Mathematics from a mathematical point of view
“The universe is written in the language of mathematics” (Galilei). Since the dawn of civilizations, mathematics has always been an indispensable tool: Pharaohs & Greeks developed arithmetic for daily use and embraced geometry as a basis of architecture (Dubby); and the foundations of modern science are, in one way or another, mathematical. However, we (humans), having taken mathematics for granted, have left the subject definitionally unclear: Is this incredibly reliable tool an invention? Or is it a discovery? From the Greeks’ perspective, mathematics is a discovery; taking Plato’s views as an example, mathematics is a world on its own, a “Platonic world”: all mathematical objects are independent of thought and language (i.e. Mathematical Platonism) (Balaguer). For modern natural philosophers, however, the topic is highly debatable. On the one hand, Professor Paul Ernest summarized his view on the matter by quoting Giambattista Vico’s “The only truths we can know for certain are those we have invented ourselves”; Ernest added, “Mathematics is surely the greatest of such inventions.” On the other hand, we have Roger Penrose sharing the platonic view of mathematics as a discovery (Closer to Truth). As it seems, all the developed understanding of the nature of mathematics is purely philosophical; one might say that we have no absolute answer. However, less debatable ideas were introduced by a 23-year-old logician. In 1931, Kurt Gödel published his "Incompleteness Theorem”. It was a paper of calculations that led to a logical statement translating to "Mathematics is incomplete"; that is, mathematical systems [amaem1] can contain non-provable, yet true statements. In our context, the implications of what Gödel developed are worth examination. His findings are new characteristics of mathematics, ones that might help us conclude more concrete statements regarding the nature of the subject. In this case, I claim that mathematics is a discovery, and I shall use Godel’s incompleteness theorem as a logical basis for my argument.
Firstly, we shall go through some historical context. In the 19th century, Georg Cantor developed his “set theory”: a theory that differentiates between the sizes of infinities. As a consequence, the mathematical society was in chaos; mathematicians were divided between proponents and opponents of the new notion of infinity. David Hilbert, undoubtedly one of the best mathematicians in human history, dealt with the situation differently: he knew that the mathematics of the 19th century was not sufficient to handle the new theories (with the precise definition of “limits”, Cantor’s set theory and many other new mathematical notions being problematic.) Hilbert was the first, since Euclid and his Elements, to develop a new mathematical methodology of proof. However, he did not only develop a new method of proof, but he designed a whole new set of notations. This new system is a way to write logical sentences using mathematics: it defines an intersection between philosophy and mathematics, a mathematical exposition. Interestingly, it could be regarded as a special new language (except that it could only be used for logical conversations, which makes it inefficient for modern societies.) Hilbert’s new notational system (formally presented in North Whitehead and Bertrand Russell’s Principia Mathematica) was powerful enough to exhaustively prove every known mathematical theorem. Although employing this new notational system might be a bit cumbersome (Principia’s proof of the statement “1+1=2” is about 700 pages long[amaem2] ), it is crystal clear and leaves no room for doubt; and this is what the mathematical community needed during that period. Gödel used this new system of Hilbert to get to his “Incompleteness Theorem”. He published a paper in which he concluded that no mathematical system can be "consistent"; that is, no mathematical system can be proven to have no contradictions. Later, Alan Turing employed similar methods to prove that no mathematical system is "decidable"; that is, we cannot have an algorithm to determine whether any mathematical statement follows from a given set of axioms (i.e., there is no algorithm for inference). By analyzing both Turing and Gödel’s ideas, mathematicians arrived at a revolutionary result: not every true mathematical statement is provable. This statement, conventionally the “Incompleteness Theorem”, is directly related to the nature of mathematics. It is essentially mathematics itself saying that it could have true, unprovable statements.
