I really wrote wrong what Marx's claim was; wrote this too early in the morning. "using limits instead of dividing by 0 is ridiculous and immaterial" is what his claim was. Really about how 0/0 could be a better basis than utilizing limits to get to a true basis. His claim was exactly what you say that nobody did; that 0/0 is meaningful and should be understood as the material/real basis of change or so
dividing by zero doesn't make sense arithmetically - in arithmetic, division is about cutting things into parts, so you can make no cuts and leave the singular whole, or you can make some number of cuts and divide the whole. algebraically, division by zero has no sensible result as it can produce any value between 0 and infinity (or negative infinity if we're approaching zero from the left). that is, it's not defined - to define division by zero as a function, we'd need to be able to assign it a particular value for any given input. if we abstract from numbers a little bit, and look at the structure that encapsulates basic arithmetic - rings - we find that the additive identity, usually denoted as 0 for the ring, never has a multiplicative inverse. the ill-definedness of division by zero is so fundamental that limits were introduced specifically so we could divide by zero rigorously - limits allow us to precisely choose a specific value out of the continuum that we expect a particular instance of zero division to evaluate to.
I really wrote wrong what Marx's claim was; wrote this too early in the morning. "using limits instead of dividing by 0 is ridiculous and immaterial" is what his claim was. Really about how 0/0 could be a better basis than utilizing limits to get to a true basis. His claim was exactly what you say that nobody did; that 0/0 is meaningful and should be understood as the material/real basis of change or so
dividing by zero doesn't make sense arithmetically - in arithmetic, division is about cutting things into parts, so you can make no cuts and leave the singular whole, or you can make some number of cuts and divide the whole. algebraically, division by zero has no sensible result as it can produce any value between 0 and infinity (or negative infinity if we're approaching zero from the left). that is, it's not defined - to define division by zero as a function, we'd need to be able to assign it a particular value for any given input. if we abstract from numbers a little bit, and look at the structure that encapsulates basic arithmetic - rings - we find that the additive identity, usually denoted as 0 for the ring, never has a multiplicative inverse. the ill-definedness of division by zero is so fundamental that limits were introduced specifically so we could divide by zero rigorously - limits allow us to precisely choose a specific value out of the continuum that we expect a particular instance of zero division to evaluate to.