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GENERAL_RUSSELL

                                             September 01, 2020

Of late I've been puzzling over the impulse of
some mathematicians to contrive "foundations"
for mathematics, which is to say to re-derive
things we already know from "fundamental principles",
which are *other* things we already know--

    Or perhaps are things we would like to
    believe we should've already known but may
    have only just adopted in order to have
    something we can call "fundamental".

    Or something like that.



Bertrand Russell tries to explain the
approach in a few places, and typically
places great emphasis on the idea of
"generality":

So the idea is to reduce the number of
things you need to assume to have a system
that can apply to the widest number of
cases?

"Introduction to the Philosophy of Mathematics" (1919)
by Bertrand Russell:

    "... instead of asking what can be defined
    and deduced from what is assumed to begin
    with, we ask instead what more general
    ideas and principles can be found, in
    terms of which what was our starting-point
    can be defined or deduced."

But then, he also seems to be conceeding that these
"starting-points" can be rather murky, perhaps more
obscure than the math that's supposedly going to be
derived from them:

    "Just as the easiest bodies to see are those
    that are neither very near nor very far,
    neither very small nor very great, so the
    easiest conceptions to grasp are those that are
    neither very complex nor very simple (using
    'simple' in a logical sense)."


    "... take us backward to the logical foundations of
    the things that we are inclined to take for granted
    in mathematics."

                                                          
And the more I look into this sort of thing it seems like       
there's something very upside-down and inside-out about         
it all.  If only we had some actual *logical foundations*        FOUND_MATH
we could be reassured that our math really is working the       
way we think it does.  But these logical foundations are            MATHISM
going to be judged by whether they're capable of getting        
to "2+2=4"-- if you've got "logical foundations" that           
can't get to arithmetic, we're not going to abandon the         
arithmetic, are we?                                             
                                                                



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