We are astonishingly ignorant about how many species are alive on earth today, and even more ignorant about how many we can lose yet still maintain ecosystem services that humanity ultimately depends upon. Mora et al.'s paper is important in offering an imaginative new approach to assessing total species numbers, both on land and in the sea. It is a remarkable testament to humanity's narcissism that we know the number of books in the US Library of Congress on 1 February 2011 was 22,194,656, but cannot tell you—to within an order-of-magnitude—how many distinct species of plants and animals we share our world with Part of the problem is that taxonomic effort is approximately divided 1∶ 1∶ 1 among vertebrates, plants, and invertebrates, whereas plant species are roughly 10 times, and invertebrates 100 times, more numerous than vertebrates PLoS Biology, Mora et al. In this issue of Mora et al.'s imaginative new approach begins by looking at the hierarchy of taxonomic categories, from the details of species and genera, through orders and classes, to phyla and kingdoms. They documented the fact that for eukaryotes, the higher taxonomic categories are “much more completely described than lower levels”, which in retrospect is perhaps not surprising. They also show that, within well-known taxonomic groups, the relative numbers of species assigned to phylum, class, order, family, genus, and species follow consistent patterns. If one assumes these predictable patterns also hold for less well-studied groups, the more secure information about phyla and class can be used to estimate the total number of distinct species within a given group.In this way, Mora et al. arrive at a global total of 8.7 million eukaryotic species, with a standard error of ±1.3 million. Most are terrestrial, with 2.2 (±0.2) million being marine.This is higher than my earlier “best guess”, but I like the simplicity of this new method.Currently, diligent field taxonomists are adding newly discovered species at the rate of very roughly 15,000 each year (when discounted for synonyms) 3±1 higher than the average extinction rate seen over the half-billion-year sweep of the fossil record But this tentative assessment makes no allowance for accelerating extinction rates. As the Millennium Ecosystem Assessment emphasized Ultimately, why should we care about how many species are alive on earth today, and about how many of them are known to us? One notable Victorian physicist (I will be merciful and not name him) opined that such a quest is little more than stamp collecting. To the contrary, we increasingly recognise that such knowledge is important for full understanding of the ecological and evolutionary processes which created, and which are struggling to maintain, the diverse biological riches we are heir to. Such biodiversity is much more than beauty and wonder, important though that is. It also underpins ecosystem services that—although not counted in conventional GDP—humanity is dependent upon.Turning from the general to the specific, I give just one among a multitude of possible concrete examples of beneficial application of taxonomic discovery. In the 1970s Yuan Longping, “the father of rice”, discovered in the wild a new variety of rice, whose cross with a conventional strain led to a new variety that is 30% more efficient. This has motivated subsequent initiatives to document and protect all wild varieties of rice, which obviously can only be done if we have the appropriate taxonomic knowledge. Given the looming problems of feeding a still-growing world population, the potential benefits of ramping up such exploration are clear.The essential fact is that, if we are to meet the challenges facing tomorrow's world, we need a clearer understanding of how many species there are—both on land and in the even less well-studied oceans—underpinning the structure and functioning of ecosystems. Mora et al.'s interesting new approach to assessing the magnitude of this task is thus very helpful.


Although Maimonides stated that perfection in the medical art, both in theoretical and in practical expertise, is very difficult to achieve, he did not accept Galen’s opinion, i.e. that perfection is beyond human capability. Any person seeking intellectual perfection should, according to Maimonides’ view, be fully trained in logic, in the natural sciences, and in theology. A physician is moreover requested to study and memorize basic medical literature; he must consider each patient as a sick individual, without neglecting the patient’s psychological disposition; and he should aim at inspiring confidence and trust, not only in his patient, but also in the latter’s environment. Even when feeling competent and trustworthy, the physician should not be conceited; here Maimonides insists on offering his personal experience, in a quite impressive way. This approach of Maimonides to the practice of medicine should be considered, even today, as a valuable incentive for patient-oriented medical education, as already expressed in the late eleventh century. Hippocrates_(fifth century BCE) must be considered first.Maimonides (1137–1204) was not only a path-breaking theologian and philosopher, he was also a well trained physician; there is, however, no detailed curriculum of his medical studies. We only know that, while living in Fes, he received some medical instruction and was in contact with other physicians, as we will document further in this essay. Among the authors he cites in his ten medical works, the most often mentioned are Hippocrates, Galen, Avicenna, Rhazes, Avenzoar, and Averroes. We will, in this essay, focus on the Greco-Roman author Galen of Pergamon (second century CE)—who remained throughout the Middle Ages the most influential source in medical education. However, makrós)—is, no doubt, the most ominous statement of the “Father of Medicine.” The adjective makrós means indeed “long,” but also “abundant” or “far-reaching,” in other words, difficult to master.The first aphorism of Hippocrates—“Life is short and the Art is long” , dietetics, follow-up of the healing process, and pharmacology (i.e. treatment). Maimonides adds terminology to these seven items, i.e. the names of diseases and drugs in various languages. We remember that he lived successively in Spain, in Morocco, and in Egypt; such migrations were then the lot of many physicians, Jewish ones in particular. Maimonides wrote a glossary of drugs,In his commentary on the first aphorism of Hippocrates, Maimonides issues a statement that may seem daring or even impossible to the modern reader. He contends that a student physician should memorize the huge corpus of theoretical and practical medicine. This does not only necessitate long and tedious work, it means that the physician will have to perform constant revision throughout his medical career.The next stage in medical education is personal experience of treating the sick, which includes experience in diagnosis, in prognosis, and in the ways of prescribing drugs and adequate diet.Mishneh Torah in particular, till late in his life.In one of his letters to a former student,Galen (131–201 CE). Most of Galen’s extensive works had been translated into Syriac and Arabic by Ḥunain ibn Yisḥaq and others; some of them only survived in their Arabic translation. A compendium of these writings, known as the Sixteen Books of Galen, was since Byzantine times in wide circulation.Let us now turn to Timaeus (preserved in its Arabic translation only),In his commentary to Plato’s On Definitions, we read: “The perfect [Gr. téleios] physician is one who has completed the whole scheme of theoretical and practical studies.” The author does not assert that perfection is beyond human reach. The next sentence reads: “The best  physician is the one who practices medicine according to the right doctrine [Gr. orthon lógon].” , Maimonides mentions four categories of perfection (Heb. shelemut). First mentioned is perfection in resources, second, perfection in health, third, in moral qualities, and, fourth, intellectual excellence. ,Fathers’ Aphorisms (Heb. Pirqei Avot), Maimonides advocates adopting the medium line regarding the moral qualities. We quote (my own translation from the Hebrew):ha-adam ha-shalem] should constantly call to mind his moral qualities [Heb. midotav], ponder his actions, and control his soul all day long. Each time he feels a propensity toward some extreme action, he should at once apply the accurate treatment in order to stop the progress of that tendency.Thus, the perfect man , although it exists in him virtually [Heb. be-koaḥ].The reasons [for the difficulties] are that a person has, at the beginning [of his studies], very limited capabilities. A man does not own initially full mastership  one.” … Nor should one say: “I have seen how my elders have treated [this disease] in such or such way.” [As a matter of fact] a physician does not treat a disease, he rather treats a sick person.15One of the central statements of Maimonides is the following:To which he adds: “Every person who falls ill necessarily requires renewed consideration and reflection.”Regimen Sanitatis (Heb. Hanhagat Ha-Beriut), Maimonides feels that a psychological assessment of the patient should even anticipate any medical intervention. “For every sick individual feels his/her heart constricted [Heb. libo tsar].”16Maimonides thus indicates that the constitution and the psychology of the patient must be taken into account. As stated in his In other words, an accomplished physician should know how to adapt his way of addressing the patient according to the latter’s psychology.Psychology was then a branch of Philosophy, and we thus understand better why even Galen said that a physician should be trained in Philosophy.Commentary on Hippocrates’ Aphorisms, Maimonides affirms that a physician who aims at doing his best for his patient’s benefit must have in view more than achieving an exact diagnosis and an adequate treatment for the disease. He must care for a full-fledged application of the treatment. Indeed, the patient might be reluctant to take a drug that is bitter or repulsive; and the care-takers might prefer taking advice from some popular quack or from a “wise woman.”17In his The physician must therefore endeavor to gain full confidence from both patient and care-takers. Moreover, he should feel responsible for the removal of any impediment to the treatment; he should even help poor patients to purchase the drugs and/or to move to some healthier accommodation.Hilkhot ‛Aniyim 10, 4–5).The duty to help poor patients applies to every individual, including physicians, but Maimonides feels a necessity to mention it here , who maintain that their knowledge is deficient even when it is perfect. Maimonides apparently feared that his readers would suspect him of being conceited and would therefore not want to accept his advice.He adds that he does not state this out of modesty, or according to the ways of the pious , Maimonides explains how someone who has a tendency toward conceit should leave the ideal middle way of virtues and adopt extreme humility, at least for some time, until he feels that he may come back to the middle way. Modesty is the right way for a Sage.In his ethical writings (cf. Aphorisms (Heb. Pirqei Moshé), he enumerates a whole list of problematic statements of Galen, particularly on Philosophy. And he remarks:[There is] a disease, which is so common that hardly one individual in a long period of time can avoid it. … This disease is that everyone imagines that he is more perfect than he really is; he wishes that all his opinions be considered perfectly true, [even when uttered] without toil or effort.The ten medical works of Maimonides may seem like a drop in the sea of medieval medical literature. In comparison with Galen, Maimonides looks like a dwarf in front of a giant. Nevertheless, in the 25th chapter of his own Galen himself—Maimonides says—suffered from that disease, not in his medical treatises, but in his philosophical and theological writings. We remember that for Galen perfection in medical practice was beyond human reach. Indeed, Maimonides asserts that he has not attained perfection, though refusing to accept the idea that perfection cannot be achieved. Did he say so out of humility, despite the fact that he openly denies it?Let us try to define humility:Humility: a state or quality of being humble in spirit; Freedom from pride and arrogance. (Compare to Webster’s definition, In other words, humility is absence of pride based on one’s own achievements. I would like to argue that Maimonides would have accepted such a definition.striving toward perfection.As he did in his other writings, philosophical or theological, Maimonides did his best in his medical works and in his practice, without a sense of pride. He therefore could not accept being presumed to be perfect. Even if he was considered as accomplished in the eyes of his patients, his ethical principles and way of life prevented him from such a belief; he was just A real Sage is one who knows that he must sustain lifelong learning in order to uphold constant progress in knowledge.


