### More about MacIntyre

In the previous post I mentioned how MacIntyre’s philosophy was considerably more integrated than is the norm in contemporary English-language philosophy, and that it is integrated around ethics. ‘Epistemological Crises, Dramatic Narrative and the Philosophy of Science’ makes an analogy between individuals facing a moral crisis and those engaged in communal forms of rational enquiry facing an intellectual crisis. It opens by considering the crisis Hamlet confronts on his return to the Danish court, and what it would mean to resolve this crisis:

MacIntyre poses the problem of someone in Ireland in 1700 who is able to inhabit both the community of indigenous Irish, and also the English community of plantation owners. To one group it must speak of Doire Colmcille - St. Columba’s oak grove – which "names – embodies a communal intention of naming – a place with a continuous identity ever since it became in fact St. Columba’s oak grove in 546". To the other it refers to Londonderry which "names a settlement made only in the seventeenth century and is a name whose use presupposes the legitimacy of that settlement and of the English language to name it" (p. 7). Londonderry was a plantation enforced on the Irish population by the English with the foreign concept of individual property rights – "what is from one point of view an original act of acquisition, of what had so far belonged to nobody and therefore of what remained available to become only now someone’s private property, will be from the other point of view the illegitimate seizure of what had so far belonged to nobody because it is what cannot ever be made into private property – for example, common land." (8)

MacIntyre’s point is that these two communities hold deeply incommensurable views about justice and the rationality of actions, and that someone coming to see this, and failing to find a neutral position from which to judge them, might well lapse into a form of relativism. One solution he rejects is that with modern languages we are in a happier position. To us in the early twenty-first century, used to reading accounts of people from different times and places, with all manner of practices and belief systems, we believe we can readily understand each side, and understand their differences. The Irish of 1700 didn’t have the concept of ‘right’. The English didn’t understand the sacred importance of the spot for the Irish. But we can congratulate ourselves today for having a language into which we can render the positions of the two peoples. However, MacIntyre then undermines our triumph by pointing out the obverse of possessing this flexible language: We can’t rationally settle major disagreements, since we don't share sufficient by way of background.

For MacIntyre, what is notable about modern languages is the lack of reference to canonical texts, and the ease of inter-translation suggestive of a presuppositionlessness. By contrast,

Now let us try to relate what we have seen so far to mathematics. Instead of someone torn between the two communities of early eighteenth century Ireland, imagine a Chinese mathematician, thoroughly versed in the Nine Chapters and other classic texts, who in 1600 comes to understand Euclidean geometry from Jesuit missionaries. They would surely experience incommensurability as they attempted to decide for themselves the superiority of one of these very different conceptions of mathematics. (See Joseph W. Dauben, Ancient Chinese mathematics: The Jiu Zhang Suan Shu vs. Euclid’s Elements. Aspects of proof and the linguistic limits of knowledge, International Journal of Engineering Science 36 (1998) 1339-1359. See also this paper.) As a mathematical parallel to what MacIntyre conceives of as a modern language the obvious choice is set theory. Today we could render each variety of mathematics as set theory. But such flexibility comes at a cost. It can’t help us choose in which direction to proceed. Rewriting mathematics within set theory we can say that something has a correct proof or that something is calculated correctly, but we can’t say that it was worth proving or calculating in the first place.

For this we need stories, and, fortunately, the condition of modern mathematics is not one of presuppositionless set theoretic universalism, but rather is scored across by countless stories, which in printed form are easiest to locate in book reviews, such as those of the Bulletin of the American Mathematical Society, or in articles in the AMS Notices. Count the number of times Atiyah uses the words ‘story’ or ‘stories’ in his Mathematics in the 20th Century, Bulletin of the London Mathematical Society 34(1), 1-15, 2002. Turning your web browser to the Arxiv every morning most closely resembles tuning into a daily soap opera. It's not easy to discern the story lines from this tapestry of articles, but if you do succeed, you find that individual stories range through: daring quests for single gems using whatever resources can be found; Hausmannian reconstructions of higgledy-piggledy slums; beautiful irrigation projects using water from distant streams; attempted take-overs, and resistance to subsumption within a broader theory.

But with all these subplots is there any danger that mathematics itself may take on the attributes of the modern condition of endless, unresolvable debates about which direction to take? Not if the traditions represented by these stories are carefully maintained and passed on, and their advocates remain open to what other traditions can offer them. I mentioned several months ago that in a Clay Mathematics Institute interview, Terence Tao speaks of the importance of "being exposed to other philosophies of research, of exposition, and so forth". The danger to ward against is dissipation in a kind of heat death.

"Epistemological Crises, Dramatic Narrative and the Philosophy of Science," The Monist, 60 (1977), 453-72. Reprinted in Paradigms and Revolutions: Appraisals and Applications of Thomas Kuhn's Philosophy of Science, Gary Cutting, ed. (Notre Dame: University of Notre Dame Press, 1980) pp. 54-74. For excerpt see previous post.

