In this paper we define n+1-valued matrix logic Kn+1 whose class of tautologies is non-empty iff n is a prime number. This result amounts to a new definition of a prime number. We prove that if n is prime, then the functional properties of Kn+1 are the same as those of ukasiewicz's n +1-valued matrix logic n+1. In an indirect way, the proof we provide reflects the complexity of the distribution of prime numbers in (...) the natural series. Further, we introduce a generalization K n+1 * of Kn+1 such that the set of tautologies of Kn+1 is not empty iff n is of the form p , where p is prime and is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of n+1. In the concluding part, we discuss briefly a partition of the natural series into equivalence classes such that each class contains exactly one prime number. We conjecture that for each prime number the corresponding equivalence class is finite. (shrink)

The conformity of mathematics and physics has so far been taken for granted. Philosophical explanations of that fundamental fact have never been satisfactory, mathematical explanations never had been attempted. In the following a fundamental theorem for the conformity of mathematics and physics will be demonstrated.Mathematics can be defined as the science of Number, physics as the science of Matter. The elementary constituents of mathematics are the prime numbers, those of matter the particles, particularly protons and electrons. The only (...) essential property of these elements as such is their existence, given within a given complex or space. Their fundamental relation is their distribution within their respective spaces. The theorem of the distribution of primes in the arithmetical series of numbers is the fundamental theorem of mathematics. The fundamental theorem of physics would be a theorem of the distribution of particles in geometrical space. If it were shown that both theorems were equal, the essential conformity of mathematics and physics would be mathematically demonstrated. (shrink)

In Big Gods, Norenzayan (2013) presents the most comprehensive treatment yet of the Big Gods question. The book is a commendable attempt to synthesize the rapidly growing body of survey and experimental research on prosocial effects of religious primes together with cross-cultural data on the distribution of Big Gods. There are, however, a number of problems with the current cross-cultural evidence that weaken support for a causal link between big societies and certain types of Big Gods. Here we attempt (...) to clarify these problems and, in so doing, correct any potential misinterpretation of the cross-cultural findings, provide new insight into the processes generating the patterns observed, and flag directions for future research. (shrink)

The subject matter of this book is the mental lexicon, that is, the way in which the form and meaning of words is stored by speakers of specific languages. This book attempts to narrow the gap between the results of experimental neurology and the concerns of theoretical linguistics in the area of lexical semantics. The prime goal as regards linguistic theory is to show how matters of lexical organization can be analysed and discussed within a neurologically informed framework that (...) is both adaptable and constrained. It combines the perspectives of distributed network modelling and linguistic semantics, and draws upon the accruing evidence from neuroimaging studies as regards the cortical regions involved. It engages with a number of controversial current issues in both disciplines. The book is intended as a tool for linguists interested in psychological adequacy and the latest advances in cognitive science. It provides a principled means of distinguishing those semantic features required by a mental lexicon that have a direct bearing on grammar from those that do not. It will appeal to researchers in neurolinguistics and lexical semantics."--p. 4 of cover. (shrink)

Baker (Mind 114:223–238, 2005; Brit J Philos Sci 60:611–633, 2009) has recently defended what he calls the “enhanced” version of the indispensability argument for mathematical Platonism. In this paper I demonstrate that the nominalist can respond to Baker’s argument. First, I outline Baker’s argument in more detail before providing a nominalistically acceptable paraphrase of prime-number talk. Second, I argue that, for the nominalist, mathematical language is used to express physical facts about the world. In endorsing this line I follow (...) moves made by Saatsi (Brit J Philos Sci 62(1):143–154, 2011). But, unlike Saatsi, I go on to argue that the nominalist requires a paraphrase of prime-number talk, for otherwise we lack an account of what that ‘physical fact’ is in the case of mathematics that seemingly makes reference to prime numbers. (shrink)

