Geometry and Philosophy
For my masters course titled ‘Architecture: Theory and Criticism’, I have been instructed to read the book Geometry and Philosophy, the 1985 book by Liaquat Ali, published by Bangla Academy. Below are some of its excerpts translated in English for my ease of understanding.
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For Plato, mathematics held so much importance that the door of his academy had a sign that said that anyone not well-versed on geometry was not welcome inside.
Both the Greeks and Plato himself considered geometry not as a branch of a mathematics, but an independent field on its own. Hence, in his book titled Republic, Plato’s master Socrates refers to geometry as a subject similar to mathematics, and made it essential for philosophers to have the adequate knowledge about it.
However, geometry began as a subject in a way that contradicted Plato’s philosophical views. He detested the practice of knowledge for worldly benefits, and yet, Babylonians and Egyptians began practicing geometry for the need of reality of existence.
Geometry:
One of the specialties of Babylonian and Egyptian civilizations was methodical observation and preservation of information. They were especially advanced in all sorts of observation related to the practice of astronomy. For this purpose, they needed to make angular measurements. Additionally, they needed the knowledge of geometry for measuring lands and for constructing houses. In fact, the original meaning of the word ‘geometry’ means earth measurement. Even in the Bangla word জ্যামিতি, জ্যা means earth and মিতি means measurement.
Gradually, they started making a set of rules for practical applications based on the experiences they had acquired over time. Apart from medicine and surgery, this was the only field in which the Egyptians were way ahead of the Babylonians. Practical necessity was the reason behind this advancement. Every year, much of the Egyptian regions would get flooded by the Nile. After the flood, they needed to make measurements to reset the boundaries of each landmass. Due to this practice, geometry as a field developed substantially among Egyptians. Their functional knowledge of geometry became evident in the construction techniques of the pyramids, excavation of canals from the Nile to the Red Sea etc.
Despite all of this, neither the Babylonians nor the Egyptians could follow the actual method of geometry. To develop that method, mankind had to wait for thousands of years for Thales and Pythagoras.
For Babylonians and Egyptians, the primary method of practicing science was from the power of observation, and some generalizations based on that experience. When they repeatedly observed the same set of conditions in some specific cases, they considered those conditions to be true for those cases.
Observation is definitely an important aspect of science – after all, it is from observation that inductive reasoning builds up. However, it became evident in many situations that observation does not become fruitful without unifying principle, hence inductive method is not an actual scientific method. In reality, deductive reasoning has a great role in scientific methods, and such deductions are provided to science through mathematics.
Geometry also follows deductive reasoning, certain conclusions are made step-by-step due to some principles and assumptions. For instance, geometry does not require any measurements for proving the Pythagorean Theorem, rather it relies on previously made conclusions based on principles and assumptions. This is the same theory that the Egyptians followed, the opposite of which they applied in practical situations. However, all their calculations were based on measurements. They believed the only way to find out the height of a pyramid was by measuring it. It never occurred to them that it was very much possible to deduce the height without having to measure it.
After a long period of time, Thales traveled from Greece to Egypt and proved that it is indeed possible to determine the height of a pyramid without measuring it. And that is how deductive reasoning originated because of Thales.
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Thales:
Thales was born sometime between 640 to 620 BC in Miletus; he died between 550 and 540 BC. He was a merchant, philosopher, astronomer and mathematician. For trade purposes, he had to travel between Egypt and Babylon, and hence the scientific knowledge of these regions influenced his imagination and cognition.
Thales primarily has two contributions in the scientific history. Firstly, he gave an idea about natural objects’ contribution in shaping the world. And secondly, he is the first known individual to use deductive reasoning applied to geometry.
It was Thales who initiated the idea of proving geometry. It is easier to understand this second contribution of Thales’ if we go back to reflect on the scientific practice of the Babylonians and Egyptians in regards to geometry.
We understand that the Babylonians’ and Egyptians’ practice of geometry was limited to generalizations based on apparent experiences. This is a lot like how children learn geometry. How does a child know that the corners of a rectangle are equal? The easiest way is for the child to draw several rectangles and measure each of its diagonals, which will turn out to be equal. Let’s say he/she measured several rectangles in this way. Naturally, he/she will start to believe that every rectangle has equal corners.
But is this the true method of geometry? Should we teach geometry this way to adults as well? Is there any certainty that if a few rectangle have equal corners, all rectangles of the world will also be the same? In that case, every rectangle have to be measured individually which is impossible. So an alternative way is needed and geometry provides that.
We understand that the Babylonians’ and Egyptians’ practice of geometry was limited to generalizations based on apparent experiences. This is a lot like how children learn geometry. How does a child know that the corners of a rectangle are equal? The easiest way is for the child to draw several rectangles and measure each of its diagonals, which will turn out to be equal. Let’s say he/she measured several rectangles in this way. Naturally, he/she will start to believe that every rectangle has equal corners.
But is this the true method of geometry? Should we teach geometry this way to adults as well? Is there any certainty that if a few rectangle have equal corners, all rectangles of the world will also be the same? In that case, every rectangle have to be measured individually which is impossible. So an alternative way is needed and geometry provides that.
The Pythagoreans:
The seed planted by Thales turned into a tree due to the contribution of the Pythagoreans. Born in 570 BC in the Greek island of Samos, Pythagoras traveled to South Italy and created a community with his followers. The adherents of his community had to follow certain rules in a strict manner. One of the rules was that no one could claim any discovery as their own; it would be a general property on which every adherent had a right. This is why it is difficult to determine how much of a contribution Pythagoras had in regards to geometry.
