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What is the different between a first and second order isomorphism (with examples, ideally)

Posted on 2013-11-14
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I'm having a philosophical discussion with a friend who is arguing that our perception of the world can be defined as a second-order isomorphism - the trouble is I'm having problems understanding exactly what that means.

I've read a paper of Shepard which in a 1970 paper he defines a first order isomorphism as "properties of real-world objects are retained in the internal representation of those objects" - for example representations of green things must themselves be green and those of square things must themselves be square.

However in a later 1975 paper he distinguishes between concrete and abstract first-order isomorphisms, the concrete case being as his 1970 definition - representations of green things being green etc - in the abstract case there is a similarity but no physical resemblance between the representation and its target, and has an example of this he cites a square whose representation contains four parts, each of which corresponds to a corner of the square.

In a second-order isomorphism there is no similarity of physical properties, instead the similarity exists between "various targets and the relationship between the corresponding representations". Here the pattern of relations between the representations mirrors the pattern of relations between the objects being represented.

This looks like a paper related to the topic:
http://cogprints.org/2281/1/choe.cogsci02.pdf

If anyone can shed some light on this distinction and how it works I'd appreciate it.
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Question by:purplesoup
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by:Fred Marshall
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"Defined" or "modeled"?  Makes a rather big difference!
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by:purplesoup
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They are both attempts to give an account of something.
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arguing that our perception of the world can be defined as a second-order isomorphism
Well "argue" and "defined" are rather absolute.  Whereas a "model" is just that and generally not expected to be exact (although maybe exact enough).  That's why I asked.

So, it appears that you want to know what a second-order isomorphism is?  Or perhaps what "our perception of the world can be defined/modeled as a second-order isomorphism"?  i.e. the whole package.

I found this:
http://www.sciencedirect.com/science/article/pii/0010028570900022
and this:
http://books.google.com/books?id=V1DFcqt1MXAC&pg=PA77&lpg=PA77&dq=psychology+cognition+first-order+isomorphism&source=bl&ots=E1-dxquvye&sig=q_4l6Qat3QtTGVs3SKkg_66XxZE&hl=en&sa=X&ei=jHWKUtSSCaiciQKG9oH4Ag&ved=0CDoQ6AEwAg#v=onepage&q=psychology%20cognition%20first-order%20isomorphism&f=false
and this:
http://www.sas.upenn.edu/~hatfield/constraints.pdf

So, it appears that all this is pretty much about models unless brain instrumentation is able to actually determine first-order non-abstract isomorphism indeed occurs.
My understanding is that the second-order isomorphisms do not deny first-order isomorphisms anyway.  So, to say that "our perception of the world can be defined as a second-order isomorphism " seems a bit narrow.  
One might say (I don't know) that:
"Our perception of the world can be modeled as a set of mappings which would perforce require the inclusion of second-order isomorphisms".  Inclusion seems a lot more tenable than exclusion of other model transformations it seems to me.
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by:purplesoup
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Thank you that's very helpful.

I'm wondering if my friend is on the wrong track in linking perception with what Shepard is saying.

Take this statement:
It is argued that, while there is no structural resemblance between an individual internal representation and its corresponding external object, an approximate parallelism should nevertheless hold between the relations among different internal representations and the relations among their corresponding external objects.

This sounds to me as if Shepard and Chipman might be talking about how we represent objects internally. I heard a good thought experiment about this.

Think of a pirate.

Now tell me - honestly - how many buttons the pirate has on his coat, if he has any fingers missing, if he has a scar and if so on which cheek, does he have a hat or a parrot? What colour are his trousers?

If we genuinely represent objects as pretty much exact representations of their external form then we should be able to answer all these questions, after all when we think of a pirate is seems to be some sort of picture but it seems a very sketchy one, with perhaps a few details, but not many.

In other words, how we represent objects internally is quite mysterious - I'm not sure the second-order isomorphism covers it - but I'm prepared to admit it isn't some sort of exact replica of an external object (although for triangles and squares perhaps it is).

But as you say, if the idea is perception - not representation - I have no idea how we are to exclude first order isomorphisms.
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by:purplesoup
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Oh - on second order isomorphism - I've seen an example of a map - where the cities are clearly not representations of the actual cities - there is no resemblance there, but the positions of the cities are correctly shown.

The explanation of first and second order isomorphisms is that a first order isomorphism would represent a city as an actual city or something which at least had the properties of a city - certainly not as a dot.

Wheres a second order isomorphism is about the relationships between the objects. On the map we are able to show City A is to the west of City B - so the relationships between the objects is accurate, and this would make something a second-order isomorphism.

Is this close?
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by:Fred Marshall
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I'd suggest that people are putting to much weight on the terms being used.  They may be handy for discussion but in themselves have fuzzy meaning - largely  because they're being used to describe fuzzy concepts.  Isn't that so?  

So, I'd recommend discounting the assertion.  Not because it's necessarily wrong but because it already seems incomplete AND because what it refers to isn't observable.
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by:purplesoup
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Thank you - I think you're right - also I came across this:

http://en.wikipedia.org/wiki/Mental_image

It makes it clear:

A mental image is an experience that, on most occasions, significantly resembles the experience of perceiving some object, event, or scene, but occurs when the relevant object, event, or scene is not actually present to the senses

and Shepard is cited in the article, so I'm now of the opinion that Shepard is talking about our inner mental images, not our experience of seeing external objects.

Thanks for your help.
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