Ancient mathematics of Mesopotamia, Egypt, and India, which preceded Greek scholarly mathematics, was practically oriented. It was done by accountants, land surveyors, and architects. These practical mathematicians were primarily interested in results of measurements and algorithms. While the masters in those fields surely had an intuition and understanding as to why their algorithms and abstractions worked, proofs of correctness were accepted via real-world experience and empirical evidence. That is why they had no need for theoretical proofs based on axioms.
Greek scholarly mathematics brought a shift to this way of thinking. The predominant mathematics done was still practical mathematics, but with Hippocrates of Chios a new tradition of mathematics started. Greek scholarly mathematics focused on doing mathematics for the sake of doing mathematics and on searching for truth. To convince others of their discovered truths, Greek mathematicians would rely heavily on diagrams. These were meant to guide readers from fundamental statements that were known and accepted to be true to the statements to be proved. Readers would be able to take these imprecise diagrams and abstract them in their minds to an idealized image. The writing style accompanying these diagrams was reflective of the objectives of a scholarly mathematician: it was simple and precise and written with grammar which highlighted the intrinsic and fundamental correctness of the proof. What further set scholarly mathematics apart from practical mathematics was the omission of numbers. The scholarly mathematicians saw numbers as imprecise, a tool of the practical mathematicians that was used for measuring, instead arguing with ratios. The correctness for these ratios would follow from the written arguments and the diagrams. Euclid especially is notable for his treatise “Elements” in which he lays the mathematical foundational truths needed for more dazzling proofs.
Euclid’s argumentative framework is based on Aristotle’s more general method of arguing. Socrates, and later his student Plato and in turn his student Aristotle, challenged the political status quo of arguing which, at the time, was about winning arguments rather than arguing for what one believed to be the truth. Socrates famously died for this stance rather than going into exile and Plato further criticized contemporary Athenian rhetoric in his dialogue “Gorgias”. These rhetorical ideals were formative to mathematical thinking and the modus operandi of mathematical argumentation. Mathematicians had the same goal of arguing that philosphers of the time were advocating: convincing of truth, not debating for the sake of winning a debate. In turn, this mathematical mode of arguing affected the methods of political discourse, attracting the social elite of the time. What set mathematicians and philosophers apart, though, was that philosophers struggled to reach concensus on many topics, whereas mathematical knowledge was, for the most part, accepted and agreed upon by the mathematicians.
While mathematics had been done for many centuries and in many places, Greek scholarly mathematics, prodded by the political and philosophical revolutions, sets itself apart from earlier mathematical practices by forming a coherent school of thought, accompanied by rigorous methodologies and a search for truth. Through the Greek Miracle, mathematics was broadend from being a tool to be used in practice to an abstract theoretical discipline as well.