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

The article highlights certain aspects of the obscured structure of the number, which occur irregularly in the teaching of numeracy in Preschool Education. Its absence, as an effect, leads to the child's misunderstanding of the concept of number. In the presymbolic stage, the number is taught through the word. Structural particularities are found in the semantics and phonetics of the number word and are substantial in the processes of speech and listening. The objectives are to make known the obscured structure (...) of the number and its elements and to analyze the nature of the name of the number. It is justified that the basis of the word "ONE" are the first sound manifestations of the infant. It states that there is the same and equal relationship between the acquisition of knowledge of language and speech with the acquisition of knowledge of number through the development of tonal auditory balance. Methodology: the theoretical analysis of the structural parts (semantics and phonosemantics) of the number and the identification of reciprocal correlation between the constructions of the knowledge of spoken numeral word in Preschool Education through the implemented technology. Contribution: the development of the method for learning the concept of number. (shrink)

In this paper, I concentrate on two themes: to what extent numbers bear on an agent's duties, and how numbers should relate to social policy. In the first half of the paper I consider the abstract case of a choice between saving two people and saving one, and my focus is on the contrast between a duty to act and a reason which merely makes an action intelligible. In the second half, I turn to the issue of social (...) policy and investigate how if at all numbers can have a bearing there, if there is no obvious duty on individuals to save the greater number. My proposal is that it is not the bare numbers themselves (or even the ratio of claimants on either side of the dilemma) which explain our intuitions in such cases, but rather considerations of the extent to which each of us can make a reasonable claim on others. In short, I argue: numbers don't count, people do. (shrink)

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the (...) sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

In this paper, I concentrate on two themes: to what extent numbers bear on an agent's duties, and how numbers should relate to social policy. In the first half of the paper I consider the abstract case of a choice between saving two people and saving one, and my focus is on the contrast between a duty to act and a reason which merely makes an action intelligible. In the second half, I turn to the issue of social (...) policy and investigate how if at all numbers can have a bearing there, if there is no obvious duty on individuals to save the greater number. My proposal is that it is not the bare numbers themselves which explain our intuitions in such cases, but rather considerations of the extent to which each of us can make a reasonable claim on others. In short, I argue: numbers don't count, people do. (shrink)

This paper aims to show that Selim Berker’s widely discussed prime number case is merely an instance of the well-known generality problem for process reliabilism and thus arguably not as interesting a case as one might have thought. Initially, Berker’s case is introduced and interpreted. Then the most recent response to the case from the literature is presented. Eventually, it is argued that Berker’s case is nothing but a straightforward consequence of the generality problem, i.e., the problematic aspect of (...) the case for process reliabilism (if any) is already captured by the generality problem. (shrink)

A central strand in philosophical debate over the just distribution of resources attempts to juggle three competing imperatives: helping those who are worst off, helping those who will benefit the most, and then – beyond this – determining when to aggregate such ‘worst off’ and ‘benefit’ claims, and when instead to treat no such claim as greater than that which any individual by herself can exert. Yet as various philosophers have observed, ‘we have no satisfactory theoretical characterization’ as to (...) how to weigh each of the three imperatives against one another, we find it ‘difficult to state... precise or comprehensive conclusions’, and we do not yet have a ‘metric for integrating the three measures’. In what follows, I offer an approach to weighing the three criteria against one another that yields resolutions – in Hard Cases of the ‘saving one infant's life versus replacing ten elderly people's hips’ sort – that are cardinally definitive, intuitively satisfactory and theoretically justified. (shrink)

We show that versions of the prime number theorem as well as equivalent statements hold in an arbitrary model ofIΔ 0+exp.

