Perhaps the earliest Greek mathematician who was fascinated by numbers was Pythagoras of Samos (circa 525 BCE). He was a highly eccentric man who started his own school in Croton in southern Italy and went on to become essentially a cult leader. He insisted that his students devote themselves not only to mathematics but also to an ascetic life devoid of personal possessions and adopt vegetarianism while also avoiding beans! Pythagoras also believed in reincarnation, and is supposed to have stopped a student from beating a dog because he recognized the voice of a late uncle in the dog’s cries!

Pythagoras believed that all real numbers were what he called “rational” numbers (that is, they can be expressed as fractions, p/q, where p and q are whole numbers). According to this definition, whole numbers are a subset of rational numbers with the denominator (q) = 1 (3 = 3/1, 5 = 5/1, and the like). Pythagoras was familiar with the role that rational numbers played in music created by stringed instruments, and he went on to claim that rational numbers would explain the workings of the entire cosmos (“the harmony of the heavens”).

Because Pythagoras and his followers believed that all real numbers were rational and controlled the workings of the entire cosmos, they received a crushing blow when, ironically, Pythagoras’s theorem itself revealed that this could not be so. It is possible that the Pythagoreans realized this quite early and tried to keep it secret. Legend has it that the Pythagorean who revealed it (Hippasus of Metapontium) was drowned by his fellow students for his grievous crime that demolished their central dogma. A less-guilty account of this drowning ascribes it to divine retribution!

With this digression, let us get back to the discovery of irrational numbers. Consider a special right-angled triangle where the lengths of the two sides forming the right angle are equal to 1. (That is, b = c = 1, in figure 1 in footnote 2 from part I¹.) Then a² = 1² + 1² = 2, and a = square root of 2 (√2). Let us now follow Euclid’s reductio ad absurdum strategy to prove that √2 is not rational (irrational). Either √2 must be rational, or it must be irrational. Let us assume that √2 is indeed rational. In that case, we can write √2 = p/q where p and q are whole numbers (integers) with (q ≠ 0). Let us also assume that p/q is simplified to its lowest terms. In that case, both p and q cannot be even (because the fraction can then be further simplified by dividing by the common factor 2). So, at least one of p and q must be odd.

Since √2 = p/q we get (by squaring),

2 = p²/q² or p² = 2q² —(1).

This implies that p² (and therefore) p is even (p cannot be odd because odd × odd is odd).

Since p = even, we can write p = 2k (where k is another whole number). Then substituting in (1) we have,

4k² = 2q² → q² = 2k². —(2)

This implies that q² (and therefore q) is even.

Thus, we have shown that both p and q are even. This contradicts our starting assumption that at least one of p and q must be odd. This means that √2 cannot be written as p/q, as assumed. So √2 is not rational. It is thus irrational.

How the unfortunate Pythagorean, Hippasus, arrived at his conclusion is not known, only that “whistle-blowing” within the cult had deadly consequences!

Incidentally, √2 is not the only irrational number; there are an infinite number of them. Clearly the square root of any number that is not itself a perfect square—such as 4 (2²), 9 (3²), 16 (4²)—is irrational (e.g., √3, √5, √6). Besides these numbers, which include the square roots of all primes, as well as higher roots (such as cube roots), there are other well-known examples. Two such are the exponential (e) that appears as the base of natural logarithms and π, the ratio of the circumference to the diameter of a circle. Ancient mathematicians thought that π must be a rational number but were unsuccessful in finding its value. Archimedes of Syracuse (287–212 BCE), who knew π was not a rational number, sought to find an approximate value by noting that π would be greater than the area of any regular polygon inscribed within a circle of diameter 1, and less than the area of any regular polygon that circumscribes it. He also realized that π would be closer to these limiting values as the order of polygons increased. Starting with a regular hexagon (six sides) and proceeding all the way to a ninety-six-sided polygon, he was able to show that π lay between 3 + 10/71 and 3 + 10/70. The latter value, which equals 22/7, is the approximate value of π that one learns in school. Indeed, it is a fair approximation. (While 22/7 = 3.142857…, the actual value of π = 3.141592….) This last value is calculated using infinite power series—an algebraic method first used by Isaac Newton over two millennia after Archimedes used his geometric method. There are now multiple power series representing π, and any one of them could be used to calculate π to any desired number of decimal places.

