Why do mathematicians always agree?
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- Published on Friday, 30 November 2012 14:48
- Written by Christophe Heintz
Question: How can you spot an extrovert mathematician?
Answer: He looks at YOUR shoes when he talks to you.
Is mathematics "less social" than other academic disciplines? Some support for a 'yes' answer can be found in a recent piece of news. A famous mathematician, Nelson, had claimed to give a proof of a rather surprising proposition: “Peano Arithmetic is inconsistent.” Two other famous mathematicians, Tao and Tausk, said the proof included one specific mistake, which they spelled out. Nelson's reaction was: "Ah, you're right. So I have not proven that Peano Arithmetics is inconsistent". End of the story. No fight, no disagreement, no formation of alternative schools of thoughts, no playing with how to interpret this or that claim. Just plain boring consensus.
(These are the axioms that Nelson claimed were inconsistent. They are supposed to express central propositions true of our system of natural numbers with addition. They are used to prove things about an object that is central in many cultures.)
Mathematics is full of that: easily achieved consensus. Everybody agrees. No debate, and yet, the consensus is not socially induced in any standard way.
In a recent blog post, The (in)consistency of PA and consensus in mathematics, Catarina Novaes takes this as a nice illustration of some points made by philosopher Jody Azzouni, who argues that mathematics is unique as a social practice.
How cultural is cultural epidemiology? 2- Cultural embedding
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- Published on Tuesday, 25 August 2009 16:47
- Written by Christophe Heintz
This is the second part of Christophe's series of posts on what culture does to culture (the first post is here).
Most cultural phenomena are embedded in other cultural phenomena. For one thing, any cultural phenomenon takes place within a community that already has many traditions, cultural practices, rituals and beliefs of its own. The important point, however, is that the embedding cultural phenomena are likely to have some effects on the embedded cultural phenomenon and to partially determine its evolution and the content of its constitutive representations. Religious beliefs can have effects on economic practices; economic practices can have effects on kinship relations; etc.
Let us call “cultural embedding” this aspect of cultural evolution. Cultural embedding is certainly what motivated some cultural anthropologists to have a holistic view of culture: every aspect of one culture will be related, more or less directly, to other aspects of the same culture.
Cultural epidemiology on promiscuous causality
For cultural epidemiology, cultural phenomena result from social cognitive causal chains that go from mental processes to social interactions (e.g. communication, imitation, the production of artefacts) to mental processes again. Some of these chains involve many members of a community, last through time and eventually have the effect of stabilising the distribution of cultural items in that community.
Read more: How cultural is cultural epidemiology? 2- Cultural embedding
How cultural is cultural epidemiology? The case of enculturation
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- Published on Wednesday, 29 July 2009 07:41
- Written by Christophe Heintz
When discussing about cultural epidemiology with informed colleagues, I often come to think that they tend to underplay the extent to which cultural epidemiological accounts can integrate enculturation and other cultural phenomena that are generative of culture.
Here is a key claim of cultural epidemiology: understanding, learning and memorising what others communicate or transmit are micro-cognitive processes at the basis of cultural phenomena, and these cognitive processes are strongly constrained by the properties of the mind. Cultural epidemiologists have especially worked on specifying the consequences of universal properties of the mind for culture: naive theories and the memorisation of religious beliefs for instance; or face recognition and the success of masks as cultural artefacts. However, what is understood, learned and memorised is also dependent on those properties of the mind caused by previous enculturation. This is nearly a truism: one can learn better to read if one already knows the alphabet, one processes differently a sentence in French if one knows the language that if one does not, and, more controversially, and can more easily learn how to build a canoe, if one already knows what each part is made for. True, most of the time these things are learned in concert - but transmitted information is nonetheless processed sequentially, and the order of the elements in the sequence is sure to matter.
This has at least the following consequences:
Read more: How cultural is cultural epidemiology? The case of enculturation
Politics and the psychology of irrational decisions
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- Published on Saturday, 04 October 2008 22:00
- Written by Christophe Heintz
In my previous post (Cases of institution that make us smart), I have been considering a proposal about increasing taxes on junk food and decreasing taxes on fruits and greens. The proposal differed from campaigns of information (see picture as an example) because it implied direct action on the relative prices of different kinds of foods: changes would have been made on the structure of incentives rather than on people's beliefs. I was considering this proposal as an attempt to frame the environment for fostering beneficial decisions and behaviour.In Mean Genes, Burnham and Phelan provide a list of tricks that could help people to have a healthy diet. Most of them consist in changing their environment. It is not groundbreaking science (it does not claim to be so) but it points to an important trend: the use of psychological research on behaviour departing from some rational norm for policy making and individual decision making. One important idea is that one can change the environment so that known psychological mechanisms lead to decisions that are advantageous to the individual or to the community.
