How does Aristotle recommend that we come to know the underlying truths of reality that are hidden by the changing flux of perception?

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In some ways, it seems as though this question addresses Plato more than Aristotle. For Aristotle, although the senses can be misleading, they are not to be distrusted absolutely in the same way that they are in Plato, nor does Aristotelian metaphysics oppose flux to knowledge. Instead, Aristotle categorizes knowledge into a hierarchy, with different things knowable to different degrees.

The truths of mathematics, as they do not depend on anything contingent, can be known absolutely and with certainty. They are by their nature unambiguous and can be discovered by reason. The laws of logic, which are similarly independent of contingency, can also be discovered by reason. We know the formal, if not material truths of syllogisms, by the pattern of their reasoning. 

While some parts of metaphysics can be known by pure reason, most of our knowledge of the natural world and its regular patterns is obtained by applying logic to sensation.

Moral wisdom (phronesis) and practical arts (techne) are skills which we can develop, but are only known probabilistically rather than with certainty.

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