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Mathematics and Religion
Our Languages of Sign and Symbol
By Javier Leach Templeton Press
Copyright © 2010 Javier Leach
All rights reserved.
ISBN: 978-1-59947-386-4
CHAPTER 1
Mathematics and Natural Sciences
SINCE THE RISE of modern science in the sixteenth century, mathematics has often been characterized as the language of nature. We often forget, however, that we have never stopped debating whether we can talk most accurately about the world by using only numbers or by also using physical models. Are the solar system and the movements in the night sky, for example, best understood by a series of numbers written on a sheet of paper, or when we view a wooden model of the planets and the solar system, so to speak, as the early scientists of the Renaissance did?
Although mathematics and natural science are closely bound together, they represent essentially two different kinds of language. Mathematics refers primarily to objects of the mind. Natural science refers to objects of our sense experience. In mathematics we use abstract formal signs (that is, the language of precise mental meaning and a language that we can manipulate mechanically). In contrast, natural science uses what we may call representational language that speaks of the physical objects which physics, chemistry, geology, and neuroscience study.
We can go deeper as well. At the heart of both mathematics and natural science lies the primary level of logic. Once we have logic, we are able to move on to mathematics and to natural science. At each of these levels, we perceive reality and then we use a type of language to express that perception.
Formal Signs in Logic and Mathematics
What we perceive at the level of logic is correct reasoning, an inference that one thing naturally leads to another. We can test such logical inference in formal models of logic or mechanically, as in a computer. But many times we perceive something as logical simply by the power of intuition: it immediately seems to be so. These are logical intuitions. They intuit that something is following the rules of logic. For example, "It is impossible that something be true and false at the same time" is a logical principle that we intuit is always valid. We call this the principle of noncontradiction.
The logic we intuit can also be put into a formal language. As evidenced by the abstract signs often seen in logic or mathematics, formal language consists of a finite series of signs that follow rules of syntax. The signs have no definite meaning until they are related to each other by these rules, and then we can interpret these strings of signs as true or false. That language of logic sets the stage for the language of mathematics.
The way to understand the relationship of logic and mathematics is to say that while mathematics includes logic, it cannot be reduced to formal logic. Mathematics has something more, a kind of mathematical intuition and freedom based on logic. In fact, if we reduce mathematics to pure formal logic, we end up with paradoxes, which amount to contradictions. The great mathematical ambition of the German Gottlob Frege (d. 1925) and the Englishman Bertrand Russell (d. 1970), both of whom wanted to reduce mathematics to formal logic, illustrated this paradox. The result, however—which they conceded—is that such an effort ends in paradoxes. So again, logic and mathematics are different despite many similarities.
Like logic, however, mathematics also begins with intuitive perceptions. Mathematics begins as a purely intellectual, intuition-driven exercise. One of the first great mathematicians, Euclid, proposed many of these natural intuitions. For example, the first Euclidean postulate expresses the mathematical intuition that between any two points a straight line segment can always be drawn. In applying mathematics, we give these intuitions another name: mathematical axioms, which amount to beliefs that we presume to be true. (Hereafter, we use the terms "axiom" and "postulate" interchangeably since they have the same meaning.)
So mathematics is built of two parts, the axioms and the mathematical statements that seem logical. However, since axioms are basic intuitions, and they are the foundation of a particular mathematical system, axioms are not valid in all systems. What remains valid in all systems is the logic of mathematical propositions. As we see later in the book, this realization has created a variety—or pluralism—of mathematical systems. Yet in each one, certain logical propositions must always be valid. We can turn again to Euclid to illustrate this point. The axioms that Euclid began with ensured that his geometry was consistent and logical. However, not all forms of mathematics begin with Euclid's axiom. Basic arithmetic does not use those axioms, and thanks to modern revolutions in math, today we have non-Euclidean geometry, which uses axioms different from Euclid's.
