from Wiktionary, Creative Commons Attribution/Share-Alike License
- adj. Of, pertaining to or based on enumeration
from the GNU version of the Collaborative International Dictionary of English
- adj. Counting, or reckoning up, one by one.
from The Century Dictionary and Cyclopedia
- Serving to enumerate; counting; reckoning up.
Months, years, and entire character transformations are dispatched in a few enumerative sentences.
A case of sleeping sickness studied by precise enumerative methods: Further observations.
So it is possible to argue that enumerative induction is unjustified while simultaneously agreeing that mathematicians are rational to believe GC on the basis of the available evidence.
As Frege's remarks suggest, one way to underpin an argument against the use of enumerative induction in mathematics is via some sort of non-uniformity principle: in the absence of proof, we should not expect numbers (in general) to share any interesting properties.
It would seem to follow from this principle that enumerative induction is unjustified, since we should not expect (finite) samples from the totality of natural numbers to be indicative of universal properties.
(To what extent enumerative induction plays a role in the discovery of new hypotheses, or in the choice of what open problems mathematicians decide to work on, is a separate issue which has not been addressed here.)
The graph suggests that the hardest test cases for GC.are likely to occur among the smallest numbers; hence the inductive sample for GC.is biased, but it is biased against the chances of GC. Mathematicians 'confidence in the truth of GC.is not based purely on enumerative induction.
One distinctive feature of the mathematical case which may make a difference to the justificatory power of enumerative induction is the importance of order.
Many less theoretical instances of inductive reasoning also fail to be captured by enumerative induction.
The point is that a full account of inductive logic should not be limited to enumerative induction, but should also explicate the logic of hypothetical reasoning through which hypotheses and theories are tested on the basis of their predictions about specific observations.
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