The Philosophy of Mathematics – Euclid, Pythagoras, Plato, Aristotle
Until late 19th century, there was basically a blurring of boundaries between different disciplines, especially mathematics and philosophy was not seen as two different subjects, mathematicians and philosophers used one single canvas to use different brush strokes.
In ancient time for a pretty long period they were working in disguise much like the chalk (blackboard) and the marker (whiteboard) doing the same job in different ways but things changed and today they are literally operate like the chalk and the cheese doing different tasks and serving a different set of audience.
These are contrasting but fascinating ways to understand the complex web of world and eventually these things that make a world of difference in our ways of thinking and simplifying our standards of living. There are signs, symbols, logic and magic in analyzing the complexity of the nature nurturing our multiple senses and there are mathematical models constructed to put these intriguing natural phenomena in its proper perspective.
Philosophers and Mathematicians as thinkers use the same hat i.e. the camera but the color of the hat i.e. the lens that they use are different, the characteristics in terms of obtaining the wide-angle view to the mechanical ability to zoom it to get the granularity in perspective and that makes all the difference in what we see and how we capture. It is the plotted in perception of our rollicking reality. All these things seem similar but the output is quite different.
The mathematicians conduct experimentation to show models.
The philosophers construct scenes to simulate their wild thoughts.
Euclid – The Mathematician
During the 300 BC as a Greek mathematician Euclid lived in Alexandria, Egypt during the reign of Ptolemy. He is the most profound mathematicians of antiquity; he left an indelible mark in the treatise of mathematics “The Elements”. Other than Bible, The Element is the most translated, published and studied book in the world.
Euclid did extensive work in geometry and he was synonymous with path breaking work and was later known as the father of geometry. Inventing new ways of looking at the objects, their shapes and sizes, and also on the space and time around these objects. Euclidean Geometry, the standard he set for deductive reasoning and geometric instructions dominated the scene for two thousand years till the advent of non-Euclidean geometry in the 19th century. Not well documented but it seems he studied for some time in Plato’s Academy in Athens.
Few of the important axioms and notions from Euclid’s Elements are as follows;
- Given two points there is one straight line that joins them.
- A straight line segment can be prolonged indefinitely.
- All right angles are equal.
- Things equal to same things are equal.
- The whole is greater than a part.
Pythagoras – The Mathematician
He is most often called the first pure mathematician. Also, he was known as the Ionian Greek philosopher. His Pythagoras Theorem defines the way we study and understand geometry. The theorem states that the square of the hypotenuse is equal to the sum of the square of the other two sides. Pythagoras was one of the most dominating pre-Socrates philosophers but there is almost no written document from his work. The question always remained whether he was more of a mathematician or more a philosopher.
Pythagoras had stated that there are three kinds of people who come to Olympic Games. First kinds are the lover of wisdom, the second are the lover of honor and the third are the lover of gains. This doctrine of tripartite soul is ascribed to Plato. Pythagorean doctrine was very much around the numbers and the properties of equations that defined the way we form shapes and work around building a structure with the interplay of lines and numbers. He established the foundation of number theory. For him number was everything and to the extent God is Number itself just the way for Euclid Geometry was God.
Plato – The Philosopher
He is the great Greek philosopher best known for his Dialogues and for founding his Academy in Athens. Highly influenced by reading the works of Socrates to Pythagoras to Heraclitus to Paramedics. For Plato he was uniquely positioned in the history of development of the world of philosophy with Socrates as his teacher and Aristotle as his student. Both his teacher and student are the greatest thinkers.
Plato’s contribution extended beyond the boundaries of few subjects to many more from mathematics to metaphysics to medicines. Things in Athens deteriorated which was one of the reasons which led to the death of his teacher Socrates and the social justice system was badly affected. There was large-scale spread of evil thoughts and ideas. And for him it was the justice system that can come to rescue of their society. The virtue of any soul is justice. He criticized the prevailing theory of justice and gave his own theory of justice stating that justice is a human virtue and makes a person self-consistent and good, socially justice is a social consciousness that makes society internally harmonious.
Aristotle – The Philosopher
He was the Greek philosopher and scientist but better known as the teacher of the Alexander the Great. He worked from the field of physics to psychology to the political discourses. He loved working on logic to biology, and from ethics to aesthetics. He invented formal logic. His theory of deduction is the basis of what philosophers now call as “syllogism” where a conclusion get inferred from two or more other premises of a certain form.
Aristotle found his own school Lyceum in Athens where he spent most of his time studying, teaching and working on his body of work. He didn’t support the theory of atom, and for him all substances were made of some elements from Earth, Fire, Water and Air. The scientific revolution credited to Galileo is based on Aristotle concepts of scientific discovery.
In his book, Metaphysics he clarified the distinction between matter and form. Matter is the physical part and form gives the part the nature.
In his book, Politics he dealt with human behavior in the context of government and society at large. He stated that the purpose of government is to make it possible for its citizens to make virtue and happiness.
In his book, Rhetoric he analyzed public speaking with scientific lens in order for the readers to be effective public speakers. He believed that it helped in politics and law, and defended truth and justice.
In his book, On the Soul he examined human psychology and how people perceive the world around them, and has been the basis for the foundation of modern philosophy.
There are mathematicians turned philosophers and there are philosopher turned mathematicians, and there are who worked on both. One of them was Ludwig Wittgenstein who worked on logic and philosophy of mathematics to mind to language. According to Pythagoras “everything is mathematics”, Plato studied the ontological study of mathematical objects and Aristotle studied logic and infinity. Greek philosophy of mathematics was largely influenced by the subject of Geometry. Hippasus, a follower of Pythagoras questioned the existing rationality and credited for the discovery of irrational number.
Mathematics and Philosophy are two of the most powerful intellectual scalpels that deftly combines to dissect and critically analyze some of the most intriguing phenomena happening around us.
The quintessential question remains, do mathematicians really care about the philosophy of mathematics or more profoundly what are philosophers got to do with mathematics.
It is the contrast, the subject of abstract vs. the subject of concrete, the analytical thinking vs. the intuitive thinking or the perceptive thinking.
These are extreme ends of our thought spectrum…
After all mathematics works towards constructing concrete models and build on the strata of abstract thoughts visualized by the philosophers, scientists then convert these conceptual models into working mathematical models to work their ways into our life.
Nihar R Pradhan
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