As a civilisation, we are surrounded everyday by an excessive use of language. First of all, it is necessary here to clarify what is meant by the term language. I will be using Wayne Weiten’s (2007)definition (1989), which is, "a language consists of symbols that convey meaning plus rules for combining those symbols, that can be used to generate an infinite variety of messages." Now, it is clear that the realm of languages can encompass almost anything that has a symbolic meaning to an individual, which explains why there are an estimated 7,000 living languages across the globe. Language, in fact, has the unique duality of both uniting us through communication and separating us through its huge diversity. What, however, we all have remarkably shared throughout past centuries is mathematics. Whilst verbal languages can vary by country, the undoubtable truth of 2+2=4 prevails unchanged. The topic can best be treated under three headings: The Philosophy of Mathematics, the Parallels between Maths and Language, and its Applications. Furthermore, I will also be emphasising the striking behaviour of mathematics as a language; in its complex foundations, or axioms, a sense of structure can be recognised. Hence, could the “symbols” Weiten discussed in his definition of language be replaced with numbers? This essay wishes to draw attention to the global underlying presence of mathematics as well as its multiple uses.
Philosophy of Mathematics:
Philosophy of mathematics addresses two main questions; what is mathematics and where does it originate. In this next section, I will be answering these two questions, as well as highlighting characteristics which are arguably essential in order to define mathematics as a language.
Mathematics is a reliable art form upon axioms: the “foundation of a formal deductive system”. Axioms were first discussed in Euclid’s ‘The Elements’ under the name of postulates and are grounded in the idea of proof, logic and mathematical truth – notion whereby statements are validated through proof. According to Reichenbach (1969) “the truth of the axioms decides the empirical truth, and every theory compatible with them which does not add new empirical assumptions is equally true.” The connection he makes between axioms and, what he calls, “empirical truth” suggest the irrefutable nature of the correlation between the two concepts, one depending on the other and vice versa, highlighting the fundamental role axioms play in defining maths. Another key notion is the impressive structure that axioms set out. The term axiom itself can be traced back to the Greek axíōma (ἀξῐ́ωμᾰ), meaning ‘what is thought fitting’, stressing the unifying power of axioms in spotting patterns and standardising them by creating rules. Returning to Euclid’s ‘The Elements’, the book is considered a mathematical ‘rule book’ and has been hailed for being one of the most influential and integral pieces ever written, claimed to be second only to the Bible in fame. What is particularly intriguing about it though, is the incredibly small number of axioms or postulates, five in total, Euclid makes use of to prove and discuss theorems, such as Pythagoras’, which he demonstrates by only using the First and Fourth Postulates. ‘The Elements’ shows the logic of the axiomatic system and was essential in its development, a real milestone in the history and philosophy of mathematics. In fact, Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Sir Isaac Newton are amongst the scientists influenced by Euclid’s work, and it is said that US President Abraham Lincoln kept a copy of it with him at all times and studied it at night.
Nevertheless, the apparent stable consistency of axioms was shaken by the mathematician Kurt Gödel in his famous paper ‘On formally undecidable propositions of Principia Mathematica and similar systems’ (1931) which later became known as his Incompleteness Theorem. This was in response to a combination of two concepts dominating the pre-1950s mathematical scene; the concept of attaining mathematical truth through proof emerged from the three-volume book ‘Principia Mathematica’ (Whitehead A. N. and Russell B., 1910, 1912, 1913) and David Hilbert’s completeness proof for arithmetical axioms. Both notions relied on logic, whereby Hilbert, Whitehead and Russell argued for the sturdiness of axioms. This was the foundation of Gödel’s essay, where he questioned the consistency of the axiomatic system itself, and whether axioms can be proven, although generally accepted as true. The paradoxical statement ‘this statement cannot be proven’, or more commonly known as the liar paradox – a statement which initially seems true but is contradictory – shows his reasoning. Gödel converted this paradoxical statement into what he claimed to be a “formal language”, that of arithmetic, by forming a string of prime numbers, whereby,
“The resulting mathematical statement therefore appears, like its natural language equivalent, to be true but unprovable, and must therefore remain undecided.”
