If you assume their existence, it works spectacularly well:
The idea of infinitesimals was immediately disliked by the Christian clergy:https://en.m.wikipedia.org/wiki/Infinitesimal
When Newton and Leibniz invented the calculus, they made use of infinitesimals, Newton's fluxions and Leibniz' differential.
Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst (1734).
A fact is, however, more important than the Lord Mayor of London:ChatGPT
Key Points of Berkeley’s Critique on Infinitesimals:
1. Metaphysical Objection:
Berkeley rejected the notion of infinitesimals as entities that are neither zero nor finite quantities, referring to them as "the ghosts of departed quantities." He argued that such entities had no clear ontological status—they could not be coherently conceived or observed.
2. Logical Inconsistency:
He criticized the way mathematicians would assume infinitesimals to be non-zero for the sake of calculation, but then discard them as zero to arrive at a final result. Berkeley saw this as a fallacy—an inconsistent application of reasoning.
3. Epistemological Challenge:
Berkeley, as an empiricist, believed knowledge must be grounded in sense experience. Since infinitesimals cannot be perceived or measured, he questioned their epistemological validity.
4. Religious and Philosophical Motivation:
Part of Berkeley’s aim was also to challenge the perceived arrogance of mathematicians and scientists who criticized religious mysteries while themselves relying on what he saw as mathematically mystical concepts like infinitesimals.
Given the fact that there was indeed no rigorous definition for the notion of infinitesimal, their axiomatization was often rejected:Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results.
It took until the 1960s to finally develop a usable definition for infinitesimals:Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts.
Bishop Berkeley was wrong.The most widespread technique for handling infinitesimals is the hyperreals, developed by Abraham Robinson in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the transfer principle, proved by Jerzy Łoś in 1955.
Mathematics is not observed with the senses and is not about the physical universe. It is about the abstract Platonic world. Alternatively, it is just symbol manipulation. No mathematical object can be observed in physical reality. At best, we can see their Platonic shadows. Most mathematical objects cannot even produce such shadows.
Hence, axiomatic knowledge is not grounded in sense experience. On the contrary, pure reason is blind and deaf. Otherwise, it is not pure.
Last but not least, mathematicians do not criticize religious mysteries for being mystical. Mathematics is itself indeed often mystical. The criticism is about Christian religious mysteries being inconsistent, contradictory, and not closed under logical consequence. Christianity and logic are like water and fire.