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.

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.