Gödel’s ideas challenged the Formalist school of thought – which believed that a mathematical statement is proven through a series of rules – and questioned the truth of a mathematical world whose foundations are shown to be, in part, “unprovable”. This brings us to the question; how, as I previously claimed, is mathematics a reliable art form? Gödel showed, through a combination of logic and pure arithmetic, that mathematics behaves like a language, and that it is defined by the process of proving, not by its axioms. Similarly to an essay, in mathematics there cannot be a conclusion without first there being an explanation and evidence of one’s process, thus there are no random statements. This very lack of randomness in the proving system is what makes mathematics a reliable art form.
Mathematics is not merely grounded on proven facts, but also on logical reasoning from which we can deduce our definitions of the mathematical world, whose interpretation differs according to schools of thought. In the case of Gödel for example, who was heavily influenced by Platonist ideas, he viewed the mathematical world as abstract; where mathematical objects, such as shapes and numbers, exist in a separate dimension or reality to ours. The quest to determine the ultimate reality is indeed a popular topic amongst Platonists. Coupled with this belief is the assumption that mathematicians merely happen to stumble on pre-existing mathematical concepts. Highly disliked words amongst Platonists are, as a matter of fact, ‘invent’ or ‘create’; instead of ‘coming up’ with a theorem, Platonists claim mathematicians deserve credit for having ‘discovered’ it. This links with the idea that mathematical objects are part of a dimension separate to ours, which we cannot influence or control. As phrased by American politician Edward Everett Hale in a speech he delivered at the inauguration of the University of Washington University in 1857,
“In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.”
In this quote, Hale eloquently summarizes this notion by suggesting that the “divine mind”, either God or a higher power – not humankind – is responsible for the creation of mathematics (Cohen, 2007). Yet, this mystical quality associated with mathematics was not an unfamiliar and unusual concept. Going back to roughly 500 BC Ancient Greece, the Pythagorean movement, formed by Pythagoras and composed of his most loyal followers, held the belief that, similarly to Hale, numbers were the “key for recognizing true divinity”. However, because of the sworn secrecy of the group, described by the British historian Connop Thirlwall as a “religious brotherhood” (1835-44), their ideas could not dominate the mathematical scene until much later, during a revival between the 2nd Century BC and 2nd Century AD, called Neopythagoreanism. This example shows the connection between mathematical ideas and the progression of time, emphasising the significance of the background historical context, either social or cultural, of the time.
This belief is at the core of the Humanist movement, which unlike Platonism, claimed that mathematics was a human creation. In his book ‘What is Mathematics, Really?’ (1997), American mathematician Reuben Hersh actively voices this opinion, whilst simultaneously drawing parallels between language and mathematics, as shown in the following quote,
“The rules of language and of mathematics are historically determined by the workings of society that evolve under pressure of the inner workings and interactions of social groups, and the physical and biological environment of earth, […] of individual humans.”
By advocating a historical appreciation of mathematics, Hersh is encouraging us to consider the impact certain events and societal beliefs had on the subject. Taking the previous example of Neopythagoreanism to illustrate the case, this revival of Pythagorean and Platonist ideas was a result of a growing desire to revive the mystical and religious undertones of Greek philosophy, which before the 2nd century BC was mainly influenced by a more Formalist school of thought. Hence, history is fundamental in determining the reasoning behind certain concepts emerging in mathematics. Another way in which Humanists oppose Platonists lies in their conflicting beliefs regarding the origins of mathematics. As mentioned previously, Humanists are firm believers in the fact the human race is responsible for the creation of mathematics, contrary to Platonists’ affirmation of its pre-human existence. Having addressed this key difference, an emphasis on the commonalities of the Platonist and Humanist movements is now necessary. Both attribute mathematics to some generating power; whether that comes in the form of the human intellect and intuition or some Godly supernatural presence is arguable. Both recognise the value of a mathematicians’ work in divulging a notion, with that very ‘eureka’ moment solely attributed to the individual. In some ways, they also jointly show the significance of the historical backdrop to different extents; whilst Humanists value this tremendously, it is also fair to say that Platonists do not completely deny it either. Conclusively, other than offering so much more freedom than any other field, allowing for varied interpretations, I’ve shown that these shared beliefs in the philosophy of mathematics are what ties maths all together, embodying the saying of ‘unity is strength’. This defining characteristic of mathematics opens up and enables global communication, the latter being at the core of language, showing the extent to which mathematics behaves as such.
