Aristotle Posterior Analytics 74a18-75b4

• And it might seem that proportion alternates for things as numbers and as lines and as solids and as times--as once it used to be proved separately, though it is possible for it to be proved of all cases by a single demonstration. But because all these things--numbers, lengths, times, solids--do constitute a single named item and differ in sort from one another, it used to be taken separately. But now it is proved universally; for it did not belong to things as lines or as numbers, but as this which they suppose to belong universally.
  For this reason, even if you prove of each triangle, either by one or by different demonstrations that each has two right angles--separately of the equilateral and the scalene and the isosceles--you do not yet know of the triangle that it has two right angles, except in the sophistic fashion, nor do you know it of triangle universally, not even if there is no other triangle apart from these. For you do not know it of the triangle as triangle, nor even of every triangle (except in respect of number; but not of every one in resepct of sort, even if there is none of which you do not know it.)
  So when do you not know universally, and when do you know simpliciter? Well, clearly you would know simpliciter if it were the same thing to be a triangle and to be equilateral (either for each or for all). But if it is not the same but different, and it belongs as triangle, you do not know. Does it belong as triangle or as isosceles? And when does it belong in virtue of this as primitive? And of what does the demonstration hold universally? Clearly whenever after abstraction it belongs primitively--e.g. two right angles will belong to bronze isosceles triangle, but also when being bronze and being isosceles have been abstracted. But not when figure or limit have been. But they are not the first. Then what is first? If triangle, it is in virtue of this that it also belongs to the others, and it is of this that the demonstration holds universally.

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• Aristotle Posterior Analytics 74a18-75b4
• To analyze this passage, I start with the second paragraph, because I think I understand it.
• Then I looked back to the first paragraph to see if it made better sense in light of the second.
• Then I analyzed the third paragraph.
• Sometimes the order of analysis has to start with what you understand rather than with what Aristotle does! (as Aristotle himself points out)
• Second paragraph:
  
• For this reason, even if you prove of each triangle, either by one or by different demonstrations that each has two right angles--separately of the equilateral and the scalene and the isosceles--you do not yet know of the triangle that it has two right angles, except in the sophistic fashion, nor do you know it of triangle universally, not even if there is no other triangle apart from these. For you do not know it of the triangle as triangle, nor even of every triangle (except in respect of number; but not of every one in respect of sort, even if there is none of which you do not know it.)
• In order to know that triangles have two right angles (i.e. that the corner angles add up to 180 degrees), you need to know it of the triangle universally.
• You can prove it of the equilateral, the scalene, and the isosceles triangle, and that is worth doing, but it is not scientific knowledge, for it only shows piecemeal what needs to be shown more generally in order to have the true explanation.
• This makes sense. Think about the following examples:
  
• we might know things about how various different things burn, how animals die in a sealed jar, how they live if you add a plant, etc., but until you know that Oxygen is the causal factor, you don't really know the scientific explanation that unites it all.
• we might know that the moon is connected to the tides in a predictable fashion, that things fall downward, and that the earth circles the sun, but until you learn about gravity, you do not know the scientific explanation for those things.
  
• You need to know the most general level of explanation that fully accounts for a phenomenon to know it scientifically.
  
