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Because chess is an abstract structure, it is possible in principle to play a game of chess that is entirely mental (provided that you and your opponent have very good memories !).
- Many chess grandmasters have been known to do this. It's not far-fetched. Revolver
But the abstract structure itself is defined in a way that is not dependent on the properties of any particular implementation.
- Hmmmm...me wonders. If the abstract structure is not dependent on the properties of any particular implementation, how do tell whether its an implementation or not? It probably sounds like I'm playing devil's advocate here, but how do you "define" an abstract structure without at some point falling back upon some concrete objects in the real (or mental) world? Take the example of a chess game and the rules of chess. Define for me the "abstract structure" of the game of chess without eventually taking recourse to some concrete objects or concrete mental constructions? The best way I can think to define chess is to say that it's the collection of all possible sequences of 8 x 8 matrices with entries in the set consisting of all the chess pieces from each side, and of course these pieces could be defined in terms of ordered pairs, one for colour, one for piece/pawn. As well as the algorithm to determine whether each term in the sequence is a "legal" state for the board. But now how do you tell that the everyday chess board we use with physical chess pieces and the rules followed by those who "know" the rules is actually isomorphic to our abstract game defined by matrices, sequences, and ordered pairs? You do it the way mathematicians would do it as well...you set up a correspondence between pieces and parts of the "abstract structure" we defined, between moves in a game and terms of our sequence of matrices, so that everything is in 1-1 correspondence and all the rules and such are obeyed.
- All fine...but of course the ordinary board and ordinary physical pieces are "obviously" like the sequence of 8 x 8 matrices, the correspondence is obvious. But there's another way to think of the game of chess, and that is as an infinite tree starting at a single vertex, and each edge goes to a possible legal move. In this case, you don't even need to talk about "pieces" or "pawns" or "rules"...everything about the game of chess -- any possible playable game of chess, is included in this immense, enormous tree. In other words, this tree MUST have the same "abstract structure" as our 2 earlier "models", the sequence of matrices, and our ordinary physical game with human players. But the correspondence is now far from obvious, and it's not even clear what the structure is after all. Now it seems that the real "structure" consists of an object in graph theory, and the isomorphism is set up by corresponding vertices of the directed graph to possible playable boards embedded in their previous sequence of moves. In this way, our first (and second) example as shown to really have the same structure, because they can be given the structure of a directed graph, their vertices and directed edges can be specified, and a 1-1 correspondence between vertices and edges can be established (in theory) that shows that the vertex-edge "structure" is preserved (and this can be done in a very precise way).
- But now have we defined the abstract structure in a way this is not dependent on the properties of any particular implementation? I don't think so, because in order to define the abstract structure, we had to define what a directed graph was, and what an isomorphism between directed graphs was. But the most important part of all was -- in order to define the abstract structure, we had to demonstrate that there was at least one particular model (i.e. example) of our particular abstract structure; we had to point to at least one worldly case of its existence to show that it wasn't a figment of our misled imaginations. This is where my problem with the above statement lies. But I'm babbling. Revolver
I agree with all of the above comments up to
- ... in order to define the abstract structure, we had to demonstrate that there was at least one particular model (i.e. example) of our particular abstract structure; we had to point to at least one worldly case of its existence to show that it wasn't a figment of our misled imaginations.
I agree that many (possibly all ?) abstract structures start out as a generalisation, abstraction or rationalisation of some real world prototype. But the abstract structure can then take on properties which can not be perfectly realised in any implementation - for example, many abstract structures in mathematics are infinite - so any proposed implementation can then only be at best an approximation to the abstract structure itself.
But I guess this leads into several other interesting questions such as what kind of 'existence' does an abstract structure have ? And how can we compare the abstract with the concrete anyway ? Gandalf61 17:04, Nov 9, 2003 (UTC)
- I think we basically see things the same way (my posting was playing the devil's advocate a bit). I think the difference between us might be in what we consider to be an "implementation". From the tone of the article and based on what you say above, you take a strictly physicalist interpretation of "implementation"; I take mental implementations into account as well, or at the very least, formal symbolic implementations of abstract structures. What is the "abstract"? My way of looking at things (and this isn't too precise, more a lifestyle choice) is that an abstract structure is not an "object" at all, it's the stuff that is "preserved" between objects (in the terminology of category theory, it's the morphisms that really count, not the objects...) But in order to know that stuff is "preserved" (i.e. that morphisms exist) you have to know that there are things between which stuff is preserved (i.e. that objects exist).
- I disagree that any implementation of an infinite object in math is necessarily an approximation. In fact, the axiom of infinity is meant to provide a solution to just this type of philosophical/ontological problem. Now, one can argue that the axiom of infinity itself isn't really "an infinite object", in the sense that it consists of only a finite string of symbols, but even if you accept this, it's difficult to argue that the axiom of infinity does not meet almost all the needs mathematicians have for the use of the infinite in working mathematics (modulo various philosophical conundrums left to logicians and set theorists). So, if one accepts say, ZFC (Zermelo-Frankel axioms plus choice) as a "physical implementation" of abstract structures, then any theory derivable from ZFC (i.e. almost all of modern math) is physically implementable (in theory), or if that seems too strong, ZFC is a physical object which appears to model infinite objects, but since a physical implementation is just a model anyways, does it matter?
- Also, the reason I made the original posting is that such mistakes (talking about abstract structures that don't exist) have been made in the past in mathematics, or at least, similar situations have occured in which the existence of an abstract object was required to be proved. Some examples:
- hyperbolic geometry -- Gauss and others played around with proving stuff by replacing Euclid's fifth axiom, and came up with remarkable "theorems", and so on, but none of it meant anything at all until someone produced an actual, specific model of hyperbolic geometry. If such a model did not exist, then all the "theorems" would have been vacuous.
