Although mathematics is commonly dealt with as a pure science, its ubiquity in nature is undeniably evident—from the equiangular spiral shape of snails’ shells, to exponential growth models of world population, to the predictability of stock market behavior, to daily weather forecasts, to determining the least expensive route when travelling through air, and to finding the most efficient way to run errands. Even to the most seemingly unquantifiable disciplines such as literature and linguistics, mathematics continues to reveal itself through imperceptible yet intelligent ways. In exploring the vast ocean of mathematics’ role in nature however, it is rather easy to surface the waters—to be simply be enwrapped in such familiar and straightforward manifestations—and cease to plunge into the heart of mathematics, from which diverse applications emanate. In other words, merely recognizing or “appreciating” these apparent day-to-day punctures of mathematics greatly undervalues its true complexity and beauty. For one to authentically develop an appreciation for mathematics, one has to deliberately probe into what makes mathematics distinctively penetrative in nature. This dilemma is best phrased by Albert Einstein in the question “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” Eugene Wigner, a Hungarian-American theoretical physicist and mathematician, posited and attempted to answer a similar question in the approximate words “why is mathematics unreasonably effective in the natural sciences?” in his article The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Although these questions may eternally remain unanswered, just like any other mystery, venturing a guess can prove to be a worthwhile intellectual pursuit. On that account, this essay will delve into the core properties of mathematics from which its descriptive and predictive power springs from, revolving on the notion that mathematics is the one objective language linking all entities in the universe.
One of the key ideas of mathematics is its recognition of abstract order. Devlin (2000), in the prologue of his book Language of Mathematics, defines mathematics as “the science of patterns” (p. 3). Although it is dangerously simplistic and misleading to give such a definition, the title he proposes reveals a fundamental component of the actualization of mathematics, that is: acknowledgement of patterns. In the first chapter of Aufmann’s, Lockwood’s, Nation’s, Clegg’s, and Epp’s (2018) textbook Mathematics in the Modern World, numerous evidences of patterns in a typical nature setting are enumerated such as spiderworts’ three-fold symmetry, starfishes’ five-fold symmetry, the hexagonal construction of beehives, the external appearances of tigers and hyenas, and the number of flower petals which follows the Fibonacci sequence. Similarly, Vila (2010), in his short animation movie Nature by Numbers, beautifully illustrates the evolution of a series of Fibonacci numbers from a mere sequence of patterned natural numbers, into a collection of squares corresponding to each value in the sequence, into a harmoniously constructed spiral formed by quarter circle arcs within each square (i.e. Fibonacci or Golden spiral), and finally into a Nautilus shell which physically occurs in nature. These examples of patterns particularly occur at a high degree of clarity and comprehensibility; however, this is not always the case. For instance, a square—despite its lack of evident repetition—still exhibits order in that squares of any form must always comprise of four equal segments and four right angles. The same applies in the context of counting as demonstrated by the fact that there is invariably only one way to choose 0 balls from a bag of n balls. An even more unnoticeable form of pattern occurs in the case of projectile motion wherein an object thrown nearby Earth’s surface, solely subjected by gravitational force, always follows a definite curved path. There are infinitely many forms of pattern in the cosmos; as expounded by Devlin (2000), “Those patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, purely utilitarian or of little more than recreational interest. They can arise from the world around us, from the depths of space and time, or from the inner workings of the human mind” (p. 3). Beyond the awareness of the existence and all-inclusivity of these patterns however, one can still endeavor and challenge oneself to ask why these repeated patterns actually exist in nature. Why is it that flowers follow a structured number of petals based on Fibonacci numbers? Why are beehives internally constructed as a densely packed group of hexagonal prismatic cells? In patterns observed among life forms, the reason may conceivably be tied with natural selection and evolution but what could possibly justify more abstract patterns such as time? Although it may seem that investigating these can possibly lead one astray from the subject matter, if executed strategically, it can disclose deeper, more dumfounding truths on the relationship between nature and mathematics.
