Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century.^[22] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.^[23] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematics as science

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".^[28] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."^[7]

History

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,^[11] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

Etymology

The word "mathematics" comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.^[9] Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.

Trend "Mental Aritmatika"

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The Method of Eigenfunction Expansion

While most introductory PDE courses spend a great deal of time investigating solutions using separation of variables, a powerful method that is good to know for studies in theoretical physics is the method of eigenfunction expansion. This method is useful because in many cases, equations cannot be solved using separation of variables. To understand the method, in what follows you can think in terms of expanding the function which solves the equation in a Fourier series. We then solve mode by mode-turning a single partial differential equation into a set of elementary ordinary differential equations that can be solved by inspection. Rather than trying to explain the techniques in words or with difficult to understand abstract theorems, let’s demonstrate how to use the method with two concrete examples.

Mathematics

Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,^[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.^[4]

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.^[5]