Group practices are increasingly common for primary care physicians worldwide. Although breakups are likely to happen frequently within group practices, their process has not been studied to date. The aims of this study were therefore to explore the reasons for breakups of group practices of general practitioners and to describe the associated feelings.We conducted a qualitative study consisting of in-depth interviews of 21 general practitioners and one secretary from past group practices in the Rhône-Alpes region, France, who experienced a breakup.When getting started in group practice for the first time, young doctors did not feel ready and supported, and did not necessarily share the same expectations as their partners. The reasons for the breakups involved imbalances within the groups, contrasting working and management styles, and breakdowns in communication. The breakup process often generated long-persistent feelings of suffering and failure for almost every partner who experienced a breakup, particularly for the partner who was leaving.Weakening factors exist from the very beginning of a partnership, and problems are likely to increase at every change or event occurring in the group. We provide several recommendations, including fair management, a shared project based on a precise contract, the consultation of third parties as necessary and, in the worst case scenario, leaving the group practice in time. Group practices are increasingly common for general practitioners (GPs) and family physicians worldwide. Private group practice has become the predominant mode of primary medical care practice in many countries, such as Australia, Canada, Denmark, the Netherlands, New Zealand, the Slovak Republic or the UK . The proWorking together indeed offers several potential advantages, such as improved quality of life for the practitioner, increased continuity of care for the patient and better sharing of experience between healthcare professionals . ActuallTo our knowledge, the breakup process within group practices has not been studied to date. The aims of this study, entitled “ICARE MG”, were therefore firstly to explore the reasons for breakups in group practices and secondly to describe the practitioners’ feelings about this experience.The study was based on in-depth interviews, conducted in accordance with the grounded theory .A purposeful sample of GPs who had been confronted with a breakup in a group practice was selected. Chosen from physicians in the Rhône-Alpes region of central France, the sample brought together male and female practitioners of various ages, group practice sizes and locations. A first sample was built by advertising in the journal of the Medical Council of the Rhône county and by sending e-mails to members of the local general practice teaching faculty. Using the snowball technique, we also asked the participants from the first sample to inform us of any other GPs they knew outside of their own group practice that had experienced a group practice breakup.Twenty-two interviews were conducted, eleven for each investigator, at the interviewed physicians' offices. The first physician interviewed in each group served as the index and we tried to retrace the breakup of their own group practice by interviewing all of their partners, including the assistant if she had worked at the practice for several years. We therefore asked the index physicians for permission to interview their previous partners and how to contact them. There were 10 people we did not manage to interview: 3 refused, 3 could not be contacted and 4 were members of a group whose interviewed physician asked us not to include his 4 former partners in the study because he feared they might harm him.A semi-structured topic guide based on a bibliographic review and discussion between the authors was developed. It included open-ended questions concerning the circumstances of the partnership setup, the organisation and rules for working together, the reasons for the breakup, and the lessons learnt from this experience. The guide was modified based on the data that had been collected from the first interviews. Data from each interview was audio taped with the participants’ consent. Information provided by partners was never disclosed to other partners by the interviewers. The interviews lasted 35 to 85 minutes.Interviews were transcribed verbatim anonymously and analyzed using QSR International's NVivo 9 software . ContentNine group practices, twenty-one practitioners and one secretary were included in the study Table . The groFour major themes were identified from the analysis, from the creation of the group to the lessons learned from the breakup experience, including the reasons for the breakup and the feelings associated with this experience.Setting up in a private practice, even in a group practice, was perceived as risky, mainly for financial reasons. Thus, young doctors did not feel ready for the challenge and expected support from colleagues who were not always eager to help. The interviewees usually joined a group practice very quickly, often seizing an opportunity. They were not very careful and sometimes had not worked at the group practice, even for a short period, before becoming a partner. For instance, the length of time the physicians had worked as a locum at the group practice prior to joining it varied from 0 to 3 years. When joining a practice, most of the practitioners had no apprehension towards the group, while a few immediately feared possible incompatibility within the group practice. Some of them accepted to join a practice located in a place where they did not want to live in, simply to avoid an area with a high density of GPs.G4: “I quickly decided to set up in a place I didn’t know, where I had never worked as a locum.”B3: “When I joined the practice, I was asked very precise questions, such as “What kind of newspaper do you read?” (…) If I’d answered “Le Figaro” (a French right-wing newspaper), I’m not sure I would have been accepted by the group!”Some practitioners joined a group practice for financial reasons, for example in order to reach a sufficient level of activity quickly within the group or to share the costs of equipment and a secretary. Others highlighted their wish to organise their practice, in order to reconcile their private and professional lives. Another expected benefit was team spirit, based on sharing of the workload, ideas and knowledge. Sometimes, the expectations were excessively high, given the actual conditions of practice, especially when the practitioner had not worked as a locum within the group practice before.B2:“I thought that, in principle, since they were apparently playing by the rules, I would have work right away, which was in fact my biggest concern.”B4 “A group practice seemed to me to be somewhat of an ideal. On many levels, such as duties, working hours, secretarial tasks, shared areas, never feeling lonely, etc.”F1: “I had big illusions, and really high expectations concerning joining a group practice.”The main imbalance laid in the uneven distribution of medical activity, which particularly affected new partners. They reproached their associates for not having helped them to develop a high enough level of activity. Some of them even reported that they left the group because they did not earn a sufficient amount of money for living. The notion that certain patients “belonged” to a doctor generated a competitive climate. Several partners found that the sharing of the costs of the group practice was unfair or that common duties were unequally distributed. While the partners wanted to be acknowledged by their associates, a kind of hierarchy often developed, especially with new partners, based on seniority or a higher level of medical activity. At worst, a partner felt dominated, manipulated or even persecuted by another perceived as powerful, particularly if nobody in the practice dared to contradict him. These imbalances sometimes generated a survival instinct in one or more of the partners.G4: “He [G3] used to see between 35 and 45 people a day. When I started, I used to see 6! (…) They didn’t help me. They even tried to make it worse by coming and getting my patients in my waiting room and saying: “Umm, I can see you quicker!”G2: "Since he was working less, he was always complaining because he had to give as much as we did for the running of the practice.”Incompatible personalities were also the cause of tensions within the group practices. Extremely different working habits, concerning for example usual length of consultations or participation in continuing medical education, were specifically difficult to deal with. They were sometimes related to differing conceptions of quality of care, and could lead to intolerance. The management of the practice led to financial or organisational disagreements. New partners particularly expected to organise the practice as closely as possible to their ideal, even by trying to change their partners' habits. The intrusion of personal issues into the practice was in some cases the main reason for a breakup.B4 – “We had arguments when he [B3] could no longer accept the fact that we didn’t try to imitate him.”F1: “When I arrived (…) I wondered: “Wait a minute, there’s a place and there’s no meeting. Should I do something about it?” I said to myself, “Well, I’m going to change things” and so I suggested having a meeting. (…) In fact, it never worked. Every time, there was only one or two of us …”D1: “We had to take on board the fact that our third colleague [D3], at a certain point in time, was no longer able to distinguish between his professional life and the problems in his personal life.”Lack of communication or misunderstandings between partners were consistently reported at one stage or another in the breakup process and were sometimes fostered by workaholism. For example, it could happen that one partner was speaking about interpersonal relationships whereas another was speaking about purely legal issues. When trust was lost, it appeared difficult to regain, especially in the event of a breach of professional ethics. The conflict worsened if the partners did not question themselves individually. The secretary interviewed gave a precise analysis of the group practice she had worked at for thirty-seven years.B1: “Whenever there was a big crisis within the group, it was when we no longer took the time to communicate.”A3: “It also helped me realise that… there are some people who don’t care… who don’t have any ethics, who are not honest and who are ready to steal patients from their colleagues.”B4: “The oldest member [B3] found himself alone with his idea of medicine, his idea of the group, and he wasn’t flexible or open enough to accept the fact that there might be other ideas, other ways of doing things, etc.”The breakup process was described as extremely painful, stressful and exhausting, particularly for the partner who was leaving, and in a few cases was even a cause of illness. These feelings were sometimes related to a humiliating relationship, where a "naive" partner faced one or more "malicious" partners, or when patients changed doctors. This suffering was so invasive that it could lead to depression, even if the partner did not necessarily realise it at that time. Indeed, this illness was sometimes revealed by family, friends or even patients. In one practice, some of the physicians even reported that several former partners had committed suicide. Feelings were so hard before the breakup that some of the physicians described a form of relief just after the breakup or when they discovered new professional horizons. Distrust was so intense that some physicians were described as "paranoid" by their partners. Resentment against partners was in some cases accompanied by anger and a loss of self-control. The psychological pain was often compared to that caused by a divorce. It could be associated with the fear of reprisals or of another breakup in a new partnership. For most physicians, that pain was still present at the time of our interview, sometimes decades after the breakup.I2: “And the fact that my partnership with I1 fell apart was so distressing for me, it was such an upheaval (…) it was the only thing I