It is widely assumed that there exist certain objects which can in no way be distinguished from each other, unless by their location in space or other reference-system. Some of these are, in a broad sense, 'empirical objects', such as electrons. Their case would seem to be similar to that of certain mathematical 'objects', such as the minimum set of manifolds defining the dimensionality of an R -space. It is therefore at first sight surprising that there exists no branch of mathematics, in which a third parity-relation, besides equality and inequality, is admitted; for this would seem to furnish an appropriate model for application to such instances as these. I hope, in this work, to show that such a mathematics in feasible, and could have useful applications if only in a limited field. The concept of what I here call 'indistinguishability' is not unknown in logic, albeit much neglected. It is mentioned, for example, by F. P. Ramsey [1] who criticizes Whitehead and Russell [2] for defining 'identity' in such a way as to make indistinguishables identical. But, so far as I can discover, no one has made any systematic attempt to open up the territory which lies behind these ideas. What we find, on doing so, is a body of mathematics, offering only a limited prospect of practical usefulness, but which on the theoretical side presents a strong challenge to conventional ideas.

Inhalt

I: Theory.- I/Introduction.- 1.A. The Concept of Indistinguishability.- 1.B. Primary and Secondary Indistinguishables.- 1.C. Classes of Indistinguishables.- 1.D. Correlation and Predication.- 1.E. The Triparity of Notations.- 1.F. Levels of Notation.- 1.G. Syntactic Specification of the Notations.- II/Semantic Theory of the Notation F.- 2.A. The Meaning of Meaning.- 2.B. The Pair Functors of U.- 2.C. Classifying Functors.- 2.D. Initial Theorems in U.- 2.E. Indistinguishable Arguments - Cases (a) & (b).- 2.F. Enchained Functors - Cases (c) & (d).- 2.G. Classifying Functors - Cases (e) to (g).- 2.H. Declassifying & Confounding Functors.- 2.J. Concurrence of Symbols.- 2.K. Concurrence in V.- 2.L. Compound Statements and Quantification.- 2.M. Comparison of Biparitous and Triparitous Quantification.- 2.N. Quantification of Definiends.- III/The Physical Relevance of Indistinguishables.- 3.A. The Concept of 'Planes'.- 3.B. The Inchoative Plane.- 3.C. Observability of the Inchoative.- 3.D. Methodology.- 3.E. Types of Indistinguishables.- 3.F. The Principle of Coherence.- 3.G. Valid Representations.- 3.H. The Construction of Representations.- 3.J. The Irreducibility of the Physical Plane.- 3.K. The Combinatorial Hierarchy.- IV/Sort Theory - Axioms and Definitions.- 4.A. Indefinables of T.- 4.B. Method of Verifying Concurrence.- 4.C. Definitions in the Inferential System.- 4.D. Definitions in T - Basics.- 4.E. The Conditional Quantification Functor.- 4.F. Definition and Classification of Sorts.- 4.G. Some Classifying Functors.- 4.H. Ordered Pairs.- 4.J. Some Confounding Functors.- 4.K. Miscellaneous Functors over Sorts.- V/Sort Theory - Mappings.- 5.A. Mappings and Functions.- 5.B. Mappings from and to Perfect Sorts.- 5.C. The Closure of a Sort.- 5.D. A Classification of Sort Mappings.- 5.E. Cardinality of Perfect Sorts.- 5.F. The Invariant Subdomain Theorem.- 5.G. Functions of Two Arguments over a Perfect Sort.- 5.H. Values of the Functions.- 5.J. Properties of the Functions.- 5.K. Two-Argument Functions over Derived Perfect Domains.- 5.L. Functions of More than Two Arguments.- 5.M. Infinite Perfect Sorts.- 5.N. Operational Tables of the Functors.- VI/Representations of Initial Sorts.- 6.A. Initial and Superstruct Sorts.- 6.B. Complexes on an Initial Sort.- 6.C. Representation of $$\bar D$$20 by Pairs.- 6.D. Representation of Functors over $$\bar D$$20.- 6.E. Representing the Symmetry of the Functions.- 6.F. Representation of $$\bar D$$30.- 6.G. Representation of Functors over $$\bar D$$30.- 6.H. Symmetry of the Functions over $$\bar D$$30.- 6.J. Representation of $$\bar D$$40.- 6.K. Representation of Infinite Sorts.- 6.L. Conspectus of Representations of Initial Sorts.- VII/Representation of Superstruct Sorts.- 7.A. Sorts of Bivalent Functors.- 7.B. Families of Endomorphisms.- 7.C. Functions over Superstructs.- 7.D. The Cedilla Functor.- 7.E. The Hex Functor X.- 7.F. The Auxiliary Functor X.- 7.G. Pair Trees.- 7.H. General Definition of X over Pair-Trees.- 7.J. Representation of $$\bar D$$22 &c., by Hex Formulae.- 7.K. Are there Alternatives to Hex?.- 7.L. Representation of the Family D3.- 7.M. Representation of the Family D?.- 7.N. Rational Sorts.- 7.O. Catalogue of Rational Sorts.- II: Application.- VIII/Hypothesis and Principles.- 8.A. The Role of the Observer.- 8.B. Formulation of the Hypothesis.- 8.C. Some General Principles.- 8.D. The Pattern of the Families.- 8.E. The Nature of Observation.- 8.F. Unconditional Observables.- 8.G. Perfect and Mixed RSs.- IX/Events in the Void.- 9.A. The Void.- 9.B. Events.- 9.C. Orderings of a Segment of U.- 9.D. Specification of Particular Events.- 9.E. Ordinators.- 9.F. Dimensions.- 9.G. The Initial RSs.- X/The Texture of Space-Time.- 10.A. The Interpretation of D?0.- 10.B. Disordinate Statistics.- 10.C. The Action Metric.- 10.D. The Distinction Metric.- 10.E. The Chain of Measurement.- 10.F. The Concept of a Particle.- 10.G. The Polarity of Time.- 10.H. The Limiting Velocity.- 10.J. The Big Bang.- 10.K. The Connectivity of Space.- XI/The Constitution of Matter.- 11.A. The Finiteness of Information.- 11.B. Quantization and Conservation.- 11.C. Some Problems of Correlation.- 11.D. Partons.- 11.E. A Rudimentary Chromodynamics.- 11.F. Interpretation.- 11.G. Parton Descriptors.- 11.H. Reconstruction of Particles.- 11.J. Summary of Kinds of Particles.- XII/States of Particles.- 12.A. The Family D3.- 12.B. States in a Disordinate Space.- 12.C. The Proton-Electron Mass Ratio.- 12.Ca. The Fine Structure Constant.- 12.D. States Distinguishing Particles.- 12.E. Location and Orientation States.- 12.F. Interaction-Field States.- 12.G. The Chromodynamic Contribution.- 12.H. The Mean Density of the Universe.- 12.J. Discussion.- XIII/General Assessment.- 13.A. Characteristics of the Theory.- 13.B. New Paradoxes for Old.- 13.C. A Mendeleevian Presentation.- 13.D. Summary of Evidence.- 13.E. Assessment of the Results.- 13.F. Limits of Interpretation.- 13.G. General Conclusions.- References.- Index of Terms Defined.- Index of Symbols.