September 28, 2009

The Mathematical Vocation

Posted by David Corfield

After a visit in 1939 to the Monastery of the Prophet Elijah on the Greek island of Santorini, the Oxford philosopher R. G. Collingwood entered into discussion with his students about the value of monastic life. It appears that the students were a little perplexed to find their prejudice that monks were “at worst idle, self-indulgent, and corrupt; at best selfishly wrapped up in a wrongheaded endeavour to save their souls by forsaking the world and cultivating a fugitive and cloistered virtue” clash with their admiration for

…the atmosphere of earnest and cheerful devotion to a sacred calling, the dignity of the services and beauty of their music, the eager welcome and the loving hospitality, and above all the graces of character and mind which the life either generated in those who had adopted it or at least demanded of aspirants to it and thus focused, as it were, in the place where the life went on. (‘Monks and Morals’, Essays in Political Philosophy, Oxford, 1989)

Collingwood then draws the students’ attention to a vocation they do value.

Suppose a man devotes his life to the study of pure mathematics. Is he to be condemned for living on a selfish principle? Not, as my friends readily admitted, on the ground that pure mathematics cannot feed the hungry. Pure mathematics, apart from any consequences which may ultimately come of it, is pursued because it is thought worth pursing for its own sake. In order to judge its social utility, then, you must judge it not by these consequences but as an end in itself.

What is more, you cannot judge the social utility of a mathematician by asking whether he publishes his results. Unless there is value in being a pure mathematician, there is no value in publishing works of pure mathematics; for the only positive result these works could have is to make more people into pure mathematicians; and a society which does not think it a good thing to have one pure mathematician among its members will hardly think it a good thing to have many.

The social justification of pure mathematics as a career in any given society, then, is the fact that the society in question thinks pure mathematics worth studying: decides that the work of studying pure mathematics is one of the things which it wants to go on, and delegates this function, as somehow necessary for its own intellectual welfare, to a man or group of men who will undertake it. A test for this opinion is that the society in question should be grateful to the pure mathematician for doing his job, and proud of him for being so clever as to be able to do it; not that every one else should rush in to share his life, but that even if his neighbours feel no call to share it they should honour him for living as he does. The fact that they do so honour him is a proof that they want a life of that kind to be lived among them, and feel its achievements as a benefit to themselves. (pp. 145-146)

Posted at September 28, 2009 4:47 PM UTC

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Re: The Mathematical Vocation

I think the debate would have seemed much clearer in 1939 than it does now. In particular, the quote “Pure mathematics, apart form any consequences which may ultimately come of it, is pursued because it is thought worth pursing for its own sake.” presupposes that for things that were called pure mathematics as they were developed (ie, where there is a strong consensus, not borderline issues) to ever have “practical” consequences is rare.

In a modern view, modern physics, computer science and mathematical modelling of all sorts, tends to bring in some results from various branches of mathematics that are called pure mathematics (eg, number theory, construction of computable reals, etc). So the argument that publishing pure mathematical research cannot be any factor in deciding the “utility” of a mathematician becomes very difficult to agree with.

Of course the question still exists in a different from: how does one define social utility for working on something that is very unlikely to consequences that have practical consequences, and vanishingly likely to have direct practical consequences?

Posted by: bane on September 28, 2009 5:32 PM | Permalink | Reply to this

Re: The Mathematical Vocation

This whole “pure mathematics is useless” thing is largely a pose that mathematicians like to adopt (for various psychological reasons I don’t dare to guess at), partly popularised by Hardy. As a result they like to greatly exaggerate the claims of uselessness. In particular they have a tendency to say “there is no use for X” when they mean “I don’t happen to know of any application of X because I don’t bother to look in on what my colleagues are doing, let alone the next department down the corridor”.

For a while I worked in computational chemistry. I was amazed how much mathematics was being consumed by chemists - especially graph theory and algebraic topology, much of it the kind of pure mathematics that would have seemed the very height of uselessness back in 1939. Some of it was a weird kind of use. For example a lot of empirical results found by running linear regression on databases of various physical properties vs. graph-theoretical and algebraic-topologcial invariants of molecules. Nonetheless, it really opened my eyes.

We have no idea what pure mathematics will turn out to be useful, but the true reason why societies tolerate and even pay for mathematicians is that in utilitarian terms the payoffs have been big.

Posted by: Dan Piponi on September 28, 2009 6:25 PM | Permalink | Reply to this

Re: The Mathematical Vocation

This reminded me of a conversation I once had with a chemist who asked me why so much effort was spent on teaching undergraduates useless mathematics, such as calculus and probability theory, instead of teaching the very useful stuff – ie, group theory and algebraic topology!

Posted by: peter on September 28, 2009 7:12 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I was amazed how much mathematics was being consumed by chemists - especially graph theory and algebraic topology.

So where does an algebraic topologist look to get hired into the chemistry biz?

Posted by: John Armstrong on September 28, 2009 7:42 PM | Permalink | Reply to this

Re: The Mathematical Vocation

New Scientist!

My wife spotted the ad for a job at a pharmaceutical company with the word ‘topology’ in the job description.

But that was nearly 20 years ago now.

Posted by: Dan Piponi on September 28, 2009 7:51 PM | Permalink | Reply to this

Re: The Mathematical Vocation

John A. wrote:

So where does an algebraic topologist look to get hired into the chemistry biz?

I don’t know much about this, but I’d guess that chemists either learn a bit of algebraic topology from books and papers like this or collaborate with mathematicians — not hire mathematicians to do chemistry.