could think of, the only thing I could see. I was talking about it all the time, I was completely overrun by it.”A1: “There were a certain number of patients who changed from me to her. (…) I knew very well it was going to be like that but, for some of them, I probably didn’t measure the pain it was going to cause. The bastards!”B4: “Over several years, B3 had become increasingly paranoid about the three of us.”D2: “I would say that it’s a bit like after a divorce. You need to leave yourself a bit of time! You try to heal the wound, as it were.”F1: “It’s one of the biggest traumas of my life. On a scale of 1 to 10, I would give it an 8. It’s a trauma I have transformed into something positive.”G4: “For me, it is over. But apparently it isn’t really over [laughs] (…) I put a big lid on top of it and I don’t want to hear about it again. (…) It still troubles me today [in a very emotional voice].”Physicians who experienced a harsh breakup usually felt disappointed and defeated, and tended to devalue themselves. They expressed regrets, particularly regarding the contract. The feelings of suffering and failure were sometimes associated with feelings of betrayal by their partners. In some cases, this extended out of the practice, to disillusion with all interpersonal relationships. The shock wave often impacted their future activity, with physicians deciding to work in a solo practice or be employed as a GP, to specialize in psychiatry or in skincare, or even to stop working altogether. In some cases, it also affected their personal life, leading to divorce.G4: “I felt completely worthless. (…) Even so, objectively, for me, it’s a real acknowledgement of failure.”A2: “(What makes me bitter is) having become disillusioned with humankind. And ever since, I don’t believe in anything anymore, I don’t believe… you see? That anyone is reliable.”The litigious temperament of some partners and the eventual involvement in a lawsuit made a breakup even more difficult to face. Four of the nine breakups have actually been managed by lawyers, one breakup leading to a trial with an appeal seven years later. The breakup experience was further complicated by the perceived incompetence of professionals involved, including notaries, accountants, lawyers and representatives of the General Medical Council. Loneliness in the practice contributed to harsh feelings and was exacerbated by the lack of communication. Resentment ran higher at practices where partners no longer respected each other. We noted that an old friendship usually did not survive a professional breakup and even worsened this experience. On the contrary, an understanding spouse was a good support for several of the physicians interviewed, who reported that their wife had helped them to analyze the situation and their own feelings. A change of job or new professional prospects helped physicians to deal with the breakup, which could then proceed peacefully. However, partners remaining in the practice, especially in larger ones, were usually less affected than their partner who left and had to find another practice. One partner was so detached from his past experience that he did not remember the name of his former partner during the interview.A3: “At the end, we even resorted to communicating by certified mail, before I left!”D2: “We had to call upon lawyers to sort out the problem. (…) And it still isn’t finished! (…) There’s going to be more fighting.”D2: “The male partner [D3], yes, he was a friend: we’d known each other for 20 years, we ate lunch at each other’s houses… Anyway! We’d gone on holiday together, you see… There we are! (…) Now, we’re at each other’s throats! I would say that he’s someone I don’t like at all! Not to say worse.”G4: “One day my wife said to me, “You have to get it together, you have to do something, because you’re becoming as much of an ass as the others (the other partners).”G3: “It (the breakup) happened without any particular problem.”To build unity within the group, some physicians suggested sharing a project with common objectives. Partners should be selected with care, based on similar personalities and professional styles, after a probationary period. It was specified that working together on a shared project required some basic human qualities such as honesty, respect, tolerance and open-mindedness. Making concessions and accepting differences as complementary was considered invaluable, as in a marriage. Constant questioning of oneself and taking special care of relationships with the partners was recommended in order to make a partnership last while allowing for change and evolution within the group.G4: “You have to require a probationary period and a right to rescind, which should be at least 6 months in my opinion, no less.”G3: “It’s just like a couple (…) You have to make concessions.”A1: “ It’s never over! (…) It’s an everyday task. (…) You have to be careful of the others.”D2: “Inevitably, one day or the other, if you have different medical practices, you’ll inevitably have different goals and after some time, to set your goals and start working towards them, the logistics won’t be the same. So there will be dissensions concerning the logistics. And, if there are disagreements about logistics, there are disagreements about financial aspects, and once you start talking money… it becomes very sensitive.”Times for meetings should be scheduled and adhered to. Discussion between partners should occur in a neutral, convivial environment and be clear enough to avoid any kind of misunderstanding. Physicians should also spend specific times together, even without practice-related problems to solve. The physicians interviewed suggested fair, concerted sharing of common duties, workload, space and secretarial time. They recommended that no partner should have excessive power or authority over the others. Personal and professional lives should be separated. In particular, some physicians insisted that spouses should not interfere in the practice. Others suggested having cordial but not overly friendly relationships with partners.B5: “The meetings were very spread out, but we felt better after a meeting. We felt the pressure rise before a meeting, then there was the meeting and it went down again, in fact.”The main suggestion was to anticipate and to state clear internal rules for the management of the practice in the contract, including clauses for a potential breakup. The group should seek assistance from competent, motivated professionals such as notaries, accountants, lawyers or representatives of the General Medical Council. The physicians considered it was important to comply with the notification period before a breakup.B4: “There have to be staff rules which are also precise, so you know how to manage holidays, how to manage absences and how to manage practice duties, if you have decided to set some up.”A3: “The contract is not for managing things that run smoothly. It’s for planning ahead when things won’t work.”B1: “Planning for the separation is extremely important!”Most general practice breakups were the result of loneliness and suffering at work, which were often present long before the breakup and tended to increase over time. The breakup process was rarely smooth, illustrating that partnerships, like marriages, can be in hell rather than in heaven . Most ofLife in group practices is punctuated by risky steps. Weakening factors exist from the very beginning of the partnership, and problems are likely to increase at every change or event occurring in the group.The physicians interviewed had not been trained at GP practices during their medical studies and had not been taught how to manage a contract. Moreover, they usually only had little experience as a locum, as observed also in other countries, even if the locum time has been increasing in the recent years ,16. TheyDifferences in style can affect the concept of partnership . PersonaEvery change in the group raises the possibility of confrontation and exacerbates pre-existing tensions. In particular, the arrival of a new partner can alter the previous balance of personalities within the group . Lack ofSome physicians decide to set up private practices in order to be their own boss, while others have no alternatives. In this private model, young physicians are typically vulnerable because they earn virtually nothing for months before making a profit. This competitive system reinforces individualism, which is also grounded in the law that makes each physician responsible for their own medical procedures . As oppoIn the French system, private practices require strong involvement from GPs, and a breakup means a great personal failure to them. Furthermore, they do not get any administrative, logistical or even medical support. These difficulties may worsen in the future, due to the planned development of many multi-professional medical centers in France, which bring together healthcare providers from various professional cultures . A surveThe change in medical demography is partly responsible for differences in styles and values between generations of physicians. In broad terms, senior doctors are used to making themselves more available for their patients, fearing to lose them, while junior doctors are more interested in organizing their time schedule and maintaining their quality of life . CompetiMoreover, older doctors tend to reproduce a hierarchical organisation, in which some privileges are given to senior partners . AccordiDue to medical demography, the sex ratio of the physicians interviewed was unbalanced . However, the purposeful sampling process we used, in order to include women, allowed for the closest possible matching with the current trend in feminisation of the profession. Actually, we could not identify any gender specificity in the reasons for, or in the feelings associated with breakups. The high median age of the physicians (57 years) can be explained by their late start in the profession and the length of time between the breakups and our interviews. The proportion of clinical trainers was higher in our sample (7 GPs out of 21) than in the Rhône-Alpes region (9%) . But traBecause the skills and resources to create a supportive working environment all come down to fair, equitable collective management , an equaA new partnership contract should be discussed in detail and customised for each new member. It forces the new partner to clarify his expectations and fears, and their partners to update their own, which can vary with age and personality . Topics Using a third party may be useful at various stages for a group practice. In France, the General Medical Council provides standard contracts for partnership, but does not offer customised agreements for special circumstances. Calling upon lawyers or notaries to establish a contract with clear rules could later prove useful, especially in the event of a breakup. An internal or external assessment of the organisation of the practice could also help to improve its management, as with the Work Relationship Assessment Form, which was specifically created for group practices in the USA . SecretaSometimes the breakup can become unavoidable or preferable, to ease the suffering and prevent festering of relationships, despite a moral obligation towards partners or patients. In such situation, relationships with partners should be rational rather than based only on emotions, and each physician should remain respectful to his or her partners, during the partnership as well as during the breakup process. However, developing new professional prospects, starting a new partnership and building a new list of patients can be difficult for a physician weakened by a breakup ,29.Breakups are mostly favored by imbalances between partners, particularly in medical activity, by incompatible personalities or working habits, and by a lack of communication within a group practice. These weakening factors exist from the very beginning of a partnership, and problems are likely to increase at every change or event occurring in the group, which can lead to a painful breakup. Competition present in fee-for-service healthcare systems may favor this process. Some recommendations on how to avoid breakups or to attenuate the related painful feelings are provided. They include fair management, a shared project based on a precise contract, the consultation of third parties as necessary and, in the worst case scenario, leaving the group practice in time.The authors declare that they have no competing interest.FM and DS collected the data. FM, DS, EL and LL collectively analyzed the data. FM, DS and LL wrote the manuscript, which was reviewed by EL. All authors approved the final manuscript.The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2296/14/53/prepub


