The theory of infinitesimals is intimately connected to how analysis (differential and integral calculus) was first formulated. Leibniz and Newton understood that you could approximate instant rates of change, or areas under a curve, by taking smaller and smaller "slices" of a function, but they did not yet have the rigorously formalized notion of limits that modern analysis depends on. So they developed an arithmetic of infinitesimals, numbers greater than zero but less than any positive real number¹, with some rather ad-hoc properties to make them work out in calculations.
Philosophical problems surrounding the perplexing concept of infinity were already hotly debated by the ancient Greeks. Aristotle made an ontological distinction between actual and potential infinities, and argued that actual infinity cannot exist but potential infinity can. This was also the consensus position of later scholars, and became a sticking point in the acceptance of calculus because infinitesimals (and infinite sums of them) were an example of the ontologically questionable actual infinities.
As I mentioned before, standard modern analysis is based on limits, not infinitesimals, and requires no extension of real numbers. Indeed the limit definition of calculus only requires the concept of potential infinities, so philosophers should be able to rest easy! But infinitesimals still occur in our notation which is largely inherited from Leibniz, however. We say that the derivative of y(x) is dy/dx, or the antiderivative of y(x) is ∫ y(x) dx, and while acknowledging that dy and dx are not actual mathematical objects, just syntax, we still do arithmetic on them whenever it's convenient to do so! For example, when we make a change of variables in an integral, we can substitute x = f(t) for some f, and then say dx/dt = f'(t) and "multiply by dt" to get dx = f'(t) dt to figure out what we should put in the place of the "dx" in the integral.
Actual infinitesimal numbers are not dead, either, they're used in a branch of analysis called nonstandard analysis which formalizes them in the logically rigorous manner that is now expected from mathematics.
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¹ Not that they had a rigorous theory of real numbers, either, that came in the 19th and early 20th century. In fact what we now understand as formal, axiomatized math didn't really exist before the 19th century at all!
Philosophical problems surrounding the perplexing concept of infinity were already hotly debated by the ancient Greeks. Aristotle made an ontological distinction between actual and potential infinities, and argued that actual infinity cannot exist but potential infinity can. This was also the consensus position of later scholars, and became a sticking point in the acceptance of calculus because infinitesimals (and infinite sums of them) were an example of the ontologically questionable actual infinities.
As I mentioned before, standard modern analysis is based on limits, not infinitesimals, and requires no extension of real numbers. Indeed the limit definition of calculus only requires the concept of potential infinities, so philosophers should be able to rest easy! But infinitesimals still occur in our notation which is largely inherited from Leibniz, however. We say that the derivative of y(x) is dy/dx, or the antiderivative of y(x) is ∫ y(x) dx, and while acknowledging that dy and dx are not actual mathematical objects, just syntax, we still do arithmetic on them whenever it's convenient to do so! For example, when we make a change of variables in an integral, we can substitute x = f(t) for some f, and then say dx/dt = f'(t) and "multiply by dt" to get dx = f'(t) dt to figure out what we should put in the place of the "dx" in the integral.
Actual infinitesimal numbers are not dead, either, they're used in a branch of analysis called nonstandard analysis which formalizes them in the logically rigorous manner that is now expected from mathematics.
________
¹ Not that they had a rigorous theory of real numbers, either, that came in the 19th and early 20th century. In fact what we now understand as formal, axiomatized math didn't really exist before the 19th century at all!