Mathematics and Faith
نویسنده
چکیده
In this reflection about mathematics I shall confine myself to arithmetic, the study of the numbers 0, 1, 2, 3, 4, . . . . Everyone has at least the feeling of familiarity with arithmetic, and the issues that concern the human search for truth in mathematics are already present in arithmetic. Here is an illustration of research in arithmetic. About 2500 years ago, the Pythagoreans defined a number to be perfect in case it is the sum of all its divisors other than itself. Thus 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are perfect. The Pythagoreans, or perhaps Euclid himself, proved that if 2 − 1 is a prime, then (2 − 1) · 2n−1 is perfect. More than 2000 years later, Euler proved that every even perfect number is of this form. This left open the question whether there exists an odd perfect number. The search for an odd perfect number or, alternatively, for a proof that no odd perfect number exists, continues today, several centuries after Euler and in the fourth millennium from Pythagoras. No other field of human endeavor so transcends the barriers of time and culture. What accounts for the astounding ability of Pythagoras, Euler, and mathematicians of the 21st century to engage in a common pursuit? Of the three schools of thought on the foundations of mathematics— Platonic realism, intuitionism, and formalism—the Platonists offer what seems to be the simplest explanation. The sequence of numbers is the most primitive mathematical structure. Mathematicians postulate as axioms certain self-evident truths about the numbers and then deduce by logical reasoning other truths about numbers, theorems such as those of Euclid and Euler. This is the traditional story told about mathematics. Can there be any philosophical—let alone theological—problem with it? Let us look more closely.
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