We know that Kurt Gödel (1906–1978) worked intensively with Leibniz (1646–1716) towards the end of his life. Gödel was obsessed with Leibniz. When some people had destroyed some of Leibniz’s writings, Menger asked Gödel, “Who could have an interest in destroying Leibniz’s writings?” Gödel would say, “Naturally, those people who do not want men to become more intelligent!” (Menger, 1994). When his friends advised him to focus on his studies, he would ignore them rather than studying and reading Leibniz’s works. Eventually, the expected happened, and Gödel continued to follow Leibniz’s footsteps and gave God’s ontological proof like Leibniz.
In this article, I will refer to Leibniz’s work and interpretations, such as his characteristica Universalis and the binary number system, to give the reader an idea of the part of his work, particularly about mathematical philosophy. I will also explain how Leibniz understood the concepts of proof and analytics. Finally, I will focus on Leibniz’s role in mathematics within the framework of theology and metaphysics/philosophy.
The Latin term characteristica Universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by Gottfried Leibniz to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create a language usable within the framework of a universal logical calculation or calculus ratiocinator. (Wikipedia, 2019)
Leibniz was aware that political or philosophical debates and research did not follow a mathematical method. According to Leibniz, mathematicians are also likely to make mistakes like everyone else, but they also have some tools to discover their mistakes. However, philosophers don’t have the same tools as mathematicians, so they tend to make more mistakes. While there are Aristocians or Platonists in philosophy, there are no ‘Euclidean’ or ‘Archimedean’ in mathematics [God and Mathematics in Leibniz’s Thought, Mathematics, and the Divine: A Historical Study, pages 485–498]. According to Leibniz, it is necessary to mathematize the thought to end the quarrels dominated by feelings rather than righteousness. In order to formalize a significant part of thought, the symbols and rules are required to emerge in mathematics. As Leibniz explains in his Preface to a Universal Characteristic, characteristica Universalis will reveal our thinking’s alphabet and analyze the basic concepts. Based on those concepts, all things will be judged definitely [Leibniz, G. W. Philosophical Essays, pages 5–10]. Thus, there will be no need for clashes between philosophers who advocate two different views; they will sit next to each other and say “calculemus!” (let’s calculate). They will be able to calculate the accuracy of their thoughts!
Leibniz’s idea of characteristica Universalis is a type of computational formulation. This thought was based on matching basic or irreducible thoughts with prime numbers. A number characterizes every basic thought: the characteristic number. Let us cite Leibniz’s example in his article on Samples of the Numerical Characteristics [Leibniz, G. W. Philosophical Essays, pages 10–18]. Suppose we are given the pairs of numbers (13, 5) and (8, 7), which refer to the basic concepts of “animal” and “rational,” respectively, in response to the proposition “Man is a rational animal.” The number that characterizes the concept of “man” will be ([13 · 8], [-5 · 7]) = (104, –35). Since there is an infinite number of prime numbers, a number or a pair of numbers or a triplet of numbers can be assigned to all basic or fundamental concepts. Other compound concepts can be obtained as prime numbers, and an entire language can be mapped.
Binary Number System
Before Leibniz, the binary number system was known, but Leibniz was the first to systematically and maturely record it. In a letter, Leibniz wrote about how he dealt with creating everything out of nothing and the binary number system. That is an example of the masterful interplay of theology and mathematics (and even physics) in Leibniz, as I shall mention later.
Leibniz designed a metal medallion (coin) on the creation and the binary system. The medal has the following expressions: Imago creationism (a picture of creation), Omnibus ex nihilo ducendis sufficit unum (In order to produce everything out of nothing, one [thing] is sufficient) and Unum est necessarium (There is need of only one thing). Leibniz, following the Pythagorean doctrine, claimed that the origin or essence of everything was a number. As is well known, in the binary number system, all numbers can be expressed using 0 and 1. Interpreting 0 as “nothingness” and one as “God,” Leibniz claimed that the binary system symbolized creation so that everything could be expressed in this system. For Leibniz, everything is a mixture of 0 and 1. According to this, all things have come from the One, God.
