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September 12, 2013

Good Mathematics

Posted by David Corfield

I’m speaking next week at a conference in London addressing the issue of values in mathematics. The thrust of my talk will be that any satisfactory discussion of this topic must include the very largest units of assessment, long-term research programmes, and these in turn can only be assessed in terms of the place they come to occupy in the history of the subject.

I’ll take this quotation from Alasdair MacIntyre as my point of departure:

Let me cast the point I am trying to make about Galileo in a way which, at first sight, is perhaps paradoxical. We are apt to suppose that because Galileo was a peculiarly great scientist, therefore he has his own peculiar place in the history of science. I am suggesting instead that it is because of his peculiarly important place in the history of science that he is accounted a peculiarly great scientist. The criterion of a successful theory is that it enables us to understand its predecessors in a newly intelligible way. It, at one and the same time, enables us to understand why its predecessors have to be rejected or modified and also why, without and before its illumination, past theory could have remained credible. It introduces new standards for evaluating the past. It recasts the narrative which constitutes the continuous reconstruction of the scientific tradition. (The Tasks of Philosophy, p. 11)

Posted at September 12, 2013 10:01 AM UTC

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Re: Good Mathematics

While your mention of Galileo is just tangential, I feel mildly obliged to link to this post at the Renaissance Mathematicus blog, one of many where the blog author reacts to scientists’ mention of Galileo much as water reacts to sodium…

Posted by: Yemon Choi on September 12, 2013 8:05 PM | Permalink | Reply to this

Re: Good Mathematics

Water reacts to sodium with run-on sentences?

Posted by: Mark Meckes on September 13, 2013 5:47 AM | Permalink | Reply to this

Re: Good Mathematics

I think that the common concept is flame.

Posted by: Toby Bartels on September 23, 2013 12:10 AM | Permalink | Reply to this

Re: Good Mathematics

Well I’m sure MacIntyre would want the appearance of Galileo in any history of science to discuss his failings, and in particular failings of any intellectual virtues. But even so, he will appear in any history of science.


his adherence to an antiquated theory held him back, an adherence that he maintained although he must have known that it was wrong

line of argument I find questionable. I don’t thing we should underestimate the power of the inherited metaphysical baggage that underpins a scientist’s work.

I have an idea from somewhere that Galileo’s fixation on circles for planetary motion fed into his observations of the moon, especially that bizarrely large circular crater he drew. Would we tack on, even here in the case of reasonably direct observation, that he drew this crater “although he must have known that it was wrong”?

Presumably another reason beyond neo-platonism for his attachment to circles was that he had had some success decomposing projectile motion into circular inertia and motion towards the centre of the Earth.

Posted by: David Corfield on September 13, 2013 9:33 AM | Permalink | Reply to this

Re: Good Mathematics

Not convinced by the idea that a successful theory is one that renders its predecessors intelligible. Rendering ones predecessors unintelligible is also a good strategy: for example, the sixteenth-century humanists destroyed a flourishing logical culture, and greatly dumbed down the teaching of logic in universities, with the result that they could say anything they liked about the (supposedly) appallingly obscure philosophy of their predecessors. Logic took until the mid-nineteenth century to recover, with a few exceptions like Leibnitz who actually read medieval logic.

The main problem with this position seems to me that it’s not clear how normative it’s supposed to be. Is the relation of a successful theory to its predecessors meant to be a concept like “has defeated its predecessors in the historically existing forms of argument”, which is not normative, or is it meant to be something like “explains its predecessors”, which is normative? If we want a historical story that rationalises modern science, then we want normative accounts: but, prima facie, all that history gives us is non-normative stories about causation. But what is the concept of an “important place in the history of science” supposed to be? Of course, Galileo figures in the hagiography. But is he more important than, for example, Oresme?

Posted by: Graham White on September 22, 2013 9:58 PM | Permalink | Reply to this

Re: Good Mathematics

If a false theory displaces a true one, then I suppose it could be considered “successful” in a Darwinian sense, but I would think not really in a scientific sense.

Posted by: Mike Shulman on September 23, 2013 12:54 AM | Permalink | Reply to this

Re: Good Mathematics

As an Aristotelian-Thomist who believes that tradition was sorely misrepresented, MacIntyre is unlikely to disagree with you.

He places a heavy weight on the intellectual virtues as required by a tradition of enquiry in order to flourish. The treatment of mediaeval logic by humanism as you describe, if it is one of insufficient attention and misrepresentation, would involve failings of justice.

Important moments in a tradition of enquiry are marked by larger changes in its continually reconstructed narrative. MacIntyre requires such a narrative to be ‘true’, a term that clearly needs some cashing out, but which clearly excludes deliberate misrepresentation.

He also points to another virtue, one it seems rarely displayed, where it is incumbent upon researchers to be very explicit about what frustrations in pursuing their goals they are currently experiencing. You should facilitate the surpassing of your current state, by other traditions if it transpires that, unlike you, they possess the resources to do so. Of course, all this recognises that intelligibility between traditions is hard-won, though efforts to achieve this should not be left merely to chance.

Posted by: David Corfield on September 23, 2013 8:54 AM | Permalink | Reply to this

Re: Good Mathematics

By the way, I see from your site, Graham, that our interests overlap.


These systems are typed, and the types represent contexts for action (propositions of those types represent knowledge explicitly available at those contexts).


Much of this work is done in typed logics, where the types are to be viewed as contexts either for action, for reasoning, or for the interpretation of language. I view all of these activities – action, reasoning, and interpreting language – as strongly contextual: contextuality gives an explanation of the way in which there seems to be a distinct lack of overt content which could reliably ground any of this stuff.

I’m interested in seeing whether traditional philosophical topics can be illuminated by typed systems, e.g., presuppositions.

Posted by: David Corfield on September 23, 2013 9:43 AM | Permalink | Reply to this

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