It seems to me that knowledge should have a fractal dimension. This dimension must have increased over the history of science. Has this fractal dimension ever been measured? How can it be determined (from publications or citations)? What is the rate of change of the fractal dimension of scientific knowledge is it increasing (or will it level out)? Does this fractal dimension tell us anything useful about innovation in a particular scientific discipline?
I am not a specialist, but I am interested as a lay-person. I'm uncertain, is yours a maths question or more philosophical one? Can you expand a bit on your questions and terms?
Fractal dimensions have been employed to describe the distinctive nature / features of a variety of abstract and practical phenomena (urban growth, networks, turbulence, market response), so intuitively it seems right that (scientific) 'knowledge should have a fractal dimension'.
If I understand correctly, the fractal dimension of scientific knowledge would be a ratio indexing complexity and comparing how detail in the pattern changes with the scale at which it is measured. So the key concepts are the scale and the measuring system. [The classic (Lewis Fry Richardson) example showing that a coastline's measured length changes with the length of the measuring stick used.]
To me, it seems that the rapid increase in the amount of published information or data is not alone a sufficient indicator, as the amount of data being made available (publications) is outstripping human ability to organize and deal with it (citations).
Maybe it will take Artificial Intelligence (AI) just to construct the fractal index!
Looking forward to learning more about this topic.
Leo
Dear Leo
This is an idea that has been floating in and out of my consciousness for many years. I don’t have a clear idea of what data I would use to determine the fractal dimension of scientific knowledge, or even if there is a better way to describe scientific knowledge. Nevertheless, it is something I thought was worth a try and asked this question to test the water and see if other people had already tried it.
Nevertheless, there are so many scientific disciplines and there are so many ways to research each discipline and then there are the interdisciplinary topics. Down to the finest scale of the individual thoughts and ideas of every researcher.
This looks Fractal to me so it must have a Fractal Dimension.
One idea to determine a Fractal Dimension might be to start on the largest scale and work to the finest scale and find a metric for each:
1. High school science subjects
2. Bachelor’s degree Scientific Disciplines
3. Master’s degree scientific Disciplines
4. Post graduate topics
5. Scientific journals
6. Publications
7. Citations
These things must change over time. If the data were gathered then we could look at the data and then ask the question can I use it to define a fractal dimension for scientific knowledge.
There must be data on these things perhaps much of the data could be found on the internet already even on RG.
Then the fun can really begin; what would these numbers tell us about research?
Yes, Leo, there are fractal models for classified literature published in different domains and their connectivity can be expressed by a self similar complex network, the degree of lacunarity is generally expressed as fractal index, lacunarity means the degree of distribution of classes in a 3D space, Suppose there are n number of journals under m number of Publishing house dealing with x interconnected subjects, so the 3D space there will be some discrete groups of subjects that can be conglomerated or classified in a sphere and distributed in 3D space of n,m,x, Now if we take the envelope of the whole cluster points representing their another property say impact fator, i, we will be able to associate some numbers or weights to each points, lets take this as their distance from the origin of n,m,x space, Now if you join the points you will be getting an irregular shape which will have a geometrical fractal dimension like any other rough surfaces like rock surface, fractured material etc. This model is applicable for any complex graph and having a property of self similarity measuremnt is magnification invariant and scale dependent
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