This article is the result of collaboration between a linguist and an anthropologist. In La Troisième planète. Structures familiales et systèmes idéologiques (The Third Planet: Family Structures and Ideologies) (Todd, 1983), anthropologist Emmanuel Todd provided a world map of family types, which he used to explain the distribution of major political philosophies around the world. However, this did not explain the distribution of the family types themselves. Indeed, a concluding chapter entitled “Le Hazard” (The Effects of Chance) stated (...) that the distribution of family types did not seem to be the result of any particular economic or ecological factor and was therefore a prime example of the uncertainty principle at work. However, Laurent Sagart, a linguist specializing in Chinese dialects, noticed that this map of family types exhibited a structure well known to experts in historical linguistics and dialectology, contrasting a large, continuous zone in the center with a number of small, independent zones located around the periphery of the central zone or in isolated enclaves within it. When such maps appear in linguistic atlases, dialectologists usually conclude that the central zone was the innovative area while the peripheral and isolated zones conserved the original features. The same analysis, if applied to the map of family types, would lead to the conclusion that the communal family system in the center of the map represented a more recent innovation than the systems around the periphery. (shrink)

On September 6, 2004, using the Isabelle proof assistant, I verified the following statement: (%x. pi x * ln (real x) / (real x)) ----> 1 The system thereby confirmed that the prime number theorem is a consequence of the axioms of higher-order logic together with an axiom asserting the existence of an infinite set. All told, our number theory session, including the proof of the prime number theorem and supporting libraries, constitutes 673 pages of proof scripts, or (...) roughly 30,000 lines. This count includes about 65 pages of elementary number theory that we had at the outset, developed by Larry Paulson and others; also about 50 pages devoted to a proof of the law of quadratic reciprocity and properties of Euler’s ϕ function, neither of which are used in the proof of the prime number theorem. The page count does not include the basic HOL library, or properties of the real numbers that we obtained from the HOL-Complex library. (shrink)

Is there any link between the doctrine of logical fatalism and prime numbers? What do logic and prime numbers have in common? The book adopts truth-functional approach to examine functional properties of finite-valued Łukasiewicz logics Łn+1. Prime numbers are defined in algebraic-logical terms and represented as rooted trees. The author designs an algorithm which for every prime number n constructs a rooted tree where nodes are natural numbers and n is a root. (...) Finite-valued logics Kn+1 are specified that they have tautologies if and only if n is a prime number. It is discovered that Kn+1 have the same functional properties as Łn+1 whenever n is a prime number. Thus, Kn+1 are 'logics' of prime numbers. Amazingly, combination of logics of prime numbers led to uncovering a law of generation of classes of prime numbers. Along with characterization of prime numbers author also gives characterization, in terms of Łukasiewicz logical matrices, of powers of primes, odd numbers, and even numbers. (shrink)

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

Savant syndrome and prime numbers Oliver Sacks reported that a pair of autistic twins had extraordinary number abilities and that they spontaneously generated huge prime numbers. Such abilities could contradict our understanding of human abilities. Sacks' report attracted widespread attention, and several researchers speculated theoretically. Unfortunately, most of the explanations in the literature are wrong. Here a correct explanation on prime number identification is provided. Fermat's little theorem is implemented in spreadsheet. Also, twenty years after (...) the report, questionable aspects were found in it. Extreme abilities became dubious. One possibility for the less extreme abilities is incomplete trial division. (shrink)

The phase φ of any wave is determined by the ratio x/λ consisting of the distance x propagated by the wave and its wavelength λ. Hence, the dependence of φ on λ constitutes an analogue system for the mathematical operation of division, that is to obtain the hyperbolic function f(ξ)≡1/ξ. We take advantage of this observation to decompose integers into primes and implement this approach towards factorization of numbers in a multi-path Michelson interferometer. This work is part of a (...) larger program geared towards unraveling the connections between quantum mechanics and number theory. We briefly summarize this aspect. (shrink)

How should we design our economic systems? Should we tax the rich at a higher rate than the poor? Should we have a minimum wage? Should the state provide healthcare for all? These and many related questions are the subject of distributive justice, and different theories of distributive justice provide different ways to think about and answer such questions. This book provides a thorough introduction to the main theories of distributive justice and reveals the underlying sources of our disagreements about (...) economic policy. It argues that the universe of theories of distributive justice is surprisingly simple, yet complicated. It is simple in that the main theories of distributive justice are just four in number, and in that these theories each offer a distinct, well-defined theoretical approach to distributive justice; yet it is complicated in that the main theories disagree at several distinct, fundamental levels, and in that it is possible to spin innumerable new theories from the elements of the four main theories. (shrink)

