This old question attracted my attention again today, and I have a proposal for how to think about it. I suggest that we think about sentences $S$ with the property that at some time $t_0$ in the past, $S$ was regarded by at least some prominent mathematicians as being correct and proved, but that at some later time $t_1$, $S$ was generally regarded as being false. (For simplicity, I'm going to adopt the fiction that $S$ can be translated unambiguously between different natural languages, such as Greek, Latin, French, German, English, etc., although perhaps a more careful account would distinguish between seemingly "synonymous" sentences in different natural languages.)

As a further simplification, let's take $t_1$ to be the present day. Given the way we think about mathematics, if $S$ is something that we regard as making a clear mathematical assertion, then there are really only three ways this sort of thing can happen.

It is not entirely clear what $S$ meant at time $t_0$; perhaps it meant something slightly different from what we mean by $S$ today and the proof they thought they had was basically correct, or maybe $S$ meant the same back then as what we mean by $S$ today, and the old argument was incorrect or incomplete.

$S$ had a clear meaning at time $t_0$ but definitions have changed over time, and it does not mean the same thing today.

$S$ had a clear meaning at time $t_0$ and it is the same meaning we assign to $S$ today, but the old argument for $S$ was wrong.

It sounds like the OP means to exclude Case 3. I gave an example of Case 2 in another answer. Another example, almost "trivial" in some sense, is the sentence "1 is prime." Today we regard this sentence as false, but for example Lehmer's famous table of primes contained 1. We would quickly say that Lehmer was simply using a slightly different definition of the word "prime."

It would probably not be too difficult to dig up more examples of Case 2, but the OP's mention of Lakatos suggests that maybe examples of Case 1 are the most interesting ones. Most mathematicians reading Lakatos's account of Euler's formula probably see it as *not* as a demonstration that there do not exist final, incontrovertible mathematical theorems (as Lakatos seemed to want to argue), but as an example of people groping toward precise definitions and theorems (which are attainable if you're careful enough). In particular, the early stages of Lakatos's fictional dialogue can be regarded as dealing with somewhat unclear statements $S$, as in Case 1.

Analysis would seem to be a fruitful area to find examples of Case 1. Greg Graviton's answer proposes $S$ to be something like "the limit of a pointwise convergent sequence of continuous functions is continuous," with there being some scholarly debate about what "pointwise convergent" means if we include infinitesimals. In a comment, I suggested something like "Every function can be represented by a Fourier series." Divergent series could perhaps yield more examples, but nowadays when we see something like
$$1 + 2 + 3 + 4 + 5 + \cdots = -\frac{1}{12}$$
we are quickly able to infer the intended meaning and would not say that the assertion is false. (Perhaps there was a time $t_1$ in the past when there was a consensus that such assertions were false?) I'm thinking that the word "curve" might also be an example of a word whose meaning has changed over time. Was there a time in the past when people accepted that "a continuous curve cannot fill space"?

With the above clarifications, perhaps people can come up with more examples of Case 1.

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