Mathematics and Religion: Our Languages of Sign and Symbol (Templeton Science and Religion) - Softcover

Leach, Javier

Softcover

ISBN 10: 1599471493 ISBN 13: 9781599471495

Verlag: TEMPLETON FOUNDATION PR, 2010

Inhaltsangabe

Mathematics and Religion: Our Languages of Sign and Symbol is the sixth title published in the Templeton Science and Religion Series, in which scientists from a wide range of fields distill their experience and knowledge into brief tours of their respective specialties. In this volume, Javier Leach, a mathematician and Jesuit priest, leads a fascinating study of the historical development of mathematical language and its influence on the evolution of metaphysical and theological languages.

Leach traces three historical moments of change in this evolution: the introduction of the deductive method in Greece, the use of mathematics as a language of science in modern times, and the formalization of mathematical languages in the nineteenth and twentieth centuries. As he unfolds this fascinating history, Leach notes the striking differences and interrelations between the two languages of science and religion. Until now there has been little reflection on these similarities and differences, or about how both languages can complement and enrich each other.

Über diesen Titel

Über die Autorinnen und Autoren

Javier Leach has been director of the Chair of Science, Technology, and Religion Department since its creation in 2003 at the Comillas University in Madrid (Spain). He is also a Jesuit priest. Currently he teaches logic and mathematics in the Department of Computing at the Complutense University of Madrid, one of the main public universities in Spain. He holds degrees in philosophy, mathematics, and theology.

 

Javier Leach has been director of the Chair of Science, Technology, and Religion Department since its creation in 2003 at the Comillas University in Madrid (Spain). He is also a Jesuit priest. Currently he teaches logic and mathematics in the Department of Computing at the Complutense University of Madrid, one of the main public universities in Spain. He holds degrees in philosophy, mathematics, and theology.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Mathematics and Religion

Our Languages of Sign and Symbol

By Javier Leach

Templeton Press

Copyright © 2010 Javier Leach
All rights reserved.

ISBN: 978-1-59947-149-5

Contents

Preface, 
Chapter 1: Mathematics and Natural Sciences, 
Chapter 2: Metaphysical Language, 
Chapter 3: Origins of Mathematics, 
Chapter 4: Euclid and Beyond, 
Chapter 5: Dawn of Science, 
Chapter 6: Mathematics Formalized, 
Chapter 7: Propositional Logic, 
Chapter 8: Language and Meaning, 
Chapter 9: Science, Language, and Religion, 
Appendix 1: Syntax of Propositional Logic, 
Appendix 2: Semantics of Propositional Logic, 
Appendix 3: Syntax of First-Order Logic, 
Appendix 4: Semantics of First-Order Logic, 
Appendix 5: Numerical Systems: Their Role in First-Order Logic, 
Appendix 6: The Paradox of Russell, 
Appendix 7: The Completeness of First-Order Logic, 
Appendix 8: Jack's Formula, 
Appendix 9: Maria's Formula, 
Appendix 10: Example of an L1 Formalization, 
Glossary, 
Essay on Sources, 

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,...

Details zur Ausgabe

• Verlag: TEMPLETON FOUNDATION PR
• Erscheinungsdatum: 2010
• Sprache: Englisch
• ISBN 10: 1599471493
• ISBN 13: 9781599471495
• Einband: Taschenbuch
• Auflage: 1
• Anzahl der Seiten: 188
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Anja's; service@anja-karge.de; ; Von-Gumppenberg-Str. 94; Pöttmes; 86554; Deutschland