As we can see, if axioms and logical principles are mixed in the wrong ways we end up with paradoxes, which means that we can deduce a proposition, but also its negation. It would seem that paradoxes would always be a bad thing, since they suggest that reality is not truly logical at all. However, the value of paradoxes is that they stimulate us to look more deeply for the logical connections in our intuitions and prove them in the language of logic or mathematics. While some paradoxes seem insurmountable, they also stimulate us to look beyond the use of the purely formal language of formal signs—used exclusively in logic and mathematics—to employ the representational language of empirical science and even the symbolic language of metaphysics (see Figure 1.1).
Representational Signs in Natural Science
The natural sciences begin with perceptions of the objects in the world, which is what separates them from the purely mental starting point of logic and mathematics, though natural science employs logic and mathematics as well. The empirical observations of natural science can be very sophisticated. Still, they are limited by the perceptions of the five senses. Once making their perceptions, scientists may certainly express them in natural languages, just as Copernicus spoke in Polish or German, but used Latin as academic language when he talked about his belief that the sun was stationary and the earth moved. Natural science ultimately seeks a high precision in its use of applied mathematics. Here mathematics becomes a privileged language; scientists understand each other and can conduct identical experiments despite their different national languages.
In practice, of course, the argument that natural science uses representational signs differently from the way logic or mathematics uses formal signs is a bit more subtle and complicated. The same signs can be used in either case. For instance, the signs E, m, and c in the equation E = mc2 can be either formal or representational. As formal signs they stand for elements of a mathematical structure, such as the system of real numbers (one of our more complex and inclusive sets of number systems). As representational signs, E, m, and c represent energy, mass, and speed of light. The difference between the two uses lies in the fact that, in the former, the semantics refer to mental objects (such as pure numbers) while, in the latter, the semantics refer to physical observations.
But we should emphasize again that logic and mathematics are at the core of natural science. Mathematics is indispensable in scientific research. The instruments that natural science uses to measure physical observations are designed based on mathematical theories. Moreover, logic and mathematics are not merely languages alone. They are the basic logical and mathematical intuitions that we cannot separate from our empirical sense experiences, and the bridge between those intuitions and our sense experiences has, in the history of science, been the building of scientific models.
Formal and Representational Models
When we create models, we have structures that help us imagine how the world works. Models are mediators between perception and theories. In science, these models designate and describe the relations between the parts of a given domain of discourse and the procedures we can use to analyze the topic of research. The domain of discourse contains all the elements to consider in a given model.
Naturally, science builds formal models of logic and mathematics, and it also builds representational models that describe empirical observations (such as the wooden solar system model of early astronomers). The first (formal) is real, but it is purely conceptual and does not have to necessarily match the reality "out there," for it only needs internal consistency. A representational model, however, must somehow match the empirical reality that an ordinary person can observe.
The models may be used together, depending on the problem that science is trying to solve. I mentioned earlier the model of non-Euclidean geometry, which essentially speaks of something we call curved space, as distinct from the normal flat space of geometry. So a model of non-Euclidean geometry can be created to talk about a reality that is not known to us; it is speculative, in this sense. But also, a model of non-Euclidean geometry can be a representational picture of the physical reality spoken of by Einstein's theory of relativity, which is a mathematical theory verified by observing the curvature of light in space.
For another illustration of how these models interact, we can turn to the story of Nicholaus Copernicus in the sixteenth century. In his day, the earth-centered model of Ptolemaic astronomy—essentially based on Aristotle's physical model—had dominated Europe for more than a thousand years because this representational model succeeded in explaining what astronomers saw in the skies year after year. However, Copernicus offered a mathematical model that explained the physical observations just as well—and more simply than Ptolemy's model of circular orbits and epicycles.
Let's reconsider Einstein's work. His theory of relativity was a purely mathematical model since he was not an astronomer (and, indeed, neither was Copernicus for the most part). Einstein built in his mind a model that tried to create a logical system to explain the universe on scales that were too large to measure physically. Einstein's mathematical model was tested in the empirical world in 1919 when the English astronomer Arthur Eddington journeyed to the Principe Island during a total eclipse to measure, by photography, whether curved space bent light as Einstein's model argued. In turn, the result was that Einstein's mathematics could explain the photographs (rough and questionable as they were). Today, our common scientific language refers to curved space and the four dimensions of space-time—representational models based on the formal mathematical concepts.