Mathematics does not conclude in recognition of these patterns. Along with the recognition of patterns comes its utilization of ever-powerful symbols to represent such structures. As articulated by astrophysicist Brian Greene, “With a few symbols on a page, you can describe a wealth of physical phenomena” (as cited in Aufmann et al., 2018, p.18). Because natural phenomena described by mathematics are relatively more formal, symbols and notations in mathematics take on more strictly structured forms— from basic equality and inequality symbols, to arithmetic operators and algebraic variables, to logic symbols, to probability symbols, to statistical graphs, and to geometric figures. Greene’s statement is best illustrated in an example provided by Aufmann et al. (2018) which focuses on world population. As studies on population dictate a larger scope of societal mechanisms including economic growth, environmental depletion, agricultural progress, and expansion of rich-poor gap, the ability of mathematics in condensing the pattern of population growth into a short equation as A=Pert takes a crucial role in the entire society. Amidst this myriad of notations however, one must not be overly absorbed and mistakenly identify symbols as the essence of mathematics. As mentioned by Devlin (2000), “…the symbols on a page are just a representation of the mathematics.” On that account, allowing abstract notations to outshine abstract patterns in one’s mind is no far from engrossing oneself to temporary and imperfect material objects rather than investing in the growth of one’s higher faculties.
Another one of mathematics’ intrinsic influential properties is the constancy of pure mathematical concepts across time. As described by Tyson (2011), ancient mathematics has journeyed through history with minimal to no alterations (e.g. the unchanging reliability of Euclid’s axioms as invented over 2,000 years ago). Owing to the elegant mathematical proofs which hold applaudable certainty, mathematics possesses a characteristic level of regularity across space and time as compared to other disciplines such as the social sciences and natural sciences. Because of this regularity, mathematics can depict nature in a more orderly and precise fashion. In a more practical setting, this uniformity is beneficial as it facilitates the predictive power of mathematical models. For instance, the potentiality of calculus to forecast tomorrow’s weather given sufficient data on air pressure, temperature, wind speed, and precipitation, lies on the applicability of the involved differential equations in nature irrespective of changes in time.
It is also worthwhile to note that mathematics holds a unique level of exactitude which makes it a convenient tool for describing abstract reality. An example of this is an equation commonly regarded as “the most beautiful feat in mathematics” (Hoang, 2014)—Euler’s identity. By assembling three of most important mathematical constants into one seemingly elementary equation (i.e. eiπ+1=0) via convergence of different subfields, Euler has opened doors into the perfection of the unit circle and inevitably touched on the equally mind-blowing concept of Fourier Analysis. In reality, these uncanny numerical relationships not only occur in the unfamiliar corners of mathematics but is ever-present in the field. An excellent typification of this would be Pythagorean’s theorem, which precisely correlates the three sides of a right triangle specifically through arithmetically relating the areas of the squares whose sides are those of the triangles’ respectively. The same beauty can also be observed in other areas of mathematics such as the fundamental theorem of calculus, central limit theorem (CLT), prime number theorem (PNT), the infinitude of primes, binomial theorem, and Fermat’s little theorem. Although employed in calculations countless times, the underlying beauty of these theorems and concepts are often overlooked and rather trivialized through mere memorization.
In a more general sense, the descriptive and predictive power of mathematics in nature’s context also lies on the ability of its methodologies to shape human critical thinking skills. As effectively solving mathematical problems demands for both creativity and formality, mathematics is able to exercise the collective mind which actualizes all its core properties. In this regard, there may be heavy implications on the ideal teaching procedures of problem-solving such as the employment of both sequential and formula-based strategies and intuitive-based strategies.
As revealed in the entirety of this essay, dissecting the very nature of mathematics might potentially be the most difficult problem a mathematician can attempt to solve. It’s ironic that mathematics, through theorems, proofs, and postulates, can provide an unyielding level of certainty about the universe yet its own identity it cannot firmly define. Through this paper, we have peaked into a small hole into this realm—gaining insight into mathematics’ ability to recognize patterns, concretize powerful symbols, remain robust through time, possess remarkable exactitude, and shape the minds of who materializes it. While we may never gain access into all these energy-providing properties, it is worthwhile to note what they collectively unravel about mathematics, that is—the absolute objectivity of the language which allows it to be comprehended by all entities in the cosmos.
References
Aufmann, R., Lockwood, J., Nation, R., Clegg, D., and Epp, S.S. (2018) Mathematics
in the modern world (Philippine Edition). Manila, Philippines: Cengage/Rex.
Devlin K. (2000). The Language of Mathematics. New York, NY: Holt Paperbacks.
Hoang, L.N. (2014). The most beautiful equation of math: Euler’s identity. Retrieved
from http://www.science4all.org/article/eulers-identity/
Tyson, P. (2011). Describing nature with math. Retrieved from
https://www.pbs.org/wgbh/nova/article/describing-nature-math/
Vila, C. (2010). Nature by numbers. Retrieved from
https://etereaestudios.com/works/nature-by-numbers/
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