| | Jens,
The foundations of math are arrived at inductively, not deductively. So are the foundations of logic.
For instance, in order to grasp the proposition that “One plus one equals two,” one must first have formed its constituent concepts. How does one form the concepts “one” and “two”? In the same way that one forms any other concept -- by observing various instances of the concept’s units and then abstracting their common feature – i.e., by observing that certain things bear a greater similarity to each other than they do to certain other thing(s) from which they’re being differentiated. In other words, one forms a concept by identifying similarity against a background of difference.
For example, one forms the concept ‘fruit’ by observing that (say) an apple and an orange bear a greater similarity to each other than either does to (say) a carrot or a beet. Similarly, one forms the concept ‘apple’ by observing that two different apples (say a MacIntosh and a Pippin) bear a greater similarity to each other than either does to (say) an orange or a pear.
The same principle applies in forming the concept of a particular number (say, two). One observes that a group of two oranges and a group of two apples, say, bear a greater numerical similarity to each other than either does to a group of three oranges or to a group of four apples. In so doing, one isolates what the groups of two have in common as against the other groups, and thereby forms the abstraction ‘two’, which one then designates by the visual-auditory symbol “two” or “2.” The same principle of concept formation pertains to numerical concepts as to any other concept.
Having formed the concepts “one” and “two” along with the concepts of ‘addition’ and ‘equality,” one can then grasp the proposition “One plus one equals two.”
The laws of logic are also discovered by observing reality – by observing that existence is non-contradictory – that existence is identity. Aristotelian philosopher H.W.B. Joseph makes this point as follows: We cannot think contradictory propositions, because we see that a thing cannot have at once and not have the same character; and the so-called necessity of thought is really the apprehension of a necessity in the being of things. This we may see if we ask what would follow, were it a necessity of thought only; for then, while e.g. I could not think at once that this page is and is not white, the page itself might at once be white and not white. But to admit this is to admit that I can think the page to have and not have the same character, in the very act of saying that I cannot think it; and this is self-contradictory. The Law of Contradiction then is metaphysical or ontological. So also is the Law of Identity. It is because what is must be determinately what it is, that I must so think. (An Introduction to Logic, Oxford University Press, p. 13.)
There is no analytic/synthetic dichotomy. All concepts and all propositions (whether logical or mathematical) are ultimately arrived at by observing concrete reality, and then applying reason to the evidence of the senses. The discovery of relations between mathematical concepts and their implications can be arrived at deductively, but the process of deduction must ultimately be based on inductive generalizations and empirical observation.
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