(no subject)
Humans existed for an inordinately long time before they discovered there was such a thing as mind/body integration, ie, that there exist a way to relate abstractions to concretes. It is a category difference similar to the difference between ethics and meta-ethics, one studies the thing while the other studies the idea about the thing. Before Thales, no one had pointed out that the universe is knowable, while, all the while, one hundred percent of the inductive evidence supported the idea that the universe is knowable. Abstractions and concretes are catagorically different, they are incommensurable. Anytime that you can honestly relate empirical evidence with the concept 'all' or its equivalent you can claim certainty. Using the same inductive data Pythagoras, on Thales heel, reversed Thales progress. "The oriental mysticism of Pythagoras, however, reversed this state of affairs and gave to mathematics a supra-sensuous reality of which the world of appearances was a counterpart." Boyer's hist of the calculus. Thales and Pythagoras were both focused on the idea that numbers could be used to prove things. Thales unified his deductions (proofs) on empirical (natural) evidence while Pythagoras did the opposite.
Numbers are so basic to our thought process that the analogy of a fish and water comes to mind. Rand has explained how mathematics and epistemology are related in her theory of concepts. All concepts are based on the idea that the mind can abstract from particular things which have particular identities by relating 'thing' to 'unit'. All concepts consist of real units whether we think of them as things or units. Things exist in relation to other things and all relations are messy, ie they have more than one perspective. A unit is a blend, a unification of two perspectives which only implies their respective differences while naming what is the same. There is nothing subjective or other worldly about this process, it is all objective.
The proof that exposes Pythagoras as a fraud is his inability to understand incommensurablity. Incommensurability is basic to concept formation because the first integrations are inductive in nature, ie messy. The fact that there exist both rational and irrational numbers and that we have learned how to integrate them can be used as proof for either mathematics or epistemology, it depends on your focus. Cardinality applies to concepts in general, exactly as it does to numbers in general.
Numbers are so basic to our thought process that the analogy of a fish and water comes to mind. Rand has explained how mathematics and epistemology are related in her theory of concepts. All concepts are based on the idea that the mind can abstract from particular things which have particular identities by relating 'thing' to 'unit'. All concepts consist of real units whether we think of them as things or units. Things exist in relation to other things and all relations are messy, ie they have more than one perspective. A unit is a blend, a unification of two perspectives which only implies their respective differences while naming what is the same. There is nothing subjective or other worldly about this process, it is all objective.
The proof that exposes Pythagoras as a fraud is his inability to understand incommensurablity. Incommensurability is basic to concept formation because the first integrations are inductive in nature, ie messy. The fact that there exist both rational and irrational numbers and that we have learned how to integrate them can be used as proof for either mathematics or epistemology, it depends on your focus. Cardinality applies to concepts in general, exactly as it does to numbers in general.