User:Marc Goossens/elements of set theory

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[edit] Constructive sets (as strings of symbols) and structuralist physics

(comment to user Tastyummy 070309)

Hi, maybe the following is of interest to you.

In the above talk, a good amount of semantics is attributed “a priori” to the notion of sets, and their possible application to model (or speak about) the world surrounding us. There is an approach which avoids this, and allows any physical semantics for sets to be recovered (or indeed “added”) later on.

As shown by Bourbaki, set theory may be constructed by adopting rules for constructing, classifying and manipulating strings of symbols from a basic alphabet. This approach has been refined by Edwards and again by Schröter. See e.g. Edwards, R.E., “A formal Background to Mathematics”, Springer 1979.

The result is like a symbolic game with symbols, which mimics (of course: by design!) the “mathematically useful behavior” one expects from classical set theory, which in turn abstracts intuitive notions or expectations. For example, like any other Bourbaki-Edwards (B-E) set, the empty set is simply defined as on particular string of symbols. And lo and behold, when this string is “inserted” into the agreed (defined) string-manipulation schemes for set operations like union, intersection, … it nicely conforms to what we want. Yet the entities (sets) built in this way, are no more than (abbreviations for) strings complying with certain agreed rules, and are at this point devoid of any further meaning.

Note: any abstraction always involves stripping off semantics and reducing, simplifying behavior, on the other hand, any such construction may also bring in some artifacts, so that the construct should never be naively identified with what it’s supposed to model: the model may work only to some extent, if at all.

B-E go on to employ this notion of sets as a basis for mathematical structure-type (or “species of structure”) theories, that cover the bulk of mathematics. Apart from naked sets, they include all sorts of stuff like the natural numbers, other “number sets”, the familiar algebraic structures (groups, fields, rings, vector spaces, algebra’s, …), topological and measure structures, and combinations, extensions and variations thereof (operator algebra’s, manifolds like pseudo-Riemannian spacetimes, etc.).

Let’s keep in mind that, again by design, none of these mathematical concepts carries any interpretation or semantics: everything is but a purely formal game. This is true, regardless of the intuition and heuristics that have guided the choice of the defining axioms for these structures. Once the structure-axioms stand, they are preferably consistent, at best suitable for doing some nice math and with some luck even fun.

Note: the structure-type notion is categorical in nature, but does not go the full length category theory does. It is, in a sense, more cautious, which seems appropriate when probing deeper philosophical questions as to the how and why, scope, ontology etc. of such a tremendously powerful human art as is physics.

This is as far as B-E mathematics goes (which is pretty far). So what about physics? A general way how physical interpretation may be appended to all this was proposed by Ludwig. This is one of the “structuralist” programs for theoretical physics, as listed in Stanford Encycopedia.

According to Ludwig, at the core of any Physical Theory (PT) resides a B-E Mathematical Theory in the above sense, which is to serve as a model for some excerpt of reality. Observe that for each (attempted) PT, its MT is chosen or “proposed”, postulated, if you like; it is never formally inferred.

Next, one goes on to specify the “known inputs” for the PT. These consist first of appropriate templates for “observational statements”, which are accepted as relevant for the intended PT. In general, any concrete (experimental) observation may be formulated in set-theoretic language as “the constant a is an element of the set B”. Each observation leads to a formal such sentence, which is appended to the list of axioms of MT, thus extending it.

Actually, in order to make the MT into a PT, one also has to specify which “basis sets” in MT may appear in the observational statements. This convention, together with the “input templates” are of course additions to the formal scheme, outside of the MT considered. As such, they are “meta” notions. Together, they constitute the PC’s “mapping scheme” or “mapping principles”.

What we also have to do, is to describe in “natural language”, which parts of nature are adopted as “known inputs” for the intended PT. This is referred to as the “(fundamental) domain” of the PT.

In other words, the basic ingredients of a physical theory PT are given by the triple (domain, mapping scheme, MT). For the most elaborate and in-depth development, see Schröter, J., “Zur Meta-Theorie der Physik”, W. De Gruyter, 1996. A nice summary is given on Martin Ziegler’s page.

Ideally, the structural axioms of the MT used, have an immediate physical interpretation. This is for example not the case for the Hilbert space axioms, as used in quantum mechanics. This indicates that Hilbert space is only an ancillary mathematical structure, possibly fine for practical calculations, less so for proper understanding. (In fact, Hilbert space is just the carrier of a – very practical - mathematical representation of other MT’s that do permit more direct physical interpretation.)

Another interesting footnote is that to the extent that the Ludwig program works (as is essentially established for physical models like General Relativity and non-relativisitic Quantum Mechanics), it also shows that traditional binary logic is sufficient for physics (as all B-E mathematical theories are based on it).