Here is Byrn, 1875, writing about French practice:
"For cooling and diluting the substances in course of maceration [mashing], we employ clear spent wash, (clear part of slops), the residuum of beer, the water of breweries, or, if none of these liquids are at our disposal, pure water.
Spent wash is never used unless in admixture with one-half, or at least one-third, of pure water.
There are two principal reasons why we prefer the spent wash to the other liquors: first, because having absorbed the oxygen of the air it helps the fermentation; second, because it marks generally several degrees of the densimeter, which shows that it still contains a certain quantity of sugar, which is thus put to account. Experience also proves that the starch suspended in the spent wash helps the fermentation."
Maybe we are all right here: using backset in 1818 in rough pioneer-like conditions assisted a natural fermentation because it had "absorbed the oxygen", i.e., had submitted to the effect of wild airborne yeasts which were working on the residual sugar in the liquid.
Oxygen is not (I believe) needed for fermentation. This work was written before yeast and its properties were fully understood. It is the absorbtion and action of wild yeast which perhaps "re-yeasted" a wash in which the original yeast would have been rendered ineffective through heating to 172 F. to vaporise the alcohol (and if the wash was very strong to start with, say, 8-10% abv., I doubt very much original yeast would survive in there even before boiling). Now, today, the backset used would be industrially monitored and processed to ensure it was not affected by wild yeasts (in fact I believe modern mashes are sterilized before the yeast is put in - Mike, maybe that's why those prewar whiskeys taste so good, did they do that before 1939? I doubt it!).
Clearly, today, there is something, apart from being a source of recyclable water and some additional fermentables, motivating distillers to use this substance in the successive mash. Along with Chuck and Bobby I always thought the reason was related to beneficial effects on the acidity level. Byrn has another passage in which he talks about desirable acidity in sour mashing, I can't find it right now but will post the wording when I do.
Again, just to plug the book, it can be bought ("The Complete Practical Distiller" by M. La Fayette Byrn, M.D., published Philadelphia, 1875) at www.raudins.com).