We show that IE1 proves that every element greater than 1 has a unique factorization into prime powers, although we have no way of recovering the exponents from the prime powers which appear. The situation is radically different in Bézout models of open induction. To facilitate the construction of counterexamples, we describe a method of changing irreducibles into powers of irreducibles, and we define the notion of a frugal homomorphism into Ẑ = ΠpZp, the product of the p-adic (...) integers for each prime p. (shrink)

We show that $\mathrm{IE}_1$ proves that every element greater than 1 has a unique factorization into prime powers, although we have no way of recovering the exponents from the prime powers which appear. The situation is radically different in Bezout models of open induction. To facilitate the construction of counterexamples, we describe a method of changing irreducibles into powers of irreducibles, and we define the notion of a frugal homomorphism into $\hat\mathbb{Z} = \Pi_p\mathbb{Z}_p$, the product of the $p$-adic (...) integers for each prime $p$. (shrink)

Research based on construal level theory (CLT) suggests that thinking about the distant future can prime people to solve problems by insight (i.e., an “aha” moment) while thinking about the near future can prime them to solve problems analytically. In this study, we used a novel method to elucidate the time-course of temporal priming effects on creative problem solving. Specifically, we used growth-curve analysis (GCA) to examine the time-course of priming while participants solved a series of brief verbal (...) problems. Participants were tested in two counterbalanced sessions in a within-subject experimental design; one session featured near-future priming and the other featured far-future priming. Our results suggest high-level construal may temporarily enhance analytical thinking; far-future priming caused transient facilitation of analytical solving while near-future priming induced weaker, transient facilitation of insightful solving. However, this effect is short-lived; priming produced no significant differences in the total number of insights and analytical solutions. Given the fleeting nature of these effects, future studies should consider implementing methodology that allows for aspects of the time-course of priming effects to be examined. A method such as GCA may reveal mild effects that would be otherwise missed using other types of analyses. (shrink)

Distribution of sufficient health care resources to the maximum number of people in LIC is the central theme of the book. Bangladesh is taken as a representative of low income countries (LIe. In LIC, there is scarcity of health care resources like other resources but the deserving persons are numerous. Therefore, it requires an efficient distribution of resources. Considering 'Inequality to get access to health care' as the basic problem in LIC, John Rawls' principle of fair equality of (...) opportunity is proposed to apply to give people access to health care resources. In this book it is critically analyzed how the principle fails to return a good result in LIC. Instead the principle of maximizing utility is proposed for the purpose. It is argued that maximum number of people can be provided sufficient health care by applying the principle. The principle has also been justified theoretically in concerned low income country and a fruitful result has been achieved from the test. It is claimed that applying principle of utility to provide sufficient health care to the maximum number of people is the most efficient way of health care resources distribution in LIC. (shrink)

For a ring R, Hilbert’s Tenth Problem $HTP$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP$, which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$. It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP$. In contrast, we show that every Turing degree contains a (...) set W for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$. (shrink)

The paper is concerned with the old conjecture that there are infinitely many Mersenne primes. It is shown in the work that this conjecture is true in the standard model of arithmetic. The proof refers to the general approach to first–order logic based on Rasiowa-Sikorski Lemma and the derived notion of a Rasiowa–Sikorski set. This approach was developed in the papers [ 2 – 4 ]. This work is a companion piece to [ 4 ].

