The Logic of Infinitesimals
Students and fans of mathematics are aware that infinitesimals have a storied history. Let's rattle off the pop version: until the second half of the 19th century, mathematicians used the notion of infinitesimal, but it was vague and not rigorous, just based on intuitions. It sort of worked though. Leibniz's differentials and Newton's fluxions got them somewhere. Then, Cauchy, Weierstrass and a bunch of others managed to remove these vague infinitesimals, instead founding infinitesimal analysis on $\epsilon$-$\delta$ definitions. Often to this is added that Robinson's non-standard analysis gives a rigorous basis for infinitesimals.
This potted history is probably broadly right. But it leaves intact the utterly simple thorn in the side of mathematics that led to all this torture in the first place, a torture that does not thereby end with Cauchy and Robinson (seriously, who is satisfied by $\epsilon$-$\delta$, and who amongst those that pay lip service to Robinson really understand his ideas?).
That thorn is the law of the excluded middle as a global axiom. If we simply recuse ourselves of the capacity to assert that every quantity is either equal to zero, or not equal to zero, we are no longer barred from thinking infinitesimals. That is not to say that there are then some magic infinitesimal quantities that are somehow in a third state that is neither equal to zero, nor different from zero. Infinitesimals are not quantities: zero is the only global element of an infinitesimal object. Infinitesimals work at the level of logic: their presence stops us from inferring that something that does not differ from zero is thereby zero.
Why would we recuse ourselves of this capacity? Yes, on the one hand, because assuming fewer axioms makes our results more general: the usual constructivist refrain. I do not think this is compelling to most classical mathematicians. What is more compelling perhaps, and rather immediate, is that by freeing ourselves of this assumption, we can properly talk of infinitesimals. And mathematics must talk of infinitesimals. Choose excluded middle, resign yourself to the misery of a fractured continuum.
In modern times, this has been brought into sharp focus by Kock and Lawvere's synthetic differential geometry. But, as Lawvere makes clear in his essay on Euler's continuum, this is something of a vindication of how mathematicians used to think outside the injunction to disjunct. The moves they made make sense in a logic more parsimonious than that of today's mainstream mathematics.