Jeff Rubard
2010-01-31 22:40:18 UTC
Structure/Model: What Truth Is Out There?
A definition was recently requested for the concept of "mathematical
structure", and I suspect the magic of this term suggested a more
searching answer than that which was initially provided (a mapping of
mathematical entities onto their formal representations). This is
structure enow: but in point of historical fact difficulties with
this
term are to be had, and gave birth to the discipline of model theory
as
we know it today. As Abraham Robinson's text has it, the term
"structure" comes out of Tarski's post-*Wahrheitsbegriff* work,
specifically "Some Notions On The Borderline Of Algebra And
Metamathematics": but in the presence of algebra which is none too
universal as a contemporary logical focus, it is important to stress
the
continuity of this work both with Tarski's earlier "philosophical"
work
and with his subsequent development of model theory as an independent
field. In that essay, Tarski defines the character of the concept
"arithmetical class" by a method few number theorists would be
familiar
with: and we could stand to consider the whys and wherefores of this.
An arithmetical class is a set of algebras which share a certain
functional property (rather than a theoretical primitive);
arithmetical
types are those sets of algebras picked out by theories composed of
first-order statements about arithmetical classes. Those familiar
with
the Tarski-Vaught test for extensions are already in possession of
the
requisite conceptual distinction for understanding arithmetical
classes
and types, the ability to discern what is invariant across models of
different cardinality (the Loewenheim-Skolem theorem being on this
understanding an "acid test" for logic rather than a interesting
result
in transfinite arithmetic). But the form of speech employed here is
already apprised of certain givens, and there is an intermediate step
(Tarski's own introductory lectures on the theory of models) which
reveals the concept of algebraic properties held in common is not the
liminal one. Rather, an "arithmetical type" or relational system
comprises all *compositional* facts about a set of algebras which may
possess rich axiomatic construals as well: there is more than a
little
of the finitism Warsaw disavowed in this theoretical couplet.
However, those ready to flee in error from an "anti-realist"
construal
of Tarski's researches ought to wait a little bit. Boolean algebras
with operators are equal to the task of "representing" logics of all
kinds by establishing rigorous standards of isomorphism to their
proof
calculi: and in truth, they form the apparatus for metamathematical
assessments of provable sentences generally, a point which has been
clarified by recent work in modal logic. This having been said, is
there any reason to concern oneself with those elements of
formalization
resistant to algebraic treatment? Well, what such creatures are not
good for is proving metamathematical results about mathematical
systems;
and a convenient way to think of Tarski's concept from the vantage
point
of the present is as the "flipside" of model-completeness, as a
method
for using metamathematical systems to prove mathematical results.
There
are points of connection between such work and the original
motivation
for model theory; the systems Tarski was concerned with were those
where
algebraic compactness *mutually implied* logical completeness; but
this
is none too nomological a net, and such issues as are traditionally
studied under the aegis of modal logic ("intensionality" or
aspect-dependence of meaning) are not excluded by Tarski's treatment.
In fact, one could say such concerns writ large form the rationale
for
the discipline of model theory as it is currently practiced (a way to
justify those features of mathematical practice which are not
accounted
for by set theories constructible using first-order logic).
This brings us to a question post-dating Tarski's work, namely the
"specific difference" of structures or models from algebras. I have
already suggested that the purpose of models is to keep various
ill-defined issues apart, and if we go so far as to call these camps
"computational tractability" and "fine structure" the *index verum et
falsum* is actually hybridity, that property exemplified by logics
which
permit object-language reference to model-theoretic "points" of
evaluation. The keener-eyed among us may suggest this is exactly the
expressive purchase of Boolean algebras with operators, but to say
this
is to close any issue as to the adequacy of an algebra *qua*
formalization: and this is actually not permitted by a model-
theoretic
construction which should be familiar enough to those with pure
mathematical interests, the ultraproduct. Ultraproducts are
summations
of those assays called ultrafilters or "maximal dual ideals", but if
it
seems that the algebraic ideal is a relatively unproblematic concept
I
invite the reader to reflect upon the problems associated with the
related set-theoretical concept of specification and that solution to
Frege's difficulties such as once appealed to the majority of
practicing
mathematicians, the theory of types.
