Cambridge Foundations, 3

I was particularly looking forward to two of the talks scheduled for the last day of the Foundations conference, those by Steven Methven and Dan Isaacson. Sadly, for different reasons, neither happened. That left just three talks. Sam Sanders spoke about his research on the reverse mathematics for a certain system of non-standard analysis. You could get a few glimmers of what might be interesting in this area from his talk, but it was hopelessly ill-judged for the occasion: but visit Sam’s web-pages if you want to know more.

Alan Weir also gave far-too-rushed a tour, this time through some of the themes of his book Truth Through Proof which I’ve blogged about at length: I think it is fair to say there was nothing new here.

The remaining session, though, was a very nice review by Mary Leng of some points of difference between the sort of fictionalism that she favours and the kind of formalism that Alan defends. On the principle ‘the enemy of my Platonist enemy is my friend’, Mary found herself wanting to minimize some apparent big divergences between their positions. But the crunch disagreement, for her, comes over their respective views about logical consequence. The fictionalist talks of inferences within the fiction, the formalist of inferences within the formal game (putting it roughly). But what makes for ‘good’ patterns of inference in either case? Mary thinks she has to appeal to a primitive logical modality here. Alan eschews such modalities, but then — Mary argued — has nothing helpful to say about logical consequence (unless he smuggles a modality in the back door). Alan of course thinks otherwise. Like a Catholic bemused by warring factions of Protestants, I’ll leave them to battle that one out …. But I think Mary probably does jab her finger on a sore spot for the formalist.

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