The exact mathematics
Mathematics is an important tool for all of science. Mathematics is exact since its numbers, figures and calculation rules are precisely defined and internally consistent constructs of the mind.
Is it true that 9 + 16 = 25? Yes, unquestionably. The calculation is exact; experience shows that it always ”works” if the abstract numbers are replaced with physical objects like stones or apples.
Scientists use mathematics. Their calculations are exact, but that is not usually true for the measurements that precede the calculations. Only numerical counts (of stones, apples…) are exact. The height of Uppsala Cathedral is said to be 118.7 metres, but the exact height is unmeasurable, with an infinite number of decimal places. Perhaps there is no exact height?
Geometrical figures such as the perfectly round circle, the perfectly straight line and the perfect 90 degree angle exist only in our minds. We strive to approach them in our drawings or in the manufacturing of objects, but fail because our measurements inevitably introduce errors. Then, when we make our calculations, we turn again to the perfect figures in our minds. We aspire to precision!