Picture: "Manger Bouger": the French city of Nîme's advertisement campaign on the benefits of a healthy diet.
Read more: Politics and the psychology of irrational decisions
Cases of institutions that make us smart
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- Published on Tuesday, 30 September 2008 20:35
- Written by Christophe Heintz
The adaptedness of cognitive processes is characterised by a fit between the process and the environment. The fit means that the processes reliably lead to positive outcomes or tend to maximise results. The processes, however, do not lead to beneficial results because they perform a comprehensive analysis of the situation and an evaluation of the each possible output. In fact, such processes work, and can be considered rational, only in a specified environment. They make the most of properties (esp. statistical properties) of specific environments. They are, as Simon and Gigerenzer put it, "ecologically rational".
There is a puzzle that comes with the above assertions: if our cognitive apparatus is best adapted to the Pleistocene environment and if our modern environment depart more and more from this original environment of evolutionary adaptedness, then we should be less and less adapted, less and less ecologically rational, ... dumber and dumber. But this does not really seem to be the case.
In the medical case nicely described by Denis, there is the added fact that such tipping is illegal. Any other important difference? We may feel that doctors should treat all patients equally well, but then we should object to private medecine or surgery, and so on when it can fix its prices and offer better services to patients who can pay more. My guess is that even people who don't object might still see tipping doctors are immoral.
The general point is this: this case might be best approached within a discussion of tipping in general, a discussion very well worth having from avariety of points of view: cultural, economic, rational choice, reputation, and so on.
Anyhow Denis, get well soon!
Denis, your story strikes a Romanian chord. The situation around here is even worse, from what I can tell. But it is quite a fascinating question, with different answers from different points of view.
For an economist, it is a matter of price formation. In the state system, Romanian doctors are paid a fixed (and miserable) wage, largely unrelated to quality or effort. The incentive to pocket bribes is huge, and patients know it so well. In the private sector (with transparent and varied prices for medical services), bribes are almost unheard of. Also, there is a more or less efficient market for bribes. Patients find out how much a doctor expects, usually from past patients, or from other doctors. Surgeons receive more than GPs, professors more than debutants, etc.
But I think there is something more about "medical envelopes", from a cognitive point of view. First of all, there is a vast asymmetry of competence between doctors and patients, which gives the former a large freedom of action. Is this pill better, or another one? Surgery or not? Home treatment or hospitalisation? To make things worse, the post-hoc reckoning is not very helpful, since most decisions may be medically justified, but you might also end up dead. The patient is at the mercy of the practitioner since she does not know what choices are better. The best way to make sure one gets the proper treatment is to insure the benevolence of the doctor, and a bribe is the simplest path to gain the doctor's amity.
Second, there is something special about this particular social exchange: the patient is dealing in an ultimate value - her health. Something everyone in Romania says is that there is no price too high to be healthy. (Paradoxically, giving up smoking somehow does not make the list - self-hint-hint-nudge-nudge). If people would risk not bribing a policeman to avoid a fine, they are extremely unlikely to jeopardise their health in this manner. One cannot afford to stick to abstract principles (like discouraging corruption) when her life is at stake.
Finally, there is something like a Maussian gift in the affair: one passes a fat envelope even without the explicit mention of an economic exchange. It is not that the surgeon would not operate without being bribed - the patient just shows gratitude without visible economic reckoning. Of course, under the veil of generosity stands the solid self-interest of the patient. The fat envelope is meant to make sure that no scalpel is lost in her belly. But no-one says it out loud. It's a "I know that you know that I know etc" which makes sure that the transaction is smooth and polite.
To end with a personal anecdote: I was (and to some extent I still am) very wary of giving out envelopes to doctors. A little bit of moral prudishness, a little bit of fear (what if he feels insulted?), a bit of monetary unsaviness. Those who are more competent in these matters reassured me: "just put the envelope on his desk - he knows what to do next" After all, he is the expert, and I am not.
"Very well-rounded analysis. A few thoughts. First, I am glad you mentioned nurses in your comment* because in the article you discount this, perhaps unintentionally. I remember my aunt consistently bribing the nurses when my uncle was recovering from a stroke for several months in the hospital. Also, I've had many conversations with my family here in Hungary about this, trying to understand the rationale behind this irrational system (I'm originally from the US). I think both motivations could be at play here. I got the impression that, in addition to the bribe, people are still very sensitive to the "wage supplement" aspect. That is, most people I've talked to find the wages of doctors and other health care providers rather deplorable. Even if GMs are a considerable expense for my working class family members, they seems to use the wage supplement as a way to render this dysfunctional reality more palatable somehow. I also think there is a third factor at work here - but I think it's linked to the others. I've witnessed situations where doctors behave very condescendingly toward patients or their families, despite a hefty bribe of some 20,000 HUF. Part of that harks back to the days of the socialist regime - when the power of public authorities was unquestioned. As one of my Hungarian friends likes to say about health clinics here: "they just want to make you feel like they still have power over you." When my aunt and I went to visit my cousin in critical care last year, the doctor didn't want to give us the time of day. We didn't give her a tip, but we kept pressing her for answers. I said to her, "is it a virus or a bacteria?" The doctor looked at me like a deer in headlights. I think she was surprised I even knew the difference. She opened up quite a lot to us after that and we never gave her a tip. Finally- and I'll get off my soapbox - private insurance systems are not necessarily more transparent. The US being a case in point. There is a great (surprisingly) 28-pg TIME article about this, "The bitter pill: why medical bills are killing us." I'm sue you'd find it relevant. Anyway, thanks so much for posting this!!"
