Leo, (01)
Thanks for the pointer to those talks. I started browsing through
the slides, and I also followed some of the pointers on that web site.
The following quotation is related to my comments about Common Logic.
It's not the same, but it's in the same ballpark. (02)
> Phase I seeks to reveal greater similarities between HOL and set theory
> than generally appreciated. Phase II explores four arguments that HOL
> collapses to first-order logic, i.e. that every higher-order entity
> defines a corresponding first-order entity. These arguments are generally
> ignored as they threaten to reintroduce the paradoxes. But we show that
> a properly circumscribed form of collapse is a valuable source of
> mathematical and semantic insight. (03)
Source: (04)
http://www.bbk.ac.uk/philosophy/our-research/ppp/description-of-research (05)
I'd also like to mention the conclusion of the following talk: (06)
http://www.bbk.ac.uk/philosophy/our-research/ppp/ConferenceAntonelli.pdf (07)
Note the following point: (08)
> Numbers... are no longer regarded as logical objects on this view.
> "Number" is not a logical notion (although cardinality might well be). (09)
This deflates Hilbert's hope of defining all of mathematics in terms
of logical principles alone. The author leaves some "wiggle room"
for the status of cardinality, but other authors would also say
that set theory (including cardinality) is part of mathematics,
not logic. (See below for the complete quotation in context.) (010)
That view is one I learned in a course at MIT on recursive function
theory taught by Hartley Rogers. In fact, Rogers claimed that
formal logic is a branch of *applied mathematics*, because it uses
mathematics to analyze the structure of something else (namely logic,
set theory, and the foundations of mathematics). That was long before
I had read anything by Peirce, but CSP would strongly agree. (011)
In any case, the lectures and tutorials provide a good overview
of many of related issues. (012)
John
_________________________________________________________________ (013)
DEFLATING THE ABSTRACTION MYSTIQUE (014)
According to the present view, abstraction principles are
extra-logical tools devised to accomplish two distinct,
but specifically mathematical, tasks: (015)
1 The selection of first-order representatives for second-order
equivalence classes. (016)
2 The imposition of cardinality constraints on the relative sizes
of the first-order and the second-order domain. (017)
The status of abstraction principles as extra-logical tools is
consistent with the failure of their logical invariance. Similarly,
any worries about the ontological status of the special objects
they deliver (i.e., abstracta) disappear, as anything — anything at
all — can play the role of these abstracta, as long as the choice
respects the equivalence relation. (018)
Numbers, as abstracta delivered by HP, are no longer regarded as
logical objects on this view. “Number” is not a logical notion
(although cardinality might well be). (019)
ALDO ANTONELLI, UC DAVIS, THE ABSTRACTION MYSTIQUE
DATE: AUGUST 2011 SLIDE: 26/26 (020)
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