It is controversial whether Wittgenstein's philosophy of mathematics is of critical importance for mathematical proofs, or is only concerned with the adequate philosophical interpretation of mathematics. Wittgenstein's remarks on the infinity of prime numbers provide a helpful example which will be used to clarify this question. His antiplatonistic view of mathematics contradicts the widespread understanding of proofs as logical derivations from a set of axioms or assumptions. Wittgenstein's critique of traditional proofs of the infinity of prime (...) class='Hi'>numbers, specifically those of Euler and Euclid, not only offers philosophical insight but also suggests substantive improvements. A careful examination of his comments leads to a deeper understanding of what proves the infinity of primes. (shrink)

In many simple integral domains, such as Z or Z[i], there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like Z[x] or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we (...) will show how to computably extend Z in such a way that we can control the ordinary integer primes in any Π20 way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable unique factorization domain such that the set of primes is Π20-complete in every computable presentation. (shrink)

ABSTRACTIn light of the large recent inflow of refugees to the EU and the Commission’s efforts to relocate them, I raise the question of what a fair distribution of refugees between EU countries would look like. More specifically, I consider what concerns such a distributive scheme should be sensitive to. First, I put forward some arguments for why states are obligated to admit refugees and outline how I believe the EU should respond to the refugee crisis. This involves, among (...) other things, resettling a proportion of refugees from countries neighbouring Syria in the EU. Second, I consider how the intake into the EU should be distributed between member states, that is, the shares different countries can be expected to admit. I discuss the relevance of a number of different factors that may be claimed to affect such shares, including population size, GDP, number of refugees admitted so far, unemployment rate, country-specific costs and cultural ‘closeness’. Third, I consider whether the distributive scheme should be restricted to reflect specific states’ responsibility for creating refugees in the first place, levels of racism and xenophobia, and whether other states are required to pick up the slack if some refuse to admit their fair share. (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during (...) the transitional period of his thought (1929-33). (shrink)

A number of (insect) parasitoids have been found to avoid superparasitism, i.e., these parasitoids distribute their eggs more evenly over the available hosts than might be expected from chance only. By doing so each parasitoid individual ensures a greater probability of survival for its offspring as a result of a reduced within-host-competition.Recently a number of mathematical models have been developed, describing the distribution of the parasitoid eggs in the hosts. This paper gives a survey of these models, placing them (...) within one and the same mathematical framework. An essential conceptual distinction, neglected up to now, emerges: parasitoids can either react to the number of previous visits to a particular host, or they can react to the number of eggs already present in that host. (shrink)

Nowadays, the development of human society and daily life are inseparable from the power supply. Therefore, people also put forward higher requirements for the reliability of distribution network, but power companies can only passively deal with distribution network failures, which is a bottleneck for the improvement of distribution network reliability. The Internet of Things is the best solution for online equipment status monitoring and basic data sharing for large, widely distributed, relatively fixed, and large numbers of (...) equipment. The construction of the IoT for power distribution equipment faces many important problems, including the selection of networking, equipment selection, and interaction standards. When researching the implementation plan, research on the distribution of IoT market was carried out. Based on the grid, the idea of optimizing the investment selection plan of the power distribution using IoT was discussed, and a result verification model was established. After the completion of the theoretical part, a case study of medium-voltage distribution grid equipment management and medium-voltage distribution network equipment management based on the grid was carried out by taking a real enterprise application situation as an example. Realizing fault diagnosis of distribution network will not only provide decision support for operation and maintenance of distribution network for power companies, but also have great economic and social benefits. Aiming at the shortcomings of single data mining method in distribution network fault diagnosis, hybrid data mining method is proposed. First, rough set theory is used to reduce the original fault data and form a simplified rule set. Because of the non-linearity of distribution network fault and the strong learning ability, adaptability, and robustness of Bayesian network, Bayesian network can be used to classify distribution network faults. Therefore, a simplified fault diagnosis system is established in this paper, and its correctness is confirmed. Then, the learning and training are carried out by using Bayesian network to call the simplest rule set, which has the characteristics of short learning and training time and high diagnostic accuracy. (shrink)

