This question on MathOverflow motivates me to ask what the reasoning is behind the conjecture that there are only finitely many biplanes. More generally, it has been conjectured that for fixed $\lambda>1$ there are only finitely many triples $(v,k,\lambda)$ such that a symmetric balanced incomplete block design with these parameters exists.
My understanding is that prior to the proof that there is no projective plane of order 10, there was some hope that the Bruck-Ryser-Chowla condition would prove to be sufficient for the existence of a design. Yet by that time the finitely-many-biplanes conjecture was already in wide circulation. Since there are infinitely many triples $(v,k,2)$ that satisfy the Bruck-Ryser-Chowla condition, it appears that the design-theory community was able to entertain two conjectures that are not merely incompatible, but starkly opposed.
Is anyone aware of any of the history behind this?
Elaboration (added later): If we think about families of symmetric designs, with $v$ regarded as a parameter and $\lambda$ some function of $v$ (which would make $k$ a function of $v$ as well), then we can talk about "large-$\lambda$" designs, where $\lim_{v\to\infty}\frac{\lambda}{v}=\frac{1}{4}$, and "small-$\lambda$" designs, where $\lim_{v\to\infty}\frac{\lambda}{v}=0$. (Of course there are other possibilities.) Constant $\lambda$, of course, fits into the "small-$\lambda$" case.
There is considerable optimism about "large-$\lambda$" designs. The Hadamard conjecture, for example, would imply that a $(4n-1, 2n-1, n-1)$ exists for every $n$. I think people have conjectured something similar for Menon designs (parameters $(4u^2, 2u^2-u, u^2-u)$), and that most people would make the same conjecture for designs with parameters $(2a^2+2a+1,a^2,a(a-1)/2)$, for which an infinite family is known.
Since my experience is mostly with the "large-$\lambda$" case, I'm curious about the reasons for the extreme pessimism in the "small-$\lambda$" case.
I do not think there was ever much hope that Bruck-Ryser-Chowla would be a necessary and sufficient condition for the existence of a symmetric design in general. For projective planes, everything is more strongly constrained, and it was possible to be more optimistic. So it would have been quite reasonable to believe that BRC was sufficient for projective planes, but not for biplanes.
Also it is possible (likely?) that there conjectures were made more in the spirit of "For all I know the following could be true", rather than from a firm conviction based on evidence and deep insight.