Posted by: John Baez on September 29, 2009 1:58 AM | Permalink | Reply to this

Re: The Mathematical Vocation

This is sort of the problem. Once you’ve studied pure mathematics you’re pretty much useless for anything else in the eyes of employers.

Posted by: John Armstrong on September 29, 2009 2:43 AM | Permalink | Reply to this

Re: The Mathematical Vocation

I find it’s more tricky than that. At least in the various areas of computing, lots of people think they want to employ people who have greater exposure to mathematics than they have. The problem tends to be that the people in immmediate charge of you don’t “deep down” understand that figuring out new mathematics (in the sense of “here’s how we can apply this existing theory to your problem”) is time-consuming. So they expect you to produce “mathematical bon-mots” at a moments notice and are very unimpressed when you can’t, and that can somewhat colour their views of future hiring.

Posted by: bane on September 29, 2009 12:01 PM | Permalink | Reply to this

Re: The Mathematical Vocation

This whole “pure mathematics is useless” thing is largely a pose

I don’t think so. You don’t get people saying “I went into nursing because there’s a small but non-zero probability that at some point, decades or centuries in the future, somebody practising a profession somewhat related to mine might do something of benefit to mankind.” I don’t think that’s how vocations work. Maybe some people feel a vocation to study algebraic topology because they want to be of help to chemists, but not many, surely…. And that’s all that’s needed for Collingwood to make his point.

the true reason why societies tolerate and even pay for mathematicians is that in utilitarian terms the payoffs have been big

Is that the reason the Ptolemies supported the library of Alexandria? Surely societies support all manner of activities that are useless in utilitarian terms. Sometimes they even invent fake reasons why these activities must secretly be useful. (“These jewels are so beautiful that they must have magic powers!” “If you can predict the positions of the planets, you must be able to foretell the fate of kings!” “This economic model is so elegant that it must hold the secret of wealth, health and happiness!”)

I don’t think everybody’s values are purely utilitarian. I’m not sure if that is even philosophically coherent (it makes it seem as though everything derives its value from being useful for something else, but nothing is of value for itself…). Well, I don’t think you meant that exactly—but I do feel that the claim that everything that people support is useful seems like just as much of a pose as that people sometimes do (or support) things because they’re useless.

Posted by: Tim Silverman on September 28, 2009 11:19 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I certainly don’t think that everyone is a utilitarian about everything, after all we have art that is paid for by private and public institutions. But art is widely appreciated, which is why every major city has galleries and museums. Mathematics has a much smaller audience on the other hand, but it has such a long history of application success it would be foolish not to support it for utilitarian reasons.

The Library of Alexandria was an extension of Ptolemaic imperialism. They wanted to be seen as the leaders of everything, including Greek culture. They had *the* copy of Homer. That gave them status and power. I’m not sure how much it had to do with love of learning or anything like that.

But going back to the pose thing: I have met so many mathematicians who take such great delight in telling people how useless their work is that it seems plainly apparent to me that there is more pleasure from this telling than simply that of relaying a statement of fact. Individual mathematicians might not take any interest in the applications, but they are often there nonetheless. And even if a mathematician is working on solving problem X, and X seems completely useless, they may be honing useful technique Y to get there.

Posted by: Dan Piponi on September 29, 2009 1:09 AM | Permalink | Reply to this

Re: The Mathematical Vocation

“They had *the* copy of Homer. That gave them status and power.”

Before pronouncing a complete dissociation from love of learning, perhaps one might ask *why* it gave them status and power.

Enough mathematicians have some feel for the tangled network of concerns linking their own preoccupation to the variety of human endeavor. In a typical cocktail-party conversation, it just takes less energy to be modest about the utilitarian weight of our own work.

Posted by: Minhyong Kim on September 29, 2009 10:16 AM | Permalink | Reply to this

Re: The Mathematical Vocation

Tim said

I don’t think everybody’s values are purely utilitarian. I’m not sure if that is even philosophically coherent (it makes it seem as though everything derives its value from being useful for something else, but nothing is of value for itself…).

Exactly the point Collingwood goes on to make. The chain of means-end justification must end somewhere in an end valued for its own sake.

I should think you mathematicians have much to fear about the development of a society which values mathematics solely as a means to external ends.

Posted by: David Corfield on September 29, 2009 10:33 AM | Permalink | Reply to this

Re: The Mathematical Vocation

Why do you think of it as a chain?

Posted by: Minhyong Kim on September 29, 2009 11:09 AM | Permalink | Reply to this

Re: The Mathematical Vocation

Why a chain? This is Collingwood’s argument against taking usefulness as the sole value of an action.

Say I take an action A to be of value solely for the value of the action B it enables. But the value of B is then that of enabled action C, and so on without termination. Or if there is termination, and an action enables no new action to be taken, it is valueless according to this account.

Posted by: David Corfield on September 29, 2009 11:49 AM | Permalink | Reply to this

Re: The Mathematical Vocation

But this is exactly the image that seems not to be realistic. Most people’s real conception of value, individual or social, is associated with at least a graph of actions rather than a chain with $A_i$ justifying $A_{i+1}$. Any given node may have high degree, and it seems quite natural to find cycles, as might occur in any eco-system. I don’t know if this helps at all with the general problem of utility and justification (of mathematics), but I think the chain idea is useless, even as a toy model. It’s obvious, for example, that nothing needs to be an end in itself.

