Answer to Question #138970 in Operations Research for Jeah Alinsalorin

Question

What role is played by mathematics in the following areas?
Economics
Psychology
Music
Data Science

Answer 1

Economics

Use of Mathematics in Economics allows to highlight and formally describe the most important relationships of economic variables and objects; gain new knowledge about the object; evaluate the type of dependencies of factors and parameters of variables, draw conclusions.

Related sciences: Mathematical Statistics, Analytic Geometry, Probability Theory.

Psychology

Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. The mathematical approach is used with the goal of deriving hypotheses that are more exact and thus yield stricter empirical validations. Quantifiable behavior is in practice often constituted by task performance.

Music

Symmetry is often used in music. A number of musical forms are built symmetrically. Pythagoras was one of the first to try to express the beauty of music with numbers. He believed that the harmony of numbers is akin to the harmony of sounds. The basis of mathematical knowledge is arithmetic counting. Similar to arithmetic, we can calculate a musical sound by moving along the musical row. Pythagoras also noticed that the frequency ratio of two neighboring notes is always different, and the frequency ratio of two notes spaced four positions apart, on the contrary, is always constant and is 3/2. This consonance is now called the fifth. Taking it as a basis, he developed the so-called "Pythagorean scale", where he derived a musical formula that allows, based on the frequency of the base note from which the counting is conducted, and the ordinal number of a given note, to obtain the desired value of the frequency of the next note. As a result of the consistent application of the formula, sounds are obtained that are separated from each other by a fifth. This row contains all the notes of the scale. And although they belong to different octaves, but by dividing or multiplying the frequency of the desired sound by two, you can transfer it to the next octave. By repeating the division (or multiplication) operation several times, you can fill the entire range of the instrument.

Data Science

Knowledge of mathematics is necessary to understand how machine learning methods work, because often ML methods use matrices to store and process input data, and matrices, linear equations and vector spaces are nothing more than Linear Algebra.

Also, Mathematics and Mathematical analysis are needed when optimizing processes, because the application of relevant knowledge increases the speed and accuracy of ML-models, reduces the number of errors. Thus, calculus is important when calibrating a model and building an optimization apparatus.

In addition, the already mentioned Linear Algebra is necessary for working with multidimensional spaces and allows you to transform data in large quantities, and not value by value.

Equally important is mathematical statistics, which is widely used today when working with linear regression models. Mathematical Statistics helps to choose the right distribution and establish dependencies between variables.