Numbers have properties and it is possible to see them in many instances. Two chairs, two mice. How is this possible? A short explanation is that we have analogies. A class of analogies are “twos”, “threes”, also “trees”, “chairs” etc. We usually cannot see analogies, because they are already always there, and are, I believe, the most primitive aspects of cognition. We usually cannot see them until we put in a form where we contrast them, for example, the form, “A is to B, as C is to D”, (A:B::C:D) eg two cheeses : three mice :: two carrots : three rabbits. Here we had to understand the analogy 2, the analogy 3, the analogy of eating and the analogy of animals and a higher order analogy all of these elements had to one another.

With the two species of inference, in order to understand claims which require inductive support, such as “All Carrots are Orange” we need analogy of ‘carrot” and the analogy “orange” along with understanding operators like “all” and “are”, and in order to understand claims which are deductively supported such as “All even numbers are divisable by two” we need to understand the analogy “two” with the analogy “even number” (ie skipping a number starting at n=0) along with operators “all” and “divisable”. Seen this way, induction and deduction only differ in that deduction occurs where we have a function for producing instances of everything to which the analogies apply. In an arithmetic, to take one example, this would be Peano’s 6th axiom, the successor function. In contrast to deduction, induction has no successor function, for one physical reason or another, and thus requires observation. In the case of Hume’s famous problem with induction, this is the reason we cannot move from induction to a deductive justification. The difference between a deductive system is that we have something like a successor function which produces all instances of the analogies we are reasoning about, where as we cannot produce ever produce all instances of rabbits. Seen this way, deductive inference and inductive inference are the same in kind, and differ only in our capacity to generate instances of that which we are reasoning about. There is no a priori truths because there is no fact we have known without first constructing the analogies essential to it’s truth. This construction is an a a posteriori activity. To the extent the a priori does exist, it is characterised by our faculty for generating analogies. But this is inaccessible to reason since there are no “good” or “bad” analogies. Analogies just are.