Transcendence, Truth and Reality: A Mathematician’s Fumble By Fr Jonathan Munn Obl OSB In which we continue some of the ideas put forward in Mathematical Sehnsucht in the first edition of Blue Flower to examine how we understand things that are transcendental to our reality using mathematical concepts and ideas and apply them to our understanding of God. Introduction We live in an age in which the existence of objective truth is disputed in order to give some intellectual status to the subjective control of one’s identity by the individual. We hear, sometimes in reputable academic establishments, that “my truth is not the same as your truth” and this is to be accepted at the risk of great offence and/or social exile. To assert an objective truth is seen by some as an act of violence upon an individual’s existence. Certainly, philosophers such as, notably, Immanuel Kant have separated the impressions of things from things-in-themselves suggesting that the thingness of things is something that can never be known by the senses. If we go that step further, as many modern philosophers and pseudo- philosophers do, and dispute the existence of things-in-themselves then we find ourselves only with the truth of sensory impressions leading to the triumph of the subjective over the objective, and thus the Marxist entitlement of the individual oppressed by a procrustean, objective Society. There is an alternative, and one that has been known both to mathematicians and also to theologians. If the truth cannot be fully grasped by individual minds, then maybe it exists as something transcendental lying beyond the grasp of sense but apprehensive in part by the rational intellect. This requires careful definition and thought and it is to be admitted that the author is merely a poor student of basic philosophy and not its master, unlike many of his colleagues. The purpose of this essay is to examine the relationship of our understanding of what is truth and our understanding of what is transcendent. We intend to use ideas from mathematics to begin a point of exploration and perhaps to stimulate some thoughts in the reader whose better learning and better grasp of philosophical thought may be more productive and bring about a clearer idea of what is the case. Transcendence in Mathematics A Square meal All things in mathematics begin with a simple idea. A farmer has a perfectly square field of area 9 square miles. If he wants to put a fence all around the perimeter of his field, how long must the fence be? Our thinking will be along the following lines. To find the length of the perimeter, we need to find the length of each side of the square. The only hard fact we know is that the area of the square is 9 square miles. However, we also know that to find the area of a square we have to multiply the length of its side by itself. The question now boils down to finding a number which, when multiplied by itself, gives us 9. That number is 3.i This is why the process of multiplying a number by itself is called squaring. Thus, the square of 2 is 4; the square of 5 is 25; and the square of two fifths is four twenty-fifths. To find the side of a square with a given area is to find the square root. A square of area 25 square miles has side 5 miles; a square of area 4 square inches has side 2 inches; a square of area four twenty- fifths of a square furlong has side two fifths of a furlong. What happens if I have a square of area 2 square feet? What’s the side length of this square? Well, this side length can’t actually be a whole number of feet, nor can it actually be a fraction of a number of feet.ii Can a number exist that is neither whole nor a fraction? The existence of such a number depends on whether we can construct a square of area 2 square units. Take two squares of side length 1 unit. They each have area 1 square unit. Put them together to create a 2 x 1 rectangle of area 2 square units. Now divide each square along its diagonal as shown. Now take the upper two triangles and put them below. Now, the diagonals of both squares are the same length and meet the base at 45° thus the figure we have constructed has four equal sides and four equal angles: it is a square. We have constructed its area to be two square units. This means that the length of the side of this new square must be the square root of two which we write as √2. Thus we have visible, constructible evidence that there is a number which is neither whole, nor a fraction. We cannot even express it completely as a decimal number. √2 = 1.4142135623730950488016887242097 … This decimal never ends and never repeats itself.iii Cubes, Duels and Ascending Dimensions What we have done is to give a physical context to the solution of the equation 푥2 = 2. This, perhaps, is the simplest example of an algebraic number which is neither whole nor fraction. It is said that the Pythagoreans, infuriated with the existence of such a number, drowned the poor soul (said to be Hippasus) who proved its existence to them.iv We can indeed construct the square root of any number we like. While we may not be able to express them in decimal form, we can understand them as numbers get an idea of their size and manipulate them in the way that we manipulate any number whose value we know. The only problem that we find is that we have to refer to them as solutions of an equation rather than by their value.v We have shown that we can double the square and essentially construct the square root of 2 geometrically. Can we double the cube in the same sort of way? What does that mean? Can we construct a cube of volume 2 cubic feet from two cubes of 1 cubic foot each in the same sort of dissection as we did for the square? Algebraically, we could say that we seek a way of finding a number which when I multiply it by itself twice, I get 2, i.e. find a number x so that 푥3 = 2. This equation defied construction for a long, long time until Evariste Galois, the night before he was to be shot in a dual, set up the mathematical framework to show that it isn’t possible. We can’t dissect two cubes to construct another in the same geometric way as we did for squares. To do so requires special techniques not available to the Greeks who were armed only with a straight edge (without markings) and a pair of compasses. Nonetheless, it’s clear that such a number must exist as we can certainly think about how we can do this with pieces of plasticine (which wasn’t available to the Greeks!). We can understand, therefore, that there are a whole myriad of numbers which solve equations which involve powers. These are called algebraic numbers and they all exist as solutions of equations of the form: 푎 + 푏푥 + 푐푥2 + ⋯ + 푥푛 = 0 where the numbers a,b,c, et c, are either whole numbers or fractions. The highest power (n in this case) is called the degree of the equation. Thus, the problem of finding the side of a square given its area is a degree 2 problem. The problem of doubling the cube is a degree 3 problem. The upshot is that there are suddenly a whole plethora of new numbers that are neither whole numbers nor fractions which must extend our thinking beyond our ideas of finite decimals. Does this mean that we have all the numbers now? Having our pi and e-ting it. Actually, we haven’t even scratched the surface. There are more numbers that exist than are algebraic. These are the famous transcendental numbers and most of us are already familiar with at least one of them. The ratio of the circumference of a circle to its diameter is the number 휋. We often approximate 22 this number by the fraction ⁄7 but its true value is much more complicated. Again, we find that it is neither a whole number nor a fraction. The question is, is it algebraic? Does it solve an equation of degree n where n is a finite number? The answer is no, and this is demonstrated by the Lindeman-Weierstraß theorem – a result formed from the work of two nineteenth century mathematicians. In fact, we can find 휋 as the solution of an equation of infinite degree but algebraic numbers solve equations with finite degree. We know that 휋 is finite and yet its very existence is bound up with all kinds of infinities within its finitude. The same is true of another mathematical number e which despite its intractable definition keeps cropping up again and again in Mathematics, Physics, Biology and Economics. It seems that we cannot adequately describe our physical reality without numbers that go beyond our grasp. What we can do, however, is approximate. The mathematical waving of hands In practice, we do not need to know a number infinitely precisely. Measuring a field’s perimeter, we can be satisfied with the nearest yard, foot or inch depending on the circumstances and depending on how picky we are.