The universal language

In his “Assayer”, Galileo describes mathematics as a “universal language”.  After a bit of contemplation, we can surely accept Galileo’s notion in a cultural sense: anywhere in the world, mathematics is the same. If you ask “What is one plus one?” in the United States, Egypt, or China, everyone will all answer with the same “Two!” In that sense, mathematics transcends linguistic barriers, being the result of pure human logic (which is shared mutually among all of humanity). However, being linguistically transcendental is a mere side effect of what lies behind the nature of mathematics. Galileo, among many others, believed that mathematics is not just culturally identical, but that it is a complete universal language. Galileo and his proponents believed that mathematics is complete in the sense that whenever math is used, it leaves no room for doubt (that is, every true mathematical statement is provable[aa1] ). They were convinced that the significance of mathematics in sciences and how math leads to powerful conclusions from simple statements imply that the universe is speaking to humans in mathematical terms; that is, they believed in mathematics as a universal language. Nevertheless, the nature of mathematics was and is still debatable. On one hand, we have Galileo’s notion; and on the other hand, many believe that mathematics is a human construct; they see it as a tool synthesized by human minds to exceed the limitations of qualitative analysis. This second group sees mathematics as a “system” created by humans to allow for more accurate classifications and measurements. The two groups cannot be both right: If mathematics is a universal language, then an alien intelligent species would also discover and understand our mathematics; however, if mathematics is a human construct, the same alien species would not develop and understand our mathematics; that is, mathematics cannot be both a discovery (as in the first example of the aliens) and an invention (as in the second example), because these two characteristics are mutually exclusive. For me, the truth is, as Galileo believed, that mathematics is a universal language, and all human contributions are discoveries of what is already “out there”.

To discuss the nature of mathematics, we must establish its definition. According to Oxford Dictionary, Mathematics is defined as “The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics) or as applied to other disciplines such as physics and engineering (applied mathematics).” So, mathematics is a science, isn’t it? In reality, mathematics does not follow the criteria of the sciences; that is, the scientific method does not apply to mathematics. We cannot observe a mathematical phenomenon in the same sense we observe physical or biological phenomena. Regardless, mathematics is of great use in science; to honor the importance of mathematics in sciences, the scientific community has created a new scientific category, namely “formal sciences”, just for mathematics (computer science was later added to this category); so, Oxford Dictionary is not totally incorrect in referring to mathematics as a science (although it is just a conventional definition). The abstract nature of mathematics, however, can make it seem like one of the humanities rather than the sciences; that is, some people might see mathematics as a human achievement, resulting from the mere thoughts of human minds. Apparently, the comparison has led us to two more specific groups (compared to the discovery – Galileo’s notion – vs. invention groups) with different views on the nature of mathematics; the first group defines mathematics as a science in the sense that it is an external study (i.e. not related to some internal property of humans) of patterns and changes, and because of the great importance of mathematics in all sciences (in physics particularly); the second, humanities group defines mathematics as a system, developed by human minds, to allow for philosophical abstraction. Abstraction, in a mathematical context, is the process of extracting the underlying structures, patterns, or properties of any concept, removing any dependence on real world objects with which it might originally have been connected; this abstraction property is very similar to the analytical nature of human philosophy, and this forms the basis of the second group’s point of view. If we look closer at how each of the two groups defines mathematics, we will find that the first (science) group believes that mathematics is discoverable in nature in the same manner as all other sciences. On the other hand, the other (humanities) group believes in mathematics as an invention of the human mind, just as all other humanities are the results of critical and philosophical thinking. Accordingly, mathematics appears to fluctuate between the discoverable nature of sciences and the inventible character of humanities. However, from a third perspective, the Cambridge dictionary defines mathematics as “the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them”. In this definition, mathematics does not belong to either category: it is a unique study on its own, not an element of sciences or humanities. These three definitions (mathematics as a science, as one of the humanities, and as an independent study) do not lead to a definite conclusion regarding our search for the nature of mathematics, but they surely add more context to our goal; now we have finite options to consider as the “identity” of mathematics: it can be a science, one of the humanities, or an independent study.

