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Knowledge and skills acquired in an educational domain, assessed through specified tasks or records. Mathematics procedures, mathematical concepts, word decoding and reading comprehension are distinct outcomes; a single aggregate score can conceal different cognitive demands.

A mathematics educator and senior contributing author of Prealgebra 2e.^[ref]

An area of mathematics groups related concepts and methods. The glossary of areas of mathematics collects terminology for these fields.

Computer science studies how information can be represented and transformed by algorithms, and what those transformations can achieve. Its central questions connect mathematics to practical design: what answer should a procedure produce, why is it correct, and how much work does it require as its input grows? Computability theory asks which tasks can be solved at all; computational complexity theory asks what resources a solution needs.

The beauty is that choosing a representation can change what is feasible. Finding a name among a million entries sounds like a million checks. Store the entries in a sorted array, and binary search needs at most twenty middle-entry comparisons. The order makes each unsuccessful comparison eliminate a whole region of possibilities. A data structure supplies useful organization; an algorithm exploits it; time complexity explains the resulting savings.

Algorithm correctness covers every permitted input, including absent names and empty collections. Time complexity predicts scaling; statistical program profiling investigates a particular implementation.

Geophysics applies physics and mathematics to the structure, composition, and evolution of planets, including Earth. It connects measurements with mechanisms: plate tectonics explains moving crust, the geodynamo explains a planetary magnetic field, and albedo connects reflected sunlight to a planetary surface energy balance. Even a small model can reveal a useful organizing principle, such as the competition between ice-ocean albedo feedback and stabilizing thermal radiation.

A mathematics educator and senior contributing author of Prealgebra 2e.^[ref]

A mathematics educator and senior contributing author of Prealgebra 2e.^[ref]

Mathematical logic studies formal languages, formal proofs, model theory, computability theory, and the foundations of mathematics. It makes the rules of reasoning explicit enough that reasoning itself becomes an object of mathematics. Proof theory studies deductions; model theory studies interpretations in which statements hold.

Three questions organize the subject: is an expression a well-formed first-order formula, is it true in a specified first-order structure, and does it have a formal proof from specified axioms? These questions are distinct. For example, $\exists n(n+n=1)$ is a first-order sentence in an appropriate first-order language; it is false in the natural numbers but true in the rational numbers, with $n=1/2$.

The beauty of mathematical logic is that the same logical connectives and rules of universal quantification and existential quantification organize arguments across mathematics. Classical propositional logic uses two truth values; Intuitionistic propositional logic interprets logical implication through constructions on proofs, without unrestricted law of excluded middle. This proof-based framework is developed in Frank Pfenning, Natural Deduction, sections 1–3.

Early modern mathematical disciplines applied to physical or practical objects, including astronomy, optics and mechanics. The social standing and explanatory claims of practitioners varied; this is not simply a synonym for the present administrative category of applied mathematics.

Official course description.

Six to eight taught units from the Part C mathematics and statistics menus. Shared branches include Category theory course of the University of Oxford, Algebraic topology course of the University of Oxford, model theory and elliptic curves. Current departmental catalogue. Compulsory dissertation worth two units.

Executive functions should contribute most when learning or assessment requires keeping several pieces of information active, suppressing an attractive error, coordinating successive operations or changing strategy. This predicts particularly strong links with multistep mathematics, mathematical problem solving and reading comprehension, alongside classroom engagement and following instructions. It predicts a smaller direct contribution to well-practised fact retrieval or fluent decoding once those operations are automated. That is a task-demand hypothesis, not a claim that every mathematics score must correlate more strongly with executive functions than every literacy score.

Working memory maintains goals and intermediate results: a child doing a subtraction with borrowing must retain both the original problem and its updated intermediate state. Inhibitory control helps suppress a larger-number answer that is tempting but irrelevant, or a familiar operation when the problem requires another. Cognitive flexibility supports switching between representations or abandoning an unsuccessful strategy. In reading, maintaining earlier propositions and rejecting an initially plausible interpretation should matter especially for comprehension and inference; knowledge of letter–sound correspondences is not itself an executive function. Repeated practice can change which process is the bottleneck, so developmental stage and familiarity matter.

