The Link Between Knowledge and Doubt in Mathematics and Ethics


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Modern philosophy often casts doubt and certainty as polar opposites: while doubt describes an uncertainty with respect to accepting a given proposition, certainty is the acceptance of a proposition without doubt. Rene Descartes, the father of this field, believed that what we regard as knowledge is only what we cannot doubt. (Newman). He described knowledge as enduring, certain and indubitable – suggesting an inverse relationship between knowledge and doubt. Many have criticised Descartes’ beliefs due to his strict definition for what constitutes knowledge, yet his work thrust the relationship between knowledge and doubt into the limelight. Goethe offered a contrasting take, claiming that knowledge creates doubt. He proposes a positive relationship between knowledge and doubt: an increase in knowledge can lead to an increase in doubt; conversely, possessing lesser knowledge allows little room for doubt and increases confidence (or certainty) in knowledge.

This belief initially roused my scepticism; however, I soon realised it possessed some kernel of truth to it. As child raised in a Hindu household, I initially possessed a staunch belief in the existence of multiple gods. As I became exposed to other schools of thought such as Christianity and Islam, which touted the existence of only one god; or atheism, which claims that there is no god, I found my faith was shaky and uncertain. When I knew more, I experienced more doubt. However, when reconsidering Goethe’s statement in contexts outside of religious knowledge systems and faith I found that it was not faultless. For example, in the case of mathematics and ethics, an entirely positive causal relationship between knowledge and doubt did not fully explain my own thought process and realisations. In this essay, I intend to evaluate whether the notion of knowledge creating doubt is accurate in different areas of knowledge. This led me to the following knowledge question: How are knowledge and doubt linked in mathematics and ethics?

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My first foray into the field of mathematics began before I could even express my thoughts fully and clearly; yet even then I was aware of the certainty of mathematics. Mathematical knowledge is established by reason. Although some conjectures stem from intuition, they must be rigorously proved with logic. Unlike other areas of knowledge, mathematical knowledge seemed to endure. In the natural sciences, for example, the proposition that protons and electrons were elementary particles of the universe was immediately falsified by the discovery of quarks, which are smaller particles yet. Learning this made me uncertain about any scientific fact: while I think that life on Earth arose from organic compounds, there is no completely logical basis for my belief and I still have doubts. Whereas, when I learn a mathematical fact, I am completely confident that it is true due to its logical basis. 1 + 1 would always turn out to be 2, and the Pythagoras’ theorem would always dictate how long each side of a right triangle was even a thousand years from now.

However, I did feel some uncertainty. When I found that Pythagoras’ theorem did not hold for curved surfaces, my doubt was roused and could only be quenched by further consumption of mathematical knowledge. When I read further on the subject, I learned about the existence of spherical and hyperbolic geometries – and while non-Euclidean, they were certainly logical and proof based mathematical subdisciplines. My certainty in mathematics was immediately restored. I came to realise that doubt was not the product of a gain in knowledge but rather an intermediate step. When I learned more, my personal knowledge increased. While I ruminate over this new knowledge, doubts in my understanding were exposed. Yet upon probing further my doubts were assuaged with more mathematics filling in the gaps in my knowledge. Pondering the new mathematics I learnt, I would find more uncertainties in my knowledge and the process would repeat. While Goethe’s description of a positive causal relationship between knowledge and doubt could describe the initial phase of acquiring knowledge, it did not paint the full picture in mathematics. In my opinion this relationship is cyclic: more knowledge creates more doubt, which is satisfied by more knowledge which in turn creates doubt, and so on.

