On the sophistication of naïve empirical reasoning: factors influencing mathematicians' persuasion ratings of empirical arguments

Abstract

This paper presents the results of an experiment in which mathematicians were asked to rate how persuasive they found two empirical arguments. There were three key results from this study: (a) Participants judged an empirical argument as more persuasive if it verified that integers possessed an infrequent property than if it verified that integers lacked such a property. (b) Participants judged an empirical argument about modular congruence as more persuasive than an empirical argument about generating primes, suggesting that empirical arguments might be more convincing in some domains than others. (c) There was a marginally statistical effect between mathematical field of study and level of persuasion, with applied mathematicians finding empirical arguments more persuasive than pure mathematicians.

Keywords:

• argument evaluation
• empirical evidence
• mathematicians

Acknowledgements

I would like to thank Sean Larsen, Kristen Lew and Iuliana Radu for helpful comments on earlier drafts of this manuscript.

Notes

1. Throughout this paper, I use ‘conviction’ to refer to individuals’ confidence in a mathematical assertion and ‘persuasion’ to describe how effective an individual finds an argument in favour of an assertion. For an individual, how persuasive an argument is relates to its efficacy in increasing his or her level of conviction in the assertion it supports.

2. In the mathematics education literature, some authors do not distinguish between naïve empirical evidence and more nuanced uses of empirical evidence. However, if these researchers regard empirical evidence as insufficient to form conviction, it follows that they regard naïve empirical evidence insufficient for this purpose as well.

3. By quasi-empirical evidence, de Villiers (2004) was including naïve empirical evidence collected with the aid of computers (see p. 398).

4. Actually, this was the contrapositive of the claim that he read. One interpretation of why he chose to examine the contrapositive is that checking that perfect squares were not congruent to 3 modulo 4 provided more information than showing numbers congruent to 3 modulo 4 were not perfect squares, since it is more common for a number not to be a perfect square than not to be congruent to 3 modulo 4. However, an equally plausible interpretation is that checking the contrapositive was computationally easier.

5. In a trivial sense, one can argue that this statement is meaningless, since lacking a specific property can itself be defined as a property. (e.g., we can define ‘non-square’ as being a property of integers). To avoid this difficulty, I operationalise an attribute P of a subset of the natural numbers if for large n, where P(n) is the number of integers less than n of having attribute P.

6. Participants who evaluated this argument were given the option to leave a comment about it. No participant commented about the existence of odd superperfect numbers being an open question.

7. Again, no participant commented that they were aware that the sequence only produced 1s and primes. In fact, two participants claimed they believed no prime number generator had yet been discovered.

8. I have published several papers using ‘think aloud’ task-based interviews (Weber, 2001, 2008; Weber and Alcock, 2004) and open-ended interviews (Weber and Mejia Ramos, 2011) with small numbers of mathematicians.