Mathematicians and Their Gods: Interactions Between Mathematics and Religious Beliefs is a fascinating anthology of scholarly articles about how various people in the past have viewed the relationship between mathematics and religion. The editors begin their preface with the following remarks.
The overlap between mathematics and theism may seem at first sight an unpromising topic for a book. But as the many contributors to this volume show, the intersection between the two provides fertile ground both for historians of mathematics and of theology, and for the interested general reader.Indeed, the topic is a rich one, and the main difficulty that the editors faced was not the lack of available material but its abundance.
It will be noted that the material covered in this book is biased towards Western Christendom. The most significant omission is that we have ignored the rich vein of Islamic mathematics that runs throughout the medieval era. This reflects, at least in part, the interests of the editors, but it was also a self-imposed restriction to help keep the book within manageable proportions. However, even in this restricted field the potential material is substantial and we have necessarily been selective. There has been room for only a sample of the historical movements and characters where God and mathematicians intersect.It is rather difficult to review an anthology of diverse articles. I will approach the task by grouping the articles into categories.
1. A property is positive if and only if its negation is not positive.
2. Any property necessarily implied by a positive property is also positive.
Definition: Something is God-like if and only if it has every positive property.
3. Being God-like is a positive property.
4. If a property is positive, then necessarily having that property is also positive.
Definition: A property is an essence of something if it is a property that the thing has that necessarily implies every other property that the thing has.
Definition: A thing necessarily exists if any essence of it is necessarily exemplified; that is, if it is necessarily true that some existing thing has that essence.
5. Being a necessarily existent thing is a positive property.
Lull aimed to unify all knowledge into a single system, and through
this to teach Christian theology so logically that Moslems could not
but see its truth and be converted. Regarding all reality as the
embodiment of aspects of the divinity, he attempted in his Ars Magna
(c. 1273) to teach the natural sciences of his day as congruent
expressions of the truths of theology and philosophy. He believed
that he had discovered ‘an Art of thinking which was infallible
in all spheres because based on the actual structure of reality, a
logic which followed the true patterns of the universe.’
Lull’s method was based on the belief that knowledge arises from a finite number of basic principles or categories. By moving through all possible combinations of these categories, we reach all knowledge: combinatorics is thus the basic tool for exploring all that can be known. In particular, he used ‘combinatory diagrams’ to present the active manifestations of the divine attributes, which he called ‘Dignities’: these included Bonitas (goodness), Potestas (power), Sapientia (wisdom), and so on.
I brought with me here the MSS, such as it is (very fragmentary and unarranged) for the book about religious difficulties, and I meant, when I came here, to devote myself to that; but I have changed my plan. It seems to me that that subject is one that hundreds of living men could do if they would only try, much better than I could, whereas there is no living man who could (or at any rate would take the trouble to) arrange, and finish, and publish, the 2nd Part of the Logic. Also I have the Logic book in my head: it will only need 3 or 4 months to write out; and I have not got the other book in my head, and it might take years to think out. So I have decided to get Part II finished first: and I am working at it, day and night. I have taken to early rising, and sometimes sit down to my work before 7, and have 1½ hours at it before breakfast. The book will be a great novelty, and will help, I fully believe, to make the study of Logic far easier than it now is: and it will, I also believe, be a help to religious thoughts, by giving clearness of conception and of expression, which may enable many people to face, and conquer, many religious difficulties for themselves. So I do really regard it as work for God.Unfortunately, Dodgson never finished writing Symbolic Logic, but what exists of Part II was eventually published posthumously.
Now that multi-dimensional geometry has been given a demonstrable existence, Christians must reassess their understanding of God. The defining attributes of a modern God must be more than omniscience, omnipotence, and omnipresence (or, as he summarizes all them in Flatland, ‘omnividence’). Any being, good or evil, who descends from an unseen fourth dimension could manifest those attributes without being Godly. In the late nineteenth century, Abbott is saying, Christians have to look beyond the scriptures for a modern foundation for their religion.Abbott laid out his proposal for a modern version of Anglican Christianity in his book The Spirit on the Waters, where among other things he laid out five new religious axioms (reminiscent of Euclid’s five axioms for geometry).