Posted by: Minhyong Kim on September 29, 2009 12:15 PM | Permalink | Reply to this

Re: The Mathematical Vocation

It’s obvious, for example, that nothing needs to be an end in itself.

I’m puzzled by this. Do you mean obvious in a useless toy model, or obvious in real life? If the latter, does the “need” operator have scope inside or outside the existential? That it may be the case for any given individual, particular action, that it need not be an end in itself, seems obvious enough to me, but that there need be no actions at all which are valuable in themselves seems … well, not impossible, but pathological.

it seems quite natural to find cycles

They might well exist as a matter of fact, but I would consider them a pathology rather than a natural part of a value system. I don’t see how such a cycle could generate utility. It would simply be a waste of time. Everybody on the cycle would think they were doing something useful, but actually they wouldn’t be.

Or have I misunderstood you?

Posted by: Tim Silverman on September 29, 2009 1:31 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Sorry. I’ll restate it in mathematics so that I don’t botch up the English again:

A directed tree is either infinite or has a node with no incoming edges. But this is not necessarily true of a general directed graph.

Here is an action graph that is quite simple really: A family whose members find great satisfaction in keeping each other happy. I don’t doubt that you can model it as (embedded in) some other tree, and that might be useful sometimes.

By the way, in case someone finds the example ‘pathologically’ Confucian, I refer him/her to O’Henry’s famous story.

Posted by: Minhyong Kim on September 29, 2009 2:29 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I think we’re talking at cross-purposes. Consider the following chain:

“What’s the point of you buying this bicycle?”
“So I can give it to my son.”
“What’s the point of giving a bicycle to your son?”
“So he can ride it.”
“What’s the point of your son riding a bicycle?”
“So he can have fun.”
“What’s the point of him having fun?”
“Because Fun is GOOD!”

A cyclic arrangement of value would replace that last statement with something like “To give me a reason to buy a bicycle.”

This obviously doesn’t exclude reciprocal or cyclical arrangements of mutual benefit, but the benefit has to exist or the arrangement seems pointless: “I’m buying a bicycle for the sake of buying a bicycle, even though the bicycle is worthless and the act of purchasing it is pointless.”

(This is much broader than just the utilitarian pursuit of pleasure: one can aim to create beautiful things because beauty is inherently good regardless of any pleasure it might give, etc. Then the beauty would be of value in itself, hence the end of the chain of value.)

It would be possible (if unlikely) for a group of people all to derive all of their pleasure from the results of actions undertaken by others (so nobody would do anything to please themself). But the fact that people were getting pleasure would presumably be the point of the whole exercise. It would only be in that sense that anybody would be doing anything “for” anyone else—because of the expectation that that person would benefit in some way, and that that benefit would be of value in itself.

(Again, pleasure need not be the point of the exercise—the goal could be a harmonious society or whatever. But there would have to be an ultimate goal for any particular action, in order for it to have value.)

Posted by: Tim Silverman on September 29, 2009 3:40 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Yes, as (more or less) predicted, you’ve constructed a larger graph in which the cycle can be embedded with a node above the whole cycle. It’s not obvious to me that consistently going for such a view is particularly illuminating.

Posted by: Minhyong Kim on September 29, 2009 3:53 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I don’t think I’ve “constructed” a larger graph—I think that the other nodes are always there; and that sometimes ignoring them leads to confusion, sometimes not.

However, I don’t think I distinguished adequately between actions and their consequences; perhaps that accounts for our disagreement. Against that, I think demands for utilitarian justification are partly a way to gloss over (or distract attention from) questions about the actual goals (e.g. to pre-empt certain kinds of answers to the question of what mathematics is actually “for”).

I confess I need to think more about what you are saying, so this is very inadequate.

Posted by: Tim Silverman on September 29, 2009 4:38 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Going off at a slight tangent here: this is reminiscent of how Google page rank works. How do I value a web page? Well if I value the web pages that point to it, then that confers on it a certain value. Those in turn are conferred value from other web pages. Ultimately we end up with circularity. But that’s not a problem, the cycles just give an eigenvalue problem for which we can solve. But with no reference to anything external from which to bootstrap, it’s surprising that it measure anything useful at all, and yet it does.

Posted by: Dan Piponi on September 29, 2009 6:12 PM | Permalink | Reply to this

Re: The Mathematical Vocation

There may be no direct reference to anything external, but of course the links are put there by people based on their own judgements of value. The links are used as a proxy for those judgements.

Posted by: Mark Meckes on September 30, 2009 3:27 PM | Permalink | Reply to this

Re: The Mathematical Vocation

This reminds me of the motto:

Work to live;

Live to bike;

Bike to work!

On a more serious note: One earns one’s living as a mathematician because someone is paying for it (for whatever reasons, ranging from “the work is needed” to “there is a budget for it”), and one does good mathematics because one likes to do this (sometimes only in one’s spare time, due to the administrative load in the official hours).

How much the two aspects correlate is probably very person-dependent.

Posted by: Arnold Neumaier on September 29, 2009 7:28 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Tim Silverman gave a chain of justifications ending platonically in the Good:

“What’s the point of you buying this bicycle?”

“So I can give it to my son.”

“What’s the point of giving a bicycle to your son?”

“So he can ride it.”

“What’s the point of your son riding a bicycle?”

“So he can have fun.”

“What’s the point of him having fun?”

“Because Fun is GOOD!”

But I can also imagine something that goes like this:

“What’s the point of you buying this bicycle?”

“So I can give it to my son.”