Formal language theory (FLT), part of the broader mathematical theory of computation, provides a systematic terminology and set of conventions for describing rules and the structures they generate, along with a rich body of discoveries and theorems concerning generative rule systems. Despite its name, FLT is not limited to human language, but is equally applicable to computer programs, music, visual patterns, animal vocalizations, RNA structure and even dance. In the last decade, this theory has been profitably used to frame hypotheses and to design brain imaging and animal-learning experiments, mostly using the ‘artificial grammar-learning’ paradigm. We offer a brief, non-technical introduction to FLT and then a more detailed analysis of empirical research based on this theory. We suggest that progress has been hampered by a pervasive conflation of distinct issues, including hierarchy, dependency, complexity and recursion. We offer clarifications of several relevant hypotheses and the experimental designs necessary to test them. We finally review the recent brain imaging literature, using formal languages, identifying areas of convergence and outstanding debates. We conclude that FLT has much to offer scientists who are interested in rigorous empirical investigations of human cognition from a neuroscientific and comparative perspective. We suggest that this grammar is appropriate for answering certain interesting questions, but has sometimes been over-extended to address issues for which it is poorly suited, for which we suggest alternative, more appropriate grammars. In the process, we highlight the need to clearly distinguish among a number of separate issues, which—although related—should not be conflated. These include notions such as hierarchical structure versus centre-embedding, context-freeness versus long-distance dependency and formal complexity versus recursion. FLT provides the theoretical concepts and terminology to clearly distinguish among all of these terms, and we argue that it should be used to do so more rigorously in the future. We then provide a detailed look at some of the recent brain imaging literature using FLT, highlighting the areas of nascent agreement along with outstanding open questions. We conclude by pointing out some areas within FLT that remain little explored, but might provide fertile ground for future research.In this paper, we review recent progress in applying FLT to empirical research in animal cognition and neuroscience, as well as highlighting some pitfalls that can accompany attempts to merge theory and practice. We start with a non-technical overview of FLT, intended to give an intuitive understanding of the theory and its significance and to provide a gentle preparatory overview for the more rigorous paper by Jaeger & Rogers . We then2.We have an intuitive sense that some cognitive computations are more difficult than others. For most people, it is harder to play chess or solve equations than to buy groceries or drive a car. For most of us, it is more difficult to parse sentences in a non-native language (regardless of our level of proficiency) than in our native language. However, a central finding of computer science is that our intuitions about complexity do not necessarily apply to computer programs. In fact, it has proved relatively easy to create machines that can play chess at a high level, but so far impossible to create adequate car-driving systems. Because of this, an important component of modern computer science is a framework for quantifying the ‘difficulty’ or ‘complexity’ of a computational problem or algorithm in terms that are explicit and unambiguous. Starting with the work of the brilliant mathematician Alan Turing, and combined with further insights owing to Gödel, Church, Post, Kleene, Chomsky and many others, FLT has grown today into one key pillar of the theory of computation (and thus compiler design and many other aspects of computer science). The other main pillars are the theory of computability (what problems can or cannot be solved) and the theory of problem complexity  ,17.The theory of computation provides the practical basis for software tools we use everyday, which thus provide useful illustrations of the core concepts of FLT. We favour such everyday examples from computers, rather than mathematical formalisms, because we expect that most of our readers will have some experience with the former but not necessarily with the latter. More detailed and mathematical treatments are easy to find ,17–20, a(a)dir command in DOS, or the ls function in UNIX, provide everyday examples. Such functions use a syntax that allows us to search for the arbitrary target pattern in a large database of words and/or numbers. Given a set of file names:We start our survey with a simple, well-defined computational system, termed a finite-state machine, which is equivalent to another simple construct called a ‘regular expression’. Search functions, such as the > filenames = {a.wav, b.doc, c.bmp, MySong.doc, MySong.wav}running the function:> ls *.wav filenames on this set will return the subset> {a.wav, MySong.wav}.ls or dir command that the string(s) can start with any characters in the alphabet, we do not care which or how many. This search string is one simple example of a general framework called ‘regular expressions’, which provide a very powerful basis for computer-based search that underlies searching, replacing and other functions in many computer programs. This ability to use regular expressions to match patterns was first instantiated in the grep function in UNIX/Linux, and has proved so useful that the term ‘grep’ has entered hacker lingo as a verb meaning ‘to search by computer’.The search string ‘*.wav’ says, in effect, ‘give me all the strings that end with “.wav” ’. The * character tells the 1 More specific operators also allow us to specify a single, unspecified character (?), a character from a particular set (e.g. numbers {0–9} versus letters {a–z}), or even a specific number of characters from a certain set. Any time you have some pattern or a set of patterns that can be captured by a regular expression, you can use grep to search an arbitrary database for that pattern. You can grep for your name or email address or telephone number in the archives of a discussion group, or grep for a particular gene sequence in the online human genome database. The search engine Google is an extended version of grep that takes the entire web as its database. Regular expressions are at the core of computer search in today's world.Regular expressions are composed using a few simple but powerful rules and operators, familiar to many computer users. The operator *, as used earlier, means ‘any string of any length’ and by appending it to our search string (e.g. ‘string’), we can find our target pattern even if preceded by anything (*string), followed by anything (string*) or buried in anything (*string*).nBn in FLT. It is easily proved that this set cannot be specified by a regular expression . We conclude from this fact that there are patterns that we can conceive of, and that we could easily (if laboriously) recognize ourselves, but that cannot be captured by a regular expression. Why does this matter? Because, as demonstrated by the mathematician Stephen Kleene in 1956 in the theorem that bears his name, regular expressions and the corresponding rule sets termed ‘regular grammars’ are exactly equivalent to one of the most ubiquitous classes of computing devices, which are termed ‘finite-state automata’ [Given this flexibility and power, we might think that regular expressions are capable of specifying any kind of pattern that we can imagine. Crucially, however, this turns out not to be true. For instance, imagine a simple symmetrical pattern where a particular number of items of type A is followed by the same number of a different type B. Examples of this set include {AB, AABB, AAABBB, AAAABBBB, etc.}, and extend indefinitely (so a string of 1346 ‘A's followed by 1346 ‘B's is still a member of the set). This pattern is notated Automata’ .(b)FLT relies on abstract models of computational systems termed ‘automata’ . Two canonical examples of such models are the finite-state automaton (FSA) and the Turing machine. Automata such as these are mathematical abstractions, not real devices designed to be manufactured. For example, the Turing machine includes as part of its mathematical definition a storage tape of infinite length, and thus we could never build a real Turing machine. Many automata, although well-defined in theory, are unbuildable in practice (a fact that has some implications that we will discuss later). Despite this, the abstract notion of a Turing machine is extremely important and powerful in mathematics and FLT: infinity is a powerful tool for mathematical abstraction, but not a real thing that we find in the world.The simplest class of well-defined automata are called finite-state automata because they have a finite number of operating states or ‘positions’. The FSA starts at a predefined start state, and then jumps between its other states, depending only on its current input and current state. For each of these jumps, it can emit an output symbol as it hops along. Thus, an FSA can be fully defined by its set of states, its input alphabet (the input symbols that it recognizes), an optional output alphabet (which might or might not be different) and a function that tells it which is the next state to go to, given its current state and current input. Any given FSA is capable of ‘recognizing’ a certain set of patterns, and rejecting others. By ‘recognize’ we mean simply that, given this pattern as input, it can generate some particular prespecified output (e.g. ‘OK’). The set of patterns recognizable by an FSA may be infinite. Because of this, and its simplicity, the FSA is a good starting point for further discussions of automata and computational complexity. Critically, as already mentioned, Kleene's Theorem demonstrates the equivalence, or interchangeability, of FSAs and regular expressions.c shows (p. 1103). However, other patterns are clearly beyond the capabilities of an FSA, because we already know that certain patterns, such as AnBn, cannot be captured by regular expressions: they are beyond the capabilities of our simplest class of automaton. Thus, something with more computational power is clearly needed.One point, overlooked by many, is that FSAs can recognize (or generate) a simple, long-distance dependency of the start-and-end sort (e.g. ab*a or cd*c) as the automaton in figure 1(c)2 Thus, in addition to possessing a large (but finite) set of states that its controller can occupy at any one moment, the Turing machine has by virtue of this tape an additional unlimited form of memory for storing past operations, intermediate results and so on. A Turing machine can easily recognize the AnBn language described earlier: it simply stores the number of times A has been repeated  and then compares that with the number of Bs (B to B jumps). Thus, Turing machines are more ‘powerful’ than their FSA component, in the sense that they can recognize patterns and solve problems unsolvable by any FSA. This is not surprising given their additional resources. What is altogether more remarkable is that the Turing machine is capable of computing ANY deterministic function whatsoever: if something is computable, a Turing machine can compute it. Thus, modern computer scientists accept the Turing machine as their very definition of ‘computability’, broadly accepting the ‘Church/Turing thesis’ that a function is computable if and only if it is computable by a Turing machine.Other automata take an FSA as their starting point, and achieve additional computational power by adding some additional form of memory. The most important and powerful such automaton is the Turing machine, which adds to an FSA (the ‘controller’) a storage tape of unbounded length, on which symbols can be written or erased.computational power: the FSA provides the lower level , while the Turing machine provides the upper limit . We will use the term ‘computational power’ in this paper in this specific sense, framed by the specific automata discussed in FLT. We do not mean to imply by this that this is the only way to insightfully characterize the power of algorithms, or that ultimately this is the best way to think about the different aspects of brain function. What this specific sense of computational power gives us is an explicit, formal axis along which any particular algorithm can be placed, which thus provides one useful dimension along which to characterize the rule-governed capacities of a machine, or a human or animal subject. Other potentially useful dimensions will be discussed briefly at the end of this paper.The Turing machine and FSA are well-defined automata that provide useful endpoints for a scale of — Are there other intermediate classes of automata, with powers greater than an FSA but less than a Turing machine?