For Leibniz, the binary number system revealed the beauty and perfection in God’s creation. Any single number in the binary system may not appear to be beautiful, but the beauty seems due to the order within the overall system when they are written one under the other. Similarly, there may be things in the world that we don’t like singularly, but when we get the right perspective, we see that it is perfect.
Leibniz’s number mysticism does not end there; he said other things such as “God loves odd numbers.” Since we do not want to extend this issue, we will be satisfied with one last example. Leibniz says that the seventh day after creation is a non-zero (“perfect”) number in the binary system, adding on to the many numerical analogies that have been made on God’s creation of the world in six days. It also states that 111 points represent the Trinity [God and Mathematics in Leibniz’s Thought, Mathematics, and the Divine: A Historical Study].
Modern Proof Concept
As science philosopher Ian Hacking has shown, Descartes did not know what proof was in a contemporary sense. Leibniz had a much closer thought to modern proof (Hacking, 2002). He considered Descartes’ mathematical accuracy independent of proof. For Descartes, even if an exact thing is not proved, it is by itself true. Therefore, the truth value of something and the proof given to it are not related.
Let us also recall that Descartes is not seeking proof, but rather practical methods that give new mathematical results. What evokes the concept of modern proof is that Leibniz realizes a proof is valid, not due to its content, but because of its form. Accordingly, the proof is a finite number of sequences of certain sentences according to specific logic rules, starting with particular identities. If we recall Descartes’ method, he attaches great importance to intuition when he is collecting new information, whereas in a Leibnizian perception of proof, what is essential is to find “mechanical” proof of the sentence that we have.
The ideas of his time probably influenced the idea of proof presented by Leibniz. As Hacking says (Hacking, 2002 pp. 202), it is customary to find and remove a person who shook the thoughts deeply before him in each period; Leibniz plays the role of such a person for his time.
There is a plausible explanation provided by Leibniz about the emergence of the idea of proof during his time. It is difficult to reach the concept of modern proof when the geometry is taken as a measure of precision: this is because geometric proofs are based mainly on their “content.” Such proofs’ validity is determined by their conformity with the known properties of the geometric object being studied. With Descartes’ algebraization of geometry, a way for the proofs to be transformed into a formal form was opened.
The predicate statements or identical to the subject or the subject containing the predicate are called analytic. For example, when we say “all people are alive,” for Leibniz, we mean that the concept of being alive is within the idea of being human [Leibniz, G. W. Philosophical Essays, page 11], so this statement is analytic. According to Leibniz, all mathematical truths are logical.
It is well known that Immanuel Kant (1724–1804) introduced the analytic-synthetic distinction by diligently transforming Leibniz’s concept of reality. According to Kant, analytic a priori knowledge is information obtained only by using logic. Synthetic a priori is the information obtained by using time and space intuition. According to Kant, arithmetic and geometric lines are synthetic a priori based on instinct. Here we want to emphasize that Leibniz’s implications for analytic and righteousness (although Kant has transformed these meanings) have shaped the basic claims of logicians, such as Frege and Russell. They attempted to reduce all mathematical statements to logic in the early 20th century. Moreover, it is likely that Leibniz’s idea that “axioms can be proved” influenced logicians. Besides, Leibniz himself tried to present sound proof of the principles used in a mathematical proof.
Leibniz’s concept of proof and analytic concept complement each other because the derivation of any statement from another statement while giving proof corresponds to the concept of analytics.
So far, we have touched on some of Leibniz’s views on mathematics. One of the issues raised in this paper is that Leibniz’s approach to mathematics cannot be distinguished from his theological and metaphysical or philosophical views. We have mentioned above that Leibniz, for example, does not understand the binary number system as an arithmetical issue. As Breger quoted, for Leibniz, mathematics and theology were like the steps of a ladder ascending to God” [God and Mathematics in Leibniz’s Thought, Mathematics, and the Divine: A Historical Study, pages 493]. To understand Leibniz, the relationships he assumes between mathematics, theology, and metaphysics are all matters that need to be addressed. Such a complex issue cannot be dealt with in detail in this short article; instead, I will merely address a few points to give the reader an idea.