Is Mathematics already a language? The Parallels between Language and Mathematics:
Throughout time, mathematics has been a useful way of representing the sizes of an object or specific quantities. This falls within the branch of arithmetic, which uses figures in the form of number for quantifying purposes. In this next section, I’ll be exploring the history of this particular field of mathematics and how the concept of number is highly intuitive to humankind, to the extent that it became charged with symbolic connotations, thus forming part of both culture and language.
What is most intriguing, perhaps, is the curious way mathematics evolved and the striking parallels between its development with that of languages. First of all, let us define the concept of ‘number’. Numbers convey meaning, notably in the form of magnitude, but even when we are not considering this application, they surround our daily lives. There is a difference between numbers and their counterparts, numerals. Numerals are a notational way to represent numbers, which ultimately is a really abstract concept – the idea of imbuing a symbol with a figurative meaning adds dimensions to the concept of magnitude, as it can be represented in ‘pictorial’ format. Moreover, numbers can mean order, as is the case with the calendar; here, days, months and years are arranged in a ‘sequence’ so as to enable us to write dates in digit format. This simple example serves to emphasise how numbers are so inherently integrated in our lives, to the extent that we forget their origins. In fact, the concept of number has been around us for longer than any written language, as exemplified by tally sticks. Evidence for this is the Ishango Bone (Figure 1), found in what is now the Democratic Republic of Congo and estimated to be 22000 years old. To put this in context, the earliest known written language, Sumerian, is dated back to 3300 BC, leading to the assumption of deeper roots of humanity’s pursuit
of mathematical representation compared to linguistic. This, of course, can only be presumed because there are no records of pre-existing written languages older than Sumerian. However this may be because evidence could not have been yet found or it could have decayed with the passing of time, as was common with Egyptian Papyrus for example. Nevertheless, the Ishango Bone although initially appearing to be a simple bone tool with a group of markings, upon closer inspection, a
sequence of prime numbers can be observed. Taking the left column to show this (Figure 2) groups of 11, 13, 17 and 19 markings are counted, which are the four ordered prime numbers between 10 and 20. Although mathematicians and historians have speculated about the meaning behind these markings, some claiming the sequence of prime numbers may form the basis of a rudimentary lunar cycle (Marshack, A. 1991) or even of a menstrual cycle (Zaslavsky, C. 1979, 1991, 1992), the most significant underlying implication of the Ishango Bone is the deep understanding of mathematics the so-called ‘primitive’ humans of the Palaeolithic Age possessed so early on – a perfect string of prime numbers is not as coincidental as it may sound after all. This ultimately shows how written mathematics predates written language, highlighting humankind’s predisposition to use symbols as a way to communicate mathematical quantities.
Numbers are not only perceived as carrying quantitative connotations but also cultural significance. Throughout the world, specific numbers are imbued with powerful connotations, studied by the field of numerology. I will be discussing two ways the symbolic meaning of a number can be classified; based on superstitious and cultural beliefs and religion. Firstly, let’s take as an example superstition in China. In particular, an incredibly loved number is 8, widely believed to have an auspicious meaning. This is due to pronunciation, where the word for the number 8 (pinyin: bā; Cantonese Yale: baat) is said to sound like (pinyin: fā; Cantonese Yale: faat) meaning “to prosper”, emphasising the strong correlation between the power of spoken language in influencing the language of mathematics, in the form of numbers symbolism. This is not only a simple result of a prolonged language-math association, but an active part of China’s customs.
