Excerpts from Books on Math

Many books on Math have interesting things to say about the subject. These are typically longer than quotations which need to be short.

In this chapter we will include such excerpts.

Perhaps we can see more easily why one should study mathematics if we take a moment to consider what mathematics is. Unfortunately the answer cannot be given in a single sentence or a single chapter. The subject has many facets or, some might say, is Hydra-headed. One can look at mathematics as a language, as a particular kind of logical structure, as a body of knowledge about number and space, as a series of methods for deriving conclusions, as the essence of our knowledge of the physical world, or merely as an amusing intellectual activity - Morris Kline from his book Mathematics for Liberal Arts

The abstractions of mathematics possessed a special importance for the Greeks. The philosophers pointed out that, to pass from a knowledge of the world of matter to the world of ideas, man must train his mind to grasp the ideas. These highest realities blind the person who is not prepared to contemplate them. He is, to use Plato’s famous simile, like one who lives continuously in the deep shadows of a cave and is suddenly brought out into the sunlight. The study of mathematics helps make the transition from darkness to light. Mathematics is in fact ideally suited to prepare the mind for higher forms of thought because on one hand it pertains to the world of visible things and on the other hand it deals with abstract concepts. Hence through the study of mathematics man learns to pass from concrete figures to abstract forms; moreover, this study purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas. Morris Kline from his book Mathematics for Liberal Arts

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics… Just as the eye was made to see color and the ear to hear sounds, so the human mind was made to understand quantity - Johannes Kepler, seventeenth century German physicist best known for his work on planetary motion

The knowledge at which geometry aims is knowledge of the eternal, and not of anything perishing and transient. Geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Therefore, nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. Plato’s Republic (Book VII)

We must endeavor that those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; … arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible and tangible objects into the argument. Plato’s Republic (Book VII)

"… the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions - Paul Lockhart in his piece, A Mathematician’s Lament

"Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they are right. - Paul Lockhart on education in schools

Math & AI in Biology - There’s an age-old adage in biology: structure determines function. In order to understand the function of the myriad proteins that perform vital jobs in a healthy body—or malfunction in a diseased one—scientists have to first determine these proteins’ molecular structure. But this is no easy feat: protein molecules consist of long, twisty chains of up to thousands of amino acids, chemical compounds that can interact with one another in many ways to take on an enormous number of possible three-dimensional shapes. Figuring out a single protein’s structure, or solving the “protein-folding problem,” can take years of finicky experiments - Google DeepMind CEO Demis Hassabis explains how its AlphaFold AI program predicted the 3-D structure of every known protein

"One enters the first room of the mansion and it’s dark. One stumbles around bumping into furniture, but gradually you learn where each piece of furniture is. Finally, after six months of so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them" - Andrew Wiles recalling his pursuit of a proof for Fermat's Last Theorem

“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.” - Pierre Simon Laplace on the Hindu Decimal Place Value System

Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly, it became a wave equation instead of a heat conduction equation. And Schrödinger found, to his delight, that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom. It turns out that the Schrödinger equation describes correctly everything we know about the behaviour of atoms. It is the basis of all chemistry and most of physics. - Freeman Dyson, about the usage of imaginary numbers

‘Mathematics is simultaneously a discipline, a method, and a language, applicable to other disciplines or methods of knowledge, but also to mathematics itself. It is a mode of organizing, validating, and communicating knowledge, which uses the principles of logic and gives it a field of manifestation and application that is “pure,” free from contingency, and free from the confusion of natural language, in which the truths of propositions are universally valid and do not depend on any interpretation. It is the only discipline entitled to investigate concepts of any nature and any degree of generality, but also to be self-referential and self-applicable, as is in fact human reason.’ - Dr. Catalin Barboianu is the author of the book What is Mathematics: School Guide to Conceptual Understanding of Mathematics

“Geometry exists in all scales of nature, from microscopic scales — such as molecules, to astronomical scales such as in the patterns of orbital resonance of celestial bodies. The geometry formed in the universe is due to the laws of spatial geometry and motion which rules how the fundamental forces of electromagnetism, gravity, strong and weak nuclear forces works. Modern physics points to the idea that everything in the universe can be explained by the interaction of the fundamental forces, and that all of them without exception are ruled by laws of geometry and motion. The laws of vibratory motion forms geometric patterns through the resonance and synchronized interaction of wave interference. Physics proves all geometry in reality, from symmetries to asymmetries, can be reduced to the phenomena of energy, frequency, vibration. Energy of what? All fundamental forces, gravity and both strong/weak nuclear forces merges with electromagnetism, which is light. Light can be converted into cycles, motions, animations, vibrations, into magnetism and electricity, into energy such as heat or cold, into matter such as particles and molecules, into life and consciousness, into intelligence and ultimately guides and forms all cycles of evolution in the known universe. This infinite mandala-creating universe is a holographic and electromagnetic fractal substance formed by geometric laws of light. Art by my artist friend João Junqueira. I imagined the concept of an art that could symbolize the symmetric relationship between toroidal dynamics, harmonic motions, orbitals, resonance and cymatics — so I asked him to create this art and he recreated my idea exactly in the way I asked him to do it: with a pentagonal cymatic resonance below the symmetry of the DNA golden decagon molecule..with a Pentagram above formed by orbital resonance... As above, so below...”- Gabriel Lyrio (@lightisconsciousness)

1. A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

2. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

3. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

4. “It (proof by contradiction) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

5. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”

6. "The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains.”

G H Hardy in "A Mathematician's Apology"