In plain English, Tarski's result with respect to the ramified theory
of
types was that the compositional method for constructing models did
not
carry over to object languages rich enough to express every feature
of
their metalanguage via an "interplay" akin to Goedel-numbering: they
are, too, and a contradiction is easily derived where the recursively
generated metatheory is outstripped by an element of its condition of
adequacy (all and only true sentences). In other words, we are not
permitted to do this for *computational* reasons, and the seemingly
irresistible choice of "incompleteness" as a motivation for
model-theoretic representation of mathematical systems may in fact be
so. If we take computational tractability as a metric of hybridity,
the
extent to which a formal system requires "autochtonous" standards of
evaluation due to metamathematical problems arising from putative
decision procedures, what is revealed is something like "Hilbertian
steppes" of logical consequence and mathematical proof: a
justification
for those informal and aspectually-understood arguments which form
the
basis of mathematical research as the natural environment of
formalization.
If this is so, a common understanding of formal systems (wherein
robustly decidable logics have a unique charm for the rigorous mind)
is
defective. Such systems as the "sentential calculus" and other
propositional logics ought to be understood as *proper fragments* of
those logics possessing no stronger criterion of isomorphism than
Tarski's: hybridizable, but no less serviceable for all that. The
bit
where we took it back earlier ensures that non-logical properties of
of
an arithmetical system do not affect conclusions drawn regarding it;
in
proof-theoretical terms this is called "closure under Cut", but in
mathematics as in life there is no guarantee this is possible in
every
interesting case or necessary in every significant case (another
standard of closure marking out that area of logic submitting to the
discipline of discontinuity). And perhaps the most elegant
definition
of "structure" is that which does not require reflection upon the
most
general features of quantitative reasoning (many recent logical
studies
resist the tripartite heuristic employed here, and quite fully). But
perhaps there is often no reason to go further than discrete
verities,
and if so Tarski's concept of structure may at times require no
metamathematical accoutrements.
-----
I wrote this a long time ago, and I'm still not quite sure what it /
ought/ to mean. Presumably it has some "semantic value" or values
tho'.
A definition was recently requested for the concept of "mathematical
structure", and I suspect the magic of this term suggested a more
searching answer than that which was initially provided (a mapping of
mathematical entities onto their formal representations). This is
structure enow: but in point of historical fact difficulties with
this
term are to be had, and gave birth to the discipline of model theory
as
we know it today. As Abraham Robinson's text has it, the term
"structure" comes out of Tarski's post-*Wahrheitsbegriff* work,
specifically "Some Notions On The Borderline Of Algebra And
Metamathematics": but in the presence of algebra which is none too
universal as a contemporary logical focus, it is important to stress
the
continuity of this work both with Tarski's earlier "philosophical"
work
and with his subsequent development of model theory as an independent
field. In that essay, Tarski defines the character of the concept
"arithmetical class" by a method few number theorists would be
familiar
with: and we could stand to consider the whys and wherefores of this.
An arithmetical class is a set of algebras which share a certain
functional property (rather than a theoretical primitive);
arithmetical
types are those sets of algebras picked out by theories composed of
first-order statements about arithmetical classes. Those familiar
with
the Tarski-Vaught test for extensions are already in possession of
the
requisite conceptual distinction for understanding arithmetical
classes
and types, the ability to discern what is invariant across models of
different cardinality (the Loewenheim-Skolem theorem being on this
understanding an "acid test" for logic rather than a interesting
result
in transfinite arithmetic). But the form of speech employed here is
already apprised of certain givens, and there is an intermediate step
(Tarski's own introductory lectures on the theory of models) which
reveals the concept of algebraic properties held in common is not the
liminal one. Rather, an "arithmetical type" or relational system
comprises all *compositional* facts about a set of algebras which may
possess rich axiomatic construals as well: there is more than a
little
of the finitism Warsaw disavowed in this theoretical couplet.
