PA Alpha, chapter 5 February 8, 2006
Posted by Michelle in Aristotle, Primary Source.trackback
PA Book 1, chapter 5
We often make mistakes. What we try to approve does not hold primitively and universally, even though we think we are proving it universally and primitively.
We make mistakes:
1) when there is nothing higher we can take apart from a particular case;
2) when there is something higher, but it is nameless and covers objects of different forms
3) when the proof applies to something that is a partial whole
Example of 1: If the only triangles we were aware of were isosceles, we might believe 2R held universally of the isosceles because we have no concept of triangle.
Example of 2: Aristotle is referring to a universal mathematics – an allusion to Eudoxian generalization.
Example of 3: If we observe that all perpendiculars are parallel and we wrongly inter that this is what we demonstrate.
When do you not know universally but do know simpliciter?
* you would know simpliciter if it was the same thing to be a triangle and be equilateral. But if it is different and if something holds of them as a triangle, then you don’t know it.
To what does the demonstration apply universally?
* to the first item after the removal of which it doesn’t hold.
— for example, 2R holds of a bronze triangle. When ‘bronze’ is removed, it is still a triangle. But when ‘figure’ is removed, it is no longer a triangle.
But ‘figure’ and ‘limit’ aren’t the first (item after the removal of which…), rather triangle is the first.
‘first’ means first terms after whose abstraction R fails to hold. The thing that holds first holds primitively; so the abstraction colds primitively of ‘triangle’.
