Lars> I think we revolve around the same basic problem: is there anything called 'abstraction' beyond instances of abstract cognition?
 
Yes. Those things are called "patterns".   And the formal study of patterns is called "mathematics".  And the foundation for cognition in every living thing from bacteria to humans is the perception of patterns.  And a pattern is a simplified image with some of the detail omitted.  Those simplifications are called "diagrams".  The simplest diagrams are called "graphs".
 
Lars> I wouldn't (and nobody could) know what this would be, as anytime one or any assumed alien looks or thinks about it, it would just be an instance of their cognition, or an instance of a mneme (if you think in terms of memory/thought processes).
 
No to the first part of that sentence.  And yes to the second part.
 
There is something fundamental that is common to all forms of cognition on planet earth:  simplified perception.  Since aliens that evolved on any planet in any galaxy in the universe would also need some form of perception, a reasonable assumption is that their cognition is based on a simplified version of their perception.
 
Since everything in our universe is based on the same physics and chemistry, it's very likely that any kind of biology and any kinds of patterns the aliens may encounter would have some similarities in the patterns they perceive.  Therefore, their cognition is likely to be based on simplified patterns that have some similarity to the patterns that scientists have observed on earth, our solar system, and our galaxy.
 
Lars> In the cognitive sciences (as I explain in my book), scientists more and more start studying the workings of cognitive development and evolution instead of thinking of language and logic...
 
That's a good start.  But I suggest that you go to a more fundamental level.  Perception is more fundamental to all living things than any kind of human cognition.  Images are a good start for anything that has eyes.  But structures are more fundamental for sentient beings without eyes.  And chemical distributions are more fundamental for single-celled beings.
 
For bacteria, a important feature of their perception enables them to swim upstream in a glucose gradient.  that  implies that chemistry plus plus a spatial direction is the most primitive form.  And that can be mapped to a simple kind of pattern.
 
Conclusion:  patterns are fundamental to perception, and the simplest patterns are directed graphs with labels that indicate good or bad directions.  That would be the simplest and most universal basis for cognition.
 
Can anyone think of anything simpler or more fundamental?
 
John