Editor of the Ancient History Encyclopedia, Cristian Violatti, claimed that there were three proof techniques that arose during the development of ancient Greek mathematics:
- The technique of abstraction
- The technique of generalization
- The art of deductive reasoning
The first technique contributed by the Greeks, abstraction, is a key quality of communicating in modern mathematics. Before the concept of abstraction was introduced to mathematics, each problem had to be approached as a new, unique situation. Problems like the length of rope needed to create a 4-sided fence or the angle to which the peak of a roof had to be constructed were seen as unique problems that had to be solved in context. However, with the development of abstraction, these every day problems could be taken out of their real world situation and be solved using visual and mathematical representation.
Another technique that was contributed was generalization. The concept of generalization allowed mathematicians to make general claims about situations that were true in all cases. An example given by Violatti was the Pythagorean Theorem. For example, if an applicable real-world situation could be abstracted as a 3:4:5 right triangle, the theorem would apply to this abstraction. However, with the use of generalization, mathematicians can now claim and prove that theorems, such as the Pythagorean Theorem, can be applied to all situations. As math students, we see the use of generalizations every day in our studies. We have the tools to prove statements with universal quantifiers and for multiple cases thanks to the progress that the Greeks made towards modern proof technique.
The last technique that was mentioned by Violatti was deductive reasoning. Since the technique of generalization allows us to prove claims for general cases of mathematical situations, we are then able to use those proven claims to produce desired conclusions. This method of deductive reasoning that was established by the Greeks has been used to perpetuate every field of mathematics since it allows us to use what we already know in order to create new conclusions and mathematical relationships.
Cristian Violatti. “Greek Mathematics,” Ancient History Encyclopedia. Last modified September 24, 2013. http://www.ancient.eu /article/606/.