Japanese prefer odd numbers, whereas Westerners emphasize even numbers, an observation that is clear from the distribution of number-related words in Japanese and English dictionaries. In this article, the author explains why these two cultures differ by surveying the history of numbers, including yin-yang thought from ancient China, ancient Greek philosophy, and modern European mathematics. The author also mentions that oddness and evenness are only mathematical concepts, but understanding the cultures and histories of individual countries contributes (...) to world peace. (shrink)

Short-term or working memory provides temporary storage of information in the brain after an experience and is associated with conscious awareness. Neurons sensitive to the multiple stimulus attributes comprising an experience are distributed within many brain regions. Such distributed cell assemblies, activated by an event, are the most plausible system to represent the WM of that event. Studies with a variety of imaging technologies have implicated widespread brain regions in the mediation of WM for different categories of information. Each kind (...) of WM may thus be expected to involve many brain regions rather than a local, uniquely dedicated set of cells. Neurons in a distributed “cell assembly” may be self-selected by their temporally coherent activations. The process by which this fragmented representation of the recent past is reassembled to accomplish essentially automatic and reliable recognition of a recurrent event constitutes an important problem. One plausible mechanism to achieve the identification of past with previous events would require that the representational system mediating WM must coexist in spatial extent and somehow overlap in temporal activation with cell ensembles registering input from subsequent events. The detection of such a postulated mechanism required an experimental approach which would focus upon spatial patterns of coherent activation while information about different events was stored in WM and retrieved, rather than focusing upon the temporal sequences of activation in localized regions of interest. For this purpose, the familiar delayed matching from sample task was modified. A series of information-free flashes, or “noncontingent probes,” was presented before an initial series of visual information items, the Priming Sample, which were to be held in WM during a Delay Period. A second series of visual information items were then presented, the Matching Sample. The task required detection of any item in the second series which had beenabsentfrom the initial series. Thirty such trials with a particular category of visual information constituted a single task. Several DMS tasks with this standardized design, but with different categories of visual information, were presented within each test session. The information categories included letters of the alphabet, single digit numbers, or faces from a school yearbook. Event-related potentials , were computed from 21 standardized electrode placements, separately for information-free probes and for information items in each interval of the trials within a task. Because each electrode is particularly sensitive to coherent activation of neurons in the immediately underlying brain regions, topographic maps were constructed and interpolated across the surface of the scalp. The momentary fluctuations of the resulting voltage “landscapes” throughout the task were then subjected to quantitative analysis. Distinctive landscapes sometimes persisted for prolonged periods, implying sustained engagement of very large populations of neurons. “Difference landscapes” were constructed by subtraction of topographic maps evoked by noncontingent probes during the Delay Period from maps of probe ERPs before the presentation of the initial information in the Priming Sample. Such probe difference landscapes displayed recurrent high similarity to momentary landscapes elicited during subsequent presentation of the information items in the Matching Sample. It seemed as if the distributed cell assembly continuously engaged by mediation of WM of the diverse attributes of the initial stimuli was being dynamically compared to the ensembles engaged by registration of the subsequent stimuli. Spatial Principal Component Analysis was applied to the sequences of momentary voltage landscapes observed throughout trials of each task. This method sought a small number of spatial patterns with which these large sets of inhomogeneous spatial distributions of voltage could be reconstructed. This is the spatial analog of the reconstruction of local ERPs by temporal principal components, as often described previously. Five Spatial Principal Components were found which accounted for about 90% of the total variance of voltage across the surface of the scalp throughout every task. Theloadings,or distinguishing topographic features, of these SPCs, were highly similar during every cognitive task for every subject. However,factor scores,or relative average contribution to the overall voltage distributions, of the different SPCs varied substantially among subjects between the tasks and momentarily within successive intervals of each task. These five SPCs may reflect coherent activation of huge distributed ensembles of neurons which comprise independent but interacting functional brain subsystems. These subsystems may correspond to basic resources available to individuals for allocation to mediate conscious evaluation of information during cognitive activity, providing a filter to bind together fractionated representations of the past to evaluate the present. (shrink)