*This is the comment by me which Eva refers to:
"I should have also added that, in fact, there is GM directed to nurses when they are perceived as the primary caretakers. Usually this is the case for families having elderly parents in retirement houses."
That GM thing reminds me of a funny routine that happens in France: around the end of the year, firemen and mailmen knock at your door to sell (ugly) calendars. Folk wisdom holds that if you don't buy the calendar, firemen will not rush if there is a fire in your house. Similarly, mailmen will be more likely to lose important mail you receive. What is striking is that this belief seems to carry on though it makes complete non-sense. I bet the situation is a bit different as for GM: the physician obviously remembers you and s/he is more likely to act benevolently towards you with a bit of extra money...
Azzouni certainly has the bona fides to weigh in on this. But it seems to me that the pure sociology of it isn't quite so simple.
Take Wiles' first proof of Taniyama-Shimura. It had an error, but it took concerted efforts by extreme experts to locate it. But that's not the end of the story. It turns out that he and Richard Taylor were able to ascertain that piecing together two parts of the theory that didn't quite seem to work on their own was in fact enough to 'patch' the proof together (Wiles himself says as much).
So, Yes, the original proof was wrong. To a much lesser extent, Perelman didn't fill in all the blanks in his landmark proof of Poincare, leading to a (minor scandal) where two other mathematicians claimed to give the "first" proof based on the "ideas of" Perelman and Hamilton.
The question is this: if someone had done the patching of Wiles' proof for him, would THEY be the prover? How large does the hole have to be? When an error is found, who gets to decide whether it is trivial, whether it wrecks the proof entirely, and who will be the one credited with the insight that makes the whole thing work?
These are not trivial matters, and the issue isn't apportioning credit, but deciding what an error truly is. Typos don't count. Proving incorrect results certainly do. But what about "generally correct" ideas that eventually lead to a proof? How loose do those ideas have to be?
I don't think there's ANY argument about when large, demonstrable errors have been found in published proofs. But there are many other cases -- like de Branges' purported proof of the Riemann Hypothesis -- that fall through these neat cracks.
In respect to kinship terminologies, Levinson's question, "What constrains this exuberant diversity of systems?", is not answered by Kemp and Regier's analysis for one simple reason: Terminologies have a structure and logic, like grammars for language, that determine the possible range of kinship terminologies. Kemp and Regier assume any partition of the space of genealogical relations is a potential terminology and then show that existing terminologies occupy only a small portion of this space due, they assert, to a tradeoff between simplicity and usefulness. This would be like saying a sentence can be any subset of all possible vocabulary words, then asserting that the realized languages have sentences that are a tradeoff between simplicity and usefulness, but ignoring the fact that the simplicity and usefulness of sentences is created through the grammar of the language that constrains what are admissible sentences. The same is true for kinship terminologies, and the answer to Levinson's question has already been made by showing that kinship terminologies have a generative structure that determines the corpus of kinship terms, starting from the primary kin terms of a terminology, along with kinship concepts that are expressed in the terminology (such as reciprocity of kin terms), and the kinship structural properties embedded in a particular terminology (Read 1984, 2001, 2007, 2009; Read and Behrens 1990; Leaf and Read 2012, among others). For example, the difference giving rise to the fundamental division of terminologies into descriptive versus classificatory (bifurcate merging) terminologies derives from two different ways that sibling relations are conceptualized in different societies: (1) a sibling is the child of my parent other than myself (descriptive terminologies) or (2) siblings are those persons who have parents in common (classificatory terminologies) (Bennardo and Read 2007; Read, Fischer and Leaf 2013). Trying to understand kinship terminologies (and hence kinship systems) without first working out the generative logic of a terminology is like trying to understand languages without working out the grammar of a language. Extensive work has already been published on the generative logic of kinship terminologies and this work makes evident what constrains the variability in kinship terminologies that Levinson asks about.
References
Bennardo, G. and D. Read 2007. Cognition, Algebra, and Culture in the Tongan Kinship Terminology. Journal of Cognition and Culture 7: 49-88.
