Re: Using the RM for ADTs

"David BL" <davidbl_at_iinet.net.au> wrote in message news:cc914f85-fc4c-438b-97c8-89067eb5f015_at_g1g2000pra.googlegroups.com...
> On Jul 7, 11:43 pm, "Brian Selzer" <br..._at_selzer-software.com> wrote:
>
>
>> Here is a case in which what defines something is how it stands in
>> relation
>> to other things rather than just a collection of properties.
>
> This idea is fundamental to the axiomatic approach to mathematics.
>
> One can draw a parallel with the natural numbers which can be defined
> using the Peano axioms. This treats the natural numbers as an
> abstract set. As such one could use abstract identifiers for each of
> the elements, and define them though their relation to each other.
>
> ISTM that we don't use abstract identifiers for the natural numbers
> because we don't need to. We can instead encode them using a physical
> representation that maps to the number of successor operations from 0.
>
> Evidently "tricks" like this cannot be used in more complicated
> examples.
>
>
>> ...,it is intuitively obvious
>> that each component or node should still be distinguishable from all
>> other
>> components and nodes in the same template;
>
> I disagree. Circuits may contain a lot of self-symmetry. One can
> investigate this mathematically by considering the group of
> automorphisms in the obvious way.
>
> As an example, consider a circuit consisting of 12 x 1 ohm resistors
> and 8 nodes wired up in the manner of a 3-dimensional cube. All the
> resistors are indistinguishable and all the nodes are
> indistinguishable, even in the context of the circuit that they appear
> in.

They are not indistinguishable. Just pick an arbitrary component lead, and the rest can be described in terms of it because each has different paths to it.

>
> Some circuits have more symmetry than others. For example, one can
> partially break the symmetry by changing only one of the resistors.
> This affects the group of automorphisms.
>
Received on Thu Jul 09 2009 - 04:11:43 CEST