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Additional info for Betrachtungen und Maximen
It is especially helpful to think of definition as elimination when we contemplate the multiplicity of ways of defining NATURAL NUMBERS, q, V,; for each way is separately acceptable but no two are compatible. All that is really afoot, we there observe, is a multiplicity of effective progressions. An even neater case to the same effect is the ordered pair (x J y), defined under COMPLEX NUMBERS; for, as there noted, 'the one and only demand upon the definition is unique recoverability of x and of y from (x J y).
The turn that one gives to one's research, and the supporting evidence that one marshals or the corollaries that one derives, are substantial indications that one holds the belief, though it be a belief on which a bet could never be settled. But beliefs grade off, as my first two examples illustrate, to where their dispositional content apart from lip service becomes tenuous to the vanishing point. What shared trait can have grouped all these extravagantly diverse states of mind, real or professed, under a single serviceable term, belief?
The system with 2 as base is thus the longest-winded of the lot, but it enjoys the slightest of multiplication tables and striking economies of a higher kind. Its only digits are '0' and '1 '. It is called the dual or, more frequently, binary notation, but a better parallel of 'decimal' would have been 'dimidial'. The decimal system breaks numbers down by tenths (partes decimae) , and the dimidial breaks them down by halves (partes dimidiae). However, I shall not persist. Let it be 'binary'. In binary notation 365 becomes 101101101, which is to say, in powers of 2, This expression illustrates a remarkable law: every posItIve integer is a sum of distinct powers of 2.
Betrachtungen und Maximen by Vauvenargues