Leaf, M. and D. Read. (2012) Human Thought and Social Organization: Anthropology on a New Plane. Lanham: Lexington Press
Read, D. l984. An algebraic account of the American kinship terminology. Current Anthropology 25: 4l7-440
Read, D. 2001 What is Kinship? In The Cultural Analysis of Kinship: The Legacy of David Schneider and Its Implications for Anthropological Relativism, R. Feinberg and M. Ottenheimer eds. University of Illinois Press, Urbana. Pp. 78-117.
Read, D. 2007. Kinship Theory: A Paradigm Shift. Ethnology 46(4):329-364
Read, D. 2009. Another Look at Kinship: Reasons Why a Paradigm Shift is Needed. Algebra Rodtsva 12:42-69.
Read, D. and C. Behrens. 1990. KAES: An expert system for the algebraic analysis of kinship terminologies. J. of Quantitative Anthropology 2:353-393.
Read, D., Fischer, M. and M. Leaf. 2013. What are kinship terminologies, and why do we care? A computational approach to analyzing symbolic domains. Social Science Computer Review 31(1): 16-44.
Yes, kinship is back -- or more accurately, it is reclaiming its original vigor. Haven't you heard of the Kinship Circle? For each of the past three years, and as part of this year's annual meeting of the Amerian Anthropological Association as well, we have had highly successful sessions on kinship. The sessions have been integrated with the themes of each of the meetings. We have had an international group of scholars from Australia, Brazil, Canada, China, England, France, Germany, Italy, Qatar and the United States, presenting a wide range of papers, ranging from more "classic" questions about kinship systems to current research that is challenging some of our theoretical ideas about what constitutes kinship. The papers from the first two sessions will be published shortly.
Dwight Read
Fadwa El Guindi
Dear learned scholar of mathematicians, I disagree with your premise that mathematicians do not disagree, and, being wonderful souls, are easily converted to consensus. No less a scholar, intellectual and role model than Von Neumann (1961), the founder of game theory, argued against your premise. In fact, he bemoaned that unlike physicists, mathematicians who don't agree behave in an unsocial manner by striking out in new directions, leaving their conflicts unresolved. In his article, the first in his collected works, Von Neumann wished that mathematicians disagreed as physicists did. Whenever conflict arose between two physicists (e.g., Bohr and Einstein), physicists refused to ignore it, often bringing their field to a standstill until a resolution was found (i.e., consensus via debate, unlike your fanciful example of consensus without debate). I have long cherished Von Neumann's insight, and his remarkable paper on mathematicians. BTW, in my research, I too have found that consensus without conflict is indeed possible, except that none of the participants can agree on the result.
Von Neumann, J. (1961). The mathematician. Collected works, Pergamon.
http://www.nytimes.com/2013/03/10/magazine/the-professor-the-bikini-model-and-the-suitcase-full-of-trouble.html?_r=3&
People concur in saying that Frampton is unusually gullible.
This story of an incredibly gullible scientist (or so it seems) might also be relevant to your remark that the optimality of epistemic vigilance can only be measured in view of its fit to the milieu. An optimal epistemic vigilance would enable people to believe most of the true things they are told and to disbelieve most of the false things they are told (especially the costly one). The inconvincible sceptic as well as the gullible has less than optimal epistemic vigilance. The optimal vigilance fall in between, but its precise position depends on whether the environment is full of false claims or not. It would be interesting to know whether there are different cognitive developments of epistemic vigilance depending on the type of environment in which a child grows up. This could account for some variability across individuals.
As for scientists, they are supposed to instantiate high epistemic vigilance. So how can Frampton be at the same time so gullible and a good physicist? I see two non-exclusive possibilities:
(1) Frampton exercises epistemic vigilance, but only in the domain of physics. This can happen because the scientific environment fosters argumentative abilities. By contrast, Frampton did not wish or need to convince others that he was having a relation with a beautiful model. He did not need to find good reasons for his beliefs and did not wish to adress counter-arguments. Hugo Mercier pointed to me that this difference in the argumentative context could explain the fact that Newton, with so great achievements in physics, did so badly in chemistry/alchemy. There was in alchemy no need to convince others; it was a secret enterprise.
(2) Frampton does not exercise much epistemic vigilance, but does well in physics nonetheless because the process of checking the plausibility of claims is distributed to others. Only very selected information arrives to his creative mind. This is thanks to the process through which scientific information comes to be distributed---the review process for instance. In science, epistemic vigilance is distributed across individuals and institutionalised. In that context, some gullibility might be an advantage. The schoolgirl, in any case, does better by believing the apparently crazy things that her teacher says (e.g. sound is the vibration of matter). At the research level also, it can pay to believe improbable hypotheses; it means pursuing a high risk, high reward research programme.