‘Undefined Terms of the E.D. First Arithmetic for Dialectic’ -- Part 08: ‘Seldon on the Record’ Series.
It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.
The eighth release in this new such series is posted below [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].
In this 8th installment, Seldon identifies two “primitives” or “undefined terms” of the axioms-systems of the various versions of the N_Q_ first arithmetic for dialectic, in the dialectical progression of the Seldonian axioms-systems for ever higher arithmetics for dialectic.
“The concept of an ‘ontological category’, as well as that of the ‘‘‘unit(s)’’’ that are implicit in any such ‘ontological category’ -- each such unit sharing in the defining quality of that ‘ontic category’ -- are “primitives” or “undefined terms” of the axioms-systems -- the ‘rules-systems’ -- that govern various variants of the N_Q_ system for dialectical arithmetic.”
“Similarly, “line” and “point” are “undefined terms” of Euclid’s ancient geometrical axiomatic system.”
“Such “undefined terms” are described to a degree, informally, but never specified exhaustively and precisely, in such systems.”
“Attempts to define all such terms impend potentially endless regresses. Such “undefined terms” are often chosen because they name concepts that are broadly understood “intuitively” by the audiences to whom the presentations of the axiomatic systems are to be directed.”
“Deductive-logic-based axioms-systems require premises, postulates, axioms – a small number of assertions that must be taken as true on faith, or hypothetically, or provisionally, or that must be taken to be somehow “self-evident”. Axioms-systems must have their [e.g., Aristotelian] “Archai” -- a few “truths” from which to begin their deductions, just as a dialectical ontological categorial progression must have an «arché»-ontological category to launch from, e.g., when formulated in the N_Q_ ‘ideographical’ language.”
“Similarly, the formal definitions of terms that a typical axiomatic system requires must have a starting point in the “undefined terms” or “primitives” of that system – terms in terms of which all other terms of that axioms-system must be defined.”
“Our old friend Morris Kline wrote as follows about the vicissitudes of “undefined terms” in axiomatic mathematics: “One reason that unintended interpretations [K.S. – i.e., “Nonstandard Models”, recalling that the N_Q_ are ‘‘‘non-standard models’’’ of the Peano “Natural” numbers, N.] are possible is that each axiomatic system contains undefined terms. Formerly, it was thought that the axioms “defined” these terms implicitly. But the axioms do not suffice. Hence the concept of undefined terms must be altered in some as yet unforeseeable way.” -- from p. 272, in Morris Kline, Mathematics: The Loss of Certainty, Oxford University Press, New York, 1980.”
For more information regarding these Seldonian insights, please see --
For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ -- see:
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.],
Member, F.E.D. General Council
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.
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