“What’s the point of giving a bicycle to your son?”

“So he can ride it.”

“What’s the point of your son riding a bicycle?”

“So he gets exercise, and has something to do with other kids.”

“What’s the point of his getting exercise and doing things with other kids?”

“It helps him become happy and well-adjusted, build his coordination and strength, and develop social skills?”

“What’s the point of being happy, well-adjusted, coordinated, strong, and socially adept?”

“It’ll help him make friends, get a good job, and stay healthy for a long time.”

“What’s the point of his having friends, having a good job, and being healthy?”

“It’ll help him contribute to society and do a good job raising kids of his own… for example, by buying them bicycles.”

And so on… I think the justifications can endlessly branch out into the future or loop around, and I think they actually do.

But what if you ask me “What’s the point of this whole complicated web???”

Well, I would have to admit that if the whole universe didn’t exist, nobody would be the slightest upset.

But if you tried to remove any one part of the universe, the nearby parts would be affected.

Posted by: John Baez on September 29, 2009 10:33 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I thought you mathematicians were supposed to understand reductio ad absurdum.

The point of Tim’s chain is not to explain how things are, but to show that there’s a contradiction in the claim that the value of an action lies solely in those its consequences.

He’s arguing that once you begin a chain of supposedly purely utilitarian actions, that you will be forced to conclude it with an action valued intrinsically.

This doesn’t mean he believes the value of actions has the form of a chain terminating in a single good action. He may well have an image close to the web you present, where intrinsic good occurs at many points in the web.

Posted by: David Corfield on September 30, 2009 9:48 AM | Permalink | Reply to this

Re: The Mathematical Vocation

To use the word ‘solely’ is dangerous in any circumstance.

Posted by: Minhyong Kim on September 30, 2009 12:06 PM | Permalink | Reply to this

Re: The Mathematical Vocation

But I guess the urge to find a single reason can be helpful in science, even if it’s occasionally misguided. I vaguely remember from long ago two essays by Einstein, the first of which concluded with the proposal of a World Government as the *only* solution to contemporary ills.(My impression is he was substantially under the influence of Russell at the time.) He subsequently received a letter from some Soviet scientists spelling out the dangers of such an entity, especially from the perspective of weaker nations. He was clearly troubled by the points they made and wrote a rather rambling reply. I think it was a short while later that he wrote the essay concluding with ‘Only *socialism*…’.

Posted by: Minhyong Kim on September 30, 2009 12:16 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I suppose I should make a serious point as well.

For example, in this portion of your Collingwood quote:

‘Pure mathematics, apart from any consequences which may ultimately come of it, is pursued because it is thought worth pursing for its own sake. In order to judge its social utility, then, you must judge it not by these consequences but as an end in itself.’

he is clearly the one making the claim for a *sole criterion* that lies above others. Normally, it’s like my $B_{691}$ below, and I can’t see the point of insisting otherwise. I really think his argumentation comes from an exclusive focus on the tree model, perhaps stemming from the contemporary interest in the axiomatic tradition.

Posted by: Minhyong Kim on September 30, 2009 12:28 PM | Permalink | Reply to this

Re: The Mathematical Vocation

he is clearly the one making the claim for a *sole criterion* that lies above others.

I think you’d have to look much more closely at his writings before you could gain a clear sense of how he takes goods to be ordered. I read the passage you quote to mean just that mathematics is thought worth pursing for its own sake, and to that extent its value should be judged accordingly.

But I see we have a copy of the The first mate’s log of a voyage to Greece in the schooner yacht Fleur de Lys in 1939 in our library, so may be able to say more when I’ve read it. It appears that he also questions the worth of philosophy.

Posted by: David Corfield on September 30, 2009 1:17 PM | Permalink | Reply to this

Re: The Mathematical Vocation

OK, we’ll see. I’ll make one ‘practical’ point to conclude my contribution to this discussion. If it comes to a question of giving justification for mathematics (or philosophy, for that matter) to the general public, we might do well to suggest, at least vaguely, a rich network (that could contain a $B_{691}$), perhaps by way of a few key examples. It seems more effective as well as more accurate than insisting on the overwhelming inevitability of intrinsic value and the stupidity of those who fail to recognize it.

Posted by: Minhyong Kim on September 30, 2009 1:41 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Is it really true that the point, the only point, of being happy, or having friends, or having a good job, is for the sake of something else? Don’t people actually think these things are good in themselves? Sure, they may also help you get other things which are also good, but is that what people are actually thinking when they feel glad that their children are happy, healthy and prosperous? Whenever people trot out these external reasons for things that seem obviously good in themselves, I always suspect these are fake reasons prompted by a utilitarian anxiety about simply saying something is good. Maybe I shouldn’t ascribe motives like this, but I just don’t think people really think like this as a rule, except when someone actually demands that they justify something that is, in reality, an end in itself. Suppose it turned out that being happy didn’t help you make friends, or that making friends didn’t help you get a job? Would being happy or having friends then become worthless?

Got to dash out now, or I’d say something more thought-out.

Posted by: Tim Silverman on September 30, 2009 10:52 AM | Permalink | Reply to this

Re: The Mathematical Vocation

I, for one, had intention of objecting to ‘good in itself.’ (Even though I was tempted to ask ‘What’s so good about fun?’)

The list in response to ‘Why $A$?’ could certainly have had one edge going to

$B_{691}=$good in itself

with other edges leading off to infinity, loops, and what not. One might even ascribe the need to find a single overarching node of ‘The Good’ to the tradition of monotheism :-).

