For most people, calculus is first encountered in a classroom that stays long in memory. It could be the scent of dry-erase markers, the whirring of the projector, or the writing of an expression like (∫x²dx) on the chalkboard as if it were a natural thing to do. For an instant, the symbol was almost ornamental, like twisted metal or an elongated letter from a forgotten alphabet.
Then the teacher begins to speak of area, motion, slope, accumulation, and change, and the symbol begins to seem like a machine.
The topic of study is regarded as an array of complex rules reserved for the intellectually gifted, as the tongue of physics, or as the academic pursuit that creates divisions between those students who get it, and those who do not believe in it. However, what if there is more to the concept of calculus than this? What if the topic is a language of signs that enabled us to perceive and understand change in a new light?
Semiotics, in its broad sense as elaborated by Ferdinand de Saussure and Charles Sanders Peirce, is the field of inquiry dedicated to the processes of meaning creation through signs. A sign signifies the other phenomenon and evokes a response in the mind. In such a way, calculus is not an algebraic construct.
The fundamental concept of Saussure was that the sign had two components—the signifier and the signified. The signifier referred to the form, the sounds, the marks, or the images. On the other hand, the signified referred to the concept that it generated. In his interpretation of signs, Pierce makes another distinction by explaining how signs are interpretive. A sign generates an interpretant, which is an interpretation in response to it.
It means that a sign sets off a chain of interpretation. When you view the idea from the perspective of calculus, then the signs used in it define the way in which the concept has to be interpreted.
Much before Newton and Leibniz codified the classical version of calculus, there were those who tackled issues such as area, volume, tangents, and curves. While Archimedes was not doing calculus as we understand it today, he was nevertheless constructing an entire world of signs about the issue of quantification of curves and continuities.
His method of exhaustion was one in which he took the curve and broke it down into manageable forms. All curves like circles, parabolas, spheres, and cones had to be translated into another language that was visually and conceptually comprehensible to the mind. The problem of math had to be symbolized and made readable first.
Here is how semiotics brings another dimension to the history of calculus. Instead of being focused on what and who was discovered, it is possible to explore the ways in which mathematical signs enabled certain discoveries. In the case of Archimedes’ polygons used to calculate the area of a circle, we can understand it through signs.
Archimedes’ method involves an intellectual habit expressed through signs. It is an understanding that a curved object may be approached by a set of linear objects, that the essence of the curve can be approached visually and systematically substituted.
Particularly, Fermat’s case is noteworthy since his technique for solving problems related to maxima, minima, and tangents is practically an experiment in semantics. Suppose the function is:
Fermat then rewrote it with the variable slightly changed:
He compared the two expressions, eliminated the common elements, divided by the small change, and allowed the difference to become zero, obtaining:
The procedure can be described with modern mathematical language as differentiation.
Semiotically, there is yet another process taking place. Namely, Fermat creates a formal link between a sign of change and the idea of change. The small change in value is a symbol of movement.
It is difficult because of the unusual duality embodied in its symbols. While it is easy enough to see the marks on the page, they are actually referring to something abstract and unpredictable. Consider the expression:
While the symbol (dx) may appear to be a minor mark in the overall expression, it does not refer to an object in the usual meaning of this word. It refers to a relation, to a difference, to a potential transformation, to something infinitesimal. Likewise, consider the expression:
The integral sign is a sign of accumulation, reminding us that countless infinitesimal quantities can be combined to produce a larger whole. Once learners begin trusting these symbols, they start seeing the world differently. Motion becomes a relationship, and area becomes an accumulated process.
From how the use of calculus in various mediums, calculus is represented using formulas and graphs in a textbook, while on a blackboard, it takes a more fluid form in terms of arrows, crossing outs, and incomplete symbols, signifying the workings of a mind. When represented through spreadsheets, the concepts of calculus are applied to the modeling of growth, decay, and costs over time.
Calculus is also employed in computer graphics to create the motion of curves, animate objects, and smooth surfaces, even in physics in terms of equations of forces, velocities, and energies. Whatever medium is used, there is always the same basic concept of employing signs to transform continuous reality into something that a mind or a machine can understand.
