Is ‘Philosophy of Mathematics’ somehow useful?

What the philosophy of mathematics is useful for?

Max Black, the author of The Nature of Mathematics (1933), thought the main task of the foundation of mathematics (and, consequently the main task of any philosophy of mathematics) would be to elucidate “and analyze the notion of integer or natural number” [1]. Is the task described by Black really so difficult to complete? Yes, it is.

Pythagoreans, for example, relied on geometrical symbolisms to defend the idea that “all is so far formed according to number” [2]. For example, for them, the Tetractys (which is an equilateral triangle composed of 10 points in 4 rows, 1,2,3,4 arrangement, and yields a total of 13 equilateral triangles among other geometric shapes that are contained within it) is the source that “generates” other less “fundamental” geometric figures.

Euclid, on the other hand, thought the “numbers” were constituent elements of the geometric space (which he tried to fully describe in his famous work “The Elements”). Other ancient philosophers like Plato and Aristotle also postulated a few thoughts about the foundations of mathematics. None of them, unfortunately, achieved any final word on the matter. Even modern philosophers like Descartes, Locke, Hume, and Kant, also wrote theories about what “mathematical entities are,” yet none of them articulated a completely successful theory.

Then, in the 19th century, “Analysis” emerged. In this discipline, new ideas such as the concept of complex numbers and “limits” emerged to help with the remaining challenges of differential and integral equations. According to Stewart and Stillwell, these new concepts would be required to calculate “the total distance traveled by a vehicle moving at varying speeds, the depth at which a ship will float when placed in the sea, or the total fuel consumption of a rocket.” [3]

Also during the nineteenth century, in the field of “Analysis”, there was a project called “Arithmetization of Analysis” which mathematicians concerned with “the nature of function and number” [4]. With that same concern, not long after, the three most famous contemporary philosophies of mathematics emerged: logicism, intuitionism, and formalism.

The current state of the art

According to Bertrand Russell, in his Recent Work on the Principles of Mathematics, 1901:

mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

This statement pretty well describes the current state of the art of philosophy of mathematics in the early 20th century. Yet, Russell wasn’t satisfied with it at all. Along with Whitehead, Russell wrote one of the 20th century’s most ambitious treatises on the subject — Principia Mathematica (1910). In this work, Russell and Whitehead sought to articulate a unified theory that could describe all elements of mathematics. Unfortunately, it didn’t work. Russell’s logicism wasn't enough for other mathematicians such as Poincaré, and in 1931 Gödel established the unreliability of the whole system postulated by Russell.

Conclusion:

Since then, if a student questions “what do numbers represent” — or “what are numbers,” unfortunately, mathematicians all over the world would not yet be able to offer a consensus opinion. Moreover, of course, they still may cite philosophers like Kant, Locke, Descartes, Hume, Pythagoras, Aristotle, and Bertrand Russell when they discuss the uncertainty surrounding the nature of mathematics.

[1] BLACK, M. The Nature of Mathematics (p. 8). Ed. The Humanities Press, New York, 1950.

[2] ZELLER, E. Outlines of the History of Greek Philosophy (p. 50). Ed. Henry Holt and Company. New York, 1889.

[3] STEWART, I.; STILLWELL J. C. Analysis. Encyclopedia Britannica. 2017.

[4] BOYER, C. and MERZBACH, U. A History of Mathematics (p. 533). Ed. John Wiley & Sons. New Jersey, 2011.