Sunday, September 13, 2015

The Influence of Greek Mathematics on Modern Proof Methods

As mathematics progressed from simply being a field of computation and quantitative observation, the techniques of proof that we use today began to develop. The first historical observation of what we today consider a mathematical proof was during the time of ancient Greek mathematicians. The contribution of proof was incredibly vital in the progression of the field of mathematics.

Editor of the Ancient History Encyclopedia, Cristian Violatti, claimed that there were three proof techniques that arose during the development of ancient Greek mathematics:


  1. The technique of abstraction
  2. The technique of generalization
  3. 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.

Without the contribution of new proof techniques by the Greeks, modern proof techniques would not have been established and the field of mathematics could not have progressed as far as it has.
Citation:
Cristian Violatti. “Greek Mathematics,” Ancient History Encyclopedia. Last modified September 24, 2013. http://www.ancient.eu /article/606/.

2 comments:

  1. I really like your explanation and description of the proof techniques that developed from Greek Mathematics. I think it is fascinating hearing how you describe the history of proof. You did a very good job summarizing the techniques and explaining them clearly overall in my opinion! If there was anything I would change, I would just consider adding more pictures into your blog and possibly links to liven it up! It reads like an essay which is not so bad, but think about making it more visually appealing! Great post!

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  2. Good analysis. clear, coherent, content, consolidated. +

    Complete: it would be nice to have an example for abstraction (which you sort of have, without detail on the abstraction) and deduction as well. These could be classical, or from your own experience. Actually, your own experience would be nice, since you claim these 3 things are an important part of math today.

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