I should take this opportunity to correct one extreme sentence from above, where I said the chain model is ‘useless.’

Posted by: Minhyong Kim on September 30, 2009 11:52 AM | Permalink | Reply to this

Re: The Mathematical Vocation

The notion of a hierarchical organisation of goods, arranged according to a single final end, is certainly there in Aristotle, and finds itself explicitly integrated with monotheism in Aquinas.

Posted by: David Corfield on September 30, 2009 12:17 PM | Permalink | Reply to this

Re: The Mathematical Vacation

Also the notion of entelechy, literally, that which has its end in itself, sometimes translated as actuality or realization and sometimes glossed as perfection, depending on the sense of the word has that one has in mind.

Like certain pinball games I used to play where the payoff of game $G$ is nothing but another play at game $G$.

Posted by: Jon Awbrey on September 30, 2009 1:48 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Here I feel I ought to observe that I don’t believe in a single overarching Form of the Good, and I would not at all represent intrinsic value as a single node, for the sake of which other things are done.

To forestall the obvious follow-up question with regard to the issue of comparing goods, as mentioned by David C below: I don’t in fact belief that these sorts of questions are resolvable in theory, although they may sometimes be so in practice.

Posted by: Tim Silverman on September 30, 2009 11:59 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Even though I’m contributing nothing at this point, I feel obligated to reply.

The reference to an overarching good came up because of your sentence

“there would have to be an ultimate goal for any particular action, in order for it to have value”(Sorry, I don’t know that trick of linking to an earlier point in a thread.)

But perhaps you didn’t mean it in any absolute sense. In any case, if you don’t believe in such a thing, there’s no reason for the mere presence of intrinsic value to lead to termination of a ‘value chain.’ Nor is there reason for intrinsic value to be the exclusive or even most important motivation for anything.

I note with dismay that my sentences are becoming progressively metaphysical.

Posted by: Minhyong Kim on October 1, 2009 2:04 AM | Permalink | Reply to this

Re: The Mathematical Vocation

I should have added that $B_{691}$ could then lead to

$C_1=$ (good in itself)’

$C_2=$ (good in itself)”

$C_3=$ (good in itself)”’

The universe of intrinsic value is itself quite rich.

Posted by: Minhyong Kim on September 30, 2009 11:58 AM | Permalink | Reply to this

Re: The Mathematical Vocation

The universe of intrinsic value is itself quite rich.

The anti-utilitarians would be with you on this. The next question is how to represent this richness. Have I accurately glimpsed your disapproval of the notion of an overarching good?

One way to think about our (implicit) ordering is to see how we would behave in various circumstances. We may take research mathematics in our community to be a part of its flourishing, but we may be prepared to sacrifice it for the continuation of some other activity, such as the provision of public green spaces, if we were forced to choose. How would we give good reason for this choice?

To read Aquinas weighing up goods, see questions 1-5 of Prima Secundae Partis of Summa Theologica.

Posted by: David Corfield on September 30, 2009 2:25 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Thanks for the Aquinas reference. The problem of adequate representation is obviously subtle in any serious inquiry.

‘Disapproval’ of overarching good seems a bit too strong for me, especially since I’m probably more religious than many other agnostics. However, too heavy an enphasis on intrinsic value does seem to lead to a certain rigidity of outlook, and perhaps inhibit sensible decisions in the kind of scenario you describe. Obviously, none of us are so partisan as to defend all parts of theoretical scholarship at all costs. So then it’s probably a good idea to have at hand some balanced pragmatism, even with regard to intrinsic values.

On the other hand, people who pugnaciously insist on the intrinsic value of their own work can also be very interesting and valuable…

Posted by: Minhyong Kim on September 30, 2009 6:52 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I have this nasty habit of afterthoughts, but perhaps I should explain just a bit more.

When asked ‘Why $A$?,’ a typical person’s considered response is `Because of $B_1, B_2, \ldots, B_n.$’ And so we take off backwards along the various edges. Why would you consider it pathological to find cycles here and there, or to have trouble finding an end?

Posted by: Minhyong Kim on September 29, 2009 2:47 PM | Permalink | Reply to this

Re: The Mathematical Vocation

This is just the kind of discussion they have in epistemology. How is belief in P justified? If through earlier premisses, how do they receive their justification in turn? Can there be an infinite regress? Is there some point where I must stop? If the latter, is that point a proposition or a state of the world?

Coherentists take all justification to arise from the coherence of the network, including cycles of justification. Foundationalists take there to be justified foundations. Infinitists argue for a justification to emerge from infinitely long chains of inference.

Hybrids include foundherentists.

Our message crossed earlier, in case you didn’t notice this.

Posted by: David Corfield on September 29, 2009 3:17 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I suspected these issues were already discussed to death by you philosophers :-). Anyways, I do think the Confucian cycle mentioned above is a genuine one.

Posted by: Minhyong Kim on September 29, 2009 3:27 PM | Permalink | Reply to this

Re: The Mathematical Vocation

By the way, in case you folks here are interested, we now have a London Number Theory Blog

Posted by: Minhyong Kim on September 29, 2009 3:30 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I’m interested. I’ve added it to the online resources entry.

I like your Non-abelian fundamental groups for the public. One small step in making the public “grateful to the pure mathematician for doing his job, and proud of him for being so clever as to be able to do it”.

I wonder if they could cope with a little more explanation of this sentence:

Here, critical use is made of the space associated to an equation, in that the fundamental group weighs the totality of paths that one might attempt to traverse through it.