— Where do human computational powers (or those of other species) fall along this spectrum?Given these two endpoints, we might immediately ask two questions:(d)In an attempt to answer the second question, the young Noam Chomsky and his colleagues built upon the framework already discussed and provided a positive answer to the first question. Chomsky outlined a set of intermediate formal possibilities, between the extremes of Turing machines and finite-state automata, and arranged them in a theoretical hierarchy that now bears his name. Because both the nature and the importance of this hierarchy are sometimes misunderstood, we will try to make clear here both what the Chomsky hierarchy is, and why it is important. First, let us consider the relationship between a Turing machine and an FSA. Because every Turing machine contains within it an FSA, anything computable (e.g. any pattern that can be recognized) by an FSA is perforce computable by a Turing machine. Thus, the set of FSA-recognizable patterns is a proper subset of those computable by a Turing machine. This is obvious from the way in which these automata are defined.nBn language discussed earlier, which is beyond an FSA. Thus, we now have a nested set of patterns, enclosed one within the other like Russian dolls. A second intermediate form of automaton is called a linear-bounded automaton and includes both the FSA and PDA within it. The Chomsky hierarchy incorporates several intermediate levels of automata, which have in common with the Turing machine an additional memory system but discard the assumption that this memory can be freely accessed. For example, a ‘pushdown automaton’ (PDA) includes an FSA and a pushdown stack . Because a stack is more limited than the infinite tape, it is intuitive  that the PDA is less powerful than a Turing machine, while being more powerful than an FSA. And as before, the set of patterns recognizable by the PDA is a proper subset of those captured by the Turing machine. In fact, a PDA can recognize the AThe Chomsky hierarchy provides a broad framework for discussing the computational power of automata, universally accepted in this role in theoretical computer science, algorithmic theory, FLT and discrete mathematics. It is well-defined, explicit and unambiguous and discussed and defined in any textbook on these topics. The Chomsky hierarchy contains no hidden assumptions about Universal Grammar, the ‘poverty of the stimulus’ argument or other of Chomsky's various more controversial ideas about language. A computer scientist using the Chomsky hierarchy need not accept other aspects of Chomsky's thought, any more than a logician using Bertrand Russell's innovations in mathematical logic need accept Russell's pacifism or atheism. While this should be an obvious point, we have been surprised by how often the distinction gets blurred. Although there are other ways to arrange automata in hierarchies of ascending power, as well as finer subdivisions of existing hierarchies , we focuany computational device: the definition is formally equivalent to a vast number of alternative implementations, in a wide variety of forms . Thus, the term ‘computation’ is used here, and in computer science in general, in a very broad and inclusive sense to capture any algorithm (information processing system). Neural networks and a vast array of other implementations are part of this classification, not alternatives to it.One might object that some models of neural computation, especially connectionist networks, offer just such an alternative framework. This is not true. In fact, neural networks are automata, like any other well-defined computational system. Indeed, surprisingly simple recurrent neural networks can be shown to be Turing-equivalent in principle  With these preliminaries behind us, and the basic aims and principles of computational theory clarified, we can now introduce the terminology and principles of FLT. On the one hand, fortunately, this terminology is simple and unambiguous, and thus quite familiar because it maps the core technical concepts onto everyday terms. On the other hand, this familiarity can be deceptive and misleading. As typical with technical terms, we must beware of unwittingly slipping from interpreting the words in their technical sense to their broader everyday sense.alphabet A is a set containing a finite number of indivisible symbols. A string  is a finite sequence of symbols, and a string-over-A is a finite sequence of symbols from A. The set of all such strings-over-A is denoted A*, pronounced ‘A star’.3 Finally, a language over A is any subset of A*. Put verbally, a ‘language’ in this abstract sense is some set of ‘legal’ strings from our alphabet A. Put concretely, lets say an alphabet A1 consists of the digits 0–9. Then any integer number is a member of a language defined over this alphabet (a member of the language denoted by A1*), but the number 1.35 is not . The string ‘cat’ is, for the same reason, not a member of this language. Similarly, if our alphabet consists of the letters of the Roman alphabet a–z, we could define some more specific languages where ‘cat’ is contained in the language but ‘katze’ is not.First, the terminology: an These terms and their definitions seem quite intuitive and easily understood. The danger is that they are so easy to understand that it is easy to forget what they do and do not involve. Most prominently, there is no discussion of meaning within this framework. Although we can easily design a grammar, in the technical sense, that can accept ‘cat’ while rejecting ‘katze’, this system has no understanding that these strings mean ‘feline animal’, or indeed that they could mean anything at all. A grammar responds purely to sequence and has no way of embodying meaning or of ‘understanding’ the signals that are fed to it. These systems are purely syntactic and have no semantics. Thus, ‘language’ in FLT is missing a substantial component of natural language . A ‘language’ in this formal context is a set of strings (a ‘stringset’), and nothing more.grammar is a finite set of rules that specifies some  language. Note that the sentences making up such a set will themselves be finite (in the same way that the set of integers is infinite but each integer is itself finite).4Continuing our exploration of FLT, the simplest way to define a finite language is to simply list all its members. This ‘brute force’ approach is a possibility as long as the language is finite, e.g. for the set containing the words of English . It will not, however, be able to deal with the integers, or the sentences of English, because the number of integers or English sentences is unlimited, and the list could never be complete . If the language in question is infinite, it cannot be listed, and our only hope is to come up with some finite set of rules to generate the language, termed a grammar. A We are finally in a position to restate the Chomsky hierarchy in explicit formal terms. The Chomsky hierarchy incorporates a nested set of automata of increasing power, each of which can generate the strings of some formally defined class of languages. The automata  are shown with their corresponding formal language classes, in sensu lattu logically includes all FSAs as a subset or special case. However, these terms are often used informally to exclude their subsets, much as the term ‘reptile’ is used by biologists to delineate all those descendants of the ancestral reptile who are not birds or mammals . Because this type of usage is quite indispensable in the research programme outlined here, we will explicitly define it. Using basic set theory, we can define any of these automata sensu strictu in an exclusive manner. Thus, we could use the term ‘PDA sensu strictu’ to delineate all PDAs that are not simply FSAs . Of particular importance in the present context are the ‘supra-regular grammars sensu strictu’, which we define as the class of all automata  that are not simply FSAs . This is the sense in which the terms ‘supra-regular grammar’ or ‘supra-regular processing’ will be used for the rest of the paper.A subtle terminological difficulty arises from the nested aspect of the rule systems just described. Because FSAs are contained within the class of PDA, the term ‘pushdown automaton’ (f)As already suggested, Chomsky's primary interest in formalizing automata and organizing in this way was to provide an initial framework for understanding human natural language . In particular, Chomsky pointed out that English cannot be captured by a regular or ‘finite-state’ grammar (FSG) because it includes structures  that are beyond any FSA's capabilities. He further argued that various linguistic phenomena of movement and sentence transformation (e.g. from active to passive) are beyond the capability of context-free languages as well, and thus that natural languages must occupy some broader subset of the type 0 grammars . However, it quickly became clear that transformational grammars are in fact too powerful, requiring a cumbersome set of constraints to make grammars with this degree of power tractable. Subsequently, theoretical linguists working within this formal paradigm have gradually honed in on the level of computational power required for natural language ,26. Afteany FSA. Similarly, just because some animal species can be shown to do various tasks at the finite-state level, we cannot assume that they can induce any FSG. Whatever class of computational systems natural language entails, it will always be some subset of the categories of automata described in FLT . The difference between a finite-state and context-free representation is also important to keep in mind when compiling computer code or analysing natural language texts.5 Knowing whether such texts demand context-free (or higher) powers, or not, is thus very valuable for anyone interested in building fast, robust computing systems. Similarly, recent years have seen an explosion of interest in complex animal signals such as bird song and whale song, and a vast amount of data have been pouring in from both laboratories and the field that needs to be processed by computer [Given this caveat about the purely syntactic nature of FLT, one might ask why anyone should be interested in the question of where human  capabilities lie in the classification system of FLT. Here are a few reasons, ranging from practical to theoretical. From a purely practical viewpoint, scientists attempting to create computer programs that deal with corpora of data are greatly aided by knowing where the signal-generating system they are studying lies in this system. For instance, the parsing and compiling of either regular or context-free languages are well-defined problems with practical working solutions, but this is not true for context-sensitive languages , there is no direct correspondence between the units of music  and the structure of thought. For all its power and greatness, music simply cannot communicate plebian