To What Extent Is the Acquisition of New Mathematical Knowledge Based on the Need to Solve Problems?

Upon first glance, it appears that “knowledge” is created mainly “to solve problems”, indirectly implying that knowledge can be used for other purposes. In the context of this investigation, “knowledge” in the Area of Knowledge (AOK) Mathematics refers to two specific areas: tacit and explicit knowledge. Tacit knowledge is skill-based knowledge gained from intuition, experience and memories (e.g. riding a bike)[1] whereas, explicit knowledge is anything based on fact or opinion and hence, can be proved either true or false[2]. While, Artistic “knowledge” refers to three specific areas: personal (gained through experiences and involvement), shared (community-based knowledge)[3] and conceptual knowledge[4] refers to one’s understanding of principles, models and theories. Furthermore, “problems” refer to any pending matter, stemming from the distant or recent past, which need to be addressed in order to improve the state of a situation. For example, the mathematical ‘The Twin Prime Conjecture’ (determining if there is infinite number of twin prime numbers)[5] or the artistic problem of a lack of exposure to wide audiences (possibly affecting an artist’s overall intentions of differentiating themselves from other artworks)[6]. Moreover, the phrase “main reason”, will be interpreted as the intended purpose of producing knowledge at the time of production. To discuss the main claim, the following knowledge questions will be considered: To what extent is the acquisition of new mathematical knowledge based on the need to solve problems? To what extent does the application of artistic knowledge solve problems?

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In mathematics, knowledge is “produced” from pure reasoning and axioms (statements taken to be true) to produce proofs[7]  of mathematical theories, without the intention to solve problems, but to further one’s own curiosity and personal knowledge. From my personal experience, I believe that this process commences with the identification of a field-based problem, a proposed method to solve this problem utilising one’s tacit and explicit knowledge to create a solution, which can later be proved true or false. Ways of knowing (WOKs) such as intuition, imagination and reason are used in solving problems through an experimental approach to the application of axioms, within the rules of mathematics. Thus, mathematical knowledge is defined as “produced” if it is validated by proof. This allows for its application in solving theoretical problems, such as the discovery of ‘Calculus’ primarily by self-taught German mathematician Gottfried Wilhelm Leibniz and Sir Isaac Newton’s synthesised observations of the natural environment[8]. Leibniz was concerned with analysing changes in graphs, leading to his belief that calculus was a “metaphysical explanation of change”[9]. However, Newton used his background in physics to determine that Calculus is a “description of motion and magnitudes” [10]. This consensus regarding the objective of Calculus describing change, was essential in allowing calculus to be applied, using one’s tacit and explicit knowledge, to theoretical problems such as ‘Fuzzy Calculus’ (study of theory and applications of integrals and derivatives of uncertain functions)[11]. While, this application may not have had any use in the real-world, through paradigm shifts, stemming from technological advancements, extensive investigation has been allowed into these theoretical problems. ‘Fuzzy Calculus’, has been used in modelling several engineering and scientific phenomena, such as creating a three-dimensional risk index model for large-scale infrastructure. Another example is the development of ‘Musical Set Theory’[12] which utilises abstract concepts for categorising musical objects and describing their mathematical relationships. It involves using probability to list the number of possible notes sets ranging from 2-11. Using their mathematical and musical backgrounds, Howard Hanson[13], Allen Forte and Milton Babbitt[14], spent time in finding these relationships through examining thousands of sets and permutations. Their invested interest and curiosity coupled with their intuitive thinking and imagination helped establish this theory, which is yet to be validated by technology. Evidently, another “main reason” why this “knowledge” was “produced”, stems from one’s mathematical curiosity and creativity.