Outside mathematics as well, there are music notation as a system of symbols which renders audible sounds visible, maps render geographical space into a set of symbols, subtitles render spoken sounds into symbols, an emoji reduces emotions to tiny images, a traffic signal halts a crowd of a hundred by virtue of having been agreed upon as symbols of power, and so on.
Hence, reality needs to be symbolized in order to be ordered in a manner that can be communicated.
Considered in such light, the history of calculus is a history of the slow creation of a language of continuity. Early Greek mathematics was heavily reliant on geometry, stable shapes, and demonstration by form and balance. Later generations of mathematicians sought to make such forms move, to make such moves towards difference, and to make the continuous become signs. It could be assembled into a new whole.
Newton and Leibniz were beneficiaries of a long string of semiotic discoveries, codified techniques into a language with grammar rules, giving stability to a symbolic language in which the truths could be articulated and advanced.
For Peirce, the sign interprets the meaning mediated by interpretation, which can be endless. Calculus does the same thing. The sign, such as the differential, allows the next level of interpretation of what is happening with respect to a function at a certain point, how the slope changes, how the motion builds up, and how one magnitude turns into another. Each sign leads to another interpretation.
The relationship between the signifier and the signified is conventional, meaning arbitrary socially and historically formed. Again, consider the expression:
You had to decide how the symbol (∫) would signify integration and the (dx) would mark an infinitesimal change. Once the conventions took hold, it became part of a powerful shared language. The symbols of calculus were invented, refined, debated, and adopted, lying partly in the fact how we learned to read it in the same way.
On the other hand, there is a pragmatic argument that must be mentioned. It will be too simplistic to say that calculus deals only with signs. As a branch of mathematics, it cannot escape material reality. Bridges can break down, planets can shift their orbits, liquids can flow in certain ways, electricity can conduct through circuits, and not all beautifully constructed signs get to survive the implementation.
On the contrary, both are inseparable. The sign system functions because it is able to structure experience. The signs it uses are bound by the task we have to accomplish. For sure, it had an elegant notation system. However, it was also efficient in its application in engineering, astronomy, mechanics, and later virtually any discipline. It required a change to be measured.
Symbols can help us visualize the invisible, transform motion into lines, accumulation into integration, change into law, and infinity into order. While it occurred throughout the long process spanning from Archimedes to Newton and Leibniz and further on to modern mathematics, as an unrefined idea, it turned into a set of symbols able to transcend centuries.
Mathematics is supposed to be the epitome of objectivity, but the history of mathematics demonstrates that the most precise of all human knowledge is still tied to the systems of meaning we construct for ourselves. We may think about how we discover the world, but we also create ways of interpreting it. A derivative is a way of thinking about the fact that movement has structure, and an integral is a way of thinking about how a whole can be composed of parts.
Initially created as symbols of space and change, calculus has evolved into the language of modern science. However, calculus has turned out to be a demonstration system. In symbols, we always receive our reality. The curve must first be written down to be understood, motion must first be labeled to be measured, and change must first be made into a sign to become knowledge.
When we learn how to read its signs, the world becomes understandable.
References
- Baron, M. E. (1969). The Origins of the Infinitesimal Calculus. Pergamon Press.
- Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development. Dover Publications.
- Cajori, F. (1993). A History of Mathematical Notation (Vols. 1–2). Dover Publications. (Original work published 1928–1929)
- Edwards, C. H., Jr. (1979). The Historical Development of the Calculus. Springer.
- Heath, T. L. (1897). The Works of Archimedes. Cambridge University Press.
- Peirce, C. S. (1998). The Essential Peirce: Selected Philosophical Writings (Vol. 2). Indiana University Press.
- Rotman, B. (2000). Mathematics as Sign: Writing, Imagining, Counting. Stanford University Press.
- Saussure, F. de. (1983). Course in General Linguistics (R. Harris, Trans.). Duckworth. (Original work published 1916)
- Stillwell, J. (2010). Mathematics and its History (3rd ed.). Springer.
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