Later researchers like Ronald Fisher or Hermann Muller had no trouble understanding the maintenance of heritable variation, because mutations had been discovered to be the ultimate source of such variation. They, on the contrary, saw sexual recombination as the problem that needs an explanation and proposed that it accelerates evolution. The difference was that Weismann needed sex for his evolutionary theory to work in the first place, whereas Fisher and Muller had no doubt that natural selection would also work in asexual organisms. They only proposed that sex accelerates evolution in comparison with asexual reproduction.
Although it is common to state that Weismann was the first to propose a theory for the evolutionary significance of sex, its ubiquity or even maintenance (e.g., Hamilton et al. 1981; Hamilton 1986, 1991 reprinted in Hamilton 2001, pp. 27+300+777; Joshi and Moody 1998; Hamilton 2001, p. 622; Burt 2000; Goddard et al. 2005; Otto 2008; Roze 2012), Weismann's problem was simply different from Muller's, Williams's, Maynard Smith's of whomever else is ceremoniously lumped together with Weismann in one string of parenthetical references. Lumping all them into one citation as if they were all struggling with the same issue is a feat of potting history leading to a Whiggish misrepresentation of Weismann.
The eldest of the above given sources that do lump Weismann with later researchers in this Whiggish way reads:
“Actually this type of ‘reassortment theory’ of sex began long before the 1960s, indeed can be dated back to Weismann in the last century (see ref. 12)” Hamilton et al. (1981) reprinted in Hamilton (2001, p. 27)This is wrong, because Weismann's theory was not that sexual reproduction merely reassorted already existing variation, but that it was the ultimate source of heritable variation. In retrospective, we can say that Weismann proposed sexual reproduction to achieve what we now know to be due to mutation. That reference 12 mentioned by Hamilton is a publication by V. Thompson that has been published in the journal Evolutionary Theory.
Weismann had a problem with the maintenance of heritable variation
Thompson (1976) starts with an accurate description of Weismann’s early context and view (the following quote contains a second quote).
Early in his evolutionary studies Weismann concluded that external factors must have no influence on the hereditary disposition of organisms, that acquired characteristics are not inherited. He championed this idea so forcefully that his name became synonymous with anti-Lamarckism in the popular lore of science. Weismann’s strong anti-Lamarckian stand raised anew a major question that Darwin never adequately resolved, the source of hereditary variation. “How,” Weismann asked, “can deep-seated hereditary characters arise at all, if they are not produced by the external influences to which the individual is exposed?” The source of hereditary variation, he answered:Anybody who took the time to merely read the preface of Weismann's essay could already have gathered as much from it.
“is to be looked for in the form of reproduction by which the great majority of existing organisms are propagated: viz., in sexual reproduction, or, as Häckel [sic] calls it amphigonic reproduction … in amphigonic reproduction two groups of hereditary tendencies are as it were combined. I regard this combination as the cause of hereditary individual characters, and I believe that the production of such characters is the true significance of amphigonic reproduction. The object of this process is to create those individual differences which form the material out of which natural selection produces new species” (Weismann, 1891, p. 279)
The transmission or non-transmission of acquired characters must be of the highest importance for a theory of heredity, and therefore for the true appreciation of the causes which lead to the transformation of species. Any one who believes, as I do, that acquired characters are not transmitted, will be compelled to assume that the process of natural selection has had far larger share in the transformation of species than has been as yet accorded to it; for if such characters are not transmitted, the modifying influence of external circumstances in many cases remains restricted to the individual, and cannot have any part in producing transformation. We shall also be compelled to abandon the ideas as to the origin of individual variability which have been hitherto accepted, and shall be obliged to look for a new source of this phenomenon, upon which the process of selection entirely depend [sic].The German original, the translation of which Thompson quoted as Weismann 1891, is actually from a lecture published in 1886. At this time, Weismann had a problem with the maintenance of heritable variation in adaptive traits after having rejected the inheritance of acquired characteristics. Inheritance of acquired modifications seems to have been thought of as a necessary source of heritable variation. In defending his claim that germ-line and soma were separated, Weismann used sexual reproduction as a stopgap – he took it as the ultimate source of variation. Today we know that mutation is that ultimate source of variation, which is only recombined through sexual reproduction. A few pages down, Weismann even claims that asexual populations cannot adapt through natural selection because they lack this source of variation.
In the following pages I have attempted to suggest such a source.
A. W. Freiburg i. Br., November 22, 1885
(Weismann 1891. "Significance of sexual reproduction." In: Essays upon Heredity and kindred biological problems, vol. 1, p. 259)
“[...] in other words, natural selection, in the true meaning of the term, viz. a process which could produce new characters in the manner above described, is an impossibility in a species propagated by asexual reproduction. If it could be shown that a purely parthenogenetic species had become transformed into a new one, such an observation would prove the existence of some force of transformation other than selective processes, for the new species could not have been produced by these latter.” Weismann (1891, p. 282, first published in German in 1886)This conclusion is of course wrong, because the premise was wrong in the first place. Asexual reproduction also has an ultimate source of variation, that is, mutation. Nevertheless, the last quote makes it quite clear that Weismann was not thinking of sexual reproduction in terms of recombining existing variants but in terms of genuinely new variation in 1886.