Posted by: David Corfield on October 1, 2009 11:20 AM | Permalink | Reply to this

Re: The Mathematical Vocation

OK. I’ve added an appendix to the original document.

Posted by: Minhyong Kim on October 6, 2009 11:28 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Speaking of hybrids, fundherentists, 3-quids, here’s a Peircean musement on entelechy.

Posted by: Jon Awbrey on September 30, 2009 2:16 PM | Permalink | Reply to this

Re: The Mathematical Vocation

But this is exactly the image that seems not to be realistic.

Collingwood doesn’t need it to be realistic. He is simply taking it to be the image held by a certain kind of utilitarian who takes the value of an action to be solely that of the action it enables. His argument is to the effect that nobody who holds this image can do so consistently.

If we accept his argument, the only move that utilitarian can make is to say that there are images which they may hold, other than the chain one, where we can understand how both (i) all value is value of consequence and (ii) actions do have values. Perhaps we could have some richer network of actions enabling other actions, none of which are valuable in themselves, but where value emerges from the complex pattern. I can’t say I can see how.

Collingwood’s own view is far from this, where actions have value in themselves as well as in terms of the actions they enable. Not only do I work to earn money, or work because I hold to the rule that able-bodied people should work if possible, but I take there to be an intrinsic good to the activities of my job.

Posted by: David Corfield on September 29, 2009 2:42 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I have the same sort of instinctive reaction to the chain viewpoint as I do to Asimov’s Three Laws of Robotics: it’s absolutely logically fine, except it breaks completely as soon as you have any degree of uncertainty at all in your knowledge about the inputs. The world is full of things in all sorts of areas that weren’t envisaged as being of practical use when they were first “developed”, and even more things where the actual use is completely different from the envisaged use, and many things that were believed would be useful turned out not to be. So any attempts to figure out in advance what things may eventually be useful are of limited efficacy, and hence any value-system predicated on that will be hopelessly inaccurate.

That’s not to say that the intuitive problem has gone away: how should one justify working on something which has a (provisional) estimated probability of being useful that’s smaller than the probability associated with some other endeavour? My personal feeling has always been partly that society should be doing many things that are not of direct utility and then basically about “aptitude”: a good surgeon is likely to do more good in the world than I’ll ever do as a computer scientist/mathematician, but I just don’t have the desire or aptitude to be a good surgeon.

Posted by: bane on September 29, 2009 1:35 PM | Permalink | Reply to this

Re: The Mathematical Vocation

“The world is full of things in all sorts of areas that weren’t envisaged as being of practical use when they were first “developed”, and even more things where the actual use is completely different from the envisaged use”

In fact, the truth is even stronger. For most hi-tech products, most ultimate applications are not what the inventors originally envisaged. This is a such a well-known phenomenon that product engineers, marketers and marketing researchers regularly make use of it, involving so-called “lead-users” of technologies in the design and marketing of new products. The pioneer in this area was Eric von Hippel, of MIT.

Posted by: peter on October 1, 2009 5:15 PM | Permalink | Reply to this

Re: The Mathematical Vocation

to a man or group of men who will undertake it.

Sheesh. Even in 1939 Collingwood should have had enough examples of female mathematicians. I’m usually pretty forgiving of “he” used for he/she, pre-1970 say, but this is just silly. Sorry to derail.

Posted by: Allen Knutson on September 28, 2009 6:55 PM | Permalink | Reply to this

Re: The Mathematical Vocation

Indeed. I stumbled over that phrase and had to try reading it again. This does not have the feel of a ‘generic’ “he” but rather seems like an overt statement that pure mathematics, as a useless venture, is only taken up by males.

Posted by: wren ng thornton on September 29, 2009 2:20 AM | Permalink | Reply to this

Re: The Mathematical Vocation

AK wrote: “Sheesh. Even in 1939 Collingwood should have had enough examples of female mathematicians.”

——————————————–

I saw a list of famous women mathematicians and there were six, ending with Emmy Noether who died 1935; only three, and in the last two hundred years, were accomplished pure mathematicians. This may be a prejudice but people think of pure mathematicians as exceptional rather than mediocre and that’s the bias I read into Collingwood. When he wrote this piece there was no living famous or highly competent woman pure mathematician.

Likely this had something to do with the prejudice of male dominated institutions of higher learning, but is that the sole cause? Women think differently than men and have different physical brain structure. I don’t think Collingwood meant an average man or group of male mathematicians elected to pursue pure mathematics, so then he wouldn’t mean an average woman or a group of women pure mathematicians. By far, there were many more accomplished male pure mathematicians at the time while women in the same class were rare, and one doesn’t generalize a rule from the exceptional female examples. So perhaps he shared in the common academic male opinion of the potential of women for mathematics which was held in the time he wrote this, but based on the evidence available to him, what he wrote is true if one accepts the premise that he was describing a group that had demonstrated abstract accomplishments (nearly all men) rather than a rare individual female, but certainly not a group of women who had demonstrated accomplishments in the abstract area of pure mathematics.

Collingwood wrote:”to a man or group of men who will undertake it.”

That portion of his remark is found in the sentence which talks about ‘pure mathematicians’. Your criticism uses the
phrase “should have had enough examples of female mathematicians.”