A counterclaim could be that the “main reason” for developing new mathematical knowledge is to solve problems in the theoretical or real-world. This is exemplified in the ‘Königsberg Bridge Problem’, set in the Prussian city ‘Königsberg’, consisting of seven bridges placed on three islands such that no bridge can be crossed just once in one route[15]. This problem led to the development of the mathematical concepts of ‘Topology’[16] and ‘Graph Theory’[17], which provided new avenues in solving this problem and other problems concerning movements through networks. Another example is the development of cryptography[18] over thousands of years, to break codes (ciphers) and access information. Applied in World War I and II, the latter successfully attempted to read Nazi Germany’s ciphers and helped shorten the war. Hence, its development has been essential in solving political “problems”. Whether solving problems is the “main reason” or not, I believe that it is the ultimate objective. When mathematical knowledge is “produced”, its utility and applicability is not a measure used to determine its usefulness, as all valid mathematical knowledge derived, is useful. Such is the case with ‘Fuzzy Calculus’, as mathematics does not always have to have an application in the present day, but it can be applied in the future to solve problems, when technology advances. This type of mathematics is deemed ‘Pure Mathematics’ as it has no current real-world application, but a possible application in problem-solving in the future. Therefore, the development of new mathematical theories intends to solve problems in the future or present.

Overall, mathematics is an important part of society, despite some arguments against its usefulness in real-life. Whether knowledge ultimately solves real-world problems will remain unknown until paradigm shifts occur, giving ‘Pure Mathematics’ the potential for higher-order use in the future and applying “produced” theoretical knowledge, in intentionally solving problems.

Conversely, the Arts are represented in any form of media (e.g. paintings) and are produced from human perspectives which attempt to evoke a response or reaction from the viewer, rather than “solve problems”. From my personal interpretation of artistic knowledge, this knowledge is acquired through sense perception and is communicated through emotion, instead of intuition and reason in mathematics. Artistic knowledge is defined as “produced” if it is deemed as a representation of the human experience, displaying unformulaic characteristics in comparison to mathematics, where strict guidelines are imposed in deeming “knowledge”. For example, the 1871 ‘Whistler’s Mother’ portrait painted by son James McNeill Whistler, embodies a son’s perception of his mother[19]. When Whistler painted this, he recognised his mother’s intricate physical features and expressions, to accurately portray what he saw. This personal knowledge is valid because it comes from a first-hand source, his mother. On the other hand, the shared knowledge, acquired from the public’s perception of this portrait was the life of US working mothers in general and the female gender roles during the Great Depression. Another similar example is ‘The Scream’ painting created by Edvard Munch in 1893, made to symbolise the anxiety of the modern-day man[20]. Its usage of colours, facial expressions and distortions help convey this message and human experience. Thus, in representing this experience, conceptual knowledge is communicated, such as feelings of sympathy or empathy, towards a certain group. While this conceptual knowledge is “produced”, it is not attempting to solve “problems”, but provide insight into one’s human experience and convey this through different media forms, to appeal to all human senses and fortify this newly-acquired Artistic knowledge.

A counterclaim could be that evoking a response through appealing to emotion is a by-product of producing Artistic knowledge, where the artist is attempting to highlight underlying problems and make a call-to-action. This is seen in the 20th Century (c. 1916 – 1924) Dadaism Artistic Movement which focuses on creating purposely not aesthetically-appealing art, targeting the upper Bourgeois classes[21]. Not only did this question the role of the artist in the aesthetic process, it challenged social norms serving as a way of venting the frustration of the social divide caused by the ongoing World War I. This use of emotion and sense perception in artistic formats have appealed to a wider audience, who have their own views. However, art’s limitation as a language, is confusing or being misinterpreted by the viewer. This could change from the audience’s age, understanding etc, as different people view art from different social contexts. Another example is the controversial 1964 Norman Rockwell painting ‘The Problem We All Live With’, depicting a six-year-old African-American girl, Ruby Bridges, being escorted to school by US Marshals during the ongoing New Orleans school desegregation crisis[22]. Rockwell’s prior experience in seeing these real-life events, provoked his blunt expression of what he saw unfolding. Given the context of the US Civil Rights Movement, this shared knowledge represents the underlying social problems faced by people of varying races. Thus, it exemplifies the power of artistic knowledge in highlighting problems, with the aim to “solve” them.

Therefore, in the arts’ contribution in society and their perspectives is important because human nature is partially based on emotion. This emotion conveyed through Artistic knowledge has an effect on the viewer, and an overall collective who decide to react to this stimulus.

