User:Mateuszica~enwikibooks/Math Quotations

Pythagoreans

 * On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt—mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician—mathematics was God!

Plato

 * Plato writes: “Geometric equality is of great importance among gods and men.”

and in The Republic, knowledge of mathematics is taken to be a crucial step on the pathway to knowing the divine forms. Plato does not use mathematics for the formulation of some laws of nature that are testable by experiments. Rather, for him, the mathematical character of the world is simply a consequence of the fact that “God always geometrizes.”
 * To Plato, the only things that truly and wholly exist are those abstract forms and ideas of mathematics, since only in mathematics, he maintained, could we gain absolutely certain and objective knowledge. Consequently, in Plato’s mind, mathematics becomes closely associated with the divine. In the dialogue Timaeus, the creator god uses mathematics to fashion the world,


 * Plato wrote in Timaeus: “As there still remained one compound figure, the fifth, God used it for the whole, broidering it with designs.” So the dodecahedron represented the universe as a whole. Note, however, that the dodecahedron, with its twelve pentagonal surfaces, has the golden ratio written all over it. Both its volume and its surface area can be expressed as simple functions of the golden ratio (the same is true for the icosahedron).


 * The note may have been written by the fourth century orator Sopatros, and it reads (in a translation by Andrew Barker): “There had been inscribed at the front of the School of Plato, ‘Let no one who is not a geometer enter.’ [That is] in place of ‘unfair’ or ‘unjust’: for geometry pursues fairness and justice.” The note seems to imply that Plato’s inscription replaced “unfair or unjust person” in a sign that was common in sacred places (“Let no unfair or unjust person enter”) with the phrase “one who is not a geometer.”

Archimedes

 * Archimedes changed the world of mathematics and its perceived relation to the cosmos in a profound way. By displaying an astounding combination of theoretical and practical interests, he provided the first empirical, rather than mythical, evidence for an apparent mathematical design of nature. The perception of mathematics being the language of the universe, and therefore the concept of God as a mathematician,was born in Archimedes’ work.

Galileo
Galileo was not satisfied with mathematics as the mere go-between or conduit. He took the extra bold step of equating mathematics with God’s native tongue. This identification, however, raised another serious problem—one that was about to have a dramatic impact on Galileo’s life. According to Galileo, God spoke in the language of mathematics in designing nature. Moreover, Galileo argued that by pursuing science using the language of mechanical equilibrium and mathematics, humans could understand the divine mind. Put differently, when a person finds a solution to a problem using proportional geometry, the insights and understanding gained are godlike.
 * Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician. First, we must realize that to Galileo, mathematics ultimately meant geometry

Descartes

 * Functions are truly the bread and butter of modern scientists, statisticians, and economists. Once many repeated scientific experiments or observations produce the same functional interrelationships, those may acquire the elevated status of laws of nature—mathematical descriptions of a behavior all natural phenomena are found to obey. Descartes’ ideas therefore opened the door for a systematic mathematization of nearly everything—the very essence of the notion that God is a mathematician.

For our present purposes the most interesting point is Descartes’ view that God created all the “eternal truths.” In particular, he declared that “the mathematical truths which you call eternal have been laid down by God and depend on Him entirely no less than the rest of his creatures.” So the Cartesian God was more than a mathematician, in the sense of being the creator of both mathematics and a physical world that is entirely based on mathematics. According to this worldview, which was becoming prevalent at the end of the seventeenth century, humans clearly only discover mathematics and do not invent it.

Newton
For Newton, the world’s very existence and the mathematical regularity of the observed cosmos were evidence for God’s presence.

Newton added yet another twist, based on the universality of his laws. He regarded the fact that the entire cosmos is governed by the same laws and appears to be stable as further evidence for God’s guiding hand.

In his book Opticks, Newton made it clear that he did not believe that the laws of nature by themselves were sufficient to explain the universe’s existence—God was the creator and sustainer of all the atoms that make up the cosmic matter: “For it became him [God] who created them [the atoms] to set them in order. And if he did so, it’s unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature.” In other words, to Newton, God was a mathematician (among other things), not just as a figure of speech, but almost literally—the Creator God brought into existence a physical world that was governed by mathematical laws.

Others

 * If you had to choose one concept of our mathematics that has the highest probability of having an existence independent of the human mind, which one would you select? Most people would probably conclude that this has to be the natural numbers. What can be more “natural” than 1, 2, 3,…? Even the German mathematician of intuitionist inclinations Leopold Kronecker (1823–91) famously declared: “God created the natural numbers, all else is the work of man.”


 * The German mathematician Jacob Jacobi (1804–51) presumably expressed those shifting tides when he replaced Plato’s “God ever geometrizes” by his own motto: “God ever arithmetizes.” In some sense, however, these efforts only transported the problem to a different branch of mathematics. While the great German mathematician David Hilbert (1862–1943) did succeed in demonstrating that Euclidean geometry was consistent as long as arithmetic was consistent, the consistency of the latter was far from unambiguously established at that point.

if mathematics wasn’t the language of the cosmos, why did it work as well as it did in explaining things ranging from the basic laws of nature to human characteristics?
 * On the relationship between mathematics and the physical world, a new sentiment was in the air. For many centuries, the interpretation of mathematics as a reading of the cosmos had been dramatically and continuously enhanced. The mathematization of the sciences by Galileo, Descartes, Newton, the Bernoullis, Pascal, Lagrange, Quetelet, and others was taken as strong evidence for an underlying mathematical design in nature. One could clearly argue that


 * To be sure, mathematicians did realize that mathematics dealt only with rather abstract Platonic forms, but those were regarded as reasonable idealizations of the actual physical elements. In fact, the feeling that the book of nature was written in the language of mathematics was so deeply rooted that many mathematicians absolutely refused even to consider mathematical concepts and structures that were not directly related to the physical world.