Leibniz hoped that his mathematical achievements would draw attention to his philosophical and theological ideas; after all, a mathematical achievement is a sign of a healthy mind. Leibniz’s “opportunism” on a personal level reflects another opportunism on the social class of his era. As it is known, the Christian missionaries who went to China used the mathematical achievements of Europe to impress the Chinese and later Christianize them. Leibniz would approve it without hesitation. In fact, for Leibniz, the characteristica Universalis method is the safest way to show the truth to those who do not believe in God because it will measure and show the accuracy value of everything like a scale. [God and Mathematics in Leibniz’s Thought, Mathematics, and the Divine: A Historical Study, page 9] In other words, the fact that the missionaries show the truth to non-Christians with this computational method will be enough to lead them to Christianity!
There is a metaphysical basis for Leibniz to use numbers for characteristica Universalis. Leibniz dealt with the belief that “God created everything according to a measure, number, and weight,” which Plato expressed. Leibniz thinks: some objects have no weight, so their weight cannot be calculated; some objects do not have dimensions, so their length cannot be measured, but anything can be counted. In summary, numbers are the essence of everything. According to Leibniz, God is a perfect mathematician. The act of creation took place with “divine mathematics” (Mathesis quaedam Divina). In his famous essay On the Ultimate Origin of Things, Leibniz says that the origin of all things is a “metaphysical mechanism” or “sacred mathematics” [Leibniz, G. W. Philosophical Essays, page 151]. Everything in the world exists according to certain measures and laws, and these laws are not only “geometric” but also “metaphysical” [Leibniz, G. W. Philosophical Essays, page 152].
For Leibniz, a world of free will, even if there exist cruelty and evil, is better than a world without cruelty, evil and free will, as mentioned in Theodicy and many other writings. That is the explanation of God’s creation of a world with evil in it. In all possible worlds, why did God create this world, not another world, in this way?
According to Leibniz, this is the perfect world! So, as an ideal mathematician, God has calculated all the possible worlds and created the best of them. An example of the best of all possible worlds is that lions are dangerous animals, but without them, this world would be less perfect. Besides, our assessment of the well-being of this world is limited to the events we have known and experienced so far. However, God has chosen this perfect world, taking into account all times and all creations [Leibniz, G. W. Philosophical Essays, page 149–155]. Another example given by Leibniz in this regard is that a person born in prison cannot judge that the whole world is evil by looking around. After all, for Leibniz, individuals see only a particular part, whereas God decides by taking everything into account.
Instead of Results
Leibniz’s dazzling characteristica Universalis program has never happened. David Hilbert defended a formal mathematical form of Leibniz’s thought and proposed a program accordingly. Kurt Gödel, who admired Leibniz, proved the Deficiency Theorem and showed that programs such as characteristica Universalis are doomed to fail, not only in philosophy but even in mathematics.
Leibniz’s partly based metaphysics and theology on a mathematical level brought about serious problems. Leibniz, in a sense, reduced everything to calculation. For example, he reduced God to a calculator that solved mathematical questions. It may seem paradoxical, but it is clear that such a God does not have a say in matters with no mathematical solution. Leibniz says that in some places, even God cannot do eternal operations. When he made God a mathematician, Leibniz understood that even God was made incapable of where mathematicians were capable. Say, for example, that God cannot perform infinite operations. Still, he can see the result (just as the mathematician does not perform unlimited operations one by one while performing limit calculations, but can calculate the outcome of those continuous operations).
Moreover, Leibniz did not think that there could be more than one consistent mathematical system in itself, taking absolute mathematical accuracy. That raises a problem that Leibniz is not interested in, which mathematics God uses. Mathematics occupies a vital place in all of Leibniz’s ideas from what we have written so far. According to him, a mathematician must be a philosopher, just as a philosopher should be a mathematician. In correspondence with L’Hôpital, Leibniz stated that his metaphysics was mathematical and could be written mathematically [God and Mathematics in Leibniz’s Thought, Mathematics, and the Divine: A Historical Study]. Moreover, according to Leibniz, mathematics is very close to logic, the art of making new inventions, and metaphysics is no different from that.
“I started as a philosopher, but as a theologian,” says Leibniz. Today, if someone wants to understand Leibniz’s philosophy, they still encounter the main issue; the relationship between mathematics and philosophy, metaphysics, and theology in Leibniz’s works.