As stated earlier, Euclid’s powerful method called the axiomatic method of deductive reasoning is now used very extensively in many areas of study. It is invariably used in all areas of higher mathematics to establish theorems. It is also used extensively in the development of modern physics (both continuum and quantum). Also, as the ancient Greeks correctly perceived, mathematics plays a central role in theories of modern physics, making physicists’ predictions verifiable both qualitatively and quantitatively.

Indeed, the modern guiding methodology of science that has been succinctly laid out by the highly influential philosopher of science Sir Karl Popper is “conjecture and refutation.” According to the distinguished mathematician and astrophysicist Sir Herman Bondi, “There is no more to science than its method, and there is no more to the method than what Popper has said.”

There is much in common between the two methods of investigation, both of which start from an assumption and use logic, essentially rigorous mathematics (symbolic logic), to arrive at conclusions. There is, however, an essential difference. In the axiomatic method of deductive reasoning used by Euclid in mathematics, all that is required is that the conclusion reached not be in contradiction of the starting assumption. This also leads to the powerful indirect method of reductio ad absurdum, which we illustrated with examples from geometry and number theory. In the method used in modern scientific analysis, which starts with a conjecture and uses logic, what is extra is the role of observation to confirm or refute the conjecture. What is most important here is the role of refutation, because it immediately shows that your conjecture is false. On the other hand, if the observation you use confirms the conjecture, it only means that it is provisionally true; subsequent observational evidence may still contradict your conjecture. This is the case because observational technology is becoming ever more sophisticated, and so over time, the conjecture can be tested in ever finer detail. Thus, this method of conjecture and refutation provides modern science with a self-correcting procedure, which is what makes it such a reliable method of acquisition of new knowledge. Indeed, this is how natural selection works in biological evolution and ultimately how the mind works in the acquisition of knowledge at a neuronal level.

It is instructive to point out that the Athenian philosopher Aristotle (384–322 BCE), who was a generation ahead of Euclid and was famously responsible for the formalization of logic, came up with a far-reaching theory of how the known world worked. He conjectured that Earth was at the center of the cosmos and did not move. Furthermore, he conjectured that all other celestial objects (the Moon, the Sun, the individual planets, and all the stars) were attached to their own “crystalline spheres,” all of which rotated around Earth along a common axis with a period of a day. According to this conjecture, the twenty-four-hour day-night cycle that we experience was not due to the spin of Earth around its north-south axis, as we now know, but rather due to the revolution of the Sun (attached to its crystalline sphere) around Earth in a period of twenty-four hours. In this connection, it is noteworthy that Aristarchus of Samos, arguably one of the greatest Greek astronomers of antiquity, got it right (at least as far as the solar system was concerned) by correctly presuming that while the Moon revolved around the Earth, all other planets, including Earth, revolved around the Sun with different periods. He further presumed correctly that the observed diurnal motion of the Sun and the stars was due to the spinning of Earth around its north-south axis with a period of twenty-four hours. Aristarchus also knew from the work of another great contemporary astronomer, Eratosthenes (278–194 BCE), that this spin axis of Earth was inclined to the axis of Earth’s orbital plane at an angle of about 23.5° (the so-called obliquity of Earth’s orbit). Aristarchus correctly attributed the seasonal variations (spring, summer, autumn, and winter) as Earth revolved around the Sun to this fact. Aristarchus’s “heliocentric” model was able to correctly explain the complicated “looping” orbits of planets as seen in the sky by Earth-bound observers by noting the different angular speeds of the planets around the Sun, while Aristotle’s geocentric model could not.

Despite the enormous superiority of Aristarchus’s heliocentric model to Aristotle’s erroneous geocentric model, it was the latter that gained general acceptance during ancient times and prevailed for almost two millennia. There were several reasons for this, including that the motion of Earth is not perceptible to our senses; so a stationary Earth, while everything outside was moving, was a highly comfortable and commonsense premise. Another reason for the longevity of Aristotle’s geocentric model was the enthusiastic support it received from the powerful Roman Catholic Church, because, according to the Bible, man (created in God’s own image) is the pinnacle of his handiwork. So what better place for man’s abode than at the center of all creation? Furthermore, anthropocentrism has always had a powerful appeal to humans.

