Wednesday, April 29, 2020

Fermat's Last Theorem and Soft Skills Development in the Modern Days


Problems are bad. This is a common-sense consensus that can be easily drawn from our life’s common experiences. However, a person that possesses the ability to solve problems can easily thrive in life. The discussion here will show how significant Problem-Solving skills are, both in academic and professional environments; this, in turn, will show that it is possible to use problems to our benefit.

We can define Problem Solving as the threefold ability to analyze complex problems and generate creative solutions supported by informed evidence. It can be derived from this definition that the complimentary skills of Critical Thinking, Creative Thinking and Information Literacy are to be adequately understood so that one can apply Problem Solving in an adequate manner.

A question that can come from this definition is why one should adopt the characteristics of Problem Solving as outlined by the definition above? To answer that, we can traverse through some other definitions of Problem Solving. Two of the three aspects – Information Literacy and Critical Thinking are found in some definitions (Cummins, Yamashita, Millar, & Sahoo, 2019); (Symons & Pierce, 2019), (Mohaghegh & Größler, 2020), but none of them brings the aspect of Creative Thinking to the discussion.

If the benefit of adding this third aspect to the table can be shown, it can be inferred that this broader definition can have an added benefit, if applied. Creative Thinking can be understood as the ability to synthesize and connect multiple ideas, whether convergent or alternate, in an innovative way, with a willingness to take risks. It is left to determine whether this understanding that Creative Thinking should be considered advantageous to students.

In order to introduce this discussion, it is relevant to outline a view of Creativity/Creative Thinking that allows to justify its inclusion within the definition of Problem-Solving cited above. A short review of some studies reveal that they tend not to define creativity per se, but instead they discuss aspects related to creativity (Keenan-Lechel & Henriksen, 2019), (Henriksen & Mishra, 2020), (Cain & Henriksen, 2017); however, two definitions fit the analytical context of this essay: one portrays creativity as “an approach to problem-solving through the imagination” (Evans, Henriksen, & Mishra, 2019) and another that views it as “the genesis of all learning, in every area and across every discipline” (Richardson, Henriksen, & Mishra, 2017).

This shows that in a scholarly point of view, at least, it is possible to see creativity as a fundamental skill linked to problem-solving. However, to complete the effort, it is necessary to verify if this view is shared in the professional environment as well.

There was an extensive discussion at the World Economic Forum in 2015 about the skills that are needed by workers in the following 5 years. A list was then compiled, containing the top 10 skills workers should possess in order to thrive in the market. The first three skills listed are Complex Problem Solving, Critical Thinking and Creativity (World Economic Forum, 2016). Creativity was added because of the understanding that it would be necessary in order to adequately deal with the innovative new technologies.

So, from this it is possible to see that in both environments – academic and professional – the role of Creativity/Creative Thinking in Problem-Solving is very relevant. Two practical examples of how problems end up being used for added benefit, with a creative approach, are shown in two famous situations: Fermat’s Last Theorem and the Clausewitzean Theory of War.
It is said that about 1637, a French lawyer called Pierre de Fermat wrote 35 words in the margin of his copy of Diophantus’s Arithmetica, unknowingly causing a ruckus that would last for almost four centuries:
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.[1] (Dixon, 1920)

There are still controversies about whether he did solve this or not, since no actual proof was found among his writings save it for n = 4, but for the general problem that became known as Fermat’s Last Theorem, the challenge had just begun. Hundreds of attempts to demonstrate the theorem were made in the subsequent 350 years (Dixon, 1920). One of these attempts is an example of how problems, even when unsolved, can be useful: Ernst Eduard Kummer, around the 1840s, after several unsuccessful attempts to demonstrate the theorem, ended up creating a definition for ideal numbers. 

From that definition arose the general theory of algebraic numbers, one of the most important branches of modern mathematics (Bell, 1986).

The beautiful demonstration of this theorem began when in 1984, Gerhard Frey hypothesized about a link between two apparently unrelated problems: the Taniyama-Shimura conjecture and the Fermat’s Last Theorem (Singh, 2013). This inspired an English mathematician, that managed to formally prove the theorem in 1995, 358 years after its conjecture (Knudsen, 2015), (Wiles, 1995).

What is the relationship between Fermat's Last Theorem and its implications shown above and the construct Creative Thinking + Critical Thinking + Problem Solving + Information Literacy? Unless we learn how to adequately confront our problems, we may end up missing opportunities to accomplish powerful results. This requires extreme effort in our part in understanding that unless we see our soft skills development in a systematic approach, the same way we pursue hard skills, we will miss opportunities to develop our abilities.


Bell, E. T. (1986). Men of Mathematics. New York: Simon and Schuster.
Cain, W., & Henriksen, D. (2017). Uncreativity: a Discussion on Working Creativity Before and After Ideation with Dr. Chris Bilton. TechTrends, 61, 101-105.
Cummins, P. A., Yamashita, T., Millar, R. J., & Sahoo, S. (2019, August). Problem-Solving Skills of the U.S. Workforce and Preparedness for Job Automation. Adult Learning, pp. 111-120.
Dixon, L. E. (1920). History of the Theory of Numbers Vol. 2. Washington: Carnegie Institution.
Evans, M. D., Henriksen, D., & Mishra, P. (2019). Using Creativity and Imagination to Understand our Algorithmic World: a Conversation with Dr. Ed Finn. TechTrends, 63, 362-368. doi:10.1007/s11528-019-00404-3
Henriksen, D., & Mishra, P. (2020). A Pragmatic but Hopeful Conception of Creativity: a Conversation with Dr. Barbara Kerr. TechTrends, 64, 195-201. doi:10.1007/s11528-020-00476-6
Keenan-Lechel, S. F., & Henriksen, D. (2019). Creativity as Perspective Taking: An Interview with Dr. Vlad Glăveanu. TechTrends, 63, 652-658. doi:10.1007/s11528-019-00437-8
Knudsen, K. (2015, August 20). The Math Of Star Trek: How Trying To Solve Fermat's Last Theorem Revolutionized Mathematics. Retrieved from Forbes:
Mohaghegh, M., & Größler, A. (2020). The Dynamics of Operational Problem-Solving: A Dual-Process Approach. Systemic Practice and Action Research, pp. 27-54.
Richardson, C., Henriksen, D., & Mishra, P. (2017, September 6). The Courage to be Creative: An Interview with Dr. Yong Zhao. TechTrends, 61, pp. 515-519. doi:10.1007/s11528-017-0221-1
Singh, S. (2013). The Simpsons and Their Mathematical Secrets. A&C Black.
Symons, D., & Pierce, R. (2019). Beyond fluency: Promoting mathematical proficiencies through online collaborative problem solving. Australian Primary Mathematics Classroom, pp. 4-8.
Wiles, A. (1995). Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics, 142, 443-551. doi:10.2307/2118559
World Economic Forum. (2016, January 19). The 10 skills you need to thrive in the Fourth Industrial Revolution. Retrieved from World Economic Forum:

[1] [It is not possible to divide] a cube into two cubes, or a fourth power into two fourth powers, or in general, until infinity, above fourth power into two similar powers. I have found a wonderful demonstration for this, but it won’t fit in this margin. In more modern words, it is to say that there are no three positive integers a, b and c that satisfy the equation an + bn = cn for any integer value of n greater than 2 (free translation by the author, with terms added for clarification).
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About Ricardo Prins
Ricardo Prins is a Software Engineer who thinks that technology is not the answer to all our problems.

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