If we seek to find which of our three options fits mathematics the best, we must examine mathematics and its history in more detail. By discussing the historical contexts of specific areas, we can gain insights into how mathematical historical figures perceived mathematics (and their perceptions may add up to what mathematics really is). In exploring the branches of mathematics, we will find the most prominent branches being algebra and geometry. Algebra is defined as “a part of mathematics in which signs and letters represent numbers”; and we will find geometry defined as “the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues” (Cambridge) – (the definitions of algebra and geometry are basically synonyms in most dictionaries, so we can adapt any definition to use in our context; the problem, however, is that algebra and geometry are always described as “branches of mathematics”, and we are trying to answer the question “what is mathematics?”) After defining the most prominent mathematical disciplines, we shall seek their historical contexts. It is noticeable that humanity’s first encounter with mathematics (according to the records) was with geometry and arithmetic[aa2] . Geometry was actually the topic of the first mathematical texts (belonging to Mesopotamia and Egypt, 1900 BC). Nevertheless, during that era, mathematics was mostly perceived as a tool to facilitate human interactions (as in trades and markets for example), not as an area of knowledge. The first rigorous mathematical scripts, however, belong to the Greeks of the 6th century BC (mathematical rigor indicates deep conceptual understanding, procedural skills, and stated applications). The Greeks were the ones to introduce mathematics as a discipline of knowledge and education. It was the Greek “Pythagoreans” who first coined the term “mathematics”, originally “mathema” (translated to “the subject of instruction”). Ancient Greeks, known for their philosophical character, focused on geometry and algebra as tools to describe the natural world. I must add that the algebra the Greeks used was “geometrical algebra” (that is, the study of algebra as an extension of geometry), not the usual algebra we study and use today. They clearly believed in mathematics as a “demonstrative discipline”, with most of their texts describing and utilizing geometry as an abstraction of the nature around them, as in “Euclid’s Elements”; that is, they usually analyzed nature as a superposition of mathematical concepts. It is difficult to find a perfect circle in nature, but it is rather simple to find a superposition of multiple geometric shapes; it is in this context that the Greeks studied and applied geometry. The way the Greeks used and studied mathematics (as a demonstrative discipline) implies the existence of the belief that mathematics is connected to nature (their study of mathematics as a demonstrative discipline mirrors that mathematics exists “out there”). The Greeks possessed rather explicit notions about the nature of mathematics and its disciplines; Plato, for example, defined geometry as “the knowledge of what always exists”, and that it “leads a human towards the truth” (this is similar to our definition of “complete” in describing mathematics as a “complete universal language”). Ancient Greeks also believed that mathematics was superior to other sciences of that time, as Aristotle stated: “A science such as arithmetic, which is not a science of properties qua inhering in a substratum, is more exact and prior to a science like harmonics (namely, music theory), which is a science of properties inhering in a substratum” (For the Greeks, science means knowledge in general, rather than the methodological definition of our modern days). Pythagoras believed that reality itself is made up of numbers (the Greeks did not know anything about atoms, but Pythagoras believed that numbers are the building blocks of the universe in the same sense we believe atoms are the building blocks of matter. He developed a model of the cosmos that fits his definition of reality and named it “harmony of heavens”; a fancy name for a fancy notion). Pythagoras’s belief survived for a while; in “The Marriage of Mercury and Philology”, Martianus Capella held the same notion of “numbers as the substance of reality”, and this was more than 800 years after Pythagoras’s era. To sum up their ideas, the Greeks believed that mathematics is superior to all other disciplines when it comes to accuracy and that it is the ultimate truth when it comes to modeling our nature; some of them, as we discussed, believed that nature is made up of numbers. This historical context is in favor of our third option: the Greeks believed in mathematics as a separate unique discipline of knowledge, and that it is superior to other disciplines (which are, in our case, sciences and humanities). While the ideas of the Greeks may sometimes sound extreme and purely philosophical, it is apparent that humanity has realized that mathematics is “special” in the very first encounter; historically, no discipline of knowledge was ever believed to be “supreme”, “ultimately true”, and “universal” except for mathematics.

