Great thinkers on the structure of the world
From Pythagoras and Plato through Kepler and Newton to Planck and Dürr. How scientists thought about the order of the world.
Many great thinkers appear scattered through the other chapters. Here they are gathered together, each with the idea that matters for the tetractys.
Pythagoras and Philolaus
All is number. The world is ordered by ratios of numbers, and to know is to count. The decad is the beginning and guide of all life. See The ancient sources.
Heraclitus
The logos is common to all, and yet most people live as though they had an understanding of their own. Nature loves to hide, and the invisible harmony is stronger than the visible. Whoever does not know his measure, falls short of it or oversteps it, comes to a fall.
Plato
In the Timaeus the world is built of geometric solids, and the world soul is ordered by the number series of the lambdoma. Over the entrance to his Academy there is said to have stood the warning that no one ignorant of geometry should enter. Plato is said to have bought the writings of Philolaus and taken them into the Timaeus.
Plotinus
For the Neoplatonist it is the idea that enters and orders what is to become one out of many parts, joining it into a whole and bringing it into accord with itself.
Albertus Magnus
The forming powers flow ceaselessly from a luminous, spiritual world into matter. They do so not to become material themselves but to accomplish something divine. They are led back to their origin only through the human mind, which recognises the divine forms in matter and so frees them from it.
Nicholas of Cusa
The first archetype of things in the mind of the Creator is number, the cardinal concludes. That is why joy and beauty dwell in all things, for they consist in proportion, and proportion in number. Number is therefore the best path leading up to wisdom.
Agrippa von Nettesheim
Everything created from the beginning is formed according to numerical ratios that lay as a model in the mind of the Creator. That is why great powers dwell in numbers, and the simple numbers in particular signify divine things.
Albrecht Dürer
For Dürer, according to the historian of mathematics Ernst Buchholz, mathematics was the science that explores the forms of the mind, seeks their connections and traces the laws of the mind as harmonious symmetries.
Galileo Galilei
The book of nature is written in the language of mathematics, and its characters are triangles, circles and other geometric figures.
Johannes Kepler
Geometry, music and astronomy are at heart one. The geometric archetypes are thoughts of God, according to which music and the cosmos are built. Kepler had himself portrayed with a pair of compasses in his hand, so important was the study of geometry to him. See Harmony of the spheres and Gap-free filling of space.
Kepler adopted a thought of the Neoplatonist Proclus here: mathematics is recollection of the invisible forms of the soul. It awakens the mind, purifies the intellect, brings to light the ideas that essentially lie within us and frees us from forgetting and ignorance. For the study of nature it achieves most, because it reveals the ordered structure of thoughts according to which the universe is built.
Isaac Newton
The physicist engaged at least as intensively with alchemy and theology as with mechanics, and also owned Rosicrucian writings. The translator Ed Dellian, who re-edited Newton’s original Latin writings, sees the geometric fourfold proportion, a tetractys, as the key to understanding Newton’s actual theory of motion.
Newton himself was convinced that this world, with all its variety of forms and motions, could have arisen from nothing other than the free will of God, who rules over all.
Gottfried Wilhelm Leibniz
He described the rhythm of six, 6n ± 1, in which the primes lie, and took up the combinatorics of Ramon Llull, from which his idea of a universal language of signs and mathematical logic emerged.
Leibniz foresaw a time when people would recognise again the value of a holier philosophy. Then mathematical studies too would take a new direction: towards sharpening judgement, towards knowledge of the nature of harmony and the archetype of beauty, and natural science would again serve the Author who shows us in the visible world the image of the spiritual one.
Novalis
“The life of the gods is mathematics. All divine messengers must be mathematicians.” And: to everything he does, a person must give his undivided attention.
Carl Gustav Carus
The Romantic physician and natural philosopher saw in all natural forms an articulation of the eternally becoming according to the eternal laws of form and number, differing from pure law only in that it is never realised without a faint admixture of the irrational.
Ralph Waldo Emerson
When reason is stimulated to more earnest vision, outlines and surfaces become transparent and are no longer seen, causes and spirits are seen through them. Emerson called this awakening of the higher powers the best moments of life.
