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Dorothy Stein, a biographer of Ada Lovelace, was pretty blunt in her assessment: Lovelace was a mediocre mathematician, for example see here.

I wonder if she's fair to her. The fact that Lovelace translated a printer's error "cos" (like cosine) literally as "cos" (instead of recognizing the French "cas" was meant, translating it as "case") could be just some careless mistake.

More serious is her trouble with functional equations, like the following exercise (English and notation slightly modernized):

Show that the equation $\phi(x+y) + \phi(x-y) = 2\phi(x)\,\phi(y)$ is satisfied by $\phi(x)=(a^x + a^{-x})/2$ for every value of $a$

So I guess that would go like this: $$\begin{align}2\phi(x)\,\phi(y) &= (a^x + a^{-x})(a^y + a^{-y})/2 \\&= (a^{x+y}+a^{x-y} + a^{-x+y} + a^{-x-y})/2 \\&= (a^{x+y}+ a^{-(x+y)} + a^{x-y} + a^{-(x-y)} )/2 \\&= \phi(x+y) +\phi(x-y)\,.\end{align}$$

Lovelace struggled with the problem and wrote:

I do not know when I have been so tantalized by anything, & should be ashamed to say how much time I have spent upon it, in vain.

Now it could be that as a beginner's mistake she misread the question, trying also to prove that this was the only solution - which would be a much, much harder mathematical problem.

These are just two examples of the controversy.

So I wonder if we can make a somewhat objective judgment about her talent for mathematics?

Wouldn't it be surprising if the evidence is compatible with both extremes:

  • Ada Lovelace simply wasn't very good at math; other factors like her heritage and social connections allowed her to take the position as Charles Babbage's assistant
  • she was a mathematical genius?
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    $\begingroup$ We can line up the facts, as Stein apparently does, but isn't post mortem "assessment of talent" inherently subjective? To some, Ptolemy was a genius, to others he was a fraud, the facts are not in dispute. Outside the aptitude tests, "talent" is a nebulous concept, so it is hardly surprising. If someone does something deemed significant, that is, naturally, because they had the "talent". It's a perfect "explanation". Lovelace was a genius assessments are not hard to find. $\endgroup$ – Conifold Mar 17 '19 at 9:30
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As mentioned in the comment by Confiold, it is very hard and subjective to assess a talent for anything.

It may be true that Ada Lovelace did some basics mistakes and she had some typos or misuderstanding in her translations, however, this does not mean that she had no talent for mathematics. Very often people with gift for science and technology are poor in languages and often they do not see solution of things obvious for normal people.

However, I do not want to dive much deep into speculation and more or less philosophy. What is fact that Ada Lovelace was visionary and she saw applicaiton of technology yet to be discovered and designed in her time.

I would recommed to read her notes to article Sketch of analytical engine invented by Charles Babbage authored by Luis Menabrea. Interestingly her notes are longer than an original article itself. The notes contain what is considered to be the first computer algorithm, in particular a method how Bernoulli numbers can be computed on Babbage's analytical engine. Interestingly, a physical implementation (or programming) of the algorithm on the analytical engine is very similar to programming on early electronic computers in 1950's (especially how variables are stored in registers) but there is more than one century gap in time. Additionaly architecture of analytical engine is very similar to von Neumann scheme - there are I/O devices, memory, controler and mill (i.e. arithmetical - logical unit in modern terms). Ada also talks about controlling card and number cards - instructions and data in modern terms.

Overall, Ada Lovelace and Charles Babbage paved a way to modern computing (Ada to software and Charles to hardware) about one century before electronic computers come to existence. I think that it is not important wheter they were geniuses but their contribution to knowledge of humankind should be taken into account.

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