Going forward in time, we will find consistent evidence for the “special” character of mathematics. In Newton’s formulation of the gravitational force, he built his mathematical model based on three-figure astronomical data (the best measuring accuracy of his time). However, in our days, the Newtonian mathematical model of gravity was found to be accurate up to seven significant figures. Newton was able to discover a mathematical model that exceeded the accuracy of the tools of his time. In this example, mathematics was obviously more than a tool to describe the natural world; if so, we would have found discrepancies in the values of Newton’s model as the accuracy of measurements increased, and we would have required a new mathematical model for every increment in accuracy (Einstein model of gravity was found later to be superior to Newton’s one; Einstein’s model is also a mathematical model, but it is a general case of Newton’s one. Additionally, the Einstein model used mathematics that was not yet discovered during Newton’s time). Mathematics, in this sense, is not subjected to the defining character of science: mathematics is not “refutable”; it is immune to time (which does not apply to science) and is immune to perspectives (which does not apply to humanities); these characteristics are also in favor of our third option: mathematics is a unique discipline of study. The immutable, timeless accuracy implies that mathematics is an inherent property of nature, rather than a human construct.

Someone might argue that if mathematics is a property of nature, why do we have mathematical concepts that are purely abstract with no connection whatsoever to our natural experiences? The answer to this question is rather simple; there have been multiple mathematical discoveries that were not involved in scientific theories until many years after their discovery. An important example is the results of Ramanujan (an Indian mathematical genius). His mathematical results were used in building models that describe black holes, while he may have never actually heard about these astronomical objects.

Our historical and logical arguments conclude that mathematics is an independent area of knowledge. Surely, mathematics, based on our discussion, is a discovery. However, I would argue that the mathematical discoverable nature is far more complex and profound than that of the sciences. At multiple points throughout history, mathematical theories seemed to be out of touch with reality. At later times, the concepts once believed to be out of touch were found to be essential in understanding complex natural concepts, as in Ramanujan’s example. Mathematics is a unique discovery because it can be found “out there”, just as in the case of Newton’s gravitational model and the Greek's approach to mathematics as a “demonstrative discipline”; but Math is also “in here”, in our minds and our logic, without the need for external validation, as in the case of Ramanujan. In this sense, mathematics is a world on its own; mathematics is a world of pure abstraction that we can access through intellectuality and logic, and this world occasionally intersects with our world to illuminate our understanding of it as a “universal language”, independent of who is using it, when, how, or where it is used. If we examine Oliver Sack’s “The Twins” essay on his “The Man Who Mistook His Wife for a Hat”, we will find an example of how mathematics is not confined to a single model of understanding, and that different people can access the mathematical realm in different ways. In fact, Ramanujan, Sack’s twins, and many other mathematical geniuses perceived mathematics in an unconventional manner, and most of them were not able to put how they approached mathematics into words; if you asked them how they discovered a certain equation, most of them would reply with “it just happened; it just works”, as with Ramanujan and the Field Medalist Vladimir Voevodsky. Interestingly, many intellectuals referred to mathematics as “music”: Pythagoras referred to mathematics as “harmonics” in his model of the cosmos; Niels Bohr (Nobel laureate), in his conversation with his student J. Robert Oppenheimer (the father of the atomic bomb), asked “Algebra is like a sheet of music, the important thing is not can you read music, it is can you hear it. Can you hear the music, Robert?” In our framework, we can say that all these great mathematical minds were able to approach the mathematical world in some unique, not necessarily methodological way. Accordingly, mathematics is a discoverable, immutable universal language, with no definite approach in terms of how it is accessed. Mathematics, as Nobel laureate Roger Penrose states, is a Platonic world of logical structures and indisputable facts. Human minds enter that mathematical world to acquire superhuman accuracy (as compared to the usual human qualitative analysis) for daily life purposes, or just to experience the joy of reaching powerful conclusions from simple facts. For some people, mathematics is a reason to continue living: “There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics”, said the great mathematician Bertrand Russel. Mathematics is a bit of and more than all the above: it is as beautiful as music and art; it is as discoverable as a science; as abstract and philosophical as humanities; it is immutable; it is a reason to live; and it is, as myself and Galileo believe, a universal language. "Mathematics, rightly viewed, possesses not only truth but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show" (Bertrand Russell, “A History of Western Philosophy”).

 [aa1]This is where the Incompleteness theorem comes into play. I did not discuss it as it would not fit unless I changed the structure of the essay and the flow of ideas. I might write another one after the finals, as I do not think I have the time to do it these days.

 [aa2]Basic algebra: +, -, *, /