6. Formalism (V); Session Dec. 19, 2014

6.1 Changed plans a bit again. Since it’s our last session this year it’s perhaps useful to summarize what we have so far.

6.2 I also want to give some ideas of practical application of what we already learned from Aristotle, before we continue in January.

6.3 We said that the Syllogism of Things depends on the Syllogism of Words (5.28 f.; see 3.10.7 ff.).

6.4 We use words to classify, sort and explain the natural world. Scientific knowledge, for Aristotle then, is explanatory understanding:

6.5 That is, not merely to ‘know’ a fact incidentally & to agree that something is true, but to know ‘why’ it is the case.

6.6 And this, Aristotle thinks, can be done syllogistically: All humans are animals; all animals are mortal; so all humans are mortal.

6.7 This, of course, only works when you have sorted out the right reference class, viz. the right natural kind terms, for your deduction.

6.8 Here the distinction between genus & species comes in again (5.15, 5.21, I was a little hopplahopp there, so here’s what it means:)

[In what follows, I pick up some ideas on reduction/reductionism from a discussion with Anita Leirfall (Bergen University/Norway)].

6.9 Take the category substance/essence again (cf. 5.14), and the question of what it is for something, say F, to exists.

6.10 For Aristotle, you answer it by placing Fs in their category, C, and apply to the Fs the general account of what it is for C to exist.

6.11 Or even more generally: Check whether F falls under C. That’s what lawyers call a subsumption.

6.12 Two problems: (1) You hardly can avoid regress. Aristotle, however, rejects that there are infinitely many predicates.

[Cf. Posterior Analytics, Bk I, chs. 19-22 ].

6.13 Aristotle’s Categories, which are meant to terminate regress, are somewhat the conclusion of a somewhat ‘decisionistic’ procedure.

6.14: (2) Same problem in other gown: When we reduce an item F to another item, say, G (Aristotle calls them accidents, as opposed to categories).

6.15 All reductions face the difficulty that we hardly ever can say that Fs reduce to Gs, or to some Category, without being caught in regresses or logical circles.

6.16 This is especially true for so-called ontological reductions to ‘abstract entities’ (5.21).

6.17 Take mathematical objects (numbers, for instance). Aristotle says (Loeb edition; see 3.8.2):

“[I]f mathematical objects exist, they must be either in sensible things […], or separate from them […]; or if they are neither the one nor the other, either they do not exist at all, or they exist in some other way. Thus the point which we shall have to discuss is concerned not with their existence, but with the mode of their existence;” Metaphysics 1076a32-36.

6.18 The last sentence is certainly a stunning claim. What are we to make of it?

6.19 We skip for now the debate about mathematical Naturalism (Mill), Intuitionism (Hilbert, Bernays), Platonism (Frege), or Nominalism (Hartry Field).

6.20 And turn instead to Aristotle’s equally stunning explanation: Mathematical Objects are what mathematicians say they are (my words; cf. Metaphysics 1077b32 f.).

6.21 That’s not as silly as it seems. Recall 6.4: We use words to classify, sort and explain the natural world.

6.22 This is even truer for abstract entities, such as mathematical objects.

6.23 Hence it’s not so much a question whether certain things exist in themselves, as it were, but rather how our reason takes them to be.

6.24 This amounts neither to relativism, nor to idealism.

6.25 The problem can be exemplified by the highly contested conception of Conventionalism in the philosophy of science:

6.26 In the otherwise most instructive Cambridge Companion to Aristotle we find the following sentence (1995, 110 f.):

“The world divides (realistically, and not as a matter of mere convention) into natural kinds, and those kinds stand in relations of greater and lesser resemblance to and distinction from one another” (original parenthesis).

6.27 Did you notice the problem? The world as such certainly does not divide into natural kinds. But it is us who use natural kind terms.

6.28 It’s us who do the dividing & designating, as Aristotle himself acknowledges, when we take his categories as predicables (5.12).

6.29 The same for resemblance relations: They are the work of thought, too, viz. of sorting the world around us.

6.30 Conventionalism was en vogue in the 1930s & was advanced esp. by the Logical Positivists in the aftermath of non-Euclidean geometry.

6.31 It was Quine’s ‘Truth by Convention’ (1935) which then turned the tide. But it rested on a grave misunderstanding.

 [Reprint in: W.v.O. Quine, The Ways of Paradox, HUP 1976, ch. 11]

6.32 For nobody ever talked about ‘Truth’ by convention: Not Henri Poincaré, nor Hans Reichenbach, or his follower Adolf Grünbaum in the 1960/70s.

[An instructive recent treatment of the subject is: Yemima Ben-Menahem, Conventionalism, CUP 2006]

6.33 Of course, the world does not care how we talk about it. But only with concepts there come judgments.

6.34 And it’s only with concepts & judgments that we can be said to have the idea of an objective, independent world in the first place.

6.35 That, and not the conventionality of ‘truth,’ was the point of the Logical Positivists.

[Unfortunately, the essays in Michael Friedman’s Reconsidering Logical Positivism, CUP 1999, have had a large impact, especially on US-American Philosophy of Science, and cemented Quine’s misconception. Not to mention Friedman’s obscure ‘relativized a priori,’ which was equally influential, but unfortunate for philosophical coherence reasons. We shall say a bit more about this at a later stage].

6.36 Is that a circle again? Well, yes, of a sort. But it’s not vicious. It’s rather hermeneutic.

6.37 The question then is not whether there are conventions in science, but rather what conventions there are & whether they are useful.

6.38 Let’s leave it at that for today. It will be part of our program too.

6.39 Together with a theory of the relationship between the Syllogism of Words & the Syllogism of Things.

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