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Mathematics, Signs and Truth

05 Apr Mathematics, Signs and Truth

Otto Maier and his theorem about waves, reality and time curves are rooted in the works of the men he looked up to, Leibniz and Descartes.

But who was Leibniz? 

Gottfried Wilhelm Leibniz (1646–1716) lived an extraordinarily rich and varied intellectual life in troubled times. Although remembered as a great thinker, he was a man who more than anything else wanted to do certain things, namely to improve the life of his fellow human beings through the advancement of science, and to establish a stable and just political order in which the divisions amongst the Christian churches could be reconciled. A surprising number of his apparently miscellaneous endeavours were aspects of a single master project tenaciously pursued throughout his life: the systematic reform and development of all the sciences, to be undertaken as a collaborative enterprise supported by an enlightened ruler. These theoretical pursuits were in turn ultimately grounded in a practical goal: the improvement of the human condition in a well-ordered world, and the celebration thereby of the glory of God in his creation. The effectiveness of natural language notwithstanding, as a tool for the advancement of science, nothing could be more powerful, Leibniz thought, than a science of signs broadly conceived. Algebra, arithmetic, and their applications to the physical world were prime examples of systems of symbols which, through rigorous rules for their manipulation, empowered discovery. Leibniz conceived his mathematical studies and his invention of the infinitesimal calculus (that is, an algorithm or sequence of rules for handling infinitesimal magnitudes) as part of this broader project.

A science of signs

While for Newton the calculus was basically a brilliant way to solve certain difficult mathematical problems, Leibniz regarded it as a sample of what he was hoping to achieve in logic for thought in general. In a letter of 28 December 1675 to the secretary of the Royal Society, Henry Oldenburg, penned shortly after his discovery of the infinitesimal calculus, Leibniz linked his new ‘algebra’ to the characteristica universalis:

This algebra is only part of that general theory. Yet it ensures that we cannot err even if we wish to … To be sure, I recognise that whatever algebra supplies of this sort is the fruit of a superior science which I am accustomed to call either Combinatory or Characteristic, a science very different from what might at once spring to one’s mind on hearing these words.

Leibniz’s discovery of the calculus matured during his period in Paris (1672–6). In the 17th century, the mathematical study of motion and, in particular, the issue of how to measure curves, had become central to the revolutionary programme of providing mathematical accounts of natural phenomena ushered in by the new quantitative physics championed by Galileo Galilei (1564–1642). The calculation of the area of the surface intercepted by a plane curve (a problem known as the ‘quadrature’ of a curve), and the determination of a tangent to a given plane curve, leading to the calculation of the velocity of a body at a given instant, took centre stage in mathematical studies aimed at providing a general method of calculation applicable to all kinds of known curves and variable quantities.

One of the milestones on the road to the discovery of the calculus was Leibniz’s generalization of the method of calculation introduced by Blaise Pascal (1623–62) in his work on the quadrature of the circle. As Leibniz triumphantly noted in a text of 1673, ‘the whole thing depends on a right-angled triangle with infinitely small sides, which I am accustomed to call “characteristic”, in similitude to which other triangles are constructed with assignable sides according to the properties of the figure. Then these similar triangles, when compared with the characteristic triangle, furnish many propositions for the study of the figure, through which curves of different kinds can be compared with one another’.

The characteristic triangle

The characteristic triangle (ds, dy, dx). Building on these results, a few months later Leibniz formulated a general ‘transmutation theorem’.


Leibniz’s commitment to the characteristica universalis and the scientia generalis ultimately rested on his conviction that logic is a mirror of the structure of REALITY. In his view, the principles which govern thought were also the principles that govern reality. In turn, reality meant for Leibniz first and foremost God, the eternal and infinite Being encompassing all perfections. It is from him and his eternal thoughts that the story of the world in which we find ourselves begins. Logic therefore led via metaphysics to philosophical theology.

According to Leibniz, the first, fundamental principle which governs both the ideal order (the sphere of thought, studied by logic) and the real order (reality, studied by metaphysics) is the principle of non-contradiction: for any proposition ‘p’, ‘p’ and ‘not-p’ cannot both be true at the same time in the same respect. In other words, it cannot both be true at time t that I am such and such (whatever that is) and that I am not such and such. It cannot both be true that I am, right now, wearing my glasses on my nose and not wearing my glasses on my nose, even if I may not know which one of these alternatives is true.

As Leibniz puts it: Primary truths are those which assert the same [thing] of itself, or deny the opposite of its opposite. As A is A, or A is not not-A. If it is true that A is B, it is false that A is not B, or that A is not-B. Also: every thing is what it is. Each thing is like itself, or equal to itself. Nothing is greater or less than itself—and others of this sort which, though they may have their own grades of priority, can all be included under the name of ‘identities’. (A VI, 4, 1644)

It must be stressed, however, that for Leibniz the mental reality of non-existing possible beings does not ultimately depend on their being thought by individual human minds. One could take the view (as Leibniz did) that there are some truths (such as mathematical truths) which exist in some sense, or have some ‘reality’ of their own, even if no human being ever thinks of them.

Leibniz wrote in another note of 1677:

It is true, and even necessary, that the circle is the largest of isoperimetric figures. Even if no circle really existed. Likewise even if neither I nor you nor anyone else ever existed. … Since, therefore, this truth does not depend on our thought, there must be something real in it.


Otto Maier and Leibniz

Otto Maier and Leibniz

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