Anaxagoras Contribution in Mathematics


Anaxagoras was born in Clazomenae on the coast of Ionia, which is now Turkey, near about 500 BC. He lived there for many years where he learnt new facts of philosophy and to do a scientific study of the world. He belonged to a very rich family, but he gave up all his wealth and went to Athens for the sake of study of science. One of the philosophers Heath writes about him, “He neglected his possessions, which were considerable, to devote himself to science”. Anaxagoras was considered to be the first person who gave ideas of philosophy to the citizens of Athens. Pericles was in ruling power when Anaxagoras stayed in Athens and was five years younger than Anaxagoras. Soon they both became friends that put Anaxagoras in trouble, as the rivels of Pericles were now the rivels of Anaxagoras. In about 450 BC, Anaxagoras was imprisoned for commenting on Sun. He said that the Sun was not a god, it was a red stone and the moon shines by its reflected light. In his view, the sun was a red stone, the moon is earthy, the stars are fiery stones, and the earth is a flat surface. He was the first who gave a correct explanation of the eclipses of the sun and moon. This shows that he possessed the knowledge of geometry and applied it to the study of astronomy. He made new theories by suggesting heavenly bodies and the fall of meteors. Plutarch says, “Anaxagoras is said to have predicted that if the heavenly bodies should be loosened by some slip or shake, one of them might be torn away, and might plunge and fall to earth. He also tried to give some scientific theories about the rainbow. While talking about the matter, Anaxagoras mentioned an infinite number of elements or basic building blocks. He claimed,

there is a portion of every thing, i.e., of every elemental stuff, in every thing, each is and was most manifestly those things of which there is most in it.’

Later on, he was forced to go from Athens to Lambsacus in Troad (the historical name of the Biga Peninsula in the northwestern part of Anatolia, Turkey). In around 428 BC, he died there. The people of Lambsacus celebrated his death anniversary for many years and declared a holiday for school children on that day. On his grave they inscribed,” Here Anaxagoras, who in his quest of truth scaled heaven itself, is laid to rest”. To have an idea about his personality we can take into account a remark made by him when somebody asked him that, what is the point of being born, he replied, ‘for the investigation of sun, moon, and heaven.’ 

1. Squaring of a Circle

According to Plutarch, Anaxagoras in his book ‘On Axile’ tried to solve the problem of squaring a circle while he was in prison. Squaring a circle means, constructing a square with the same area as the given circle using a limited number of steps by compass and straightedge. With the help of a compass and straightedge, only algebraic numbers can be drawn. So it was impossible to draw √π because it was considered a transcendental number until 1882. In modern theory, approximations of squaring a circle were made by constructing lengths close to π.


Other Works

1. Anaxagoras’s two levels of Metaphysics

According to Anaxagoras, unlike human-made products, he considered plants, animals and heavenly bodies are natural constructs. They are called natural because they are constructed by the natural processes, and they are called constructs because they depend on the ingredients of which they are constructed, for their survival. This gave him two levels of metaphysics. Things like earth, fire, water, stone, and wood are genuinely real and the things that are made up of these ingredients are not genuinely real.

2. Everything is Everything

Anaxagoras claimed that “All things were together”, “In everything, there is a share of everything.” He said that all the things are mixed but as they grow they are dispersed. In the same seminal fluid, there are hairs, nails, bone, and veins together but as they grow they are separated off. He further added by saying that, black is white, white is black and also light is mixed with heavy and vice versa. This principle of everything is everything somehow states that in every space, all ingredients should be present at all times. Scholars thought that Anaxagoras mentioned this principle about small particles and how small a particle is, it will acquire some space all by itself and will not allow any other ingredient inside it. Anaxagoras’s theory needs to consider ingredients as fluids that can be properly mixed and can be differentiated based on their characteristics like density.

3. Principle of No Smallest No Largest

To counter the view that the entry of a new entity into the mixture will lead to the destruction of the previous mixture, Anaxagoras introduced the principle of no smallest no largest. He claimed that if there is no lower bound on the density of an ingredient, then no ingredient can be completely removed from the mixture. He states,

Nor of the small is there a smallest, but always a smaller (for what-is cannot not be) — but also of the large there is always a larger. And the large is equal to the small in extent, but in relation to itself each thing is both large and small.’