Historically, the unity-and-diversity framework helped replace an undifferentiated “executive ability” with correlated but distinguishable processes. Akira Miyake and colleagues used several tasks per proposed component and latent variable analysis to separate common executive variance from individual task demands. Their influential study tested adults: its factor structure cannot simply be assumed to describe preschool children. It nevertheless supplies a crucial methodological lesson. A single executive task measures the intended control process together with perception, language, motor speed and task-specific strategies. Treating a raw score as a pure capacity risks attributing the wrong source of variance to school achievement.^[ref]

More directly, Clancy Blair and Rachel Peters Razza followed 141 children from low-income homes, measuring preschool self-regulation and later kindergarten mathematics and literacy. Their inhibition measure required children to tap once when the experimenter tapped twice and twice after one tap: the apparatus is simple, but the rule requires suppressing imitation and maintaining an opposite response mapping. Inhibitory control was associated with emerging mathematics and reading after accounting for intelligence. This supports the importance of regulation during early learning, but the design was observational, not random assignment to different inhibitory capacities. Shared language, family circumstances or classroom experience could influence both performances.^[ref]

Lucy Cragg and colleagues provided a more differentiated mathematical analysis. Their sample of 293 participants aged eight to twenty-five completed executive tasks and measures of factual knowledge, procedures and conceptual understanding. Working memory had both an association with overall mathematics and indirect associations through mathematical component skills; inhibition measured in a numerical context was linked to factual and procedural knowledge. This is more informative than treating “mathematics” as one outcome. However, statistical mediation in an age-spanning observational sample does not establish that changing the putative mediator would change attainment. Numerically framed executive tasks can also share content with the achievement measures, increasing the apparent specificity of the association.^[ref]

Kate Cain and colleagues supply a corresponding literacy result. In assessments at ages eight, nine and eleven, working memory and comprehension components such as inference and monitoring predicted reading comprehension after controls for word reading and verbal ability. Sentence-span and digit-working-memory measures helped avoid defining the entire construct by one task. This supports a contribution beyond decoding, but the reported concurrent relations at successive ages are not an experimental demonstration of causal direction; comprehension itself can affect performance on verbally loaded memory tasks.^[ref]

Evidence across populations qualifies the temptation to make executive functions the master explanation of achievement. Greg J. Duncan and colleagues compared six longitudinal datasets while controlling earlier child skills and family background. School-entry mathematics and reading were strong predictors of subsequent attainment; attention added prediction, whereas several socioemotional measures did not. Attention ratings are not interchangeable with laboratory inhibition or working memory measures. The result supports a classroom-regulation pathway while showing that already acquired domain knowledge remains central. Learning also exercises executive control, so baseline-adjusted longitudinal associations do not eliminate reciprocal influence or unmeasured confounding.^[ref]

Intervention offers a sharper causal test, but its target and outcome must be distinguished. Darren L. Dunning and colleagues randomly allocated seven- to nine-year-olds with low working memory to adaptive computerised training, low-load nonadaptive training or no training. The adaptive condition improved several untrained working memory tests, with some gains maintained over time, but did not produce corresponding academic or classroom-analogue benefits. The active comparison helps separate memory-load effects from attention, computer practice and expectations. Crucially, improvement on an untrained but structurally similar memory test is near transfer; improved reading or mathematics would be far transfer. One does not entail the other.^[ref]

This trial does not show that working memory is irrelevant to achievement. A resource can constrain performance even when practising isolated tasks fails to change how it is deployed in a classroom. Nor does one intervention test all executive components, age groups or teaching methods. It does undermine the stronger inference from correlation to a promise that generic executive training will raise school attainment. Supporting instruction through shorter directions, external reminders and staged problems can reduce executive demands; whether any particular classroom support improves learning still requires an appropriately controlled outcome study.

The most defensible answer is therefore conditional and specific. Executive differences are especially plausible influences on managing novel, attentionally demanding, multistep learning and demonstrating that learning under load. They operate alongside knowledge, language, motivation and opportunity. Strong evidence would combine reliable multi-task executive measures, differentiated academic outcomes, repeated measurements with comparable meaning across ages, and interventions with active controls and delayed achievement tests. Existing prospective and experimental evidence supports executive functions as contributors to learning, while limiting claims that they are a single cause of educational success.

A mathematician in Galileo Galilei's world was a practitioner situated within a particular division of knowledge and work. The word could embrace demonstrative geometry and arithmetic, but also astronomy, optics, mechanics, surveying, calendar calculation and the design or use of scientific instruments. Such mixed mathematics treated features of natural or practical objects through mathematical relations. It was neither identical to a present-day university department nor confined to calculation without claims about nature.

The distinction from a natural philosopher was important. Within a traditional university hierarchy, natural philosophy investigated the causes and principles of physical change. Mathematics could establish a relation between abstract quantities with great certainty, while opponents doubted whether it captured the constitution of a real body. Showing what follows for an ideal inclined plane, for example, leaves a further question about whether a particular apparatus realizes the assumptions closely enough. That question is not an objection which can be answered merely by producing a longer geometrical proof. The relation of abstraction, measurement and physical explanation was itself contested.^[ref]

There were also several routes to employment. A university teacher, an scientific instrument maker and someone serving a court or military patron might all perform mathematical work while possessing different income, status and audiences. Practical expertise in fortification, surveying or optics could bring rewards without establishing that its holder had authority over every question of natural philosophy. Conversely, a learned author could rely on the skills of people whose work was not credited equally in print. The important historical questions are what tasks were recognized, who paid for them, and whose testimony carried weight. A modern boundary between theoretical research and technical service would obscure these connections.

