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With the earliest recorded origins of written mathematics thought to be lying in prehistoric African artifacts that date 20,000 years into the past, mathematics has been progressing and integrating itself into human society since the birth of civilization. It is now fundamental to all of science and our daily lives, having found its way into architecture, design, computer science and dozens of other areas of study. Math is so incredibly diverse in its applications, but what does it encompass? How does one even define mathematics?

Firstly, mathematics, to this day, does not have a complete and majority- accepted definition, which is due to its constant evolution and sheer scope of its study. There have, however, been attempts to construct definitions in various manners, ranging from prosaic and rigorous, to philosophical and diverse, to comic and humorous. For instance, Aristotle’s definition, stated in ~300 BC, “The science of quantity”, would only have been relevant in his time frame, since algebra and advanced mathematical abstraction were not developed until the 9^th century.

Looking at a much later written example from 1851 by Auguste Comte, who defined mathematics as “The science of indirect measurement”, it can be seen that definitions had to broaden and increase in vagueness to remain relevant.

This pattern remained for a fraction of subsequent definitions, which would become more abstract, difficult and possess multiple connotations:

“Mathematics is the science that draws necessary conclusions” - Benjamin Peirce, 1870

“All mathematics is symbolic logic” - Bertrand Russell, 1903

“Mathematics is the classification and study of all possible patterns” - Walter Warwick Sawyer, 1955

“Mathematics is mental activity which consists in carrying out, one after the other, those mental constructions, which are inductive and effective” - Ernst Snapper, 1979.

Such definitions include all mathematical activity and sometimes beyond, making them most persistent over time, yet causing them to lack specificity.

Alternatively, while not being formally relevant, there are also poetic and humorous definitions, valued for their artistic inclinations:

“The subject in which we never know what we are talking about, nor whether what we are saying is true” - Bertrand Russell, 1901

“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas”- G.H Hardy, 1940.

It can hence be concluded that mathematics is deprived of a definition not due to a lack of contemplation and ideas, but rather and more likely due to their excess and diversity.

As mentioned above, mathematics is incredibly diverse in its scope and application, but what exactly does it study and encompass?

Mathematical study can be expressed as the study of quantity, structure, space and change, each of which respectively correlate to arithmetic, algebra, geometry and calculus (mathematical analysis), wherein:

• Arithmetic is the branch that studies numbers and operations with them, such as addition, subtraction, multiplication, division, exponentiation and the extraction of roots;

• Algebra is concerned with the manipulation of abstract entities, such as variables, fields, groups and polynomials;

• Geometry defines properties of objects and spaces via distance, shape, size and relative position;

• Calculus, also referred to as Mathematical Analysis, studies the properties of functions, such as differentiation, integration, limits and infinite series..

In the end, mathematics is an exceptionally broad and evolving area of study that is being subjected to constant change and innovation, making it unlikely that it will ever have a concise and complete definition. It remains the reader’s choice to find the most suiting definition, or even produce one of their own, now being somewhat familiar with the extent of its reach.

By Michael Anisimov

Philosophy, when translated from the Greek word “φιλοσοφία”, stands for “love of wisdom”. However, is there any wisdom to speak of when analyzing the achievements, or lack thereof, of philosophy in the context of scientific advancement?

Firstly, let us recall the history of philosophy to better understand its scope and the evolution of its function throughout the last 2,500 years of human history. The formal birth of philosophy is thought to lie in the 6^th century BC, having been popularized by the efforts of Thales, an ancient Greek astronomer and mathematician. Diogenes Laërtius, the first biographer of Greek philosophers, classified philosophy into three fields of thought:

• Natural philosophy, or physics (τὰ φυσικά) - the study of the physical world;

• Moral philosophy, or ethics (ηθική) – the study of morality and justice;

• Metaphysical philosophy, or logics (μετὰ τὰ φυσικά)- the study of logic, causality, existence and abstract notions.

With the advancement of technology and civilization, each of these branches eventually evolved into various sciences. Natural philosophy has since been divided into physics, astronomy, cosmology, chemistry and biology; moral philosophy has become ethics, aesthetics, sociology, economics, anthropology and psychology; while metaphysical philosophy still only features philosophy of science and epistemology.

However, within this article I will only analyze the impact of natural and metaphysical philosophy and the act of philosophical thought on scientific study based on genuine examples and make a conclusion on the capacity of philosophy to supplement or hamper scientific progress.

The separation of modern opinions on philosophy can be expressed through quotations of Stephen Hawking and Albert Einstein. Hawking argued that “…Almost all of us must sometimes wonder: Why are we here? Where do we come from? Traditionally, these are questions for philosophy, but philosophy is dead”. In contrast to Hawking, Einstein argued, “…A knowledge of the historical and philosophical background gives that kind of independence from prejudices of his [the scientist’s] generation from which most scientists are suffering. This independence created by philosophical insight is - in my opinion - the mark of distinction between a mere artisan or specialist and a real seeker after truth”. It is clear that there are two conflicting opinions among scientists: one stating that philosophy has become irrelevant that science and philosophy are to be kept apart; and the other stating that philosophy offers numerous advantages when used in conjunction with science.

A situation which confirms the validity of Einstein’s view on the significance of philosophy is the famous EPR paradox, stemming from a 1935 collectively written paper by Albert Einstein, Boris Podolsky, and Nathan Rosen. In the paper, it is mathematically and logically argued that quantum mechanics is an incomplete theory due to it deeming possible a situation which breeches the principle of locality. Looking at the manner in which the argument is put forth, the necessity of the philosophical element within the extent of the article becomes evident. The mathematical analysis present in the article is used in conjunction with the logical sequence based on subjective premises, making them mutually dependent. For instance, in the introduction of the article, Einstein argues that “Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates”, which is a conclusion that could have only been achieved by previously conducted thinking on the philosophical topic of “What are the criteria for the validity of a physical theory”. It can hence be argued that philosophical thinking was integral to the formulation of the reasoning presented in the paper.

Alternatively, there is an example which demonstrates the ability of philosophy to conflict with scientific progress by presenting incomplete and unproductive paradigms. Rene Descartes’ personally crafted theories of gravity based on “aethereal vortices” and magnetism based on screw-shaped particles serve as a good example: while being were unorthodox and imaginative, they lacked a mathematical formulation, making them futile for producing predictions. It took 85 years of disproval attempts until Isaac Newton finally published his “Principia”, defeating Descartes’ theories and establishing a new order in the realm of physics. This particular situation demonstrates that philosophy should not be used for scientific modelling.

Plausibly concluding with the examples we have at hand, I believe that philosophy is inherently present in and essential to formulating frameworks of knowledge and creating new theories or paradigms. Most importantly, philosophy, as argued by Einstein, deprives scientists of generational prejudice and allows for genuinely original thought. However, the role of the philosopher must remain to question and explain, made distinct from the scientist, who is there to analyze, formulate and validate. Philosophy is not there to theorize, but is the perfect tool for placing a valid foundation and utilizing logical thought for assessing and providing coherence to a string of thought.

By Michael Anisimov