Kindly see the embolded sentence below.

      To write Cauchy’s definitions down precisely takes a bit more work. This was especially true for Cauchy himself, who had not quite phrased the ideas in their clean, modern form.* (In mathematics, you very seldom get the clearest account of an idea from the person who invented it.) [Emphasis mine] Cauchy was an unwavering conservative and a royalist, but in his mathematics he was proudly revolutionary and a scourge to academic authority. Once he understood how to do things without the dangerous infinitesimals, he unilaterally rewrote his syllabus at the École Polytechnique to reflect his new ideas. This enraged everyone around him: his mystified students, who had signed up for freshman calculus, not a seminar on cutting-edge pure mathematics; his colleagues, who felt that the engineering students at the École had no need for Cauchy’s level of rigor; and the administrators, whose commands to stick to the official course outline he completely ignored. The École imposed a new curriculum from above that emphasized the traditional infinitesimal approach to calculus, and placed note takers in Cauchy’s classroom to make sure he complied. Cauchy did not comply. Cauchy was not interested in the needs of engineers. Cauchy was interested in the truth.

*If you've ever taken a math course that uses epsilons and deltas, you've seen the descendants of Cauchy's formal definitions.

Ellenberg, How Not to Be Wrong (2014), p 49.

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2 Answers 2


The "clean, modern form" of Cauchy's ideas were arrived at after many generations of refinement. This process necessarily takes time: the ideas need to propagate throughout the research community, have to be understood, assimilated, and put to use. Then, simplifications, generalizations and variations of the idea are found, often by incorporating insights from fields other than the one where the idea originally sprung from. As this process keeps iterating, the original ideas get distilled thoroughly and we are left with their "clean, modern form".

This entire process is far too complex for any one person to complete before sending their idea out into the world. I am not even sure that this is somehow unique to mathematics. Look at what is considered standard undergraduate fare in any subject of your choice. At some point, these ideas were complicated, mysterious, or ground-breaking ideas in that field. But those ideas have now been distilled into something that we can explain to a teenager today and expect them to understand. The form and presentation of the idea will almost certainly be very different from how it looked when it was first propounded.


Inventing new things is difficult. Explaining your inventions to other people is difficult as well. In many cases, the way you think about your solution is quite different from how you explain it to other people so that they can understand. Sometimes, it requires a significant amount of effort and ingenuity to come up with the "right" way to explain it. Do you have any experience writing a computer code? If you do, try to explain it to somebody else in words. See how easy it is to give a clear explanation. If you never wrote a code, ask a friend who did to explain their code to you.


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