facts such as ‘the lion is in the third cave from the right’ or ‘you need to soak that nut in water and ashes for two days before you eat it’, nor can you use a musical phrase to represent those thoughts to yourself. Indeed, if (as in a few isolated cases) we use musical means such as drums or whistles to communicate thoughts, it ceases to be music and becomes language . Thus, an intimate correspondence between signal and meaning is the must have such a structure [et al. [There are many grounds for suggesting that thought itself has a tree-like, hierarchical structure . Researctructure . But if tructure  and Hail [et al. ), we nee3.x can be trumped by simply generating ‘John thinks that x’. Although each of these sentences is of a fixed and a finite length (there are no infinite sentences), the set of sentences is infinite. From this mathematical perspective, we should no more doubt the infinity of English than we doubt the infinity of the integers.Before we turn to experimental work grounded in FLT, we will briefly discuss the controversial issue of ‘infinity’ in discussions of language. This is an old issue, nicely encapsulated in Wilhelm von Humboldt's suggestions that human language makes ‘infinite use of finite means’ . ClearlyHowever, there are more subtle arguments for and against the importance of infinity in natural language . For exaSome system of more general or abstract rules is necessary to account for this ability. As Chomsky notes [However, no one supposes that the child simply memorizes all heard sentences: any language user can generate and understand novel sentences, beyond the finite input they received in childhood. ky notes , ‘a gramky notes , and to could be modelled by a FSA, augmented with transition probabilities (a form of Markov process). For this model to have any psychological validity, a child would need enough data to infer these probabilities from the input. So we can ask what order of Markov approximation would be needed for typical sentences. This reduces to the question ‘how many words can separate two words that are dependent upon one another in a sentence?’ In the grammatical English sentence ‘The woman you recently invited to come to New York and give a lecture in our department seems to be sick’, 15 words intervene between the inter-dependent words ‘woman’ and ‘seems’. An attentive English speaker will certainly notice if this pair were incorrectly inflected (e.g. ‘The woman … seem’ or ‘The women … seems’). Hence, we would need a k-limited Markov source with k = 15 to capture this dependency reliably. But Levelt shows that even with unrealistically lenient assumptions [15, or more than one billion. The busy child would need to set about 30 parameters per second, throughout all of childhood, to assimilate such a model. Thus, a finite-state model must be limited to be learnable  or it could be theoretically adequate but, owing to the huge number of parameters, practically useless as a model of the child. This argument can be made from many different perspectives, but will always come to the same conclusion: given realistic assumptions, no regular grammar can adequately model English or any other natural language. This is both a practical and a theoretical conclusion.Another convincing counter-argument, owing to Levelt , goes asany model of human cognition will make simplifications, and thus will be inadequate in certain ways. This is intrinsic to model-building. What we seek are models that make the right generalizations, and that fail in reasonable ways. For example, we can make a simple modification of the weakest supra-regular system, a push-down automaton, in which the stack memory is of a fixed, limited depth. Such a model has no problem with long-distance dependencies, but it will have problems with multiple levels of embedding. This is precisely what is observed in humans experimentally, in abundant psycholinguistic research [This combination of arguments has led most commentators to accept the supra-regular hypothesis for humans. Of course, research ,51. In oproves that any complex signal can be built up by a series of sine and cosine waves. Unfortunately, the proof requires an infinitely periodic signal (which continues unvarying from the infinite past to the infinite future), as well as an infinite set of sine waves. Despite these unrealistic assumptions, the discrete Fourier transform, applied to real signals, turns out to be an incredibly powerful tool at the heart of every mobile telephone and spectrographic programme on the planet today. In the same way, we can readily assume that PDA's stack or our Turing machine's tape will be of limited depth, and try to match this to empirical observations of finite humans. Although in doing so we lose the ethereal certainty of theorems, we lose few if any of the practically relevant insights of FLT.In conclusion, we should not conclude from the importance of infinity in formal mathematical proofs that infinity plays a central role when we turn to practical empirical issues. Infinity is a powerful tool for abstraction, and its judicious use in mathematics allows a kind of certainty that is wonderfully satisfying. For example, Fourier's Theorem 4.nBn, which has been the focus of numerous recent studies.In the rest of this paper, we explore how FLT can be used, practically, by biologists, psychologists and neuroscientists, to design and execute experiments and analyse the resulting data. Often, such studies use artificial grammar learning (AGL) paradigms, see . Because(a)nBn, in which the number of ‘A’ units is precisely matched by the number of ‘B’ units, has played a prominent role in the development of FLT. It is a textbook example of a simple language that cannot be captured by a regular grammar, as already discussed. Despite its ubiquity in the theoretical literature, to our knowledge, the first use of this grammar in experiments was that by Fitch & Hauser [n, which entails any number of ‘AB’ units, and the other was AnBn. In both cases, the units were consonant–vowel speech syllables, with the A units spoken by a human female and the B units by a male. Fitch & Hauser found that, while college undergraduates were able to master both grammars, the monkeys only showed above-chance rejection of non-grammatical stimuli for the regular grammar. The monkey's success on the regular grammar showed that the techniques were adequate to elicit rule learning, with generalization, from this species. Fitch & Hauser concluded from this pair of results that monkeys ‘can spontaneously master’ the regular grammar, but are unable to cope with the supra-regular grammar, and thus that ‘tamarins are unable to process a simple phrase structure’ (where ‘phrase structure grammar’ was explicitly defined to mean a supra-regular grammar sensu strictu). This conclusion is clearly consistent with the supra-regular distinctiveness hypothesis discussed in Fitch et al. [The stringset defined by A& Hauser , who comh et al. .nBn language [nBn provides a litmus test for recursion was further perpetuated by a second study testing for recognition of the AnBn grammar, this time in starlings. Gentner and co-workers [nBn and titled their paper ‘Recursive syntactic pattern learning by songbirds’ . In a commentary on the starling paper, Gary Marcus [nBn language is generally assumed to be recursive’. As a result of these multiple characterizations, there is now considerable confusion in the literature about what, exactly, mastery of AnBn (or other supra-regular grammars), by humans or any other species, is supposed to indicate. We now discuss the possibilities.Unfortunately, this conclusion was immediately misinterpreted as concerning ‘recursion’, in a commentary in the same issue by David Premack, which stated ‘In a paper on page 377 of this issue, Fitch & Hauser report that tamarin monkeys are not capable of recursion. Although the monkeys learned a non-recursive grammar, they failed to learn a grammar that is recursive. Humans readily learn both.’ , p. 318.language , only so-workers  found coy Marcus  stated t(b)nBn tells us one thing, and one thing only: that the system is supra-regular (beyond finite state), and therefore has some form of auxiliary working memory, such as a push-down stack, counter or tape, that is not available to an FSA. Although a weak automaton that can recognize AnBn is a PDA, with a stack depth limited to the maximum value of n, any more powerful automaton (such as a linear-bounded automaton or a Turing machine) can also recognize (or generate) this stringset. Thus, simple mastery of this grammar by some system is not sufficient to tell us where in the nested class of systems occupying the supra-regular portion of the Chomsky hierarchy it lies: only that it is supra-regular.From the viewpoint of FLT, a system's ability to recognize the stringset generated by AnBn strings, as illustrated in the structural diagrams of nBn language. The top-most, which is the most obvious and appears to capture what humans spontaneously do when confronted with this stringset, could be called ‘count and compare’. This involves simply tallying the number of ‘A's, storing that number, tallying the number of ‘B's that follow and tallying that number and then comparing the two. This could be implemented by an integer register that is incremented by one for each A, and decremented for each B . Two registers could also be used, one holding each of the two counts, and comparing them (a). A PDA could recognize the same stringset by storing the number of ‘A's as a series of marks, pushed one by one onto a stack, and then ‘erased’ by removing them from the stack for each corresponding B. An MCSG compatible solution would be to write a 1 for each A on a tape. Once the B phrase starts, this system would rewind, and then cross off a one for each successive B. In either of the last two cases, an empty stack or a blank tape would be required for acceptance. Crucially, all of these alternative algorithms require supra-regular processing resources , and each suggests a different order of processing. What is relevant then is that there is some way of representing the exact number of ‘A's, and if this value is not bounded a priori to any fixed number, the system is perforce supra-regular. No conclusions about recursion are warranted.This ambiguity has important implications for the parsing of Aing them a. A PDA (c)nBn language? As nBn as a subset, as does the set of all strings that start with ‘A's and end with ‘B's (written A*B*). Similarly, the union of two regular grammars, A2B2 and A3B3, accepts all AnBn strings where n = 2 or 3. These and other regular grammars provide potential ‘cheats’ that would allow a regular system above-chance performance in experiments like these.What evidence is needed for us to conclude that a system ‘recognizes’ the Apossible regular grammars, we can eliminate most of the reasonably simple ones by employing foils  and extensions (to exclude the overly specific grammars). Thus, after exposure to AnBn strings where n = 2 or 3, we can then test our candidate system with A4B4 (n = 4) strings. If the system has induced the supra-regular rule, it should accept these generalizations. In contrast, the ‘regular union’ grammar given earlier would reject such extensions. So this provides one crucial test, allowing us to empirically exclude overly specific regular grammars. Both humans [2B2 [While it is difficult to exclude all h humans  and starans [2B2  accept sans [2B2 .nBn is the regular language A*B*. A system implementing this grammar would accept all AnBn strings and correctly reject ABAB or BAAB strings. The crucial test in this case is the ‘unmatched foil’ AnBm, where n ≠ m. Such strings will be accepted by A*B* or similar variant regular grammars, but clearly rejected by any system implementing AnBn. Although Fitch & Hauser did not test for this, several later studies [A second possibility is an overly lenient grammar that accepts the target strings but many others besides. A particularly crucial superset of A studies ,57 showe studies .et al. [However, there is a third and more subtle possibility, noted by van Heijningen et al. , that diet al. , these aet al.  argued tet al. . While iLonchura striata domestica), and again provided evidence for learning of the AnBn grammar in this species. In this case, rather than operant training, a mere exposure paradigm was employed, and vocalizations produced to different grammars and their violations were used as a dependent variable. The authors found that listening birds chirped more to novel AnBn strings, including novel extensions to n = 4. Unfortunately, they do not appear to have tested their finches with ‘unmatched foils’ AnBm, where n ≠ m, and thus we cannot exclude the regular grammar A*B* based on these data. The reason, presumably, is that the authors were focused on a different question: item-wise dependency in relation to ‘centre-embedding’. For further critique of this study, see Beckers et al. [Abe & Watanabe  used a hs et al.  and ten s et al. .