The geocentric cosmological theory was not the only wrong one proposed by Aristotle, although it is the best known. Moving on to physics, he premised that different physical laws held in different regions of space. Everything was perfect (unblemished and unchanging) in the “supra-lunar” region outside the crystal sphere where the Moon was located. Imperfection and change occurred only inside this sphere. Because comets appeared infrequently and were also seen to change, Aristotle relegated them to the sub-lunar sphere and regarded them as mere “exhalations” from Earth. This view of comets, which also became widely accepted, held up the understanding of comets for millennia. Further, Aristotle premised that heavier bodies fell to the ground faster than lighter bodies. He could easily have determined the falsity of this empirically, in the same way that Galileo did early in the seventeenth century. But empirical verification seemed not to have occurred to the great logician, who was perhaps dazzled by the power of his own invention.

Moving on to human physiology, Aristotle claimed that adult women had fewer teeth than adult men! He could have disabused himself of this absurd conclusion by a bit of counting, which he did not do. However, this contention was consistent with his general view that women were inferior to men in general while being somewhat superior to slaves! This subversive contention, which like his erroneous theories on cosmology and physics was also enthusiastically embraced by both the Christian and Islamic religions, contributed to the holding back of the progress of women even to the present day.

Euclid’s axiomatic method of deductive reasoning is the essential method used to acquire abstract mathematical truths to this day. Here all that is needed is that the conclusion reached be consistent with the starting proposition. When it comes to the real world, more than mere consistency between the starting premise and the conclusion is required, as is clear from Aristotle’s false theories of the cosmos. Also essential is observational verification of the conclusion. This is what is at the heart of the modern scientific method of investigation (conjecture and refutation) that has led to its spectacular success.

The modern scientific method of investigation, which is an extension of Euclid’s axiomatic method of deductive reasoning, has been steadily extended from physics and astronomy to chemistry and other areas of study such as biology, medicine, social sciences, and economics as each became more exact and quantitative and its predictions became verifiable. Thus, all scholarship owes a profound debt to Euclid, whose Elements showed us how to think rationally.

In this regard, one area where Euclid’s axiomatic method of deductive reasoning was used with devastating effect was in the demolition of the “logical” arguments proposed by several Christian apologists to support their belief in the existence of God. I first encountered these when I read the book Why I Am Not a Christian by the prominent British mathematician and philosopher Lord Bertrand Russell. Let me demonstrate this with a very straightforward demolition of the earliest of such arguments, the so-called “first cause” argument, which starts with the assertion that everything must have a cause. Therefore, the known world must have a cause, and that cause is God. Now assuming this proposition is true, one is forced to the logical conclusion that God, too, must have a cause, leading to a never-ending “causal chain.” This conundrum was overcome by arbitrarily stating that God needs no cause (in other words, he is the “cause-less” cause). So, here is a clear example of Euclid’s reductio ad absurdum; one’s conclusion directly contradicts one’s initial premise. Also leaving aside the initial premise, if one simply assumes that there is an un-caused first cause, then one is free to assume that the world itself is the first cause, without having to introduce an unnecessary intermediary (God).

Religious apologists have attempted to wiggle out of this first-cause conundrum by arguing that the first cause (God) is supernatural and therefore outside space and time. What they don’t seem to realize is that cause is necessarily defined in terms of time; cause precedes effect in time. So outside of time, the very concept of cause disappears, and one merely jumps from one logical conundrum to another.

There are several excellent recent books wherein these apologist arguments are systematically taken apart. My favorite is a thin book by mathematician John Allen Paulos titled Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up (2008). Written with clarity and wit reminiscent of another mathematician, Lord Bertrand Russell, Paulos deconstructs irrationality not only in religion but in many other areas of human affairs as well

One of the lesser-known areas indebted to Euclid is statecraft. Consider the American Declaration of Independence, written in 1776 by a group of well-educated and well-traveled men of the “Age of Enlightenment” that included Thomas Jefferson and the scientist Benjamin Franklin. Their familiarity with Euclid’s Elements is apparent in the structure of the celebrated statement: “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain inalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.” There is of course a clear difference from what Euclid claimed as self-evident (axioms) such as: “Things that are equal to the same thing are equal to one another.” This is a logical fact on which all rational discourse is predicated. On the other hand, the aforementioned statement in the Declaration of Independence “that all men are created equal” is no more than a human opinion, which, despite its nobility, is demonstrably wrong on both counts: creation and equality.

