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The Complicated Relationship Between Philosophy, Mathematics, and Science

October 9, 2011

The contributions to philosophy are immense, and in some cases immeasurable, yet scientists and academic researchers of nearly every field of study more often than not discount philosophy as a comparably meritless subject, being variously characterized as “the prototype to science”, “an obsolete field”, or even “mere speculation”. Indeed, if philosophy were compared to science in the terms only of scientific merits, science has done more for academia than philosophy ever has. However, as Albert Einstein (who was both a philosopher and a scientist) put it, “Everybody is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.” Philosophy may appear to scientists as being mere speculation, but it is speculation that laid the foundation of science, and continues to expand to boundaries of what we know and can know, both philosophically and scientifically.

The first scientists were philosophers; indeed, there was originally no such thing as science originally, nor was there a means to empirically produce evidence using the scientific method, save for the limits afforded by observation of the physical world. Mankind observed, thought, deduced, and built upon previous thought in such a manner as this, and so it is that the first scientists were observers and thinkers, and they became as philosophers, from sophia, meaning “wise”.

The terms “science” and “scientist”, as it so happens, did not come into being until fairly recently, even after the foundations of classical physics had been consolidated by Sir Issac Newton, in his treatise “The Mathematical Principles of Natural Philosophy”; even when the term science came into common use in the modern context (in the 20th century, three centuries later), it was used not to separate philosophy from science, but to separate empirical philosophies from non-empirical ones.

But what determines what is modernly considered philosophy, and what is considered science? To simplify this inquiry: in philosophical terms, if “philosophy” is the discovery of the “potential” (the possibilities) of an academic field (via speculation, for example), science is the “actuality” of such possibilities– that is, the defining and validating of the given possibilities by providing concrete evidence and conformance to proven theory. In other words, if philosophy is the “why” (things work), science tells us how things work, and also proves that things work the way they appear to. Philosophy is observation, and science is the confirmation, categorization, and replication of philosophy. In this sense, a “theory” can be considered a philosophical axiom for the science(s) to which it applies.

Philosophy then, is observations, reasoning, and original thought directed towards the explanation of phenomena in the world, and also to explain non-phenomena. Science, by contrast, aims to validate, standardize, and replicate phenomena, usually obtaining results via the scientific method, and the rigorous testing of the results acquired; while philosophy tells us what to know, science tells us if we really know it, how we know it, and– most importantly– how to apply this knowledge to the real world. It is this latter purpose that most dramatically separates philosophy from science, and the reason why philosophy is often under-appreciated or even taken for granted: Philosophers don’t actually do anything– they observe, think, and convey, but none of their thought actually does anything in the real world– this is after all the task of scientists. Philosophy thinks, but Science acts on thought.

Mathematics, by comparison, is right smack dab in the middle of these two, trying to reconcile philosophy and science, and providing a means by which these two opposing approaches to study can coexist on common ground. Mathematics is purely Philosophical in nature, but it is also purely Scientific. Yet at the same time, it also exist as neither of these, but as its own entity, forging alliances where otherwise there would be no cohesion, and more often than not, Mathematics ends up being the one to steal the show, where the end-product is not expressed in philosophical or scientific terms, but in mathematical language.

The reason for Mathematic’s uncanny role in these matters, lies in the most basic of function of Mathematics: communication! Mathematics is, in both purpose and substance, the platform, protocol, and procedures that define and structure how science, philosophy, and mathematics communicate! By facilitating communication between philosophy and science, mathematics ensures that the balance and cohesion between these two approaches is maintained; when mathematics is properly harnessed in this manner, it allows for the maximum amount of potential (philosophy) to be actualized (scientifically validated and applied to real life). In this sense, mathematics can be considered the most important philosophy, the most important language, and (in a more broad sense of the word) the most important language, to nearly every field of academia.

Most interestingly, Philosophy, Science, and Mathematics correspond, with an astounding level of affinity, to the Freudian Id, Ego, and SuperEgo:

Philosophy (Id): Provides the desire (potential) of the given field of study.

Science (SuperEgo): Converts the desires provided by philosophy into universally validated and replicable evidence, and provides a means for such thought to be applied to the real world (i.e. Society).

Mathematics (Ego): Is the intermediary between Philosophy and Science, and ensures that philosophy is accurately and efficiently converted into an scientifically acceptable form, and that science can be converted back into a philosophically appreciable form.

Of course, the complexities of each of these fields extend well beyond my analysis, and their relationship is by extension more complex, with many more relation-defining variables at play, but let it be recognized that philosophy, mathematics, and science are heavily interdependent and inter-operable in their relationship to each other, with a proper appreciation of the merits of each (and particularly of mathematics, the “stain” by which philosophy and science can be appreciated) proving fundamental to academic advancement.

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