(d)nBn grammar: that it necessarily involves centre-embedded dependency relationships between particular ‘a’ and ‘b’ items [nBn requires a supra-regular system, we have no basis for assuming any particular supra-regular automaton must be used to do so. While one might suggest that a context-free grammar is the most parsimonious assumption, and therefore that nested matching would be most natural (b), a system that possessed a tape (like a linear bounded automaton or Turing machine) might just as well implement a cross-serial matching as in figure 3c. This seems particularly likely in the case of humans because we know that cross-serial dependencies are required in some languages such as Dutch or Swiss German . So if the system did infer dependencies between items, there is no compelling reason to assume that these would be nested rather than crossed.This brings us to a second widespread misconception about the Ab’ items . There a natural b, a systnBn language make no demands that particular A items should match particular B items. Indeed, one simple way to write this grammar involves a random selection of A and B terminals (a). A different version that would entail dependencies between specific A and B items seems, in principle, more complicated (b). Put in terms of the various mechanisms discussed earlier, there is no reason for an automaton recognizing AnBn to write individual ‘a's or ‘b's to its stack or tape memory: it suffices to simply put any mark (e.g. a 1) for any A, and then subsequently count or erase them for each B. It is thus not surprising that humans exposed to AnBn strings do not keep track of or notice any particular correspondences, even if the experimenter employed a grammar like figure 3b to generate them [figure 3b induce the grammar in figure 3a, because both are fully adequate grammars for recognizing AnBn. The assumption of a centre-embedding item-wise dependency appears to rest on  confusion between phrasal dependency (which AnBn obviously has) with item-wise dependency (which it does not necessarily have).A more important reason is that most grammars capable of recognizing the Aerminals a. A diffplicated b. Put inate them . Since nnBn strings attend to, or can learn, and a considerable literature has grown up exploring this topic, further discussed by several of the papers in this issue. Several commentators have concluded that the ‘count and compare’ option is not particularly relevant to human language, and so although this potential strategy would be supra-regular, it would be of less interest than the centre-embedded or serially linked options [nBn strings generated with item-wise dependencies (as in b) failed to notice these dependencies [n nor ‘mismatch’ foils, and thus the conclusions they can draw from their study about supra-regularity are weak. In contrast, while the Dutch subjects in de Vries et al. [either nested or crossed dependencies in an AnBn framework [Of course, it remains an interesting question what kinds of dependencies humans (or birds) exposed to A options ,63. Veryes as in b failed ndencies ,62. It is et al.  did succramework –67, and ramework . Thus, anBn stringset spontaneously appear to adopt the simplest strategy—matching the number of ‘A's with that of ‘B's—rather than inferring item-wise dependencies. However, with adequate training, humans can induce grammars over any of the three possible structures in nBn grammar is well suited to test: other supra-regular grammars seem much more suited to address this question. In particular, the mirror grammar  is another supra-regular grammar, recognizable by a context-free grammar, well suited to examine pattern-based centre-embedding. A mirror grammar over {A,B} generates strings such as ABBA, BAAB, BABBAB, etc., in which the right half mirrors the left half .In summary, humans exposed to A(e)nBn language is useful. This of course depends on the questions one is attempting to ask. Those who have employed it with human/animal comparisons have, for the most part, been focused on the ‘supra-regular distinctiveness’ hypothesis, and for this, the AnBn grammar is and remains a valid tool [Assessing this ongoing debate, it seems reasonable to ask whether a further study of the well-studied Alid tool ,9,11,58.lid tool ,62,63,68can count, exactly, for small integers up to four or five [nBn requires more than simple counting: the system must count and compare across phrases. The evidence from animals, thus far, suggests that this computation, unlike counting, is difficult or impossible for most tested non-human species. This failure seems very relevant to any detailed analysis of the computational capabilities of different species’ brains.There are two answers to this question. The first is that, if one is focused on the ability to infer grammars beyond the regular or finite-state level, ‘count-and-compare’ is just as squarely beyond this level as is the mirror grammar. Crucially, a substantial animal cognition literature demonstrates that many vertebrates  or five –73: one nBn or mirror grammars that have been used to generate written or spoken stimuli  symmetry is not obvious to animals, and indeed may be beyond reach, even with training ,75. In Fe et al.  provide nBn grammar seems well suited for testing the ‘supra-regular distinctiveness’ hypothesis.If there is a fundamental computational restriction that prevents most species from accessing even bilateral symmetry or ‘count-and-compare’ strategies, this is surely relevant to these species' inability to acquire the syntax of natural language, which by all accounts require supra-regular capabilities of at least this level of computational power. We certainly encourage the testing of multiple species with many other supra-regular grammars, but as a particularly simple starting point, the AnBn grammar (even if implemented by ‘count-and-compare’) activates different neural processing routines from the (AB)n and similar regular grammars [A variant of this positive answer is provided by multiple brain imaging studies in humans that suggest that the use of the Agrammars ,66,76. Wgrammars ,78.nBn grammar. From the viewpoint of FLT, the question about whether a species (or a brain region) can cope with supra-regular stringsets needs to be separated from questions of centre-embedding  or recursion (which is not a question that can be answered with this type of experiment). While the AnBn grammar is simple and well suited for investigating the basic issue of supra-regularity, those interested in issues of dependency might benefit from branching out to other grammar types. For example, the ‘mirror grammar’ wwR is not just supra-regular, but its recognition requires long-distance dependencies between classes, and these dependencies are centre-embedded. In contrast, the ‘copy grammar’ ww has cross-serial dependencies, which require a supra-context-free computational capacity . We suggest that pitting the mirror and copy grammars against one another may be more rewarding than continuing to apply AnBn to questions it is not the best tool for.This literature also illustrates a potential pitfall of the A5.nBn grammar, stimuli have been presented in the domains of written, spoken or synthesized syllables, have involved explicit training with feedback or virtually no instructions (‘mere exposure’) and have employed explicit yes/no answers . verbally .nBn, without training or reinforcement in ‘mere exposure’ paradigms using only positive examples. In contrast, most animal studies have involved tens of thousands of reinforced trials over months or years [Regarding animal experiments, there is so much variability that few fair comparisons can be made across species. One key difference concerns training and reinforcement. Humans readily acquire multiple grammars, including Aor years ,58,80, aor years . Each ofAnother important difference between techniques is whether single grammars or dual grammars are presented. In traditional human AGL work, as in spontaneous techniques using animals, positive exemplars from a single grammar are first presented in the exposure phase, and then single-test stimuli are presented to be accepted or rejected . This alnBn [In general, there are too few studies in which different species are given the same stimuli in comparable tasks to permit fair comparisons. The failure of tamarins in a spontaneous task involving AnBn  cannot bnBn , given tnBn ,81 or whnBn . Comparin task indicates that they do pay attention to patterns in human voices, which is unsurprising given captive animals’ close and dependent relationship to humans for feeding and care). Human voices have also been used successfully with rats [One final issue of comparability is particularly salient in auditory AGL tasks using vocalizations. Fitch & Hauser  used recith rats . Howeverith rats ,58,60, gith rats ,82 to al6.The second major wave of research capitalizing on the AGL/FLT combination is in neuroscience, and particularly human brain imaging research. An important research paradigm using the FLT framework is the AGL learning paradigm introduced more than 40 years ago by Reber . In thesIn contrast, a new and fast-growing field has used FLT to design many different grammars, including supra-regular grammars, to probe the neural mechanisms that underlie abstract pattern recognition abilities in humans, and compare them with those involved in natural language processing. Such studies involve brain imaging technologies such as electroencephalography (EEG) and magnetic resonance imaging (MRI). EEG measures brain electrical activity, while functional MRI (fMRI) images blood flow. Both techniques thus index cognitive function in the brain. Transcranial magnetic stimulation (TMS) uses a powerful magnetic field to perturb neural function, allowing experimental evaluation of the role of particular brain areas. Finally, diffusion-weighted MRI or diffusion-tensor imaging (DTI) can be used to image both grey matter anatomy and the white matter fibre tracts connecting different brain regions that constitute a network. A crucial aspect of the new brain imaging studies lies in the comparison of the brain activation for artificial and natural grammars to investigate whether particular types of artificial grammar recruit brain regions and networks used in processing natural language grammar.