However, those ready to flee in error from an "anti-realist"
construal
of Tarski's researches ought to wait a little bit. Boolean algebras
with operators are equal to the task of "representing" logics of all
kinds by establishing rigorous standards of isomorphism to their
proof
calculi: and in truth, they form the apparatus for metamathematical
assessments of provable sentences generally, a point which has been
clarified by recent work in modal logic. This having been said, is
there any reason to concern oneself with those elements of
formalization
resistant to algebraic treatment? Well, what such creatures are not
good for is proving metamathematical results about mathematical
systems;
and a convenient way to think of Tarski's concept from the vantage
point
of the present is as the "flipside" of model-completeness, as a
method
for using metamathematical systems to prove mathematical results.
There
are points of connection between such work and the original
motivation
for model theory; the systems Tarski was concerned with were those
where
algebraic compactness *mutually implied* logical completeness; but
this
is none too nomological a net, and such issues as are traditionally
studied under the aegis of modal logic ("intensionality" or
aspect-dependence of meaning) are not excluded by Tarski's treatment.
In fact, one could say such concerns writ large form the rationale
for
the discipline of model theory as it is currently practiced (a way to
justify those features of mathematical practice which are not
accounted
for by set theories constructible using first-order logic).
This brings us to a question post-dating Tarski's work, namely the
"specific difference" of structures or models from algebras. I have
already suggested that the purpose of models is to keep various
ill-defined issues apart, and if we go so far as to call these camps
"computational tractability" and "fine structure" the *index verum et
falsum* is actually hybridity, that property exemplified by logics
which
permit object-language reference to model-theoretic "points" of
evaluation. The keener-eyed among us may suggest this is exactly the
expressive purchase of Boolean algebras with operators, but to say
this
is to close any issue as to the adequacy of an algebra *qua*
formalization: and this is actually not permitted by a model-
theoretic
construction which should be familiar enough to those with pure
mathematical interests, the ultraproduct. Ultraproducts are
summations
of those assays called ultrafilters or "maximal dual ideals", but if
it
seems that the algebraic ideal is a relatively unproblematic concept
I
invite the reader to reflect upon the problems associated with the
related set-theoretical concept of specification and that solution to
Frege's difficulties such as once appealed to the majority of
practicing
mathematicians, the theory of types.
In plain English, Tarski's result with respect to the ramified theory
of
types was that the compositional method for constructing models did
not
carry over to object languages rich enough to express every feature
of
their metalanguage via an "interplay" akin to Goedel-numbering: they
are, too, and a contradiction is easily derived where the recursively
generated metatheory is outstripped by an element of its condition of
adequacy (all and only true sentences). In other words, we are not
permitted to do this for *computational* reasons, and the seemingly
irresistible choice of "incompleteness" as a motivation for
model-theoretic representation of mathematical systems may in fact be
so. If we take computational tractability as a metric of hybridity,
the
extent to which a formal system requires "autochtonous" standards of
evaluation due to metamathematical problems arising from putative
decision procedures, what is revealed is something like "Hilbertian
steppes" of logical consequence and mathematical proof: a
justification
for those informal and aspectually-understood arguments which form
the
basis of mathematical research as the natural environment of
formalization.
If this is so, a common understanding of formal systems (wherein
robustly decidable logics have a unique charm for the rigorous mind)
is
defective. Such systems as the "sentential calculus" and other
propositional logics ought to be understood as *proper fragments* of
those logics possessing no stronger criterion of isomorphism than
Tarski's: hybridizable, but no less serviceable for all that. The
bit
where we took it back earlier ensures that non-logical properties of
of
an arithmetical system do not affect conclusions drawn regarding it;
in
proof-theoretical terms this is called "closure under Cut", but in
mathematics as in life there is no guarantee this is possible in
every
interesting case or necessary in every significant case (another
standard of closure marking out that area of logic submitting to the
discipline of discontinuity). And perhaps the most elegant
definition
of "structure" is that which does not require reflection upon the
most
general features of quantitative reasoning (many recent logical
studies
resist the tripartite heuristic employed here, and quite fully). But
perhaps there is often no reason to go further than discrete
verities,
and if so Tarski's concept of structure may at times require no
metamathematical accoutrements.
-----
I wrote this a long time ago, and I'm still not quite sure what it /
ought/ to mean. Presumably it has some "semantic value" or values
tho'.