However, the above perspective is that of a student learning existing mathematics. At the frontiers of mathematics this situation is starkly different, as seen in a foundational crisis in mathematics in the early 20th century. I first came across Gödel’s Incompleteness Theorems when I read a book called Fermat’s Last Theorem (Singh), and was shocked to read about the limitations in mathematical certainty. German mathematician David Hilbert proposed a program (Zach) to formalise all mathematics in an axiomatic form and prove that these axioms were consistent (i.e., they do not give rise to contradictions). When Kurt Gödel, an Austrian logician, was attempting to solidify the axiomatic basis of mathematics with logic in response to Hilbert’s program, he instead found a paradox and proved that that Hilbert’s objectives were unachievable: no axiomatic system could avoid contradictions, and some statements could not be proved within a system. The theorems do not disprove any major mathematical truths – both arithmetic and geometry remain as logically sound in their axiomatic systems as they were when first discovered – but undermined the infallibility of mathematical knowledge. The pursuit of more knowledge led to uncertainty – while Gödel attempted to solidify the basis of mathematics, he found cracks in the foundation which could not be fixed. This supports Goethe’s statement of knowledge creating doubt.

But depending on how we define knowledge and doubt, Gödel’s theorems can also be considered as knowledge – they are simply statements which draw recognition to the fact that mathematics is uncertain. In this regard, Goethe’s statement does not suitably describe the development of knowledge. However, while it is not a complete explanation for mathematical knowledge and doubt, it does make us aware that while we can be largely certain in our mathematical knowledge, our certainty must not be absolute and we must allow some doubt to exist.

Now I turn to the field of ethics to examine the relationship between knowledge and doubt, using a famous model – the prisoner’s dilemma. I first read about it in a childhood book called The Mysterious Benedict Society (Stewart). While the model is often analysed with game theory, it also possesses important ethical implications

Consider the following example (Rapoport): Two prisoners are accused of committing the same crime. Kept in separate cells, each prisoner is asked to confess with the following caveats: If both parties confess, they receive a 5-year sentence. If neither confesses, they both receive only a 1-year sentence. If only one confesses while the other remains silent, the first goes free while the other receives a 10-year sentence. The crux of the issue is that both parties would be better off if neither confessed, yet from an individual perspective it is better to confess to the crime. This depicts the dichotomy between self interest and collective interest.

I will not be focusing on this basic framework of the prisoner’s dilemma and will instead pose a twist. When I delved into this dilemma by inserting myself into the role of one of the prisoners, my first realisation was that neither party will know the other’s decision prior to making their own – they lack knowledge. But if I were to know the other prisoner’s choice before I made my own, how would I act? Assuming the other prisoner chose to remain silent, it meant I could either do the same (with the result of a 1-year sentence) or confess (in which case I would be free). This made me hesitant about my actions, and I felt conflicted by emotion and faith. The reduced sentence for both of us did was not as unbearable as a 5-year sentence and would also allow me to escape the guilt of condemning my companion. Yet the prospect of freedom seemed equally enticing. Knowledge made me more uncertain about my actions – which aligns with Goethe’s statement.

However, my perspective was that of someone who lacks rigid moral convictions. The game theoretic ideal of acting purely in one’s own self-interest is known as ethical egoism (Rachels). If were an egoist, I would always confess regardless of how the other prisoner acts. But if my ethical beliefs were utilitarian, I would always try to promote collective good (Driver) – which means I would not confess in the above scenario. My actions ought to be dictated by my morals. If I experience doubt, it is simply because my beliefs are shaky. This means that knowledge of the other’s decision did not create doubt but rather made me aware of its existence. Rather than a positive causal relationship between knowledge and doubt as described by Goethe, this suggests a positive correlation between the two. Pondering more knowledge reveals our pre-existing doubts and can even prompt us to fill in the gaps in our knowledge – especially in non-absolute areas like ethics. This allows us to become more certain in our ethical beliefs.

While Goethe’s proposed relationship between knowledge and doubt can describe some scenarios in mathematics, ethics and other fields, it is not the only explanation for our perceptions of doubt and knowledge. It is both a curious and interesting claim with some degree of accuracy, but ultimately is only one of many ways to observe the links between knowledge and doubt.

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