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I've been revisiting calculus for the fun of it and trying to "reconstruct" over history to understand what forces led to the creation of various topics. The reason for rigorously defining limits is still alluding me.

The story of limits has its genesis with Newton's fluxions and Leibniz's infinitesimals. I understand the criticism of "ghosts of departed quantities" and how it was mostly hand waving (considering it zero in one place but feeling free to divide by it elsewhere and assuming otherwise). We got results but the math process seemed absurd.

However, if I understand correctly, Cauchy did give a definition albeit not as "precise" as the $\epsilon-\delta$ given in the modern form by Bolanzo and Weierstrass, but a definition nevertheless which may have worked (or would it not?)

My question is thus this: Why was this necessary? What do we lose if don't have such a definition?

A few counter questions that I'm mulling about:

  • Does having this definition really change anything in the world of physics/applied math? If so, any examples?
  • What changes in the world of pure math? Do we just get a definition because we like to precisely define things in the mathematical world? Is that it?
  • Or was it more to "break away" from the geometric intuition of continuity and define it from a more algebraic/arithmetic POV to finally break away from the shackles of geometry and bring calculus on a more analytical footing and thus real analysis too?
  • Any other rationale?
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  • $\begingroup$ Mathematicians do not only "like" to define things precisely. Logic in math is 100% precise, 100% of the time. $\endgroup$ – Matt Samuel Nov 12 '18 at 23:50
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    $\begingroup$ My guess (as somebody who's not a mathematical historian) would be that they were worried about making too many axiomatic assumptions and ending up with an inconsistent system, where two methods for example might give two contradictory answers. With a precise definition that seems to match "intuition" about limits, the theory of limits becomes as consistent (or inconsistent) as the foundation this precise definition is built on. $\endgroup$ – Daniel Schepler Nov 13 '18 at 0:34
  • $\begingroup$ In general it is necessary to define a thing (ie give a precise unambiguous meaning) before using it. Calculus was and is a unique subject which manages to achieve too much without proper definitions and rigor. $\endgroup$ – Paramanand Singh Nov 13 '18 at 4:02
  • $\begingroup$ You can see e.g. The True History of the Concepts of Limit and Shadow (AMS, 2014). $\endgroup$ – Mauro ALLEGRANZA Nov 14 '18 at 15:07
  • $\begingroup$ First step: avoid the (at that time) unsatisfactory notion of infinitesimals. $\endgroup$ – Mauro ALLEGRANZA Nov 14 '18 at 15:10

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