In 2003, Sichuan Airlines was reported to have bought the phone number "+86 28 8888 8888" for CN¥2.33 million (approximately US$280,000). The repeated number 8s, considered an extremely lucky omen, coupled with the sheer unusualness of the combination, arguably justifies the enormous sum of money spent by the airline to secure that phone number. This would be treasured and looked at enviously by any superstitious Chinese person. This example further illustrates Chinese society’s acceptance of what can be deemed by many sceptical people as dubious allegations to simple numbers. Yet, the key message to be taken away is that numbers here transcend their regular roles of representing a quantity and instead, similarly to allegorical language, assume an identity. Conversely to number 8, Chinese society views the number 4 with disdain, referring to the fear of it as ‘tetraphobia’. As with the number 8, 4 (pinyin: sì; Cantonese Yale: sei) is phonetically similar to ‘death’ (pinyin: sǐ; Cantonese Yale: séi). This belief in an ill-omened quality being attached to the number, to the extent of omitting it in lifts and buildings (as shown in Figure 3), adds dimensions to the original concept of number, reiterating its depiction as an entity capable of influencing one’s life. This fits with Plato’s conception of an abstract world monopolized by mathematical objects and numbers, as it’s in this world where maths truly becomes alive. Secondly, religion plays a role in numerology. For example, in Taoism, religion practised in countries such as China and Taiwan, the number 5 denotes the five elements of the cosmos. Here, the number represents a connection between nature and the human race, as it provides us with a sense of order and apparent understanding of the universe. Furthermore, in the Bible’s Book of Revelation there is a repeated mentioning of the fraction 1/3 when referring to punishments or curses. This can be observed in the following extracts,
“and the third part of trees was burnt up, and all green grass was burnt up.” (Rev 8:7)
“and the third part of the sea became blood…” (Rev 8:8)
“and the third part of the creatures which were in the sea, and had life, died; and the third part of the ships were destroyed.” (Rev 8:9)
Despite the lack of a distinct explanation for the association of the fraction with such catastrophic events, it is thought that it may be due to it being the reciprocal or inverse of 3. According to Biblical scholar Adele Berlin, the number 3 is representative of “perfection”, hence the complete opposite of 1/3. The Bible, is to an extent, a piece about opposites; good and evil, innocence and sin, light and darkness. This trend of opposites extends to numerology, showing once again the powerful correlation between language and mathematics, as numbers are used in conjunction with literary devices and description to convey contrasting themes.
Moreover, it is a widely-held view that mankind’s desire to quantify objects is the very crux of mathematics. Yet, what is overlooked is mathematics’ ability to qualify as well as quantify. Numbers have been closely linked with alphabetical letters on many occasions. First of all, their ability to communicate intention and meaning at the same time must be noted. Let us consider the way in which we are taught to count; with our fingers. Here, numbers transition from a simple word (‘four’) to a magnitude (‘four fingers’) to a symbol (‘4′). As noted by linguistics author James Hurford, “without language, no numeracy” (1987). But can this statement be reversed, as to equate mathematics and language? Recent evidence indicates that this cannot be done, as there exists a lack of inherent anatomical correlation between mathematics and languages. In an experiment conducted by neuroscientists Marie Amalric and Stanislas Dehaene (2016), it was discovered that,
"The brain regions found to be engaged by expert mathematicians during the reflection on mathematical statements lie outside areas typically associated with language.”
This suggests that the brain does not catalogue mathematical information as a language, but instead it is stored and processed entirely separately. Whilst this does not equate mathematics to a language on a neurological level, there are many other factors that suggest that mathematics behaves like a language and that, to an extent, the pursuit of mathematical development is more innately motivated than that of linguistic development. Turning now to the experimental evidence that illustrates this, according to an article by scientists Starr, Libertus and Brannon (2013), “Human infants in the first year of life possess an intuitive sense of number”. A possible explanation for this may be their heavy reliance on visual information. In a series of experiments conducted by Elizabeth Spelke and Fei Xu (2000, 2005), they showed 7-month old babies an array of 8 dots, which they initially showed interest in, and recorded the time it took for them to lose this interest, or become habituated. Once this occurred, scientists replaced the 8 dots with an array of 16 dots and immediately the infants were observed to be once again fascinated by the dot array, as their looking time increased. The revival of the infants’ interest in the dots when the number of them was doubled shows their high numerical sensitivity to change. This is indicative of a potential underlying mathematical intuition prevalent in the human mind, as from an early age we can perceive changes in magnitude. Ultimately, whilst the brain does not process mathematics as a language, it occupies an equally important place nevertheless.
