt
In this important essay, Leibniz tries to exploit a phenomenological characterization of parallel lines, in connection with the combinatorial techniques of his characteristica geometrica, to prove the Parallel Postulate. He starts with a new definition of parallels (“ubique eodem modo se habere”), and characterizes the definition perceptually by imagining that an observer moves along the straight lines X̅ looking at the line Ȳ, while another observer moves along Ȳ looking at X̅, the both of them being in uniform motion. He remarks that everything is the same: the two observers perceive no change while moving (each of them always look at the same straight line from equivalent points of view), nor are the perceptions of one observer different from those of the other. This should be the very meaning of the definition of parallels. The grounds of this indiscernibility may be found in the similarity of all straight lines to each other; but also their reciprocal position (situation), that we have to investigate. Thus, if we consider metric properties and distances, we should say that in case X̅ and Ȳ are parallels (under this definition), and if the first observer moves from X1 to X2 with uniform motion, and the second from Y1 to Y2 again uniformly (and thus the lengths of their trajectories are equal, X1X2=Y1Y2), then we have: X1.Y1.X̅.Ȳ≃X2.Y2.X̅.Ȳ. This last formula should express the peculiar property of symmetry that we find in parallel lines, stating that the reciprocal situations of the two observers among themselves and in relations to the two whole lines that they are tracing (their trajectories) are indiscernibles, and thus congruent (≃). Starting with this key-formula derived by his consideration about indiscernibles, Leibniz goes on to state a common axiom of his characteristica geometrica, that similar determinants produce similar determinates. Then he goes on, claiming that the above formula can be reduced to X1.Y1≃X2.Y2, through a kind of simplification of equal situational relations (eliminating X̅.Ȳ from both sides, thanks to the axiom), and the congruence itself implies the equality of the lengths of the segments: X1Y1=X2Y2; which means that the lines are equidistant. Thus, starting from the definition of parallel straight lines as lines that have everywhere the same reciprocal situation, passing through some phenomenological considerations about the indiscernibles and a couple of combinatorial passages, Leibniz proved that two such lines are in fact equidistant. This amounts (as we know) to a kind of proof of the Parallel Postulate, applied to straight lines identically situated (instead of to non-intersecting lines). In fact, Leibniz immediately deduces (again through combinatorics) the parallelism of the two equal transversals X1Y1 and X2Y2 and the identity of the internal and external angles of these with X1X2 (Elements I, 29, equivalent to the Parallel Postulate), the existence of rectangles (a statement again equivalent to the Postulate) and Playfair’s Axiom on the uniqueness of the parallel line through a point. He concludes with a restatement of the formal definition of parallel lines, saying that if X̅ is a straight line and A a point outside it, then if A.X̅≃Y.X̅, the set Ȳ is the straight parallel line passing through A. Leibniz has proved that two straight lines which are parallel by his new definition are also equidistant; and thus the Parallel Postulate can be deduced from their definition (as it should be, Leibniz thinks). Once again, Leibniz should prove (as in the case of the definition through equidistance) that straight lines enjoying this new property of parallelism are possible and real. It should be clear, however, that Leibniz’ final goal is now closer, as he has a definition of parallel lines which is half-mathematical (or kinematical) and half-perceptual, and it fits very well with his plans of proving the Postulate through a property of space itself. In other words, the internal possibility of straight lines having those (still a bit vague) properties of uniformity needed to ground the Parallel Postulate, seems to follow from a property of symmetry of space itself, or perhaps (which is not too different for Leibniz) from a feature of our perceptual abilities. It is interesting to remark, even though Leibniz does not further develop this connection with physics and phenomenology, that the symmetry of space hinted at by his definition of parallel lines is that two observers in inertial motion (e.g. uniform and straight motion) may have such trajectories (e.g. parallel lines) as to be reciprocally undistinguishable.