Part 8 (2/2)

Interestingly, the answers to both of these questions may be related. Mathematics itself could have originated from a subjective human perception of how nature works. Geometry may simply reflect the human ability to easily recognize lines, edges, and curves. Arithmetic may represent the human apt.i.tude to resolve discrete objects. In this picture, the mathematics that we have is a feature of the biological details of humans and of how they perceive the cosmos. of the biological details of humans and of how they perceive the cosmos. Mathematics thus is, in some sense, the language of the universe-of the universe discerned by humans. Other intelligent civilizations out there might have developed totally different sets of rules, if their mechanisms of perception are very different from ours. For example, when one drop of water is added to another drop or one molecular cloud in the galaxy coalesces with another, they make only one drop or one cloud, not two. Therefore, if a civilization that is somehow fluid based exists, for it, one plus one does not necessarily equal two. Such a civilization may recognize neither the prime numbers nor the Golden Ratio. To give another example, there is hardly any doubt that had even just the gravity of Earth been much stronger than it actually is, the Babylonians and Euclid might have proposed a different geometry than the Euclidean. Einstein's theory of general relativity has taught us that in a much stronger gravitational field, s.p.a.ce around us would be curved, not flat-light rays would travel along curved paths rather than along straight lines. Euclid's geometry emerged from his simple observations in Earth's weak gravity. (Other geometries, on curved surfaces, were formulated in the nineteenth century.) Mathematics thus is, in some sense, the language of the universe-of the universe discerned by humans. Other intelligent civilizations out there might have developed totally different sets of rules, if their mechanisms of perception are very different from ours. For example, when one drop of water is added to another drop or one molecular cloud in the galaxy coalesces with another, they make only one drop or one cloud, not two. Therefore, if a civilization that is somehow fluid based exists, for it, one plus one does not necessarily equal two. Such a civilization may recognize neither the prime numbers nor the Golden Ratio. To give another example, there is hardly any doubt that had even just the gravity of Earth been much stronger than it actually is, the Babylonians and Euclid might have proposed a different geometry than the Euclidean. Einstein's theory of general relativity has taught us that in a much stronger gravitational field, s.p.a.ce around us would be curved, not flat-light rays would travel along curved paths rather than along straight lines. Euclid's geometry emerged from his simple observations in Earth's weak gravity. (Other geometries, on curved surfaces, were formulated in the nineteenth century.) Evolution and natural selection definitely played a cardinal role in our theories of the universe. This is precisely why we don't continue to adhere today to the physics of Aristotle. This is not to say, however, that the evolution was always continuous and smooth. The biological evolution of life on Earth was neither. Life's pathway was occasionally shaped by chance occurrences like ma.s.s extinctions. Impacts of astronomical bodies (comets or asteroids) several miles in diameter caused the dinosaurs to perish and paved the way for the dominance of the mammals. The evolution of theories of the universe was also sporadically punctuated by quantum leaps in understanding. Newton's theory of gravitation and Einstein's General Relativity (”I still can't see how he thought of it,” said the late physicist Richard Feynman) are two perfect examples of such spectacular advances. How can we explain these miraculous achievements? The truth is that we can't. That is, no more than we can explain how, in a world of chess that was used to victories by margins of half a point or so, in 1971 Bobby Fischer suddenly demolished both chess grandmasters Mark Taimanov and Bent La.r.s.en by scores of six points to nothing on his way to the world champions.h.i.+p. We may find it equally difficult to comprehend how naturalists Charles Darwin (18091882) and Alfred Russel Wallace (18231913) independently had the inspiration to introduce the concept of evolution itself-the idea of a descent of all life from a common ancestral origin. We must simply recognize the fact that certain individuals are head and shoulders above the rest in terms of insight. Can, however, dramatic breakthroughs like Newton's and Einstein's be accommodated at all in a scenario of evolution and natural selection? They can, but in a somewhat less common interpretation of natural selection. While it is true that Newton's theory of universal gravitation had no contending theories to compete with at the time, it would not have survived to the present day had it not been the ”fittest.” Kepler, by contrast, proposed a very short-lived model for the Sun-planet interaction, in which the Sun spins on its axis flinging rays of magnetic power. These rays were supposed to grab on the planets and push them in a circle.

When these generalized definitions of evolution (allowing for quantum jumps) and natural selection (operating over extended periods of time) are adopted, I believe that the ”unreasonable” effectiveness of mathematics finds an explanation. Our mathematics is the symbolic counterpart of the universe we perceive universe we perceive, and its power has been continuously enhanced by human exploration.

Jef Raskin, the creator of the Macintosh computer at Apple, emphasizes a different aspect-the evolution of human logic. In a 1998 essay on the effectiveness of mathematics, he concludes that ”Human logic logic [emphasis added] was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. This is why mathematics is consistent with the physical world.” [emphasis added] was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. This is why mathematics is consistent with the physical world.”

In the play Tamburlaine the Great Tamburlaine the Great, a tale about a Machiavellian hero-villain who is at the same time sensitive and a vicious murderer, the great English playwright Christopher Marlowe (15641593) recognizes this human aspiration for understanding the cosmos: Nature that framed us of four elements, Warring within our b.r.e.a.s.t.s for regiment, Doth teach us all to have aspiring minds: Our souls, whose faculties can comprehend The wondrous Architecture of the world: And measure every wandering planet's course, Still climbing after knowledge infinite, And always moving as the restless spheres...

