Infinity and foundations · Perspectives
Was calculus knowledge before it was made rigorous?
Newton and Leibniz used calculus to get correct answers in the 1660s to 1680s, but its foundations were not made rigorous until the nineteenth century. This question asks whether a method that works, but cannot yet be fully justified, counts as knowledge.
Claims
- Calculus produced consistent, correct and testable results for more than a century, so mathematicians knew those results even without rigorous foundations.
- Rigour is one standard of justification among several; reliability and agreement with physics were strong justification too.
Counterclaims
- Without rigour, mathematicians also produced wrong results from the same methods, and could not always tell which were which, so their knowledge was incomplete.
- The demand for proof is what separates mathematics from other areas; by its own standard, early calculus was well-founded belief rather than knowledge.
Real-life situations from mathematics
Berkeley's criticism
In The Analyst (1734) the philosopher George Berkeley mocked the infinitely small quantities of calculus as quantities that were neither something nor nothing, and argued that mathematicians accepted on faith what they would criticise in religion.
Euler's daring series
Leonhard Euler worked freely with infinite series, sometimes assigning sums to divergent ones. Many of his results were correct and some were later justified rigorously; others needed careful reinterpretation.
Check dates and figures in a reliable source before you use them, and cite that source.
Use this in your TOK work
Essay. Fits titles about justification, about how knowledge changes, or whether usefulness is enough.
Exhibition. A school calculus textbook alongside a definition of a limit could support a prompt about how knowledge is justified.
Link it to the prescribed title or the exhibition prompt you are working on, in your own words. See using maths examples in your essay and choosing exhibition objects.
Themes and study heading
Knowledge and the knower Knowledge and religion Perspectives
Different views of what mathematics is and whose it is.