Another example of mathematical models rivaling representational models came about in the debate over the smallest scales of matter, as was seen in the difference of opinion between two of our greatest modern physicists, Niels Bohr and his student Werner Heisenberg. They both tried to address the problem of the uncertainty of the position and velocity of particles at the level of quantum physics. Bohr preferred a visual model of the solar-system atom, and to resolve quantum uncertainty, he ended up with a somewhat paradoxical image of the atom: he said that we can have two opposing yet complementary, visual models, one with the particle as a wave and one with the particle as a tiny corpuscle. On the other hand, Heisenberg preferred a single mathematical model of probability to explain how a real particle can exist without clear coordinates in the physical atom—a model of how it can be a wave and a particle at the same time.
The lesson here is that in the history of science, empirical observations have usually been interpreted in more than one way. We refine our knowledge by trying to reconcile a mathematical model with a physical model that we can observe. We are most satisfied when this harmonizing of math and observation works very well, but in the mysteries and complexities of the universe, there is no guarantee. In Figure 1.2, we see how we begin with both physical and mathematical perceptions, and then work our way up to models and languages, and then try to reconcile the languages.
Though we speak in Figure 1.2 about mathematical, empirical, and academic languages, we are also bound by our natural languages. We are all born into different cultures that have long tried to describe what we perceive in the world, whether that description is spoken or written in German, Hindi, or Chinese. As noted earlier, scientific language helps us transcend our local languages, although the transcendence can seemingly never be complete. Even in scientific culture, a pluralism of points of view exists—a variety of cultural languages. These many scientific communities have their own journals, congresses, and rituals. At times each scientific community seems to be, in effect, a different country.
This situation reminds us of the difficulty in achieving pure objectivity in our perceptions, language, and model building. Even the purest formalism of logic cannot escape a degree of subjective interpretation. As we shall see next, even mathematicians disagree on what is absolutely logical.
Formalism and Objectivity
Although we may affirm that logic and mathematics are the most objective knowledge, they are not totally objective or totally independent of the knowing subject. The view of what is logical and what is mathematical depends on the principles of logic that we accept. Some communities of mathematicians accept certain logical principles that other communities do not accept.
Even though the logical processes of deduction—with formal syntax and formal semantics—are objective and automatic routines that a machine can execute, several different possible views are available of what logic is. Accepting one or another view of logic can depend on personal tastes and preferences for what counts as valid logic. One can assume different views of logic, but not at the same time if they are not consistent. Once a view is assumed, one must maintain consistency.
Surprising as it may seem, not all logicians accept the famous principle called the excluded middle, for example. This principle states that "all propositions are true or false." Classical logic is strongly rooted in the principle of the excluded middle. But other schools of thought in mathematics, such as the constructivist or intuitionist schools, do not accept this as an absolute premise. This debate over the excluded middle is a disagreement about the existence of what we call mathematical objects. The classical view is that to prove the existence of a mathematical object it is enough to derive a contradiction from the assumption of its nonexistence. According to the contrary (constructivist) view, one must find (or construct) any mathematical object in order to prove its existence.
The classical approach to mathematics strongly defends the principle of the excluded middle by reducing the contrary view to a position of absurdity, a common method of proof we call "reduction to the absurd" (or reductio ad absurdum). That is the idea behind the following scheme:
A or not A (Logical principle of excluded middle)
A implies B (Premise)
not B (Premise)
* * *
[therefore] not A (Conclusion)
In the case that A is true, B should be true, because A implies B. But B's existence with the premise not B would be absurd. Therefore if the logical principle of excluded middle is accepted, not A should be our conclusion.
(Continues...)
Excerpted from Mathematics and Religion by Javier Leach. Copyright © 2010 Javier Leach. Excerpted by permission of Templeton Press.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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