• If you prove of every sort of triangle that each one individually has angles =180, Aristotle seems to think you do not yet know that of triangle. Why? Because you have not found one unifying explanation. You have found piecemeal explanations. You know it of scalenes. You know it of rights. You know it of isosceles. You know it of equilateral.
• First Paragraph:
• And it might seem that proportion alternates for things as numbers and as lines and as solids and as times--as once it used to be proved separately, though it is possible for it to be proved of all cases by a single demonstration. But because all these things--numbers, lengths, times, solids--do constitute a single named item and differ in sort from one another, it used to be taken separately. But now it is proved universally; for it did not belong to things as lines or as numbers, but as this which they suppose to belong universally.
• Since the second paragraph starts out "For this reason,...", I know that the first paragraph is meant to give a reason why the second paragraph makes sense somehow. So I have a good clue about the first.
• What does it mean for "proportion" to "alternate"? I don't know. Do I need to know? Maybe, maybe not. Let's see.
• I know that alternating proportion applies to four things: numbers, lines, solids, and times.
• I know that "proportion alternates" used to be proved of each of those thing separately.
• Sounds like equilateral, scalene, isosceles, etc., triangles: it used to be proven of each separately that their angles =180.
• I know that now it is proven by one single demonstration.
• And I know that numbers, lines, solids, and times somehow fit under one genus according to Aristotle.
• So what Aristotle is claiming here is that proportional alternation (whatever that is) applies to numbers, lines, solids, and times, but not to each separately. Rather proportional alternation belongs to them all qua species of one genus and that there is one general explanation for proportional alternation in that genus.
• So what is proportional alternation?
• as A is to B as B is to C as C is to D
• and therefore as A is to C, so B is to D.
• A:B :: B:C :: C:D and therefore A:C :: B:D
• use 2, 4, 8, 16.
• it does not matter if it is 2 minutes, 2 miles, 2 gallons, or "just 2."
  
• So when do you not know universally, and when do you know simpliciter? Well, clearly you would know simpliciter if it were the same thing to be a triangle and to be equilateral (either for each or for all). But if it is not the same but different, and it belongs as triangle, you do not know. Does it belong as triangle or as isosceles? And when does it belong in virtue of this as primitive? And of what does the demonstration hold universally? Clearly whenever after abstraction it belongs primitively--e.g. two right angles will belong to bronze isosceles triangle, but also when being bronze and being isosceles have been abstracted. But not when figure or limit have been. But they are not the first. Then what is first? If triangle, it is in virtue of this that it also belongs to the others, and it is of this that the demonstration holds universally.
• Aristotle wants to drive the nail in here.
  
• You know universally (aka you know simpliciter, aka you know without qualification), when your explanation of a phenomenon rises to the highest level of generality possible.
  
• I.e. Aristotle would wholly approve of the idea of a unified field theory as the goal to strive toward.
• He goes thru the triangle example again.
• First off, you have to begin with the full knowledge that triangles have angles = 180, and there is no more general level above this, so an explanation of why triangles have angles that = 180 is knowledge without qualification.
• How can Aristotle require that? Doesn't that require that you have knowledge before you can know that you have knowledge, or perhaps that you know you have knowledge before you have knowledge?
• NO, because Aristotle is using this as an example. He needs an example that is clear. He needs an example of a case where we are rather sure that we know something, so that he can explain the theory behind why we consider that knowledge but not other things that are very similar.
  
• In other words, Aristotle is trying to explain the theory of knowledge, not a particular bit of knowledge.
  
• So, returning to the example, Suppose you know of an equilateral triangle that its angles =180.
• IF the only kind of triangle were the equilateral, then you would have knowledge without qualification that a triangle's angles = 180.
• But if it's the case that there are other sorts of triangles, then you don't have knowledge that a triangle's angles = 180 until you know that =180 applies to all of them qua triangles and not qua isosceles or qua equilateral or qua scalene, etc.
  
• When you know (i.e. have the explanation of why) =180 applies to all triangles qua triangles, then you know that =180 applies to triangles primitively (aka primarily, aka immediately).
• You can do various sort of mental tests to see if you have a primitive explanation: try abstracting away from the thing you are explaining various aspects of it:
• take a bronze isosceles triangle
• its angles = 180 considered qua bronze isosceles triangle, right?
• do its angles =180 considered just qua isosceles triangle? YES, so bronze is not a factor: it's not just bronze triangles or just bronze isosceles ones.
• do its angles = 180 considered just qua triangle? YES, so isosceles too is not a factor.
• can we abstract away anything more?
• do its angles = 180 considered just qua closed shape? NO. There are many closed shapes whose angles >180.
• anything else we can consider abstracting away? I can't think of anything...
• Therefore mybest candidate for the "first," the "primitive" thing that =180 applies to is triangle qua triangle.
• SO, here's an important observation about Aristotelian scientific knowledge: it is not enough to know for a fact that something is the case. You have to know WHY, and you also have to know WHY at the most general level possible.