Having studied both subjects, I can conclude that “the main reason knowledge is produced is to solve problems” in light of theoretical and real-world problems in mathematics and the increased awareness of social problems through art, which is necessary to “solve problems”. Nevertheless, it can be seen that knowledge production is a consequence of one’s curiosity and passion, evident in mathematical knowledge (e.g. ‘Fuzzy Calculus’). While one’s display of the human experience, serves as another “main reason” for artistic knowledge, the aforementioned examples have acquired knowledge and solved real-life issues but most importantly, have benefitted society. Therefore, it is evident that one’s intention to solve problems is an integral factor in knowledge production.

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[1] IBO. (2013, April). Theory of Knowledge Guide. Le Grand-Saconnex, Geneva, Switzerland: International Baccalaureate Organisation.

[2] Heydorn, W., & Jesudason, S. (2013). Decoding Theory of Knowledge. UK: Camebridge University Press.

[3] IBO. (2013, April). Theory of Knowledge Guide. Le Grand-Saconnex, Geneva, Switzerland: International Baccalaureate Organisation.

[4] Training Industry. (2019). Retrieved from Conceptual Knowledge: https://trainingindustry.com/glossary/conceptual-knowledge/

[5] Weinstein, E. (2019, July 25). Wolfram Alpha. Retrieved from Twin Prime Conjecture: http://mathworld.wolfram.com/TwinPrimeConjecture.html

[6] Acharki, A. (2018, September 26). Lowercase. Retrieved from The 3 Biggest Problems in the Art World: https://www.lowercasemagazine.com/blog/2018/9/25/the-3-biggest-problems-in-the-art-world

[7] IBO. (2013, April). Theory of Knowledge Guide. Le Grand-Saconnex, Geneva, Switzerland: International Baccalaureate Organisation.

[8] Robertson, J. O. (1996, February). History Topics. Retrieved from History Of Calculus: https://www-history.mcs.st-and.ac.uk/HistTopics/The_rise_of_calculus.html

[9] Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development: (The Concepts of the Calculus). Dover Publications.

[10] Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development: (The Concepts of the Calculus). Dover Publications.

[11] Arqub, O. A., Pinto, C., López, R. R., & Ertürk, V. S. (2018, June 5). Fuzzy Calculus Theory and Its Applications. Retrieved from Hindawi: https://www.hindawi.com/journals/complexity/2018/5463920/

[12] Tomlin, J. (2014, January 10). All About Set Theory. Retrieved from What is Musical Set Theory?: http://www.jaytomlin.com/music/settheory/help.html

[13] Hanson, H. (1960). Harmonic materials of modern music; resources of the tempered scale. New York: Appleton-Century-Crofts.

[14] Talk Classical. (2015, March 2). Retrieved from What Is Musical Set Theory?: https://www.talkclassical.com/36875-what-musical-set-theory.html

[15] Carlson, S. C. (2010, July 30). Encyclopaedia Britannica. Retrieved from Königsberg bridge problem: https://www.britannica.com/science/Konigsberg-bridge-problem

[16] Carlson, S. C. (2017, February 8). Encyclopaedia Britannica. Retrieved from History Of Topology: https://www.britannica.com/science/topology/History-of-topology

[17] Carlson, S. C. (2019). Encyclopaedia Britannica. Retrieved from Graph Theory: https://www.britannica.com/topic/graph-theory#ref909878

[18] Ray, M. (2018). Encyclopaedia Britannica. Retrieved from Cryptography: https://www.britannica.com/topic/cryptography

[19] Tan, M. (2016, March 29). The Guardian. Retrieved from How Whistler’s Mother became a powerful symbol of the Great Depression – in pictures: https://www.theguardian.com/artanddesign/gallery/2016/mar/29/how-whistlers-mother-became-a-powerful-symbol-of-the-great-depression-in-pictures

[20] Affatigato, C. (2018, April 25). Auralcrave. Retrieved from The Scream by Edvard Munch: analysis and meaning of the painting: https://auralcrave.com/en/2018/04/25/the-scream-edvard-munch-and-the-despair-that-swallows-everything/

[21] The Art Story. (2018). The Art Story. Retrieved from Dada – History and Concepts: https://m.theartstory.org/movement/dada/history-and-concepts/#concepts_styles_and_trends_header

[22] Esaak, S. (2019, January 26). Thought Co. Retrieved from “The Problem We All Live With” by Norman Rockwell: https://www.thoughtco.com/the-problem-we-all-live-with-rockwell-183005