Female mathematicians are not necessarily pure mathematicians. So “enough examples”? There was not one living example of a gifted female pure mathematician at the time Collingwood expressed his opinon. And only 3 such in the two hundred years preceding his statement. That he means he could have made such a statement using “he” based only on the evidence and without any prejudice of political correctness at all. Perhaps your hunch is right, blatant political incorrectness, but that is to be inferred from his era and not from what he wrote. Was he supposed to claim that potentially women should be included without substantial evidence, but on egalitarian grounds?! You are going uphill and against the evidence to assert that the discrepancy of male genius pure mathematicians, there are more than women, is due solely to environmental factors.
http://www.nybooks.com/articles/2176
“Women have proportionately smaller brains than do men, but apparently have the same general intelligence test scores. Thus, I have proposed that the sex difference in brain size relates to those intellectual abilities at which men excel. Women excel in verbal ability, perceptual speed, and motor coordination within personal space: men do better on various spatial tests and on tests of mathematical reasoning. It may require more brain tissue to process spatial information.”

Posted by: Stephen Harris on September 29, 2009 5:54 AM | Permalink | Reply to this

Re: The Mathematical Vocation

I have to say that much as I’m irked by Collingwood’s apparent sexism in the quoted passage, I don’t find it surprising. This is pre-WW2 Oxbridge after all. Perhaps I’m too defeatist, but I find myself just taking it in stride en route to the more interesting points he makes.

Short rejoinder to Stephen Harris’ comment - I can’t work out if he is saying we should evaluate Collingwood in context, or that Collingwood was justified in his turn of phrase or supposed view: you do realise that the quote you give, which comes from a a letter sent to the NYRB, is given a pretty good kicking by the author of the original piece? which makes me wonder why you added it?

Posted by: yemon choi on September 29, 2009 8:43 AM | Permalink | Reply to this

Re: The Mathematical Vocation

Even taking into consideration the idea that ‘man’ may have been used to denote an individual of our species, rather than a male, as we do with ‘dog’ and ‘fox’, we may continue to detect a note of sexism here. And it’s wonderful that we’re all now indignant about the prejudices underlying such modes of expression from earlier times. But to feel surprise about these modes of expression can only result from a lack of historical knowledge. You only have to read a little to know that that was a common way of speaking back in the 1930s.

And people continued to speak this way, and not just in Oxbridge. Listen to Feynman lecturing in 1964 and you hear him use similar expressions when he explains what the physicist does.

Posted by: David Corfield on September 29, 2009 10:13 AM | Permalink | Reply to this

Re: The Mathematical Vocation

I think it’s inefficient to get worked up over how people used to talk. In this case, suppose Collingwood had added as an aside to justify choosing to say “he”. Suppose he said most women are predisposed to become poor pure mathematicians and then he predicted that no woman would win the Fields Medal in the 20th Century. There is a difference between making a sexist or any “ist” remark and observing some true fact that sheds a poor light on some group.
Often there are 4 winners of the Fields Medal. 30% of Math PhD’s were earned by women in 2006, 70% by men. What are the chances that no woman will win a Fields Medal if they are just as gifted as their male counterparts but not as many of them. I think the a priori odds of a woman winning the Fields Medal in 1998, or 2002, or 2006, or 2010 with 4 slots open, is much better than 20 to 1 that a woman should have won a prize. Because winners have until 40 to achieve the prize it reduces the 20 to 1 odds somewhat. What is the reason no women have won the FM? Is it all sociological: Old white men serving as FM judges; dolls instead of computer toys; teachers and counselors steer them all towards Home Economics classes? Or is their a physical reason. This has been a controversy for many years. For instance women have a thicker bundle of their corpus callosum and the effect is not well understood.

On another note I agree with bane about the lack of prediction. There is no absolute ethical standard (not even evolution) which guides a society’s choice about the usefulness of pure mathematics. There is no way beforehand to determine what is ultimately best. So that leaves a culture with pragmatic choices even about supporting the fine arts. We put a man on the moon in 1969 and were on track to send a man to Mars ten years later. Political expediency derailed that goal. I read that Australia Maths departments are under pressure to be consolidated into the Computer Depts.

Posted by: Stephen Harris on September 29, 2009 4:44 PM | Permalink | Reply to this

Re: The Mathematical Vocation

yemon choi wrote: “I can’t work out if he is saying we should evaluate Collingwood in context, or that Collingwood was justified in his turn of phrase or supposed view:” …
————————————–

I pointed out that there was insufficient evidence to regard Collingwood as a sexist because he used the word “he” which seemed to view women as inferior pure mathematicians. There is more to it than the simplistic noticing of “he”. —

“Minds and Bodies: philosophers and their ideas” By Colin McGinn Page 244

CM writes: “The true enemy of democracy is the anti-intellectual, the brain- washer, the prejudice-pumper, since she undermines what alone makes democracy workable. The forces of cretinization are, and have always been, the biggest threat to the success of democracy as a way of allocating political power: this is a fundamental conceptual truth, as well as a lamentable fact of history.” ———————

SH: Next McGinn quotes Collingwood(C), he evidently holds C in high reqard.