When an epistemological crisis is resolved, it is by the construction of a new narrative which enables the agent to understand both how he or she could intelligibly have held his or her original beliefs and how he or she could have been so drastically misled by them. The narrative in terms of which he or she at first understood and ordered experiences is itself now made into the subject of an enlarged narrative. The agent has come to understand how the criteria of truth and understanding must be reformulated. He has had to become epistemologically self-conscious and at a certain point he may have come to acknowledge two conclusions: the first is that his new forms of understanding may themselves in turn come to be put in question at any time; the second is that, because in such crises the criteria of truth, intelligibility, and rationality may always themselves be put in question – as they are in Hamlet – we are never in a position to claim that now we possess the truth or now we are fully rational. The most that we can claim is that this is the best account which anyone has been able to give so far, and that our beliefs about what the marks of "a best account so far" are will themselves change in what are at present unpredictable ways. (p. 455)MacIntyre continues by contrasting Hamlet with Jane Austen’s

*Emma*, in which the eponymous heroine comes to realise the error of her interpretation of the social position of her friend Harriet, but does so only to arrive at what she conceives to be the right interpretation, that of Mr. Knightly. No suggestion is given in the book that this new view may later find itself challenged.Philosophers have customarily been Emmas and not Hamlets, except that in one respect they have often been even less perceptive than Emma. For Emma it becomes clear that her movement towards the truth necessarily had a moral dimension. Neither Plato nor Kant would have demurred. But the history of epistemology, like the history of ethics itself, is usually written as though it were not a moral narrative, that is, in fact as though it were not a narrative. For narrative requires an evaluative framework in which good or bad character help to produce unfortunate or happy outcomes. (p. 456).This was written in 1977 and seems to be much influenced by debates which took place in the philosophy of science in the late 60s and early 70s between Popper, Polanyi, Kuhn, Lakatos, and Feyerabend. This is the kind of connectivity that interests me. Not that science through its theories – genetics, cosmology, etc. – has a bearing on philosophical theses, but that a moral philosopher may learn from philosophers of science, as may a philosopher of mathematics from moral philosophers. Let’s see what we can glean from ‘Relativism, Power and Philosophy’ (details in previous post).

MacIntyre poses the problem of someone in Ireland in 1700 who is able to inhabit both the community of indigenous Irish, and also the English community of plantation owners. To one group it must speak of Doire Colmcille - St. Columba’s oak grove – which "names – embodies a communal intention of naming – a place with a continuous identity ever since it became in fact St. Columba’s oak grove in 546". To the other it refers to Londonderry which "names a settlement made only in the seventeenth century and is a name whose use presupposes the legitimacy of that settlement and of the English language to name it" (p. 7). Londonderry was a plantation enforced on the Irish population by the English with the foreign concept of individual property rights – "what is from one point of view an original act of acquisition, of what had so far belonged to nobody and therefore of what remained available to become only now someone’s private property, will be from the other point of view the illegitimate seizure of what had so far belonged to nobody because it is what cannot ever be made into private property – for example, common land." (8)

MacIntyre’s point is that these two communities hold deeply incommensurable views about justice and the rationality of actions, and that someone coming to see this, and failing to find a neutral position from which to judge them, might well lapse into a form of relativism. One solution he rejects is that with modern languages we are in a happier position. To us in the early twenty-first century, used to reading accounts of people from different times and places, with all manner of practices and belief systems, we believe we can readily understand each side, and understand their differences. The Irish of 1700 didn’t have the concept of ‘right’. The English didn’t understand the sacred importance of the spot for the Irish. But we can congratulate ourselves today for having a language into which we can render the positions of the two peoples. However, MacIntyre then undermines our triumph by pointing out the obverse of possessing this flexible language: We can’t rationally settle major disagreements, since we don't share sufficient by way of background.

For MacIntyre, what is notable about modern languages is the lack of reference to canonical texts, and the ease of inter-translation suggestive of a presuppositionlessness. By contrast,

The Attic Greek of the fifth and fourth centuries, the Latin of the twelfth to fourteenth centuries, the English, French, German and Latin of the seventeenth and eighteenth centuries were each of them neither as relatively presuppositionless in respect of key beliefs as the languages of modernity were to become, nor as closely tied in their use to the presuppositions of one single closely knit set of beliefs as some premodern languages are and have been… Such languages-in-use, we may note, have a wide enough range of canonical texts to provide to some degree alternative and rival modes of justification, but a narrow enough range so that the debates between these modes is focused and determinate. (p. 18)It is in languages of this kind that we can expect debates to be eventually decisive.

Now let us try to relate what we have seen so far to mathematics. Instead of someone torn between the two communities of early eighteenth century Ireland, imagine a Chinese mathematician, thoroughly versed in the Nine Chapters and other classic texts, who in 1600 comes to understand Euclidean geometry from Jesuit missionaries. They would surely experience incommensurability as they attempted to decide for themselves the superiority of one of these very different conceptions of mathematics. (See Joseph W. Dauben, Ancient Chinese mathematics: The Jiu Zhang Suan Shu vs. Euclid’s Elements. Aspects of proof and the linguistic limits of knowledge, International Journal of Engineering Science 36 (1998) 1339-1359. See also this paper.) As a mathematical parallel to what MacIntyre conceives of as a modern language the obvious choice is set theory. Today we could render each variety of mathematics as set theory. But such flexibility comes at a cost. It can’t help us choose in which direction to proceed. Rewriting mathematics within set theory we can say that something has a correct proof or that something is calculated correctly, but we can’t say that it was worth proving or calculating in the first place.