Application of Mathematics:
A subject heavily influenced by mathematics is physics. Its applications are endless but in the following section I’ll be referring to two examples; Chaotic motion and Newtonian physics. The core of chaotic motion is inherently mathematical. It is governed by the principles of deterministic chaos, meaning that said object behaves according to a set of initial rules and conditions but the outcome is completely random. This is evident in the case of dice. The apparent pure randomness of the resulting number is actually partially determined by laws of mechanics as the die itself is a cube which, when rolled, bounces on a surface, behaving like any other physical object. There is a misunderstanding about randomness and chaotic motion, in that the two concept seemingly equate. The die is not completely random but the result of it is since we do not know its initial conditions when rolling it; we randomise them by shaking it. But, the laws of mechanics are not random, hence chaotic motion, as phrased by Ian Stewart (2017) “inhabits the twilight zone between regularity and randomness”. Physics is responsible for the parenthesis stage between the initial conditions and the outcome(s). This is where the collaboration between mathematicians and physicists lies. In Newtonian physics, mathematics’ applications are much more obvious, whereby I’ll be focusing on Newton’s Law of Gravitation. The law is as follows,
“a particle attracts every other particle in the universe using a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.”
Newton proved it by claiming that if the inverse square law – which describes the inversely proportional relationship between a specified intensity or physical quantity and the square of the distance from the source – is applicable to small particles, then it must also be applicable to large spherically symmetrical masses because all the mass would nevertheless still be concentrated at the object’s centre. Newton’s use of inductive reasoning, coupled with a simple proof for the inverse square law brings clarity and logic to humanity’s ceaseless desire to understand the universe. This can only be achieved by using mathematics, whereby as said by Feynman in ‘The Character of Physical Law’ (1992),
“Newton’s statement of the law of gravitation is relatively simple mathematics […] mathematics is a language plus reasoning.”
This shows the extent of mathematics’ applications in physics, the two fields sharing a bond which is not one of mutual dependence, but rather one of cooperation. SOMETHING RELATING IT TO LANGUAGE
Biology, although seeming less apparently influenced by mathematics than physics, is where the most mathematical models are used. Ranging from observing bacterial development in viruses to reproducing the “various cardiac dynamical regimes” (David Ruelle in the Chapter on ‘Turbulence, Strange Attractors and Chaos’), mathematical models can generally emulate most biological processes, highlighting the underlying power of mathematics in this science. Another significant aspect of biology is the discovery of fractal patterns, notably in the form of geometrical figures, in nature as well as in different organisms. Such structures present a certain degree of self-similarity, meaning that the shape is made up of increasingly smaller copies of itself, and can be simulated by computers,
whereby they become abstract objects. However, when magnifying on an atomic scale such a shape existing in nature, the intricate self-similarities eventually blur out, which conversely, in a computerized fractal, does not happen. As phrased by Ian Stewart (2017), fractal geometry is indeed “a mathematician’s idealization”, in that it perfects nature’s imperfections. For example, the Koch Snowflake (Figure 4), although being a fractal made up of smaller triangles, mirrors the typical snowflake structure but to an infinitely accurate extent, which nature is incapable of reproducing. Hence, whilst nature does exhibit evidence of fractal-like patterns, these shapes cannot be considered fractals. Nevertheless, after having clarified that, these structures found in nature are responsible for importing an element of beauty through symmetry due to their mathematical self-similarity. For instance, upon closer inspection of neurons, the cell body splits into dendrites which then branch into smaller dendrites exemplifying the fractal ‘tree-like’ structure to an extent. This is because the dendrites do not infinitely branch out, as instead would happen with a computerized fractal. The same occurs in the airways of the lung, as they split into bronchi and bronchioles29. Ending with a quote from Richard Feynman (1992),
“if you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”
The language Feynman is referring to could be the one expressed by the structures exhibiting fractal geometry, which is that of mathematics.