(a)One focus of particularly intensive, and controversial, research in recent neurolinguistics concerns Broca's area, or the left lateral prefrontal cortex more generally, and its role in processing linguistic syntax and sequential patterns. The term ‘Broca's area’ refers anatomically to the pars opercularis and pars triangularis in the left inferior frontal gyrus (LIFG) and cytoarchitectonically to Brodmann's areas BA 44 and 45 ,88. In BThe debate concerns the degree to which Broca's area is specialized for syntax processing or even linguistic processing more generally, or rather subserves some domain- and modality-general computations such as hierarchical planning, working memory or selection among competing alternatives. Advocates of different models often have different theoretical backgrounds, and adjudication is made difficult by the theory-specific characterization of each different model. One of the first models  assignedwh- word ‘what’ is linked to the empty slot at the end of the sentence, where the direct object of ‘like’ would normally go (this is thus termed ‘wh- movement’). Grodzinsky & Santi [In a related, but more specifically linguistic, model, Grodzinsky  suggeste & Santi  reviewed & Santi .6 and ‘nesting’ revealed a clear involvement of Broca's area as syntactic complexity in these constructions increases. Across these different studies, ‘movement’-related activation is localized in the more anterior-ventral part of Broca's area (BA 45), whereas ‘scrambling’ clusters in the more posterior part of Broca's area (BA 44). ‘Nesting’, i.e. the processing of centre-embedded structures, was only investigated in a few fMRI studies. One study that evaluated the processing of centre-embedded sentences found activation in BA 44 increased with the number of embeddings [A neuroanatomically more fine-grained model suggested by Friederici  argues fbeddings . The thrbeddings –94 all fThese three models posit an exclusive role for at least some portion of Broca's area in processing syntax. An alternative framework has been suggested by Hagoort , who offall of the earlier-mentioned characterizations of the role of LIFG share the necessity of supra-regular processing resources.Thus, all of these models have in common that they attempt to single out a certain component of syntactic complexity for which Broca's area plays an important processing role. In each case, sophisticated models have been proposed to ground the neuroscientific results in linguistic theory, but in each case the terms used are specific to a given theoretical framework (even though they can cope with the same syntactic phenomena). Progress in testing these different ideas requires an over-arching framework broad enough to encompass all of these possibilities, and precise enough to specify their differences from one another and from other hypotheses about the computational role of Broca's area. FLT, and the theory of computation more generally, seems well suited to this role. In particular, it is noteworthy that (b)Neuroimaging studies on AGL have tried to elucidate the neural mechanisms of language learning and have provided insight into the nature of knowledge that is learnt in artificial grammars of a given type. Different studies have focused on different aspects of rule and grammar learning.The first generation of AGL neuroimaging studies focused on the neural correlates of implicit leaning using Reber-type FSGs. They revealed that two aspects of learning, namely similarity-based learning and abstract rule learning, were correlated with two different brain systems. The former learning type was seen to involve the left hippocampus, whereas the latter was found to activate anterior prefrontal cortices bilaterally . Other AThe second generation of neuroimaging studies tried to relate AGL research more to natural language research, testing artificial grammars more similar to natural grammars, for example, a grammar named BROCANTO ,117,118 Another interesting fMRI study compared the learning of a natural and an unnatural language . Italianet al. [nBn grammar with the regular (AB)n grammar in a between-subjects fMRI design. They found increased activation of the frontal operculum  for syllable sequences of both grammars, suggesting a role in immediate sequencing. Additional activation of BA 44, in the heart of Broca's area, occurred only with the supra-regular AnBn grammar. They concluded by hypothesizing that BA 44 is particularly activated in tasks requiring ‘the computation of hierarchical dependencies’. In a further important finding, this study used DTI to examine the white-matter connections stemming from these two different areas, and uncovered two separable neural networks. While the frontal operculum had preferential connections to the anterior temporal lobe via the uncinate fasiculus, BA 44 connected via an independent dorsally located white matter fibre tract to the posterior and middle temporal lobe: the Broca-to-Wernicke connection via the superior longitudinal fasciculus and the arcuate fasiculus. For further discussion of these tracts see later text.An early neuroscientific study explicitly adopting an FLT framework, published by Friederici et al. , contrasnBn grammar with no item-wise dependencies, these experiments were not getting at the linguistically interesting aspect of centre-embedding [This paper prompted several further studies working with the same grammar in different ways. One critique suggested that, by using a simple Ambedding ,62. As dnBn language, it remains interesting to investigate which strategies are used under what circumstances. A considerable literature has now developed that explores this question in detail, often including a brain imaging component [nBn strings with predictive dependencies (e.g. A1A2B2B1), subjects notice a violation of this dependency . After initial results suggesting that ‘mere exposure’ to centre-embedded dependencies in an AnBn grammar is not enough for subjects to recognize violations of those dependencies [can learn such centre-embedding, and extend it correctly, but that this takes additional training, exposure or prosodic structure in the training stage [Nonetheless, when subjects master the Aomponent ,76,103. ndencies ,62,63, ing stage ,123,124.ng stage ,66. Intrng stage .1A2B2B1 whereby A1 and B1 do not represent particular items, but a class of items) again showed activation of Broca's area (BA 44 in particular) together with motor cortical (SMA) and subcortical areas  for the processing of AnBn grammar sequences compared with (AB)n sequences [1A2B1B2 dependencies [An fMRI study designed to force the processing of centre-embedded relations . As Uddndencies  review ln and AnBn grammars, given that these two grammars have the same number of very similar rules [Other recent fMRI studies using AGL further investigated Reber grammars, challenging the view that Broca's area is particularly involved in processing non-adjacent dependencies. These studies reported activation in BA 44/45 for a regular grammar learning and classification task –127, butar rules .Thus, the combined data from the fMRI studies in artificial grammar learning and from non-language domains suggest that Broca's area supports the processing of structured sequences, and of supra-regular sequences in particular.nBn) and recursion . While the AnBn grammar is simple and well suited for investigating the basic issue of supra-regularity, those interested in issues of dependency might benefit from exploring other artificial grammar types including, for example, the ‘mirror grammar’ wwR. This not just supra-regular, but its recognition requires long-distance dependencies between classes, and these dependencies are centre-embedded. But the other way to approach this issue is to examine particular sentence structures in natural language.Although the literature reviewed earlier illustrates the value of FLT in designing neurolinguistic experiments, we need to separate the question of whether a species (or a brain region) can cope with supra-regular stringset, from questions of centre-embedding With respect to the identification of the brain basis of supra-regular stringset processing and the possible underlying processing strategies, a direct comparison between AGL and natural language processing may be useful.nBn in artificial grammar by directly comparing it with the processing of centre-embedded structures in natural language. In natural language, the respective long-distance centre-embedded dependencies are not dependencies in a symmetrical structure (as in AnBn [nBn structure this is not necessary [nBn in AGL [nBn structures.Such studies shed light on possible strategies underlying the processing of A in AnBn ), but in in AnBn ). In theecessary ,62. Thusn in AGL  and in nn in AGL  suggestsnovel rules must be learned, whereas linguistic studies examine pre-existing rules from native language processing. In most AGL studies, performance is only 70–80% correct, indicating that the learned rules are not well established. Performance by native speakers in natural language experiments is usually much higher. The performance in AGL experiments rather is more comparable to the performance level to language learners, be it in first language (L1) or second language (L2) acquisition. The brain activation pattern observed for L1 learners [One additional issue needs to be considered when comparing AGL and natural language. In the AGL paradigm, learners  and L2 llearners  who are This broad activation pattern bears some similarity to the activation reported for some recent AGL experiments ,127, whe(d)As already mentioned, one study used an MRI technique that allows one to make images of white matter fibre tracts . In thisdifferent dorsal pathways can be distinguished: one connecting the temporal cortex and the premotor cortex (possibly supporting sensory-to-motor mapping) and one connecting the temporal cortex and BA 44 (possibly involved syntactic processing) [Despite a long history of associating the arcuate fasciulus with language ,134, an cessing) ,140 et al. [nBn grammar [A useful perspective on the human connectivity data for regular and supra-regular grammars may be provided by adopting phylogenetic and ontogenetic viewpoints. Phylogenetically, recent DTI data show that non-human primates differ from humans in their connectivity pattern. Comparing macaques, chimpanzees and humans, Rilling et al.  found th grammar . This raA complementary perspective is provided by studies of human populations that have problems with the processing of syntactically complex structures in natural language, such as patients with brain lesions , or chilIf the functional interpretation of this latter fibre tract as part of the auditory-to-motor mapping system is correct, we would predict that very young infants should be able to recognize simple regularities in the auditory input. There is good evidence that this is indeed the case, for both AGL  and natu7.nBn grammar introduced by Fitch & Hauser [Although the marriage of AGL and FLT has already produced some new and important results, in both the animal and neural domains, this research programme remains in its preliminary stages. Current empirical research provides a detailed exploration of only two particular grammars: the complicated regular grammar introduced by Reber  and sinc& Hauser  and sinc& Hauser ,148 and & Hauser  may prov& Hauser . For hum& Hauser .Two further directions immediately beckon. FLT has developed considerably in the last decades in two important directions, both of which offer rich opportunities for empirical research in the ‘Grammarama’ tradition . The firThe second major advance in FLT concerns probabilistic models of syntax ,155. In In the longer term, we can anticipate that this research programme will be able to replace the highly stylized models of computation used in FLT with more biologically grounded computational primitives . While wn or AnBn). FLT also, and more controversially, provides a set of ‘fixed points’ of computational complexity, such as push-down stacks versus the endless tape of Turing machines, that represent abstract representations of different types of working memory. While future research may show that such abstractions are too artificial to usefully characterize neural computational primitives, performing experiments using them will be the surest and fastest way to find out. Currently available alternative models of neural or ‘natural’ computation [FLT both allows us to characterize such experiments in more general terms  and provides an explicit language for notating and reporting different string types or grammars (as in (AB)putation –162 offeThus, we conclude that the empirical research programme combining FLT with AGL will continue, and will continue to be exciting and controversial. Such research will be particularly valuable in uncovering relevant differences among species, and in helping us to characterize the neural mechanisms underlying these differences. This research programme can help itself to a well-developed pre-existing mathematical/computational framework and a rich body of formal understanding and important theorems. If such research eventually leads to a replacement of current formalisms by a computational theory more firmly grounded in biological and neuroscientific reality, it and thus paves the way for its own demise, it will be a welcome sign of scientific progress.