Dimensions for qualitative properties –
A maths algorithm transferred to general purpose
Often, I have the impression that a mathematical concept would be incredibly helpful here – Especially in discussions outside of maths. And don’t worry: I don’t mean everyone should be an expert in maths – on the contrary: Let me share here an idea, in a way that may help a lot of people.
The problem
Let’s look at complex, multilayered problem. E.g someone (in this case gemini) wants to discuss the “The Digital Divide” and notes several facets:
Economic: The cost of access to high-speed internet and digital devices varies significantly across different socioeconomic groups, creating a digital divide between those who can afford it and those who cannot.
Geographic: Rural and remote areas often have limited access to high-speed internet infrastructure, further exacerbating the digital divide.
Social: The digital divide can contribute to social inequality by limiting access to education, employment opportunities, and social services.
Cultural: Different cultural groups may have varying levels of familiarity and comfort with digital technologies, which can further widen the digital divide.
Ethical: The digital divide raises ethical concerns about the distribution of information and resources in the digital age, as well as the potential for further marginalization of already disadvantaged groups.
But the question that comes up is: Is the problem so well discussable, the parts clearly determinable? Or otherwise, do these criteria help us to describe it? Not really. Because, somehow, all seem to be dependent on each other …
What we would need are well-chosen dimensions, or criteria, which make it tangible. Ideally, these would be independently describable and adjustable (mathematically called, orthogonal). But how would we find those?
The idea
The Gram-Schmidt orthogonalization transforms a set of linearly-independent vectors into a orthonormal basis – Or in proper English: How do you determine from some criteria, the most differentiating dimensions to describe it (better).
The Gram-Schmidt procedure goes through following steps:
1. Select a first dimension.
2. Select another dimension.
3. Remove the common features between the new and old ones.
4. (Norm it) and back to step 2.
How could we transfer this to our example?
We would start with a first (cardinal) dimension. In principle, this can be done arbitrarily. But most helpful is to select the most central dimension.
I select Economic.
As next, we could take Geographic. Now the question occurs, what have the categories in common – or most different? Rich and poor regions shouldn’t be needed to discussed separately here. Most interesting is discussing, something that is independent from this: e.g. rural/ remote areas vs. urban areas. Those exist in richer and poorer variants.
Let’s name this dimension Urbanization/ Remoteness.
The forth step to normalize the dimension is in this context perhaps unnecessary, but perhaps, we removed so much that it becomes insignificant and would need to be amplified/ extended.
No go to the next one: Social.
Common with our existing dimensions, is social inequality in economic regards. As well as social background based on rural or urban localization. Before mentioned job opportunities social services have to be removed from the urban context.
As independent dimension, one could focus on Educational and professional background.
As you do this more often, there is more and more overlap to the existing dimensions to consider. It could instead be useful to just try to imagine new dimensions from the remaining ones, that are independent from the existing ones. E.g. I could read from the last two dimensions something like Digital experience – Although this is still possibly influenced by your education…
As an exercise for the reader, you could try instead to come up with two remaining useful dimensions.
Some final remarks
For those, who find this interesting and want to learn more:
It is further useful to think about what kinds of dimensions we are using. E.g. there are categories that are partitioned into discrete cases and some that are describable in a continuum, so fluidly/gradually changing.
For these dimension also some are clearly measurable, some very subjective and less graspable.
Also, this topic is a smooth transition to data science, where very complex topics are described by a lot of data with different values in several categories (dimensions). Not a surprise that this hand-in-hand connects to orthogonalization in linear algebra, and is, in turn, a necessary step in the optimization of nodes in neural networks, which analyse the data…