Galileo Galilei's career makes the contested boundary visible. The Sidereus Nuncius by Galileo Galilei communicated new observations through a manufactured telescope, successive drawings and an account intended to make unfamiliar objects credible. The work also addressed a courtly audience: naming the bodies accompanying Jupiter was part of presenting an achievement to a patron, not an incidental ornament added after a purely private discovery. The title and dedication situate mathematical work within relationships of employment and reputation.^[ref]

His mathematical treatment of motion likewise pressed beyond the role of calculating celestial appearances. In Two New Sciences by Galileo Galilei, a definition of uniform acceleration leads to a geometrical result in which travelled distance varies as the square of elapsed time. An inclined plane stretches the time of descent, making a comparison more manageable. In modern notation the relationship is $s=a{t}^2/2$ for release from rest. The notation is ours; the historically significant combination is an idealized proof, a proposed physical regularity and a reported experimental test.^[ref]

The label mathematician therefore cannot be read as a declaration that Galileo Galilei made no physical claims. Nor does his insistence on mathematical explanation prove that other occupations or disciplinary standards were irrelevant. His 1610 transition into Tuscan service, with the combined style of mathematician and philosopher, is evidence of the status being sought, rather than proof that the existing hierarchy had already disappeared. The case shows an investigator using scientific instruments, patronage and mathematical demonstration to enlarge the jurisdiction of his expertise. It also explains why a seemingly terminological distinction could become consequential in disputes over heliocentrism.^[ref]

Francis Bacon's originality lay chiefly in the reconstruction and coordination of inquiry, rather than in the invention of observation or experiment. Earlier natural philosophy, practical crafts and natural history already contained empirical knowledge. Aristotle's investigations of nature, for example, cannot be reduced to drawing conclusions from words without reference to living things or physical change. What requires explanation is why Francis Bacon thought established practices were failing and what he proposed to alter.

The methodological target was a premature leap from a few familiar cases to very general principles, followed by deductions protected from serious revision. His alternative was a graduated movement from particulars through intermediate generalizations. This was more demanding than collecting favourable examples. Negative and contrasting instances had to constrain the proposed account. The comparison of conditions in which a phenomenon appears, is absent, or varies was intended to exclude unsuitable candidates for its underlying nature.^[ref]

A useful example is heat. Merely finding several hot substances does not establish which property explains their temperature. A candidate property has to survive comparisons with cold cases and with cases where one feature changes while others do not. These requirements point toward a disciplined search for causal explanation. They should not be equated with a modern randomized trial or with the claim that every successful science follows a single mechanical rule. The construction of the categories and the assessment of a comparison already require judgment. Francis Bacon's preliminary interpretation of heat was an attempt within a particular research programme, not a complete anticipation of present thermodynamics.

He also made intellectual habits part of the problem. His account of the idols of the mind directed attention to recurrent distortions associated with human tendencies, individual training, language and accepted systems. This is a programme for correcting inquiry, rather than a story in which the investigator simply empties the mind and thereafter reads facts without interpretation. His comparison of the investigator with a bee, combining collected material with active transformation, is especially awkward for the familiar caricature of Baconian inquiry as indiscriminate fact accumulation.^[ref]

The institutional proposal was equally consequential. New Atlantis by Francis Bacon presents organized work distributed among collectors of information, experimenters and interpreters. Scientific instruments, gardens, workshops, records and travel supply the material of inquiry. The imagined institution aims at useful operations, not simply the production of elegant arguments. However, it also has an internal hierarchy and controls disclosure. It is not a straightforward prospectus for today's open research university. Its religious language and political organization belong to the vision and must not be removed to turn it into a secular manifesto.^[ref]

There are substantial continuities. Craftspeople already altered materials and procedures to obtain desired effects; natural history already collected and compared particulars; learned reformers already criticized authority. There are also distinctions from other contemporary innovators: giving mathematics and idealization the central role found in Galileo Galilei's treatment of motion was not the same programme as Bacon's organized natural and experimental histories. A history of one uniform seventeenth-century method would lose that difference.

The strongest judgment is that Francis Bacon offered a distinctive reform of the scale, discipline, purpose and organization of knowledge. His work combined inherited resources in an ambitious programme which could be used to legitimate later collaborative inquiry. Later invocation of his name, however, does not prove that a particular discovery was produced by following his prescriptions. Originality in proposing a reform, effectiveness of a procedure, and influence of a public ideal are three historical claims requiring different evidence.