- the real number field -- one can axiomatically define the real numbers as the unique complete ordered field, and one can show uniqueness without worrying about existence. Then all of (real) analysis can be derived, in theory. But what if there is no such thing as a complete ordered field? This is the reason why the reals have to be constructed from Cauchy sequences or Dedekind cuts -- again, to make sure the concept of a "complete ordered field" isn't just something our minds have tricked us into believing exists.
- an anecdote -- there is a famous mathematical anecdote about a ph.d. defense John Milnor attended, where the candidate gave an elaborate definition about a certain class of functions, then went on to prove lots of statements about them. At the end of the talk, Milnor supposedly (it's anecdotal) arose and proved in 5 minutes that the elaborate definition simply defined precisely the constant functions...oops. This isn't quite the same, but the idea is...you can think you're making grand statements about abstract structures (in this case, defined by elaborate rules) but unless and until you check to see what lots of examples look like, it may all just be vacuous (or nearly vacuous, in this case). Revolver 10 Nov 2003
Responding to some of Revolver's comments ...
- From the tone of the article and based on what you say above, you take a strictly physicalist interpretation of "implementation"
Yes, when I said "implementation", I should have been more specific and said "physical implementation". As you say, one abstract structure could be a (more or less faithful) implementation of another abstract structure - I suppose this would be an "abstract implementation".
- ... none of it meant anything at all until someone produced an actual, specific model of hyperbolic geometry. If such a model did not exist, then all the "theorems" would have been vacuous.
Yes, it could have been a very complex way of describing the empty set. But it would still be an abstract structure - just not an interesting one. I was trying to avoid raising questions about whether any of these abstract structures "exist", and, if so, in what sense.
- unless and until you check to see what lots of examples look like, it may all just be vacuous
From a practical point of view, I would agree with you. But I think an abstract structure can still be valid even if no models of it are known - and even if it is shown that there can be no models of it. Gandalf61 13:38, Nov 12, 2003 (UTC)
I think the article is a bit inexact, I try to explain in english why. Well, when we talk about algebraic structures,
- we generally define them as a system of sets containing an "universe", and sets of relations, functions (operations).
- So, this way of definition, a structure is a set-system of objects. This is an example for that what you call "implementation".
- We define the relation "isomorphy" then: two structure (-implemenmtation) is isomorphyc, when we can't see a serious difference between them. A class of isomorfhic structures - sadly-badly - often called as structures, too. It sholud be called "type of structures", "structure-class" or somewhat like this.
- This is the traditional (Bourbakist) way of definition.
- There is an other possible approach,
- when we call structure classes as "(abstract) structure", and a representative of a class as "representation", "implementation" etc.
- Unfortunately, this two approach are often mixed;
So I think your approach is acceptable, but your implementation of this approach is inaccurant a bit. I can't imagine what does it mean "(any) physical object" and so on, I think your definition is not mathematical, it is too metaphysical (and quite senseless). The typic symptom of antiformalism: we throw out Bourbaki's technical and hard definitions, but can't give something better. So i think your way of thinking is acceptable, but you've formulated it into words hastily. Maybe the difference between structure representation and structure (class/type) should be mnore specific and the concept of "structure representation" would be defined too (e.g. as Bourbaki did it). The present definition is not a definition, it is roundly meaningless. Gubbubu 12:09, 29 Dec 2004 (UTC)
- I don't think the abstract structure article is trying to provide a formal mathematical definition - it is trying to give a general sense of what the various uses of the term abstract structure in the fields of philosophy, computer science and mathematics have in common. More formal mathematical definitions are given in the algebraic structure article and other linked articles. But the term abstract structure is not confined to mathematics, so I think it is appropriate to give it this general (and perhaps less formal) treatment. Gandalf61 11:13, Dec 31, 2004 (UTC)
- As I said, I'm satisfied with the ground conceptions of this article - more exactly, I am not fully satisfied, but it's quite right; I don't want to attack it. Only the first one or two sentences disturb me seriously: I think the expression "independently of any phisycal objects" is senseless. You hardly can define "phisycal object", but if you can, the expression "independently" would be a completely weasel term then. Gubbubu 11:30, 31 Dec 2004 (UTC)
An abtract structure is a class of isomorphic „concrete structures”, where
- a concrete structure is a set of rules, properties, relationships, operations. Elements of a concrete structure can be phisycal or mental, we call it an implementation (or realisation, representant etc.) of an abstract structure.
- isomorphy is a relation between concrete structures. We say that two structures are isomorphic, when all their important properties are the same, where important is a(n inter)subjective term for properties that are relevant to the object of examination.
For example, chess, the whole game can be considered as an abstract structure, its implementations could be whether a set of rules or an infinit directed graph, too. But a particular chess match can be considered as an abstract structure, too - in this case we must define isomorphy in an other way then we did when examined the whole game.
Mathematical system, inference rules, and references
Hello. I am trying to decipher a book by Johnsonbaugh, which at one point mentions the concept of "mathematical system", as being composed of "axioms", "definitions" and "undefined terms". It further mentions that we can "derive theorems within a mathematical system", and then goes on to talk about mathematical proofs.
I would like to know how does that relate to the concept of a formal system, where we have a formal language, axioms, and inference rules... Is a formal system just another kind of "abstract structure"? And is it a "mathematical system" too?
Is a formal system just a mathematical system where some proof techniques were transformed into mere inference rules, becoming a kind of transformational grammar? Is this separation a way to run scared from Gödel, implying that there is more to it than formal systems, and that actual mathematical proofs can't be grasped by mere inference rules?...