## Wednesday, February 18, 2009

### Knowing Each and Knowing All

Hence, even if one prove of each kind of triangle that its angles are equal to two right angles, whether by means of the same or different proofs; still, as long as one treats separately equilateral, scalene, and isosceles, one does not yet know, except sophistically, that triangle has its angles equal to two right angles, nor does one yet know that triangle has this property commensurately and universally, even if there is no other species of triangle but these. For one does not know that triangle as such has this property, nor even that 'all' triangles have it-unless 'all' means 'each taken singly': if 'all' means 'as a whole class', then, though there be none in which one does not recognize this property, one does not know it of 'all triangles'.

Aristotle, Posterior Analytics, I.5. Likewise, one could show that 'able to laugh' or 'tool-using' applies to every kind of human being; this does not show that human beings are defined by these characteristics, even partially. Or to stay in mathematical bounds: defining something by induction does tell you something -- for instance, you can determine in this way each thing that is a natural number. But this doesn't tell you what a natural number is; you learn what counts as a natural number, but even if you've covered them all, you aren't simply by that fact any more enlightened about what it is to be a natural number. The two are closely related, of course, and one can lead to the other, but conflating them is sophistry -- related, I think, to what Aristotle elsewhere calls the sophistical mistake of thinking that to know is to possess knowledge (we can know things, which is an act of acquaintance, without possessing knowledge of them, which is a disposition of being able to draw conclusions based on understanding of the things we know).