What if I tell you that during the course of history, a measly study of the area of curved figures led to the discovery of how the world came to be?
The study of calculus, as “a part of mathematics that deals with limits” (Stewart, 2016, p. 8), was unknowingly founded by one of its pioneers from antiquity, the Pythagoreans, because of the motivation of getting an understanding of nature in man’s terms (Heath in Boyer, 1949/1959). They dealt with areas involving curved figures, through the method of exhaustion, and this in turn, gave them good ‘approximations’, a notable term used in this field of study, on their values (Boyer, 1949/1959). It was this line of thinking that led mathematicians to deal more with these ‘immeasurable’ objects, along with others like changes in a single point in time (Swetz & Vistro-Yu, 1999), and infinite series (Stewart, 2016), until it was made more formal by Barrow with his concept of the slope of the tangent line on a single point, and Newton and Leibniz, who introduced the concepts of differentiation and antidifferentiation (Swetz & Vistro Yu, 1999).
These mathematical breakthroughs paved the way for imaginative thinking to take fruit in mathematics; instead of pure precision, there was more room for innovation, as in the concept of differentials which were defined by Cauchy as a variable instead of fixed quantity (Boyer, 1949/1959). Another idea that calculus made possible is the general idea of mathematics not as “a description of nature nor a metaphysical generation of quantities”, but rather as “the symbolic logic of possible relations” (Cohen in Boyer, 1949/1959, p. 308). These clarificatory concepts were some of the ideas that helped push mathematics towards abstraction.
Moving away from the conceptual waves that calculus has brought into the world, its applications were also nothing to scoff at. Differentiation and antidifferentiation were highly instrumental when dealing with the measurement of seemingly immeasurable data. From the basics such as the calculation of areas and slopes to the complex ones such as in spectroscopy (like the development of Beer-Lambert Absorption Law with the use of integration) in analytical chemistry (Nix, 2013) and from the seemingly simple such as the optimal dimensions for a maximized area (Stewart, 2016), to the more pressing and obviously significant, like in population growth models (such as the Verhulst population model) (Joyce, 2013).
In order to see the effects of calculus in a better light, it would be best to look at how it affects one major field of study and observe how its reach possesses the capability to expand exponentially. Take chemistry for example, and see how a bit of incorporation of calculus-driven thinking and formulation benefitted many other bodies of knowledge.
When a person thinks of chemistry, what most commonly comes to mind is its dealings with the physical and chemical properties of matter. However, that conception is very limited as it excludes the fact that chemistry also deals, in arguably more significant manner, with changes in matter, and what better way to study those changes than to employ the mathematical tools concerned with change, calculus.
Before the introduction of calculus, chemists have been studying all about the properties of, and reactions between and among elements and compounds, the constitution of the matter, and a bit of ‘alchemy’. That tradition of chemistry lasted until the dawn of modern chemistry which was characterized by higher degrees of formality and experimentation. It was in this period where the subatomic particles were found (University of Columbia, 2003).
The chemists of the time were highly fascinated with the properties and capabilities of the new-found subatomic particles, and one of the things they have observed is the radioactive properties of isotopes of elements (Flowers et al., 2015). There was this concept of decay wherein an unstable isotope of an element would expel a particle and become a stable isotope in itself, and the function for the amount of time it requires for an unstable isotope to decay was formulated with the use of the tools provided by none other than calculus. They were able to observe the proportionality of the rate of radioactive decay with the amount of radioactive material present, from which they derived the half-life or the amount of time it takes for a specific number of atoms of a radioactive isotope to be reduced to half (Nix, 2013).
This discovery of radioactive decay became the root of further applications such as radiometric dating where the amount of radioactivity contained in dug materials are checked to determine how long they have been decaying. Hence, telling the age of the material, as well as the objects found with it (Wiens, 2002). This technique was then employed by paleontologists, geologists, and even some forensic experts in analyzing a multitude of things like the chronology of the history of the earth, dating of historical and religious documents (Jull, 2013), and even for solving crimes (Bulman & McLeod-Henning, 2012).
We started off of very abstract and random concepts and ended up studying the history of the world. That, pretty much sums up how broad the scope of the applications of calculus is. Its causal chain runs deep within the different bodies of knowledge which humanity then puts to use. In the end, calculus in itself ended up nowhere but in the academe, as apparently all theoretical fields of study do. However, its applications, cannot be more pronounced as it planted its seeds in the various aspects of human life.
References
Boyer, C.B. (1959). The history of the calculus and its conceptual development (The concepts
of the calculus). New York, NY: Dover
Bulman, P. and McLeod-Henning, D. (2012). Applying carbon-14 dating to recent human
remains. National Institute of Justice, (269). Retrieved from
https://nij.gov/journals/269/pages/carbon-dating.aspx.
University of Columbia. (2003). History of chemistry. Retrieved from
http://www.columbia.edu/itc/chemistry/chem-c2507/navbar/chemhist.html
Flowers, P., Theopold, K., Langley, R., Robinson, W.R., Hooker, P., Frantz, D., …, Carpenetti, D.
(2015). Chemistry [pdf document]. Houston, TX: OpenStax
Joyce, D. (2013). The logistic population model [pdf document]. Retrieved from
https://www2.clarku.edu/faculty/djoyce/ma121/logistic.pdf
Jull, A.J.T. (2013). Some interesting and exotic applications of carbon-14 dating by accelerator
mass spectrometry. Journal of Physics: Conference Series, 436. doi: 10.1088/1742-
6596/436/1/012083
Nix, R. (2013). Chapter 5: Calculus. In Quantitative methods in chemistry [pdf document].
Retrieved from http://www.chem.qmul.ac.uk/software/download/qmc/ch5.pdf
Stewart, J. (2016). Single variable calculus: early transcendentals (8th ed.) [pdf document]. Boston, MA: Cengage Learning
Swetz, F.J., and Vistro-Yu, C.P. (1999). The history of mathematics: a study guide. Quezon City: Ateneo de Manila University – Office of Research and Publications
Wiens, R.C. (2002). Radiometric dating – a Christian perspective (2nd ed.). Retrieved from
https://www.asa3.org/ASA/resources/Wiens.html#page%204
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