Theorist Problem Solvers
Lilly Solo
I consider this to be one of the major problems in combinatorics and have devoted manymonths of my life unsuccessfully trying to solve it. And yet I feel almost embarrassed towrite this, conscious as I am that many mathematicians would regard the question as moreof a puzzle than a serious mathematical problem. (p. 11)
if one is trying to maximize the size of some structure under certain constraints, andif the constraints seem to force the extremal examples to be spread about in a uniformsort of way, then choosing an example randomly is likely to give a good answer. (p. 6)
If one is according importance to mathematical activity in terms of its impact on mathematics as a whole, then rather than the transfer of theoretical results and apparatus between fields, it may be necessary to look to more subtle relationships, such as when:
Finally, there remains the question of whether there are missed opportunities arising from the presence of a barrier between the two cultures. Gowers ends his essay by encouraging dialogue. Perhaps blogs are the right arenas in which such dialogue might take place.
Cauchy-Schwarz is pretty much fundamental, not only in extremal combinatorics, but in virtually every area of analysis; it is the assertion that self-interactions control cross-interactions, regardless of how unrelated the two interacting objects are. One could almost define analysis as the branch of mathematics which uses the Cauchy-Schwarz inequality and its relatives (e.g. the triangle inequality).
If we additionally assume that the category is extensive, then we can certainly do some things that are usually proved using negative numbers. For example, we can show that there is a monomorphism
2. To my mind, this approach greatly clarifies the relation between labeled and unlabeled enumeration, and also greatly clarifies the role of symmetry in defining and studying combinatorial structure. This helped me (at least) better understand connections with some of the most interesting topics in combinatorics.
My interest tends to be in better understanding connections I might otherwise only vaguely sense should exist, so I am untroubled by the fact that I made no attempt to alter the classical techniques discussed by Wilf for studying asymptotics of the terms in ogfs and egfs.
There is a famous distinction in prime number theory between the number theorists who like to multiply primes, and the number theorists who like to add primes. As the primes are very heavily multiplicatively structured, the mathematics of multiplying primes is very algebraic in nature, in particular involving field extensions, Galois representations, etc. But the primes are very additively unstructured, and so for adding primes we see the tools of analysis used instead (circle method, sieve theory, etc.).
What then is necessary to get the field FF? One deep ingredient is a theorem of Neukirch that says that a subgroup of G FG_F that looks like a decomposition group of a prime is a decomposition group. This allows us to recover the primes of FF. The other part, regarded as more standard, is the Cebotarev density theorem, which is of course the non-abelian extension of the theorem on arithmetic progressions. So from the point of view of the previous paragraph, the non-abelian theorem on arithmetic progression plays a key role in recovering the additive structure of FF, albeit indirectly. Note, however, that the Cebotarev theorem by itself is only weakly non-abelian, since it follows rather easily from the abelian case.
Minhyong, do you know what the situation is for function fields over finite fields? Is there an anabelian theorem, and if so, is there some kind of direct reconstruction of the additive structure of the field?
where FinVect(q)FinVect(q) is the category of finite-dimensional vector spaces over the field with qq elements, and FinVect(q) 0FinVect(q)_0 is the corresponding groupoid. Just as ordinary species are a categorification of the Hilbert space for the harmonic oscillator, qq-deformed Hilbert spaces are a categorification of the Hilbert space for the qq-deformed oscillator!
Even better, when we replace the groupoid of finite sets by the groupoid of finite-dimensional vector spaces over the field with qq elements, we get a categorified version of the qq-deformed Fock space!
Part of my self-identity is as a Mathematician, even though I am one of the least expert of anyone on this blog. I have several times pontificated on narrative structure in mathematical Proof. Stepping back, here is a short article on the Philosophy of Narrative Identity, complete with intriguing references.
In their conceptualisations of critical theory, however, Cox and Horkheimer differ slightly but in an important way. While both are concerned to defend theory as an approach to a dynamic and interconnected totality, Cox does not foreground the status of the theorist, while for Horkheimer the critical theorist must engage with theory as a productive process. Cox does take neorealism to task for neglecting the production process in the constitution of national interest (Cox 1981: 134-135) but Horkheimer goes further: it is not a matter of adding another parameter or variable to the theoretical enterprise; it is a matter of understanding the theoretical enterprise itself in relation to and as a part of a general production process and division of labour. When Cox wrote in 1981, the prevailing epistemology in IR and the epistemological commitment of problem-solving or traditional theory was realist: the world exists independently of our thoughts about it and the task of theory is to make thought adequate to reality. What Horkheimer shows is that there is no neat division between thought and reality that can justify the privileged position of the theorist in the social division of labour: our thoughts are part of reality, as real as the city you live in or the job you work at and they must be analysed as part of the general social division of labour and of social reproduction.