The Golden Ratio is a product of humanly invented geometry. Humans had no idea, however, into what magical fairyland this product was going to lead them. If geometry had not been invented at all, then we might have never known about the Golden Ratio. But then, who knows? It might have emerged as the output of a short computer program.

APPENDIX 1.

We want to show that for any whole numbers p p and and q q, such that p p is larger than is larger than q q, the three numbers: p p2 q q2; 2pq; p2+ q2 form a Pythagorean triple. In other words, we need to show that the sum of the squares of the first two is equal to the square of the third. For this we use the general ident.i.ties that hold for any form a Pythagorean triple. In other words, we need to show that the sum of the squares of the first two is equal to the square of the third. For this we use the general ident.i.ties that hold for any a a and and b b.

Based on these ident.i.ties, the square of the first number is:

and the sum of the first two squares is:

The square of the last number is:

We therefore see that the square of the third number is indeed equal to the sum of the squares of the first two, irrespective of the values of p p and and q. q.

APPENDIX 2.

We want to prove that the diagonal and the side of the pentagon are incommensurable-they do not have any common measure.

The proof is by the general method of reductio ad absurdum described at the end of Chapter 2.

Let us denote the side of the pentagon ABCDE ABCDE by by s s1 and the diagonal by and the diagonal by d d1. From the properties of isosceles triangles you can easily prove that From the properties of isosceles triangles you can easily prove that AB = AH AB = AH and and HC = HJ. HC = HJ. Let us now denote the side of the smaller pentagon Let us now denote the side of the smaller pentagon FGHIJ FGHIJ by by s s2 and its diagonal by and its diagonal by d d2. Clearly Clearly

Therefore:

If d d1 and and s s1 have a common measure, it means that both have a common measure, it means that both d d1 and and s s1 are some integer multiple of that common measure. Consequently, this is also a common measure of are some integer multiple of that common measure. Consequently, this is also a common measure of d d1 s s1 and therefore of and therefore of d d2. Similarly, the equalities Similarly, the equalities

and

give us

or

Since based on our a.s.sumption the common measure of s s1 and and d d1 is also a common measure of is also a common measure of d d2, the last equality shows that it is also a common measure of s s2. We therefore find that the same unit that measures We therefore find that the same unit that measures s s1 and and d d1 also measures also measures s s2 and and d d2. This process can be continued ad infinitum, for smaller and smaller pentagons. We would obtain that the same unit that was a common measure for the side and diagonal of the This process can be continued ad infinitum, for smaller and smaller pentagons. We would obtain that the same unit that was a common measure for the side and diagonal of the first first pentagon is also a common measure of all the other pentagons, irrespective of how tiny they become. Since this clearly cannot be true, it means that our initial a.s.sumption that the side and diagonal have a common measure was false-this completes the proof that pentagon is also a common measure of all the other pentagons, irrespective of how tiny they become. Since this clearly cannot be true, it means that our initial a.s.sumption that the side and diagonal have a common measure was false-this completes the proof that s s1 and and d d1 are incommensurable. are incommensurable.

APPENDIX 3.

The area of a triangle is half the product of the base and the height to that base. In the triangle TBC TBC the base, the base, BC BC, is equal to 2a 2a and the height, and the height, TA TA, is equal to s. s. Therefore, the area of the triangle is equal to Therefore, the area of the triangle is equal to s s a. a. We want to show that if the square of We want to show that if the square of the pyramid's the pyramid's height, height, h h2, is equal to the area of its triangular face, s s a, then a, then s/a s/a is equal to the Golden Ratio. is equal to the Golden Ratio.

We have that

Using the Pythagorean theorem in the right angle triangle TOA TOA, we have

We can now subst.i.tute for h h2 from the first equation to obtain from the first equation to obtain

Dividing both sides by a a2, we get:

In other words, if we denote s s/a by by x x, we have the quadratic equation:

In Chapter 4 I show that this is precisely the equation defining the Golden Ratio.

APPENDIX 4.

One of the theorems in The Elements The Elements demonstrates that when two triangles have the same angles, they are demonstrates that when two triangles have the same angles, they are similar. similar. Namely, the two triangles have precisely the same shape, with all their sides being proportional to each other. If one side of one triangle is twice as long as the respective side of the other triangle, then so are other sides. The two triangles Namely, the two triangles have precisely the same shape, with all their sides being proportional to each other. If one side of one triangle is twice as long as the respective side of the other triangle, then so are other sides. The two triangles ADB ADB and and DBC DBC are similar (because they have the same angles). Therefore, the ratio are similar (because they have the same angles). Therefore, the ratio AB/DB AB/DB (ratio of the (ratio of the sides sides of the two triangles of the two triangles ADB ADB and and DBC) DBC) is equal to is equal to DB/BC DB/BC (ratio of the (ratio of the bases bases of the same two triangles): of the same two triangles):

AB/DB = = DB DB/BC.

But the two triangles are also isosceles, so that DB = = DC DC = = AC AC.

We therefore find from the above two equalities that AC/BC = = AB AB/AC,.

which means (according to Euclid's definition) that point C C divides line divides line AB AB in a Golden Ratio. Since in a Golden Ratio. Since AD = AB AD = AB and and DB = AC DB = AC, we also have AD/DB = AD/DB = . .

APPENDIX 5.

Quadratic equations are equations of the form

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