Epilogue. Conclusions and Curtain Call
Our journey through geometric spaces of words and their meanings draws to a
close, and it is time to meet some of the main characters one last
time, and to reconsider the progress that has been made in the broader
setting of science at the beginning of a new century. The linguistic
challenges that still need to be addressed are enormously diverse and
elusive, and our theoretical models still only capture a fraction of
the intuitive adaptability used by humans in perfectly normal
communication. My aim is not to leave the reader falsely confident,
but hopeful, curious, and excited at the thought that the some of the
most important scientific discoveries still await us, whatever they
may be.
Sections
A Conceptual Structure of Conceptual Structures
In this section we try putting our money where our mouth is and using
the techniques from the book to describe the book itself. We
compare the properties of the different models we have built by
putting them into the following concept lattice of their own.
This results in the following structure:


Crosstable showing which models have which properties. 
Concept lattice built from these relationships. 
As you can see, at the time of writing there was as yet no model that
has all the desirable properties. It is hoped that this will stimulate
readers to try and build one!
Last Linguistic Points
What has the book taught us about linguistics? What parts of language
are adequately modelled by our geometric spaces? What aspects remain
currently beyond our grasp?
Last Mathematical Points
What has the book taught us about mathematics? Is mathematics serving
the need of linguistic modelling? As well as being an elegant art of
its own, mathematics traditionally provides models and techniques to
help describe phenomena in other sciences. Is mathematics serving
linguistics well? What new mathematics would we need to do a better
job of modelling something as subtle and complex as human language?
Epilogue  Aristotle the Mathematician
I never expected to find myself writing a book on geometry in which
Aristotle turned out to be the hero. But that's just what happened.
Aristotle is usually acccepted as a great empirical scientist,
logician, and philosopher, but thought of as a poor mathematician. In
researching the story of "Geometry and Meaning", I gradually found
that nothing could be further from the truth. If anything, Aristotle
anticipated the invention of many mathematical concepts that have only
been rediscovered in the 20th century, and there may be many more to
come.