Mathesis in ancient Greek means to learn, to teach. (Teaching and learning are exchangeable). The use of this verb here is not accidental. The understanding of the meaning of the mathematical knowledge is part of the foundation of science, and a key idea for the overall comprehension of mathematics as a subject. The German philosopher Martin Heidegger, in his essay “Modern Science, Metaphysics and Mathematics” wrote:
“In its formation the word mathematical stems from the Greek expression ta mathemata, which means what can be learned and thus, at the same time, what can be taught; manthanein means to learn, mathesis the teaching, and this is a twofold sense. First, it means studying and learning; then it means the doctrie taught.”
In a broader sense, mathematics is the symbolic language we use to describe many of the more complex ideas about the physical world.
The everlasting question still lingers:
Mathematics prepares us to think logically about abstract concepts.
Mathematcis is, or can be seen, as a logical discourse.
In Mathematics we learn the rules and methods of representing problems of the physical world into symbols and equations in order to discover the hidden structures and patterns of nature and phenomena in general. This is the essence of mathematical modeling: the use of mathematical language to describe the world.
Mathematics gives us the tools to study random events, and to predict the most likely outcomes both in the physical world and human society. Inference and predictions in science are always related to mathematics.
Mathematics develops the theoretical basis of understanding the fabric of the space, and the inner shape of the Universe we live in.
Mathematics is so powerful that not only allow us to study the patterns of what is predictable (Probability) but also the patterns of the unpredictable (Chaos Theory) or the limits of axiomatic systems (Godel). Mathematics studies static and dynamical systems alike. Not in vain the great mathematician Johann Carl Friedrich Gauss called mathematics the Queen of the Sciences.
Mathematics, the quest for patterns
A very well known assay from the relevant British mathematician G.H. Hardy, make a clear and distinct reference to the topic of patterns in Mathematics and the Arts: A mathematician, like a painter or poet –wrote Hardy-- is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
Hardy's essay -- A Mathematician's Apology, may be found on the web published by the University of Alberta Mathematical Sciences Society
http://www.math.ualberta.ca/mss/ or click here to download the ebook.
Note that the word “apology” is used in this context meaning praise, eulogize. This is a must read article for all those interested in the Mathematics field including teachers and instructors in general.
“Plato's influence on mathematics was not due to any mathematical discoveries he made, but rather to his enthusiastic conviction that the study of mathematics furnished the finest training for the mind and, hence, was essential for the cultivation of philosophers and those who should govern his ideal state. This explains the renowned motto over the door of his Academy: Let no one unversed in geometry enter here".
Citation taken from An Introduction to the History of Mathematics by Howard Eves, (6th Ed., pg. 106).