If mathematics is a human construct, it follows that it was invented as a logical model that we could employ as a tool for sciences and daily activities; this is the only reason to invent such a tool, as any other philosophy would imply that mathematics is more than a tool, which distances it from the domain of inventions. Consequently, the incompleteness theorem, by fracturing the image of our supposedly perfectly logical tool, is evidence that mathematics cannot be an invention of humans. What the incompleteness theorem implies is not only that we did not invent mathematics, but that we were far from knowing its essence. It is like staring at the night sky, believing that all the stars are hanging lamps that illuminate the world during the night. The truth is these stars are not lamps and that this view is far from what the stars really are. Mathematics, just like the stars of the night sky, is far from being a mere tool that occasionally illuminates stuff for humans. Mathematics, in that sense, must be a discovery.
From another perspective, Paul Ernest argued that after humans establish the rules for something's existence, such as chess, number theory, or the Mandelbrot set, the resulting implications may consistently astonish us. Nevertheless, he believes that this doesn't alter the reality that we initially created the "game." Rather, it underscores the depth and richness of our invention (Ernest). However, I would argue that the implications of our inventions follow logically from the axioms of their construction; the astonishing consequence that the number of different possible chess games is more than the number of atoms in our universe follows from the architecture of the chess board and the allowed dynamics of chess pieces. However, in our case, the incompleteness theorem is not an implication of the designed rules of Helbert’s system; it is not a result; it is an axiom itself. It was inferred by analyzing the theorems of Gödel and Turing. The incompleteness theorem is a characteristic regarding the nature of mathematics that is never apparent in our developed notion of the subject (with mathematics being synonymous with logic and inference, no one would believe that this same mathematics is sometimes not guaranteed to be logical and decidable). In this sense, mathematics is a discovery, as we are discovering its rules, not unraveling their implications.
In conclusion, the debate over whether mathematics is a discovery or an invention has persisted throughout history. While some argue that mathematics is a human invention, akin to a tool created for specific purposes, others, including Plato and Roger Penrose, view it as a discovery, a realm of truth independent of human thought. Kurt Gödel's exploitation of Hilbert's new mathematical language, alongside the work of Alan Turing, introduced a revolutionary perspective on mathematics; by demonstrating that not every true mathematical statement is provable within a given system, the incompleteness theorem shattered the notion of mathematics as a purely logical construct invented by humans. Instead, it suggested that mathematics possesses inherent discoverable truths beyond what we know of the subject, which is similar to discovering the nature of the stars in the night sky. Unlike the implications of human inventions, which follow logically from their constructed rules, Gödel's theorem is an inherent characteristic of mathematics itself, challenging our fundamental understanding of the subject. Thus, the incompleteness Theorem serves as evidence that mathematics is not an invention, but a profound entity that has always been out there. This entity is not a human construct that is restricted by what we define as axioms and rules, but it is rather a free entity. "The essence of mathematics is in its freedom." (Georg Cantor)
Works Cited
Balaguer, Mark. “Mathematical Platonism | Definition, Metaphysics, Philosophy of Mathematics, Abstract Objects, and Facts.” Encyclopedia Britannica, 6 Oct. 2023, http://www.britannica.com/topic/mathematical-Platonism#:~:text=mathematical%20Platonism%2C%20in%20metaphysics%20and,true%20descriptions%20of%20such%20objects.
Closer To Truth. “Roger Penrose - Is Mathematics Invented or Discovered?” YouTube, 13 Apr. 2020, https://www.youtube.com/watch?v=ujvS2K06dg4.
Dubbey, J. M. “Mathematics of Ancient Egypt.” Mathematics in School, Jan. 1975, https://www.jstor.org/stable/30211437.
Ernest, Paul. “IS MATHEMATICS DISCOVERED OR INVENTED?” PHILOSOPHY OF MATHEMATICS EDUCATION JOURNAL, Sept. 1996, https://webdoc.sub.gwdg.de/edoc/e/pome/pome12/article2.htm.
Galilei, Galileo. The Assayer. Translated by Stillman Drake, https://web.stanford.edu/~jsabol/certainty/readings/Galileo-Assayer.pdf.
Gödel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. United Kingdom, Dover Publications, 1992.
Whitehead, Alfred North, and Russell, Bertrand. Principia Mathematica. United Kingdom, Cambridge University Press, 1927.
[amaem1]A mathematical system is a set of elements that can be acted upon by one or more arithmetic operations (almost all of mathematics)
[amaem2]The book is in the Panitza Library for those who are interested.