In 1891, Weismann extended the realm for his law against the inheritance of acquired characteristics to include unicellular eukaryotes. Beforehand he allowed for the inheritance of acquired characteristics in unicellular organisms reasoning that an organism consisting of only one cell that divides, changes to that cell must be inherited by the descendant cells. In the wake of this extension, he seized on conjugation and other forms of genetic exchange as further sources of variation, which he now subsumed under amphimixis. His own studies of parthenogenetic organisms might have changed his view about the impossibility of adaptation through natural selection in asexual species, however (I found it is hard to determine whether he still firmly regarded amphimixis as the ultimate and only source of heritable variation).
In 1902 (translated in 1904), in the midst of the rediscovery of Mendel’s laws and the discovery of mutations, Weismann took one step back from his previous claim about sexual reproduction and amphimixis as the ultimate source of variation:
“Thus amphimixis, together with the preparatory reduction of the ids, secures the constant recurrence of individual peculiarities through the ceaseless new combinations of individual characters already existing in the species. When sixteen years ago I first inquired into the actual and ultimate significance of sexual reproduction, I thought I had found it in this ceaseless production of new individualities. This seemed to me a sufficient reason for the introduction of amphimixis into nature, since the difference between individuals is the basis of the process of selection, which we may refer to natural or sexual selection. […] I still regard amphimixis as the means by which a continual new combination of variations is effected a process without which the evolution of this world of organisms so endlessly diverse in form and so inconceivably complex, could not have taken place.That is, he admitted that he was wrong to regard sexual reproduction as the ultimate source of variation. While this late evolution theory of Weismann was up to date in respect of mutation, he kept working with terms like ids and idants, which were probably dated at the time already. As an ironic plot-twist, just when Weismann corrected his views about sexual reproduction and amphimixis in such a way that they are still true today, despite the dated jargon, a large and vocal part of his colleagues, the mutationists, have taken the discovery of the ultimate source of heritable variation not as vindicating him but as refuting him and Darwin as well (Weismann hints at it in the preface). This was of course due to the fact that the mutations discovered were gross aberrations of individual morphology, whereas Weismann and Darwin always demanded small and gradual variation and selection.
But I do not regard this amphimixis as the real root of variation itself, for that must depend not on a mere exchange of ids, but rather upon a variation of the ids. […] Amphimixis, that is, the union of two germ-plasms, does not of itself cause variation of the determinants, it only arranges the ids (the ancestral in ever-new combinations.” Weismann (1904, p. 193ff, first published in German in 1902, my emphasis)
Fisher and Muller had a problem with recombination
For Fisher (1930) and Muller (1932) sexual recombination was the problem in need of an adaptive explanation. They both state that natural selection will readily occur in asexual species because they knew mutation to be the ultimate source of genetic variation (Fisher 1930[1999, p. 121]; Muller 1932, p. 118).
Williams and Maynard Smith had a problem with asexual mutants
Likewise, Fisher and Muller did not see the problem of sexual reproduction through the eyes of theorists after the late 1960s. For example, Williams's problem was the cost of meiosis (halving relatedness, which he later re-formulated as the cost of gamete cooperation), whereas Maynard Smith's and Hamilton's problem was the cost of lazy males (for the sociobiological roots of the current problem with sexual reproduction see here, here, and here). In both cases, an asexual or parthenogenetic mutant should get an immediate twofold advantage given even sex allocation.
The retrospective distortion
Nevertheless, Thompson concluded: “The reassortment theories date directly back to Weismann” (Thompson 1976, p. 135). Surely, he cannot have meant to say that Weismann struggled with the same problems as Muller, Williams, or Hamilton. Nevertheless, Hamilton et al. (1981) picked that up and Hamilton later wrote of the “Weismann-Muller-Fisher theory” (see 1986, 1991 reprinted in Hamilton 2001, p. 300, 777; Hamilton 2001, p. 622) as though they were of one piece.
[Update] Roze (2012) even includes Morgan in this contracted, abridged and potted history:
"The oldest hypothesis on the evolutionary significance of sex was formulated by Weismann in 1889 (earlier edition of ref. 1891 below) and elaborated during the first part of the 20th century by Morgan, Fisher, and Muller: according to this hypothesis, sex is beneficial because it increases genetic variation, allowing faster rates of adaptation by combining different beneficial mutations into the same genome."Wrong again. Weismann did not see sexual reproduction as a problematic trait in need of an adaptive explanation, but as a solution to a fundamental problem of the theory of natural selection—the ultimate source of variability to be plugged into the hole left by uprooting the Lamarckian source. [/update]
[Update] For an excellent exception and good review of the historical development of the evolutionary significance of sex see: Meirmans S (2009) The evolution of the problem of sex. In: Schön I, Martens K, van Dijk P (eds) Lost sex. Springer Science.[/update]