CM: “Collingwood identifies a deeper problem: [quotes C] “It is much easier for any kind of man known to me to doze off into daydreams which are the first and most seemingly innocent stage of craziness.” …

SH: Did you notice that CM used the word “she” in “since she undermines” in a critical sentence? That is not evidence of sexism. Did you notice that C used the word “man” in a critical way? That indicates that he is using “he” and “man” in a generic way not a sexist way. About the 1980’s Feminists made an issue over using he generically and the rules were changed to be what is now politically correct. Before then, you had to read what a person wrote to decide if they were sexist. Now if you read “Monks and Morals” where he wrote the challenged “he” you will see that Collingwood is championing erasing preconceived prejudices. Finally, I quote both sides of an issue because I’m interested in presenting both sides of an argument. As in whether _men_ are predisposed to be better at mathematical reasoning. You, and others might not know it, but this has been a controversial issue for some time, there is no such thing as of course men and women have equal areas of expertise in their thinking. It happens that the brain is strongly influenced by hormones. They gave a number of women IQ tests. Then they injected them with testosterone and gave them another IQ test. Their areas of strength were altered by the testosterone. More like males with higher scores in mathematical reasoning than in the first test they took. So this is not proof. It is the reason why it is ill-assumed that Collingwood made a sexist assumption by elevating males to better pure mathematician status; it might be factual on his part, rather than sexist. But mainly, I think his remark is not sexist because using “he” was politically correct when he wrote that article. I think people jumped to the conclusion that because C used the word “he” that he was impugning women. I think you have to look at other things people wrote back in that era to decide if they had a sexist attitude or were just following the normal mankind practice. It looks to me like C deplored all kinds of stereotypical prejudice. Of course you can’t correct some fault that you are not aware of such as a hidden (to oneself) prejudice. CM is quite outspoken. Also C did not particulary associate with his Oxford colleagues, so there is no guilt by association. No, Oxford academics were typically sexist so likely C was too.

Posted by: Stephen Harris on October 1, 2009 9:07 AM | Permalink | Reply to this

Re: The Mathematical Vocation

The latest trendy idea in public research funding in Britain is ‘impact’. When applying for funding you now have to fill in a large section on the social and economic impact of the research. The responsibility for higher education and research funding were recently moved into the Department for Business, Innovation and Skills, headed up by our so-called Prince of Darkness Peter Mandelson (the Rt Hon Lord Mandelson, First Secretary of State) — a very powerful and, by most accounts, extremely intelligent, though much despised man. The word on the street is that the move to this emphasis on impact comes from directly from his lordship.

Our much hated national Research Assessment Exercise, an enormously bureaucratic procedure by which ‘research outputs’ are measured to establish levels of university funding is about to be replaced by the Research Excellence Framework in which a large weighting will be given to — yes, you guessed it — the social and economic impact of our research.

I was talking to a historian friend about this and she said that mathematicians would be alright because people like Einstein have lots of impact. (Actually she said “Ask the people of Hiroshima how much impact he had.”) I had to point out to her that the four or five-year window of the Research Excellence Framework would make it impossible to link nuclear technology with Einstein’s original ‘research outputs’.

Posted by: Simon Willerton on September 29, 2009 11:43 AM | Permalink | Reply to this

Re: The Mathematical Vocation

Oddly, I think Collingwood could be read as saying there is a light in which the movement towards an appreciation of ‘social impact’ could be seen as positive, hard though it may be to do so for a directive emanating from the Prince of Darkness.

Can you say of the citizens of Sheffield that they are

…grateful to the pure mathematician for doing his job, and proud of him for being so clever as to be able to do it; not that every one else should rush in to share his life, but that even if his neighbours feel no call to share it they should honour him for living as he does. The fact that they do so honour him is a proof that they want a life of that kind to be lived among them, and feel its achievements as a benefit to themselves.

Posted by: David Corfield on September 29, 2009 3:25 PM | Permalink | Reply to this

Re: The Mathematical Vocation

A conspiracy to queer the data seems to be a delightfully mischievous endeavor. Those of us in the colonies who want to help our colleagues across the pond can simply throw in a few random citations. We could even include a sentence in the introduction, “As is standard within this area, the authors point out that this work has nothing whatsoever to do with the \cite{FavBrit1,FavBrit2}.”

Posted by: Scott Carter on September 29, 2009 5:15 PM | Permalink | Reply to this

Re: The Mathematical Vocation

To read Tim Gowers wrestling with similar issues see The Importance of Mathematics, which is the subject of a post at God Plays Dice.

Posted by: David Corfield on September 29, 2009 5:13 PM | Permalink | Reply to this

Re: The Mathematical Vocation

I really enjoyed this paper; Gowers seems so grounded including his choice not to introduce the word “fractal” which might have proved to be a distraction. Thanks!

Posted by: Stephen Harris on September 30, 2009 3:46 AM | Permalink | Reply to this

Re: The Mathematical Vocation

The most useful thing about utilitarianism, at least to politicians, is that so few people stop to ask who decides what “useful” means.

Posted by: Gavin Wraith on September 29, 2009 9:53 PM | Permalink | Reply to this

Mathematical Mystery Tour

Somewhat in the same ballpark (cricket green), did anyone see the play “A Disappearing Number”, weaving its dramatic tapestry on the web of mutual musement that Hardy and Ramanujan spun around and between themselves?

Posted by: Jon Awbrey on September 30, 2009 1:04 AM | Permalink | Reply to this

Re: Mathematical Mystery Tour

I did, and was disappointed. They took as their starting point the story of Ramanujan and Hardy which had heaps of dramatic potential, and who were already fascinating characters in their own right, and danced most post-modernly and brazenly about several timeframes trying to dramatise the nature of inquiry itself, with characters it proved hard to care a jot about. Too much spectacle, and way too little of the real meat of drama - exegesis of character and motive.

Posted by: Mozibur Ullah on October 5, 2009 5:53 PM | Permalink | Reply to this