Truth by Close Proximity
At the end of the spring 2020 semester, when teaching was concluded online, I had an unusual conversation with a student via Microsoft Teams. I will try to reproduce it as accurately as I can, using the direct dialog style to eliminate redundant verbosity
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At the end of the spring 2020 semester, when teaching was concluded online, I had an unusual conversation with a student via Microsoft Teams. I will try to reproduce it as accurately as I can, using the direct dialog style to eliminate redundant verbosity. Since we, mathematicians, aim to make things as simple as possible, I will abbreviate myself to P (for professor) and my interlocutor to S (for student). Here is the gist of the discussion. S: P: S: P:
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There has been some controversy lately about mathematics, its nature, origin, and interpretation, and I would be interested to have your views on the subject. What controversy? I see no controversy where mathematics is concerned. Can you be more specific? For example, I read that 2 + 2 should not always be taken to be equal to 4. It may, perhaps, be equal to 5? It may, if we ban the use of the number 4, counting the positive integers as 1, 2, 3, 5, 6, . . . . They do that with the thirteenth floor in some hotels. It’s not as if there is no thirteenth floor—they just call it the fourteenth. It’s stupid, and no educated person should condone this practice. But can we not find a different designation for the answer of 2 + 2 without having to delete the number 4? We certainly can. If we counted in base 4, then we would have 2 + 2 ¼ 10, pronounced ‘one-zero’ and not ‘ten’. In class, you told us that mathematics is the combination of three fundamental ingredients. . . Yes. Numbers, logic, and the power of abstraction. I did say that, and I stand by it.
C. Constanda (*) The Charles W. Oliphant Professor of Mathematics, The University of Tulsa, Tulsa, OK, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Wonders (ed.) Math in the Time of Corona, Mathematics Online First Collections, https://doi.org/10.1007/16618_2020_11
C. Constanda
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You gave me a glimpse of the flexibility of mathematics when it comes to labelling numbers. Can we not also have some degree of flexibility where logic is concerned? Logic is defined as a system of principles governing correct and reliable inference, a particular method of reasoning or argumentation that can be applied to any branch of knowledge. Flexibility? In mathematics? You mean, applying a different set of rules than those we have always accepted to prove or disprove conjectures? Yes, in a certain sense. You just said that logic is a “particular method of reasoning”. If this ‘particular’ method is replaced by another ‘particular’ one that is, well, complete and free of self-contradiction, shouldn’t we accept its conclusions, even though they may appear to differ from what we hold to be established truth? I cannot say unless I am given the opportunity to examine such a different system. Do you happen to have one in mind? I do, and it’s very simple to explain. Let’s hear it, then. In the current grading scheme, a final average of 90 is awarded an A, is it not? It is. But what would you do if someone
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