So it is not possible to separate anything and all things are in everything because small and large are equal in number. This principle is somehow similar to the ‘everything is everything’ principle.

4. Importance of Nous

Aristotle and Plato both appreciated Anaxagoras for utilizing nous as the first principle of motion, but both were disappointed as Anaxagoras was not consistent with his statement. Initially, he mentioned that everything is everything, but later he denied the fact in the context of nous or mind. He said that the principle of everything is not applied to nous. It is mixed with nothing and is alone by itself. If it is mixed with everything, then it would not be able to control and rule anything the way it does now. In his view, mind or nous starts the rotation of mass of ingredients, then it controls that rotation and the local rotation that takes place in the large circle is the whole cosmos. Anaxagoras mentioned,

Nous controlled the whole revolution, so that it started to revolve in the beginning. First it began to revolve from a small region, but it is revolving yet more, and it will revolve still more. And whatever sorts of things were going to be, and whatever sorts were and now are not, and as many as are now and whatever sorts will be, all these nous set in order. And nous also ordered this revolution, in which the things being separated off now revolve, the stars, sun, moon, air, and the aether. This revolution caused them to separate off.”

5. Anaxagoras’s Theory of Nous

In the view of Anaxagoras, it was nous or mind that drives our day to day processes and had created the whole world. He said that the growth of living organisms depends on the power of the mind that helps them to extract nourishment from their surroundings. His theory was praised by Aristotle, but both Plato and Aristotle were concerned about the fact that the driving force of the mind was not ethical. They wanted it to act in favour of the interests of the world. Also, Newton’s mechanical universe was similar to Anaxagoras views of the universe instead of ethical intelligence proposed by Plato and Aristotle.

6. Predominance Principle

After mentioning the principle of everything, he claimed that all nous are alike. In his view, something is hot, if hot predominates over the cold in it. This principle does not mean that a hot thing does not possess a cold thing. Same as, doghood is not absent in humans, rather humanity predominates doghood in humans. The problem was how to determine the predominant ingredient after giving everything is everything and no smallest-largest principle. He gave the example of a gold ring that, gold in a gold ring predominates as an ingredient over other ingredients and we can ask the same question about the ingredient of the ingredient, and so on. Anaxagoras said that if I cannot determine what predominates in it, I cannot determine what anything is.

7. Cosmology

Anaxagoras gave a complete picture of the universe, heaven, earth, and meteorological phenomena. He explains this picture by saying that the rotation of a mixture begins in a small area and then it spread throughout the mass. When the mixture is infinite, the process of rotation and expansion will continue forever and bring more ingredients into the circle. At the edges, the speed of rotation is faster where the expanding rotation moves the yet unmoved ingredients. As time passes, the continuous rotations throw the lighter ingredients to the edges of the whirl and the lighter ones to the centre and thus putting earth at the centre and air and aether (fire) away from the centre. This gives the traditional Greek picture of our Earth (itself a mixture of all ingredients, with earth, heavy ores, and minerals predominating) covered in many places by the water, with air and the fiery reaches of the heavens. The force of rotation can cause slippage of stars towards the earth that seems like meteors. A detailed view of Anaxagoras on meteors was found in Plutarch’s ‘Life of Lysander’. He also gave explanations for the light of the Milky Way, the formation of comets, the inclination of the heavens, the solstices, and the composition of the moon and stars.

8. Structure of Earth

According to Anaxagoras, the earth is a flat surface that rests on air and is stable because of its heavy size. He assumed the earth to be stationary and because of its flatness, water is spread all around it with mountains and plains rising above the level of water. Water that is trapped below the surface of the earth, by rotation goes into the sea through rivers. Also, the air moving below the earth’s surface sometimes cannot make its way because of getting trapped in crevices of the earth’s surface thus resulting in earthquakes. Anaxagoras tried to explain everything. Other sources revealed that he also explained thunder and lightning, the source of Nile, why seawater is salty and made the first correct account on the nature of hail.

He was the first Pre-socratic philosopher to settle in Athens and was a significant figure, not only for later philosophical thinkers but also for the wider civic culture of his time.






















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