Pythagoreans believed in a certain kind of mysticism. They wished to understand everything in the universe through numerals. They considered the pursuit of knowledge and asceticism to be superior to their work lives and they forbade any intellectual person to be involved with trade. According to them, the world existing in their consciousness is more real than the sensory world; the former is the ideal world and the latter is not. In fact, much of Plato’s thoughts about ideologies, general ownership of properties in ideal worlds and its strict regulations, banning of physical labour for rulers were influenced by Pythagoreans.
The Pythagorean’s way of thinking became an obstacle for experienced-based scientific practices. Since they believed that knowledge originated from the inner self, which can be obtained through insight, hence they disregarded the importance of the power of observation. When in reality, science cannot begin without observation.
Nonetheless, they made a big contribution in science. They gifted the field of science with a crucial tool by developing the first real mathematical method.
Plato:
The Pythagoreans’ way of thinking and traditions greatly influenced Plato and much of it is reflected in his philosophy. The regeneration of decaying idealism which his master Socrates began, was brought to fruition by Plato. Plato applied an assumed master plan regarding the universe, mankind, society, morality, art, education etc. In doing so, he shunned all kinds of knowledge based on observation and experience. Among the sciences, he gave importance only to astronomy, mathematics and geometry in his thinking; especially the latter two fields were used as important tools for defending his master plan.
One of the fundamentals of Plato’s philosophy is theory of ideas. He believed that the objects we see in natural environment are not real; all of these exist as ideas in an ideal world. All sensual objects are mere reflections of their ideal versions. The real object exists only in our imagination.
Plato developed his knowledge by keeping it compatible with his ideologies. He believed that the human soul is immortal and is thus the originator of knowledge. According to him, knowledge is nothing but a series of recollections.
Aristotle:
The fact that Aristotle was not a mathematician is simultaneously a fortunate and an unfortunate incident for mankind. It is fortunate because he was able to make such fruitful contributions in the fields of biology, medicine and political science, as he did not remain occupied with deductive reasoning of geometry. In these fields, to some extent, he had to rely on observation and inductive reasoning. It was known that he traveled from one island to another in order to collect samples related to his study of biology.
In political science, while Plato imagined an ideal state based on a set of principles derived from his insight, Aristotle went around collecting constitutions of over a hundred city-states, analysed them entirely and attempted to develop the principles of state based on that. For the next 1500 years, Europe continued to ignore this side of him and kept its focus on his work regarding metaphysics and other fields.
This was unfortunate for mankind. Perhaps a genius like Aristotle would have been able to understand other specialities of mathematics if he had studied it deeply, which would have made him an important figure in the field of scientific knowledge. Due to his lack of knowledge in the field of mathematics, he was completely influenced by the ideals of Plato and the Pythagoreans.
Euclid:
Euclid was the first divisional head of the mathematics department of the Museum of Alexandria. Little has been known about his life. Presumably, part of his teaching career was spent at the Athens Academy. He was in Alexandria around 300 BC.
Alexandria was the world’s first centre of knowledge where mathematics and geometry were given their due roles and what we know as actual scientific methods originated from there. Opposite to Athens’ anti-observation environment, extensive observatories, museums and laboratories were built in Alexandria. There, mathematics was prioritized to a great extent, however, it should be noted that the museum had no section reserved for philosophy. Which meant that there was no scope of turning mathematics and geometry into tools for the study of metaphysics.
And yet, perhaps because of his education at the Athens Academy, Euclid was not fully able to come out of Plato’s field of knowledge. Overall, his well-known book titled Element follows the rules of Platonism. Like Plato, he believed that the importance of mathematics lied in the fact that it advanced our thought process. But many mathematicians of that time considered one of Plato’s principles to be against conservative thinking; Eudoxus was among them.
Euclid showed how farsighted he was by opposing Plato’s opinions and writing a book on conic sections. Plato considered every line apart from a straight line and a circle to be mechanical, and banned the practice of them all. This was a backward way of thinking in the development of geometry.
Middle Ages:
There are plenty of social and political reasons why the practice of science ended in Alexandria; additionally the limitations of the methods practiced there accelerate its decline. For instance, despite the massive contribution they made in the field of geometry, it was not possible for them to make progress from then onwards. In Ancient Greek mathematics, algebra had a small role; their knowledge regarding mathematical numbers was also basic. As a result, very few methods based on numbers of arithmetic and algebra were applied to geometry. But geometry made little progress during the Middle Ages.
While Athens and Alexandria had very different perspectives, in both cities the people’s reasoning and logic was very much valued. There were disagreements about whether that reasoning was derived from experience or insight, but there was faith in the power of people.
The main difference between the thought process of Athens and Alexandria and the Middle Ages was that a lot of value was given to revelations and authority over people’s judgement. A process of gaining knowledge through meditation and mystic ways became popular. This is how the advancement of seeking knowledge was halted in Europe by closing off all ways of questioning. By reviving the paranormal world and prioritizing it over the real one, the progress of mankind fell very much behind in the Middle Ages .
Renaissance:
The Renaissance brought mankind out from the darkness of the Middle Ages. Copernicus, Kepler, Galileo and many others began a revolution in the field of science.