No natural cause by itself could have produced the harmonious arrangement by which each planet and its satellites was endowed with the precise locations, masses, and velocities that it now had. Nor could it have given rise to the mathematically precise laws that governed their interactions. Indeed, if any of these values had been awry by even a small amount, then without some external intervention, the system would have become chaotic. Recalling the claim he had made in the draft of the Principia, Newton argued that making such a system work required a ‘Cause’ that understood and compared the masses, gravitational forces, interstitial distances, and velocities of all the bodies in the solar system. Only a being highly skilled in mechanics and geometry could ‘compare & adjust’ all these elements—and perhaps only a super-intelligent mortal could understand them. Divine foresight was nicely shown in Newton’s favourite case of the gravitational interaction between Saturn and Jupiter. Having presumably been created before the solar system took its present form, they had been placed by the creator at the positions from the sun that they now occupied to prevent their mutual attractions from destabilizing the system.Category 2: The mystical significance of mathematics
That some Pythagoreans were interested in what we would call numerology is undeniable. They did attribute non-mathematical properties to numbers. So 2 and 3 were associated with male and female, while 5 was associated with marraige and 10 was seen as a divine or special number. There are some important points to make here, though. Numbers do have properties and it would not have been easy for the Pythagoreans, in the context of what was known in ancient Greece, to distinguish what we would consider mathematical and numerological properties. There was a wide spectrum of interest in number among the Pythagoreans. Some would have been what we would see as mathematicians, some what we would see as numerologists and some would have mixed aspects of these two extremes together.Later on in the same article, Gregory continues:
They use privileged numbers and attempt to say how these led to, or constitute a good arrangement of, the world. However, this was not a simple or primitive numerology. They had important philosophical reasons for their application of number. They needed to explain how the cosmos was good, that is in good order and aesthetically good, a standard assumption among the ancients, and how humans could have knowledge of the cosmos. From Plato onwards, there is an assumption of a god organizing the cosmos. For each of the actions of that god there has to be a reason and some of those reasons are supposed to be mathematical or geometrical. This is a far cry from the modern view of an accidental universe where we fit mathematics to what we observe as best we can. It is important to understand that the ancients asked a different question about cosmology, which was how has this all come about for the best, given that the cosmos is so congenial to human beings and appears comprehensible to humans as well?b. Mystical arithmetic in the Renaissance: from biblical hermeneutics to a philosophical tool, by Jean-Pierre Brach. Brach gives a rapid overview of many different writers during the Renaissance who employed “number symbolism” or “arithmology” in biblical interpretation and speculative theology.
The story of Hiram Abif goes like this. Hiram was a master mason and as such knew secret geometrical knowledge. Apart from the practical geometrical knowledge of stone-cutting masons, there is a reference to a sacred and secret word, teh interpretation of which is diverse in Masonic scholarship. Hiram was at one point asked to divulge the secret word, but as he refuses, he is killed and buried by his killers at a secret place. When Solomon hears of the murder of his architect, he orders the execution of the killers and a search for Hiram’s body ensues. This story is re-enacted in masonic ritual for the Master Mason, which is a third degree in the Masonic system of initiation, the first two being that of Apprentice and Fellow Craft.After a description of the re-enactment ritual, Lawrence continues:
In the midst of this decomposing imagery of death, geometry is invoked as a life force and one by which the dead can be raised to the ‘living perpendicular.’ In this story of resurrection, it seems that the two most important elements that make the dramaturgy of the ritual are linked to architecture and geometry. The first is the initiation into the understanding of the role of Master Mason in the preservation of the sacred and secret knowledge. The second element is the geometrical knowledge—by the right fellowship (the right grip), acceptance of one’s place in the hierarchy of the lodge (the right degree) and the right conduct—one can become again a ‘living perpendicular’ which is of course preferable to remaining a rotting cadaver.d. Newton, God, and the mathematics of the two books, by Rob Iliffe. This article was already mentioned in the previous category. Back in Newton’s day, two of the most influential books on biblical apocalyptic interpretation were Joseph Mede’s Clavis and Francis Potter’s Interpretation of the Number 666. Newton was influenced by these works but had his own way of interpreting the symbolism in the Revelation of John, paying special attention to the significance of the numbers 2, 7, and 12.
Many early modern devout authors—most notably Malebranche—used the term ‘attention’ to describe a particular state of mind, a form of intense concentration that was seen as a necessary condition for contemplative practices, and more generally for a fulfilling spiritual life. Malebranche would refer to attention as a ‘natural prayer.’ At the same time, this state of mind is described as a prerequisite for the investigation of nature, and in particular for the study of mathematics. Wary of baroque piety and what Muratori called ‘disorderly devotions,’ Agnesi highly valued the capacity of concentration and a well-trained intellect. Superstitious devotions stimulated and relied upon fantasy and imagination, inducing a credulous and fatalistic attitude rather than a clear understanding of one’s religious duties. Believers should rather train their intellect and ground their spiritual experience upon it. And which is the best way to exercise the intellect? The study of mathematics, geometry in particular. Agnesi believed that calculus was the subtlest branch of mathematics, and therefore the one that required the highest level of concentration and the strongest intellect.This view of mathematics reminds me of Simone Weil, who also emphasized the spiritual importance of the faculty of attention.
The philosophy of Copernicus counts up the principal parts of the world by dividing the form of the universe into parts. For in the sphere, which was the image of God the Creator and the Archetype of the world, there are three parts, symbols of the three persons of the Holy Trinity—the center, of the Father; the surface, of the Son; and the intermediate space, of the Holy Spirit.It is rare to find examples of religious beliefs having a direct influence on mathematical discoveries; the book Naming Infinity, which I have also reviewed, gives another example.