Thus the problem-solving theorist becomes a functionary in the maintenance of social order. The critical theorist must understand the role of theory in social reproduction in order to break down the divisions between theoretical reflection and the making of the world.
Matt Davies lectures in International Political Economy at Newcastle University and is the Degree Programme Director for the MA in World Politics and Popular Culture. He is also a co-editor of the Popular Culture and World Politics book series, published by Routledge.
E-IR is an independent non-profit publisher run by an all volunteer team.Your donations allow us to invest in new open access titles and pay ourbandwidth bills to ensure we keep our existing titles free to view. Anyamount, in any currency, is appreciated. Many thanks!
There are innumerable theories of dream function (Dallett, 1973). All of them are highly speculative and difficult to refute in a definitive way, and they therefore linger despite a lack of evidence for any of them. This situation also provides a fertile terrain for new and unlikely theories based on analogies drawn from each development or discovery in other areas of research. This search for a function seems necessary and sensible to most people, but it rests on the false "adaptationist" assumption that "all the things that have form have function" (Thompson, 2000, p. 1014). In fact, many structures and processes persist even though they have no function, and dreaming may be one of them (Flanagan, 1995; Flanagan, 2000a).
Aside from Freud's guardian-of-sleep theory and Jung's compensatory theory, which we have refuted elsewhere, the most prominent theory of dream function is that dreams provide solutions to current problems, especially emotional problems (Barrett, 1993; Greenberg, Katz, Schwartz, & Pearlman, 1992; Greenberg & Pearlman, 1993). In one variant, Fiss (1993) suggests that dreams are especially good at registering subtle internal and external signals that often go undetected in waking life, making them potentially useful for picking up early signs of physical illness.
There are many empirical findings about dreams that do not fit well with any problem-solving theory. To begin with, the idea that dreams have a purpose originated at a time when it was thought that people rarely dream. In that context, it was plausible to believe that the occasional recalled dream could be a reaction to a specific event or emotional problem. But if most adults dream at least four to six times per night, then most people are recalling less than 1 percent of their dreams. Even the best dream recallers only remember a few percent of their dreams. This lack of recall suggests that dreams in general are not an evolutionary adaptation to provide information or insight to people when they are awake.
In addition, only about half of recalled dreams seem to have even the slightest connection to the events of the previous day (Botman & Crovitz, 1989; Harlow & Roll, 1992; Hartmann, 1968; Marquardt, Bonato, & Hoffmann, 1996; Nielsen & Powell, 1992). Kramer (2000) claims on the basis of one small clinical study that the concerns of the day are incorporated into dreams, but more recent and larger studies, in which judges try to match expressed daytime concerns with dream reports from laboratory awakenings, have proven unsuccessful (Roussy, 2000; Roussy et al., 2000a). It is unlikely that dreams very often deal with immediately relevant issues, although they do dramatize ongoing emotional preoccupations in many instances.
If dreams contain important information for consideration in waking consciousness, then it might be predicted that those who do not remember or pay attention to their dreams might suffer some disadvantages. But those who rarely recall dreams do not differ in terms of personality or mental difficulties when compared with those who recall dreams regularly (Antrobus, 1993; Blagrove & Akehurst, 2000; Cohen, 1979; Goodenough, 1991; Tonay, 1993). Generally speaking, it is very difficult to distinguish "recallers" from "non-recallers" with either personality or cognitive tests. If incorporating and dealing with the content of dreams mattered for psychological well being, a different set of findings might be expected. In fact, contrary to any theory that emphasizes the problem-solving nature of dreams, dream recall is often more disturbing than it is helpful, as shown most dramatically with people who suffer from post-traumatic stress disorder. Many people also suffer from their recurrent dreams (Zadra, 1996; Zadra & Donderi, 2000a).
c80f0f1006