Max Planck
In an often-quoted lecture the founder of quantum physics took the view that there is no matter as such. All matter, he said, arises and exists only through a force behind which stands a conscious, intelligent mind. Not visible, perishable matter is the real, but the invisible spirit.
Albert Einstein
A human being is part of the whole we call the universe. That he experiences himself, his thoughts and feelings as separate from everything else Einstein called a kind of optical delusion of consciousness, a prison restricting our affection to a few people. The real task is to free ourselves from it by widening our compassion to all living beings and the whole of nature. Also well known is his remark that science without religion is lame and religion without science is blind.
James Jeans
The astrophysicist wrote in 1930 that the stream of knowledge is heading towards a non-mechanical reality: “the universe begins to look more like a great thought than like a great machine”.
Erwin Schrödinger
In everyday life we must distinguish subject and object, in philosophical thinking, Schrödinger held, we should give it up. World and mind are formed from the same data, there is not an existing world and a perceived one, only one. The barrier between subject and object has not broken down under the new physics, it never existed.
Werner Heisenberg
The variety of phenomena can be understood because it rests on unified formal principles that can be represented mathematically. The root of phenomena is mathematical law, which fixes fundamental symmetries and so the frame of all events. In Physics and Beyond (Der Teil und das Ganze) Heisenberg recalls that since Planck the quantum conditions brought a strange element of number mysticism into atomic physics: certain quantities must be whole multiples of the quantum of action, just as for the Pythagoreans two strings sound in harmony when their lengths stand in a whole-number ratio (see The monochord). Scientific concepts, Heisenberg said, always fit only a limited part of reality, and the part not yet understood remains infinite. He is also credited with the saying that the first gulp from the glass of natural science makes you an atheist, but at the bottom of the glass God is waiting.
Hermann Weyl
For the mathematician, the mystery of number, number magic, rests on the fact that in the number series the mind produces from itself an infinite variety of distinctive individual beings, perceptible for instance in the impenetrable law of the distribution of primes. Of the ornaments of ancient cultures Weyl wrote that their geometric imagination can hardly be overestimated and their construction is far from trivial: the art of ornament contains the oldest known piece of higher mathematics (see Ornament in four-four time).
Andreas Speiser
For the Swiss mathematician, geometric symmetries are the true form-giving powers. Invoking geometry and number at once is not only possible, it is the whole purpose of our life.
Carl Friedrich von Weizsäcker
The physical world view, according to Weizsäcker, is not wrong in what it asserts but in what it keeps silent about.
Hans-Peter Dürr
The physicist distinguished knowledge for use, which asks what one can do with something, from knowledge for orientation, which asks what the world is and how we stand in it. See Number theory + geometry = philosophy.
In a 2007 interview Dürr went further: basically there is no matter, only a web of relations, constant change, aliveness. Primarily only connection exists, what links without a material basis, one could also call it spirit. Matter and energy appear only secondarily, as congealed spirit, comparable to the software of a computer.
Sri Aurobindo
The life you lead conceals the light you are. Behind every thing stands an absolute that it seeks in its own way, everything finite strives to express an infinite it feels to be its true reality. The logical intellect, however, copes only with what is fixed and finite.
Erich Neumann
For the psychologist, the human being as homo creator is the decisive concern of our time: its healing depends on whether the individual can again experience himself as creative, connected with his own nature and that of the world.
Regine Kather
The philosopher describes the recognition of beauty as perception of form and at once of the forming power. Beauty sharpens our eye for the hidden, structure-giving sides of reality, and light, image of timeless presence and perfect awareness, becomes the mediator between the material and the spiritual.
Hans-Georg Gadamer
In The Beginning of Philosophy the Heidelberg philosopher studied the Presocratics. He observed that ancient biographies present almost every early thinker, from Anaximander to Parmenides, as a follower of Pythagoras. For him this shows that Pythagoras united the great basic motifs in himself: the riddle of numbers and the riddle of the soul. See Death and reincarnation.