Maths/CogSci/MathPsych lazyweb: Are there any algebras in which you have subtraction but don't have negative values? Pointers appreciated. I am hoping that the abstract maths might shed some light on a problem in cognitive modelling.
The context is that I am interested in formal models of cognitive representations and I want to represent things (e.g. cats), don't believe that we should be able to represent negated things (i.e. I don't think it should be able to represent anti-cats), but it makes sense to subtract representations (e.g. remove the representation of a cat from the representation of a cat and a dog, leaving only the representation of the dog).
This might also be related to non-negative factorisation: https://en.wikipedia.org/wiki/Non-negative_matrix_factorization
#mathematics #algebra #AbstractAlgebra #CogSci @cogsci #CognitiveScience #MathPsych #MathematicalPsychology
@RossGayler @cogsci modulo arithmetic
4 - 6 = 6 (mod 8)
or
scissors, paper, rock where rock -1 is clearly scissors
Thanks. Modulo arithmetic is actually of interest for other reasons but I think it's not quite what I'm after here.
Using your arithmetic example and assumming rep(cat) = 1 and rep(dog) = 2 I would want behaviours like:
rep(dog and cat) = 2 + 1 = 3
3 - 2 = 1
3 - 1 = 2
2 - 2 = 0
2 - 1 = 2
1 - 2 = 1
I suspect that means that the objects of the algebra have to be multidimensional, rather than unidimensional (as numbers appear to be).
@RossGayler @Heterokromia @cogsci to me it seems you need to be more clear on your requirements. Are your non-negative and multidimensional requirements independent, as far as you can tell?
If so, a multidimensional (do you know how many dimensions/animals you have?) modulo space sounds a viable solution. That'd be something denoted as https://www.HostMath.com/Show.aspx?Code=%5Cmathbb%7BZ%7D_k%5En , with k being the cardinality of one dimension (would they need to have different cardinalities?), and n being the number of dimensions.
