What are examples of theorems which were once "valid", then became "invalid" as standard definitions shifted?

Question

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in later times, be interpreted as constituting a false claim, due to changing fashions as to how to standardly formalize some of the relevant concepts.

I imagine this sort of thing has happened often (e.g., with shifting accounts of "polyhedra" a la Lakatos' "Proofs and Refutations", or a motley of different definitions of "continuity" before standardization on the one we use now), but I do not have enough awareness of history to be able to provide solid examples (e.g., it seems plausible to me that Darboux may have considered himself to have proven that every derivative is continuous, taking the intermediate value property to be defining for continuity, but I do not know if this is an accurate account of what he claimed).

Answer 1

(This is basically a copy of my answer to "Widely accepted mathematical results that were later shown to be wrong?")

A prime example for a theorem that was considered "valid" but later became "invalid" is the following:

Theorem (Cauchy)
Let $S_m(x) = \sum_{n=0}^m f_n(x)$ be the partial sums of a series on the interval $a \leq x \leq b$. If

1. $S_m(x)$ is continuous for all finite $m$
2. and $S_m(\xi)$ converges to $S(\xi)$ for all numbers $\xi$ in the interval

then the sum $S(x)$ is also continuous.$\square$

From the modern (Weierstraß) point of view, this theorem is wrong. A well-known counterexample is the trigonometric series ("sawtooth") $$ \sum_{k=1}^{\infty} \frac{\sin(kx)}k $$ which is not continuous at $x=0$.

However, this is not a counterexample to Cauchy's theorem as Cauchy understood it. His definitions of continuity and convergence were based on infinitesimals and the series violates condition 2. The point is that $\xi$ may be an infinitesimal.

In particular, let $n=\mu$ infinitely large and $\xi = \omega := \frac1\mu$ infinitesimally small. Then, the residual sum is $$ S(\omega) - S_{\mu-1}(\omega) = \sum_{k=\mu}^{\infty} \frac{\sin(k\omega)}k = \sum_{k=\mu}^{\infty} \frac{\sin(k\omega)}{k\omega}\omega \approx \int_{\omega\mu}^{\infty} \frac{\sin t}{t} \ dt = \int_1^{\infty} \frac{\sin t}{t} \ dt $$

Clearly, the integral is finite and not negligible; hence, the series does not converge for $\xi=\omega\approx 0$.

Put differently, condition 2 in Cauchy's sense is actually equivalent to uniform convergence. (I think)

I have taken this discussion and example from Detlef Laugwitz's paper "Definite values of infinite sums: Aspects of the foundations of infinitesimal analysis around 1820" (in particular pages 211-212).

Answer 2

I am still not 100% sure I understand what the question is asking for, but it occurred to me today that one example might be the following statement:

$(*)$ If every S is P then some S is P.

Today, we would say that $(*)$ is not valid, because if S is vacuous then "every S is P" is true but "some S is P" is false. However, for most of the history of Western civilization, $(*)$ was considered valid. This is usually explained by saying that the "classical" statement that "every S is P" really means, in modern language, "there exists some S and every S is P." This point is discussed in detail in the article on The Traditional Square of Opposition in the Stanford Encyclopedia of Philosophy, where it is also pointed out that if we additionally translate "some S is not P" into modern language as "if there exists some S then some S is not P" then we can recover the entire traditional square of opposition.

This seems to meet Sridhar Ramesh's request for an example not of changing standards of rigor but of "shifting standard formalizations of preformal concepts" (in this case, the concepts of "every" and "some").

Answer 3

A fancier example is Kazhdan's proof of a conjecture of Langlands on conjugation of Shimura varieties. This appeared in the Budapest conference volume in the 1970s. Perhaps the sketchiness of the proof made comprehension of it non-trivial in the first place. In the year 1972 when I think the Budapest conference actually took place, it was not at all de rigeur to conform to Grothendieck (et al.)'s terminology and viewpoint, in part because that project had not been completed in all the aspects that might be convenient. Further, many people had grown up having to "improvise" an in-between viewpoint on algebraic geometry, especially rationality properties.

So, in short, perhaps that proof was correct in a certain context, became incorrect when the terminology was given extended sense under Grothendieck, but then became correct again when more things became known.

I believe Michael Harris and some collaborators may write up something about this sequence of events, or use it as an example.

Answer 4

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.

1. 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.

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

3. $S$ 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.