I’ve just revised my paper on Euclidean Geometry and the A Priori. According to my statistics, the previous version has been downloaded a good few times, so apparently there is some interest regarding this topic. It might still require some further work, but my argument should be easier to follow now — the paper previously suffered from a lack of a clear target. My target is effectively contemporary views on a priori justification, such as the ones familiar from Albert Casullo and perhaps Laurence BonJour. I don’t tackle their views in any detail, but rather argue generally against the idea that a priori justification and knowledge are empirically defeasible, that is, the idea that further empirical evidence could defeat a priori justification. The single most influential problem for the opposing view, namely that such empirical defeaters are not possible, is the case of Euclidean geometry. A fairly commonly accepted view at the moment is that Euclidean geometry was indeed justified a priori, but once it turned out that the actual geometry is Riemannian rather than Euclidean, the original justification was defeated by empirical evidence. My argument can be outlined as follows:

1. A priori justification (and knowledge) is empirically indefeasible.
2. Cases like Euclidean geometry appear to suggest that this is not the case, hence the commonly accepted conception of a priori justification takes it to be empirically defeasible.
3. I want to keep (1), so (2) needs to be addressed somehow.
4. To do this, we must distinguish between the apriority of a proposition and the truth of a proposition.
5. Even though the judgements that we make concerning the truth of a proposition are empirically defeasible, a priori justification is not.
6. A priori justification concerns the metaphysical possibility of the proposition.

In motivating my case, I use some material which I developed in my draft ‘The Notion of Logical Truth’; mainly in the form of an analogy between the distinction of pure/applied geometry and truth in a model/truth in the world. Anyway, see the actual paper for the details of the argument. The ideas that I develop here go back to my very first published journal article, ‘A New Definition of A Priori Knowledge: In Search of a Modal Basis’, where I used Euclidean geometry as an example. This time I’ve actually looked at the details of Euclidean geometry to back up my case.

1. makislav says:

   August 4, 2010 at 18:47

   Thanks very much for this. Your paper gave courage to me because I’m trying to develop a same kind of distinction between a priori an a posteriori.

   Your concepts of pure and applied geometry and truth in a model and truth in the world sound really accurate. Have you invented them by yourself?

   They come really close to Leibniz “truth of reason” and “truth of fact” as well as Hume’s fork’s “relations of ideas” and “matters of fact” and A. J. Ayer’s “formally truth propositions” and “empirical hypothesis”. These views pretty much share the same content but I was glad to find out a more contemporary verification of the defence of the distinction from you (especially after reading Quine, who did not, in my opinion, go trough the best arguments for the notion of a priori in his “Two Dogmas of Empiricism”).

   One thing came to my mind: what do you think about the role of tautologies in a priori? To A. J. Ayer this was extremely central in his Language, Truth and Logic where he typified formal logic and pure mathematics and geometry as a system of tautology where we can move to next step deductively. This meant that we would not have any knowledge concerning the world a priori because tautologies do not tell us anything about the reality (“2x=10″, “A and A”, “Tomorrow it may rain or it may not rain” eg.) At least to me this kind of usage of tautologies seems very central in aprioriness.

   But: Keep on going with your research. I have been inspired by following your work (especially about the principle of non-contradiction and the a priori) and I will continue to do it in the future.

   Cheers,
   Matias
2. Tuomas says:

   August 5, 2010 at 09:54

   Thanks for the comments Matias.

   I wish I could take credit for the pure/applied geometry and truth in a model/truth in the world distinctions, but they have been around for a while. For some good discussion of the previous notion, I recommend Graham Priest’s Doubt Truth to be a Liar (ch. 12). The latter distinction comes from Davidson (1973), who talks about absolute truth and truth in a model in his ‘In Defense of Convention T’, although the manner in which I use them is closer to John Etchemendy’s definition in his 1990 The Concept of Logical Consequence. The ideas behind such distinctions are much older though, as you note yourself. Well, in any case, I know of no other attempts except my own to use these notions for a new definition of apriority, although already Russell talked about these issues.

   I should remind myself of Ayer’s work, but for reason or another his name doesn’t pop up much in the contemporary discussion. The role of tautologies in all this is an interesting topic, but I suspect that I disagree with Ayer about this. It seems to me that both logic and mathematics do tell us something about reality, although it may be that this is only apparent when they are applied to the world. Still, I consider both of them as synthetic rather than analytic, whereas tautologies would appear to be analytic. In short, I do think that all tautologies are a priori, but not that all a priori truths are tautologies. Having said that, you may very well be right that there is at least an operational similarity between the distinctions discussed above and those of Ayer’s — I’ll have to look his stuff up again though to be able to tell.

   I appreciate your interest towards my work. In fact, there is a potential project on the law of non-contradiction which I’m currently looking into.. stay tuned!