Clyde Greeno says: >Humbug! Russell so went way out on the limb of "pure formalism", well before the essence of mathematics was discerned. "Pure" mathematics entails descriptions of KINDS of things that are being attended, together with whatever logically substantial concepts and conclusions are derived from those descriptions.
Russell was simply trying to rescue what Frege had done (which had then been undone the by Russell.) And if you can throw out actual infinities, then it kind of works. It is a deeply involved (convoluted) way to look at "numbers", but it was done for technical reasons, which most people today need never concern themselves with if they do not want to. But I don't think most people ever need concern themselves with set theoretical constructions of numbers either, which is as "standard" an account as you are likely to get.
I think most people accept (counting) numbers as a kind of intuitive given*, that don't really need further "foundations" or justifications, except as yet another game to be played in the field of mathematics. There is no reason I can think of that children should be taught that the "number 3" is *really, really, really*, ....
The logicists, including Frege and Russell, were simply trying to find a way out of the challenges and paradoxes that the emerging set theory and new acceptance of actual infinities had thrust on the scene.
I see no need to get emotional about it - Russell didn't do bad things to young children. In fact, the distinctions and deeper understandings that arose from the whole "crises in foundations" period and its reverberations are numerous and amazing. Godel's incompleteness theorem being just one, and justly famous. But in fact, almost the whole of computability theory can be considered part of the reverberations. That Godel took particular interest in "Principia Mathematica" says a lot for is pedigree, I think.
I also think your claim that since then the "essence of mathematics discerned" is a fait accompli, all clear and luminous, whereas before all those old humbugs were just deeply in blackest error, is completely fatuous.
Cheers, Joe N
*There is some renewed interest in "how numbers arise" in our minds, a sort of neuro-cognitive approach like Dehaene, which, if you can't say its a continuation or even a reverberation, still would not be possible without what had gone before.