Mathematics, though, presents problems of a different kind: objects that exist in the mind but cannot possibly derive from the senses, and yet have a manifest relationship to the real world.
The ancients and the Scholastics fought about abstractions. After the fifteenth century, though, metaphysics was compelled to respond to physics. Curiously, the first intimation of the actual infinite appeared in music. During the s, musicians began tempering musical intervals, using string lengths that corresponded to irrational numbers. This provoked a crisis in mathematics as well as metaphysics. Aristotle knew that an irrational number could be represented as an infinite series of rational numbers, and that the irrationals therefore implied the existence of an actual infinite, which of course could not be grasped by the senses.
He rejected the concept, and under his influence, fifteenth- and sixteenth-century mathematicians and music theorists agonized over whether to admit the irrationals into musical tuning. The calculus gives us the exact sum of an infinite number of infinitesimal quantities, which by definition are imperceptible. We cannot perceive vanishingly small quantities, yet in the calculus their sum is a definite number. The physics that issued from the work of Newton and Leibniz transformed the world. That made it more difficult if not quite impossible to dismiss infinitesimals as the mere imaginings of mathematicians.
At issue was not a mythical bird, but rather the precise calculation of ballistic trajectories and planetary orbits. If God is nature, there can be nothing in nature except God, and individual objects cannot exist. Leibniz removed God from nature and re-situated Him outside it, where He creates an endless multiplicity of infinitesimal monads that comprise a coherent world through a pre-established harmony.
As Leach points out, there still are dissenters among mathematicians. But the revolution in mathematical physics and physics made for a different sort of debate than had occurred among the ancients or the Schoolmen. The infinitesimals, by contrast, were not simply a new sort of Platonic number mysticism, but rather a working principle that transformed the world. The new mathematics of the sixteenth century roused the philosophers to explain the existence of objects in the mind that were not in the senses.
Mathematics and Religion: Our Languages of Sign and Symbol
These pebbles, notches, and knots represent the first formal signs of mathematics. Numerical Amounts and Geometric Structures The archaeological record suggests that mathematical language became increasingly formal and abstract. In the Blombos cave of South Africa, two hundred miles from Cape Town, researchers have found two stones dated to about eighty thousand years ago that are engraved with geometric figures. At other sites in Africa, researchers have found bones inscribed with arithmetical calculations. A bone from the Lebombo Mountain caves in Swaziland dating from thirty-five thousand years ago bears twenty-nine notches that probably represent the number of days in a lunar cycle.
Another bone showing calculations was found at the Ishango site in the Upper Nile Valley of the Democratic Republic of the Congo and is dated to twenty thousand years ago. The ancient Incas of South America also had a form of calculation, called quipu, that may have preceded their written language system.
Sometime in the distant past, the relationships between these numbers were analyzed and put to good use. One of the first uses was the selection of a natural number to be the base, or basic unit, for calculations. In the modern field of discrete mathematics—which is popular in computer science and focuses on discrete structures that can be operated through finite processes—we can find proofs of the theorem that states that any natural number a can be represented by taking any other natural number b as its base. Before Greek civilization, we have no clear evidence of proofs of mathematical theorems.
However, earlier accounts suggest that the idea of the base number was an aspect of intuitive knowledge that was put into practice. The sexagesimal system uses the number 60 as the base.
Sixty has twelve factors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 , three of which 2, 3, 5 are prime numbers. This system is very suitable for representing fractions. The twelve factors of 60 make it possible to divide the hour into half hours, periods of twenty minutes, quarters of an hour, periods of ten minutes, and so on. In our modern world, the base 10 is most well known, a decimal system that arose from the number of fingers on our hands. However, from ancient times, the numbers 2, 3, 4, 5 on up were used as a base.
In Egyptian hieroglyphs from BCE, such decimals are used to create very large numbers. Calculations Based on Numerical Relations The ability to calculate was the next step in primitive mathematics. A single calculation involves a mechanical procedure; we say it is mechanical because it can be executed by a machine, such as a computer. Children at school learn calculations in order to find the sum or the product of two numbers. The power of calculation can also be applied to great realms of complexity in the world, going from simple numbers to shapes in space.
Mathematics and Religion
We can calculate the volume of a sphere simply by knowing the radius, the distance from the center to the edge. Even in primitive mathematics, some calculations were relatively complex. For example, problem number 10 on the papyrus calculates the area of a curved surface. Problem number 14 calculates the volume of a frustum of a pyramid on a square base with a height of 6, whose lower base is a square with a side measuring 4 and an upper base whose side measures 2.
A diagram of this pyramid is shown in Figure 3. Intuitive-Empirical Theorems Before Greek civilization, knowledge of mathematics was empirical. These earlier cultures, based on an intuitive sense of mathematical relations made empirical calculations that were correct, despite the lack of a formal mathematical language. Well before the proof of important theorems attributed to the Greek thinker Pythagoras, sages in China, India, Babylon, and Egypt understood aspects of what came to be known later as Pythagorean triples.
Before Pythagoras developed the deductive means of proving this theorem, earlier cultures had arrived at knowing its reality by induction—that is, by experience and experiment. We can almost imagine the ancient thinkers putting objects, such as squares and triangles, on the ground and looking at how their sizes, angles, and lengths compared. This was the experimental inductive study of mathematics. The Infinite in Mathematics and Religion As discussed in the previous chapter, the infinite is perhaps one of the most basic human intuitions, and it naturally played a role in the rise of primitive mathematics.
The pre-Greek epoch offers evidence of testimonies of the use of the infinite that carried both religious and mathematical connotations. Many centuries later, the mathematician Georg Cantor d. In other words, the concept of purna has been interpreted both religiously and mathematically.
As we noted earlier, we can avoid confusion between the religious and mathematical infinite by a clear distinction between formal signs and metaphysical symbols. Faith is expressed by saying that the transcendent God is infinite because God has no limits. Symbolically, this statement means that we see the world as limited and God as unlimited.
Mathematics and Religion
In mathematics we state that a set is infinite if we cannot finish counting elements—something the ancients understood quite well. Naturally, the ancient founders of mathematics were fascinated by both kinds of mysteries, and in Greek and medieval mathematics we again see this interplay of the formal and the religious—as the next chapter shows. But mathematics was also a key component of this remarkable period, and what the Greeks put down in writing might have been lost, had Roman and Arabic chroniclers not rescued some of these materials. After the rescue, the materials surfaced again in the twelfth century CE in Europe, stimulating a renaissance in mathematical exploration after roughly fifteen hundred years.
Eventually these proofs were applied to deductive science, in which a science begins with a first principle and then tests that premise against the varieties of phenomenon in the world. In the medieval period, deductive science was expanded in its application to algebra.
In both the Greek and medieval periods, as we will see, the formalism of math invariably overlapped with the religious culture of the times. The great transition of mathematics from trial and error to a deductive science came with the work of three primary thinkers: Thales of Miletus d. The kinds of new mathematical proofs that we attribute to Thales and Pythagoras, for example, likely have roots in Babylonian and Egyptian culture. By making mathematics deductive, the Greeks established a stable point from which to explore complex mathematical realities that otherwise would boggle the mind.
Here we find a compendium of proofs of arithmetical and geometric theorems based on axioms and elementary principles. Until that point, all math was based on calculations that followed mechanical rules, which, when applied to some data, produced other data. The calculation of the square root applied to the number 25 produces the number 5. Not much has changed in the nature of calculations such as this. Because they are strictly formal processes, these rules amount to algorithms—a mechanical sequence—that today we put into language that a computer understands.