I learned another interesting snippet about the influence of Euclid’s Elements on American politics from my late professor of mathematics at the University of Ceylon (C. J. Eliezer). It concerns the great American president Abraham Lincoln. While running for the presidency, he had listed among his other qualifications that he had read and mastered the first six books of Euclid’s Elements while a member of Congress. Professor Eliezer was of the opinion that if all politicians in the world would take the time to emulate President Lincoln, it would lead them to think more clearly, and the world would not be in the sad mess it is in. I could not agree more, particularly as I write this when the U.S. president (Donald J. Trump) and numerous lawmakers are proud to proclaim that they do not believe in science and their actions are guided by faith.

Returning to Euclid’s geometry, it largely deals with figures lying in a plane and is referred to both as plane geometry or Euclidean geometry. Over two millennia after Euclid, it was realized that Euclidean (plane) geometry was not the only possible (self-consistent) geometry, but that there were two other (non-Euclidean) geometries. It is instructive to outline how this came to be. It started with Euclid’s fifth postulate (the parallel postulate), which stated that through any point (P) not lying on a (straight) line (L), there was one and only one (straight) line in the plane (containing P and L) that could be drawn parallel to L.

Many geometers after Euclid felt that the fifth postulate was one too many and that it could be derived from the previous four, but all their efforts failed. Finally, they tried using Euclid’s own reductio ad absurdum to do so. Noting that there are only three possibilities for the number of parallels—namely zero, one, or many—they attempted to eliminate both zero and many by the reductio ad absurdum method and failed in both cases. This means that both were possible alternatives to Euclid’s plane geometry. The first, with zero parallels, defined a space with positive curvature (such as a sphere). The second, with multiple parallels, defined a space with negative curvature (such as a saddle). Incidentally, in a triangle drawn on a sphere with positive curvature, the sum of the interior angles is more than 180°, whereas in the case of negative curvature, the sum of the interior angles of the corresponding triangle is less than 180°. Of course, we already know in the case of Euclidean flat space, the corresponding sum equals 180°.

Until the advent of Einstein’s theory of general relativity, these abstract non-Euclidean geometries were only of theoretical interest. All that changed when Einstein’s theory showed that space can have both positive and negative curvature, besides being flat. Observations of deep space have made it possible for us to infer the nature of the space we live in. Cosmological mathematical models, which assume that the universe as a whole is both isotropic and homogenous, contain a dimensionless parameter denoted by the Greek letter Ω, equal to the ratio of the observed mass/energy density (ρ) of the universe to some critical theoretical value, (ρ_c) (i.e., Ω = ρ/ρ_c), which is the required value to keep the universe exactly balanced between collapse due to the dominance of gravity and runaway expansion due to its gravity’s weakness. If Ω were even slightly greater than unity initially, the universe would have collapsed immediately. Contrariwise if Ω were even slightly less than unity, the universe would have expanded exponentially. So, for a “stable” (albeit expanding) universe, Ω has to be exactly equal to unity. We now know this to be the case to a high degree of accuracy if we include not only the density of normal directly observed matter but also the so-called “dark matter” and “dark energy” that have been inferred indirectly from several observations of deep space. Incidentally, when Ω = 1, the universe (or more correctly, the “observable” universe) becomes not only stable, but its geometry becomes flat—not in the sense that it becomes like a pancake, but rather that its overall geometry becomes “Euclidean,” as that of a flat surface. Why this is so is neatly explained by the current “inflationary model” of the origin of the universe, which I will not go into here. One can read about it in any modern book on cosmology.

We have now come full circle, starting with the axiomatic method of deductive reasoning that Euclid introduced in his monumental work Elements. While this method has had resounding success in the advancement of every branch of mathematics, besides geometry and number theory that Euclid dealt with, it eventually led, with the use of his fifth postulate and his powerful reductio ad absurdum method, to an entirely new class of geometry called non-Euclidean. While the theory of general relativity showed that the overall geometry of space could be Euclidean or either of the two non-Euclidean geometries, observations of deep space indicate that the geometry of space is indeed Euclidean.

What has enabled present-day scientists to reach these profound conclusions about the real world is the modern scientific method of conjecture and refutation that incorporates Euclid’s method of axiomatic deductive reasoning, while adding on the extra role of observational verification, which makes it self-correcting.

Euclid has often been referred to as the “father of geometry.” I think it would be appropriate to regard him more generally as the “father of rational thought.”

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