Abstract
Two questions about certainty in mathematics are asked. First, is mathematical knowledge known with certainty? Second, why is the belief in the certainty of mathematical knowledge so widespread and where does it come from? This question is little addressed in the literature. In explaining the reasons for these beliefs, both cultural-historical and individual psychological factors are identified. The cultural development of mathematics contributes four factors: (1) the invariance and conservation of number and the reliability of calculation; (2) the emergence of numbers as abstract entities with apparently independent existence; (3) the emergence of proof with its goal of convincing readers of certainty of mathematical results; (4) the engulfment of historical contradictions and uncertainties and their incorporation into the mathematical narrative of certainty. Individual learners of mathematics internalize ideas of invariance, reliability and certainty through their classroom experiences and exposure to such cultural factors. Lastly, with regard to the first question, it is concluded that mathematics can be known with a certainty circumscribed by the limits of human knowing.
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Notes
I am discussing mathematics as a written discipline, for we know little about the oral mathematics that preceded it.
Some developments in mathematics do not assume conservation, such as Cantor’s (1999) theory of transfinite set theory (ℵ0 = ℵ0 + ℵ0) and Boolean algebra (1 + 1 = 1).
These assumptions are an application of the Sapir-Whorf hypothesis, namely that our language “cuts up” the world into the way we conceptualize it. “We dissect nature along lines laid down by our native language.” Whorf (1956) p. 212.
Number mysticism survives to this day, for example in hotels having no floor labelled 13.
Learners have to overcome unavoidable “epistemological obstacles” (Bachelard, 1938) in learning of mathematics, for example, when numbers are expanded from the natural numbers (with a least number) to the integers (no least number).
Previously, I rejected the claim that mathematical knowledge is objective and superhuman and can be known with absolute certainty (Ernest, 1991, 1998). By redefining objectivity and certainty as culturally circumscribed by the limits of human knowing, I am happy to acknowledge the certainty of mathematical knowledge.
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Ernest, P. The problem of certainty in mathematics. Educ Stud Math 92, 379–393 (2016). https://doi.org/10.1007/s10649-015-9651-x
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DOI: https://doi.org/10.1007/s10649-015-9651-x