For this we need stories, and, fortunately, the condition of modern mathematics is not one of presuppositionless set theoretic universalism, but rather is scored across by countless stories, which in printed form are easiest to locate in book reviews, such as those of the Bulletin of the American Mathematical Society, or in articles in the AMS Notices. Count the number of times Atiyah uses the words ‘story’ or ‘stories’ in his Mathematics in the 20th Century, Bulletin of the London Mathematical Society 34(1), 1-15, 2002. Turning your web browser to the Arxiv every morning most closely resembles tuning into a daily soap opera. It's not easy to discern the story lines from this tapestry of articles, but if you do succeed, you find that individual stories range through: daring quests for single gems using whatever resources can be found; Hausmannian reconstructions of higgledy-piggledy slums; beautiful irrigation projects using water from distant streams; attempted take-overs, and resistance to subsumption within a broader theory.

But with all these subplots is there any danger that mathematics itself may take on the attributes of the modern condition of endless, unresolvable debates about which direction to take? Not if the traditions represented by these stories are carefully maintained and passed on, and their advocates remain open to what other traditions can offer them. I mentioned several months ago that in a Clay Mathematics Institute interview, Terence Tao speaks of the importance of "being exposed to other philosophies of research, of exposition, and so forth". The danger to ward against is dissipation in a kind of heat death.

"Epistemological Crises, Dramatic Narrative and the Philosophy of Science," The Monist, 60 (1977), 453-72. Reprinted in Paradigms and Revolutions: Appraisals and Applications of Thomas Kuhn's Philosophy of Science, Gary Cutting, ed. (Notre Dame: University of Notre Dame Press, 1980) pp. 54-74. For excerpt see previous post.

## 2 Comments:

David you write: “the condition of modern mathematics is not one of presuppositionless set theoretic universalism”.

What does this mean? Isn’t set theory a presupposition of modern mathematics?

David, you write:

“But with all these subplots is there any danger that mathematics itself may take on the attributes of the modern condition of endless, unresolvable debates about which direction to take? Not if the traditions represented by these stories are carefully maintained and passed on, and their advocates remain open to what other traditions can offer them.”

A view perhaps in opposition to yours has mathematics doing just fine without philosophy. Suppose, for the sake of argument along this line, the following observation is true. (It’s at least plausible in my opinion.)

There is a definite trend toward digital computation, even in well-established arenas such as algebraic geometry. Part of this trend has mathematics set in the context of the “mathematical sciences” and sees mathematics reaching beyond academia in business ventures, etc. This trend results from a natural progression of sophistication and power of mathematical software, coupled with more or less collective, conscious recognition in the mathematics community that this is the way to go (with some opposition, diminishing with time).

Where, then, is a place for philosophy? Mathematics – it might be argued – seems to have the resources to move ahead, resolve debates, keep itself alive to developments, etc.

D.Lomas

The point I'm making about set theory is the obvious one that even if a piece of mathematics may be translated into set theory, and this would typically involve a fair deal of violence, not all of the story-telling that goes along with it can be similarly rendered. And I would also maintain that this story-telling is an essential part of mathematics. If a dictator were to decree that no such stories were to be told, then mathematics would die.

The Atiyah article I mention uses the word 'story' fifteen times and the word 'stories' twice in fifteen pages. 'History' appears a few times too.

Let's look at the ArXiv offerings today. Leaving aside the first article which is on the applied side, the second is Volumes of highly twisted knots and links, which introduces us to previous work in this field then presents the author's work as an extension of earlier results. These brief comments can only be a small part of the author's sense of what her research is about.

Concerning your second point, I do say I take mathematics to be flourishing. Personally I'm not so excited by advances in digital computation. If I had unlimited time, I'd wander about the realms of higher dimensional algebra, noncommutative geometry, Brave New Algebra, and the Langlands Program.

It's not that I expect philosophers to resolve debates within mathematics, but just as philosophy can learn from mathematics what allows a discipline to maintain itself for so long, we can try to make mathematicians aware of our findings. We can encourage them to be more thoughtful about their evaluative language, prompt them to consider rival points of view, etc. Inevitably such a philosophy of mathematics must involve a considerable immersion in various research areas to get the feel of them.

At one conference I attended, I was thanked for having made the participants think harder about what they were doing. One of the highest pieces of praise for my book comes on p. 18 of "Why Mathematics?" You Might Ask:

"...there is no question that Corfield likes mathematics, and for the right reasons; his

book, unlike the normative treatise in philosophy of mathematics, is definitely part of the “conversation.”"

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