Our attempt here has been to open for consideration some new ways of envisaging and analyzing simple formal systems. What these approaches have in common is a clear emphasis on visual and spatial instantiations of systems, with perhaps an inevitable affinity to fractal images. Our hope, however, is that in the long run such approaches can offer more than a visual glimpse of systems as infinite wholes; that new perspectives of this type might suggest genuinely new results. In the manner of the three simple examples offered in our final sections--the Sierpinski map of tautologies in value space, formal parallels between value solids for systems of propositional logic and the quite different value solids appropriate to infinite-valued connectives, and an approach to the values of propositional calculus in terms of cellular automata--our hope is that visual and spatial approaches to formal systems may introduce the possibility of approaching some logical and meta-logical questions in terms of geometry. Number-theoretical analysis of logical systems forms a familiar and powerful part of the work of Godel and others. Analysis in terms of geometry and fractal geometry, we want to suggest, may be a promising further step.
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