Lastly, research shows that it may be possible to communicate with extra-terrestrial beings mathematically, implying the universality of the language. In 1974, the SETI (Search for Extra-terrestrial Intelligence) Institute sent an encoded message called the Arecibo Message (Figure 5) from Earth to the globular star cluster M13 in the hope of being received and decoded by aliens. Intriguingly, the message is made up of a sequence of prime numbers, with encrypted information ranging from Earth’s location and its chemical composition to our physical appearance and the average human population. It is thought that prime numbers, described as “the atoms of arithmetic”, are charged with symbolic significance but it is generally unknown why the SETI Institute chose to use them as a method of interstellar communication. Addressing this question, Carl Sagan, co-creator of the Arecibo Message, wrote in his book ‘The Cosmic Connection – An Extra-terrestrial Perspective’ (1973),
“We think we have written the message – except for the man and woman – in a universal language. The extra-terrestrials cannot possibly understand English or Russian or Chinese or Esperanto, but they must share with us common mathematics and physics and astronomy. Those commonalities are, of course, not any spoken or written language or any common, instinctual encoding in our genetic materials, but rather what we truly share in common – the universe around us, science and mathematics.”
This emphasises the ability mathematics has to convey information concisely and reliably, as well as the dominant grip it has on, literally, the universe. Sagan’s belief in it being a shared field throughout the cosmos is ultimately justifiable. The range of skill mathematics demands, from the elementary simple to the infinitely complex, cannot be purely created and mastered by humans. Contrarily, to accept it as a fully human invention would be to negate its underlying power of communication and beauty it provides.
Conclusion:
Ultimately, throughout the essay, mathematics has been shown as behaving like a universal language to an extent. Firstly, its axiomatic system, despite being challenged by Gödel in the first half of the 20th century, has remained fairly steady since the publication of Euclid’s ‘The Elements’. It inspired both highly logical and inductive reasoning, giving maths a firm practical and theoretical basis, making the field both stable and reliable. This also allowed for the development of different schools of thought, such as Formalism, which was heavily influenced by the belief that any mathematical statement can be proven by applying a series of axioms. Others include the Platonist and Humanist movements, whose debate over the creation of mathematics still dominates the mathematical scene. Despite their disparities, their shared beliefs exemplify the confluence of philosophical schools of thought, which conclusively renders mathematics simultaneously diverse and united, which is arguably what the wide variety of spoken languages lack in. Moreover, it is clear that mathematical notation evolved at a faster rate than that of written linguistics. Evidence for this is the Ishango Bone, archaeological artefact from the Upper Palaeolithic Era with signs of premature numerical representation as well as a potential understanding of prime numbers. Conversely, the first written language, Sumerian, has been dated back to around 4000 BC, centuries after the creation of the Ishango Bone. Other than mankind’s inherent desire to quantify objects through the use tally sticks, as is the case of the Ishango Bone, and numbers, the latter are also charged with symbolic meaning, which is what the field of numerology studies. They can be given auspicious or inauspicious connotations, as for example in Chinese culture with the numbers 8 and 4, or be read in a religious light, as with the numbers 5 in Taoism and 1/3 in Christianity and Judaism. Both ways of interpreting numbers, from cultural and religious perspectives, emphasise a human longing to understand the world around them; numbers reside, rather abstractly, as powerful entities because of the connotations given to them by us. Despite recent research showing that the brain does not process mathematics as a language, I do not believe that mathematics is meant to replace languages, but rather to remain a universally constant yet expanding field; as exemplified by numbers, they can change symbolic meaning but their value as a magnitude will remain the same, showing mathematics’ endurance throughout time. Mathematics’ applications further reiterate mathematics’ importance and immense universal impact. Mathematics is, in fact, an integral part in two sciences; physics and biology. Without mathematics, would Newton have been able to prove or even discover the Law of Gravitation and how would biology be represented with a lack of mathematical models? This not only implies mathematics’ versatility but also highlights its role as both the infrastructure and foundation layer for these two sciences. Nevertheless, mathematics transcends this role as subsidiary subject with the Arecibo Message, an encoded message of prime numbers written for extra-terrestrial species. However dubious and debatable it may sound, to place mathematics at the centre of an abstract universe with humanity solely rotating around it and nothing else would be too egotistical; whilst there is no concrete evidence confirming both extra-terrestrial existence and their ability to comprehend mathematics, there is neither evidence against it. Conclusively, this shows the great extent of mathematics’ influence, ranging from philosophical to cultural to, quite literally, universal. In my opinion, the combination of these three factors renders mathematics the most indisputably reliable of fields and I admit I would not be surprised to see a future dominated by mathematical communication