RB 65

the two types of induction,we see that the conclusions of mathematical induction, due to the fact that this type of induction is deductive by nature (cf. its recursive nature),are guaranteed to be true provided that the premisses are true, while the conclusions of enumerative induction are only contingently true, even if the premisses are true. Since the enumerative induction’s conclusions cannot rely on a recursive method of verification (validation) then the enumerative conclusions must rely on an empirical method of verification, which in turn demands that the material basis for the inductive inferences must be verified in each and every instance in order to guarantee valid inferences and certain knowledge.This will also affect the general validity and reach of synthetic judgments and propositions, for they will never be more certain than their empirical basis, nor will they be more absolute than their formulation allows. For example, even if it is true that all swans are white, then it is not necessarily true that the next swannecessarilywill (prediction) or must (definition) be white (unless the definition of swans used in this case states that the property of being white constitutes a necessary condition for swanhood, whereby the big black long-necked bird swimming in the pond cannot be a swan but must be something else, perhaps a “non-swan” or a “black-swan”). Closer analysis shows that enumerative induction only states that it is more or less probable that, for example,a) the next observed swan will be white, or b) that our characterization of swans as generally being white will not be falsified by future observations, which are two propositions whose truth value rests upon contingent factors, but not a recursive definition of swans. On the other hand, it is necessarily (demonstratively) true, based upon past instances (individual numbers 0, 1, 2, 3, … n) and a recursive definition (of an “individual number”, integer), that n+1 (the mathematical induction) holds for any number, that there is no number so great that nothing can be added to it. So induction as a general term allows us to establish two different types of truths. a ca l l f o r s c i e n t i f i c p u r i t y 121