There were major transformations worth calling revolutionary, but not one event that created a single modern method everywhere at once. The term Scientific Revolution is useful if it identifies a connected set of changes in problems, instruments, explanations and intellectual authority. It becomes misleading when it substitutes a sudden victory of reason over an undifferentiated earlier ignorance for an explanation of those changes.

One substantial transformation concerned the relation between mathematics and physical explanation. Newton's laws of motion and Newton's law of universal gravitation connected terrestrial and celestial motion within the same quantitative framework. The importance of this achievement was not simply a more accurate table of planetary positions. Kepler's laws of planetary motion could become consequences of a dynamical account, while deviations from simple planetary paths became problems of interacting bodies. A theory could now generate further quantitative questions about the Moon, comets and the figure of the Earth. This was a change in what an adequate physical explanation might accomplish.^[ref]

A second transformation concerned evidence. In 1610 Galileo Galilei presented a telescope as a means of investigating objects that ordinary unaided observation could not resolve. The changing appearances of bodies near Jupiter supported the existence of satellites accompanying that planet; the disposition of light and shadow on the Moon supported an irregular surface. These observations challenged the claim that all celestial bodies displayed a uniquely perfect constitution. They also required arguments about the reliability of the instrument and the interpretation of what appeared through it.^[ref] They did not, by themselves, uniquely establish the Earth's motion: a system in which planets circled the Sun while the Sun circled the Earth could accommodate satellites of Jupiter. Distinguishing a result from the stronger worldview later associated with it prevents a misleading account of instant conversion.

Programmes for producing knowledge also changed. Francis Bacon criticized premature generalization and proposed organized movement between observations, experiments and increasingly general claims. His comparison of the ant, spider and bee rejected both the mere accumulation of facts and reasoning spun entirely from the mind. The ambition was a cooperative, productive investigation of nature, with procedures for correcting habitual intellectual mistakes.^[ref] This programme is evidence for a methodological reform movement, not proof that every successful seventeenth-century investigator followed one Baconian algorithm.

The discontinuities were nevertheless built from inherited resources. Aristotelian natural philosophy already asked about causes and systematic knowledge; astronomy had long depended on demanding mathematical and observational practices. Instrument making depended on craft skills. Galileo Galilei's discovery claims also depended on patronage: his dedication and naming of the newly observed bodies in Sidereus Nuncius by Galileo Galilei explicitly linked celestial investigation to the Medici court. A socially situated discovery need not be scientifically false, but its circulation cannot be explained solely by the logical content of a theory.^[ref]

Nor did change imply a uniform retreat of religion. The General Scholium by Isaac Newton coupled the rejection of Cartesian vortices with an argument about divine government. That addition first appeared in 1713, so it should not be silently placed in the 1687 first edition. Its combination of physical argument and theology nevertheless rules out a simple narrative in which successful mathematical physics necessarily became religiously indifferent.^[ref]

Steven Shapin's challenge to a single, self-evident Scientific Revolution is therefore a useful warning about the historian's category, rather than a reason to deny every substantial change.^[ref] The strongest qualified verdict is that a revolutionary reorganization occurred in several influential European traditions, especially mathematical natural philosophy and instrumental observation. Its chronology, geographical reach and relation to other disciplines must be demonstrated separately. The category explains something only when those differences remain visible.

There is no convincing single birthplace of science unless the definition is narrowed in advance. Systematic astronomy and mathematics preceded classical Greece; Greek natural philosophy developed distinctive explanatory and demonstrative ambitions; early modern Europe transformed the relationship between mathematics, experiment and institutions. These are different historical achievements. Treating any one of them as the creation of everything now called science confuses a particular form of inquiry with inquiry's entire history.

A good test begins with surviving practices rather than a list of supposed founding geniuses. The Babylonian astronomical diary VAT 4956 records positions of the Moon and planets relative to named stars, together with other observations, for 568–567 BCE. Such a record requires trained observers, a conventional vocabulary and a means of accumulating observations over time. Its intellectual value is not annulled because astronomy was entangled with divination and royal administration. However, this particular diary should not be credited with everything achieved by later Babylonian predictive astronomy: an observational record is evidence of systematic recording, not by itself evidence of a particular predictive algorithm.^[ref]

Egyptian mathematics provides a different kind of evidence. Problem 50 of the Rhind mathematical papyrus gives a procedure for the area of a circular field: remove one ninth of its diameter and square the remainder. Written in modern algebraic notation, the procedure isComparing this with the modern circle formula $A=\pi d^2/4$ gives the effective approximation $\pi \approx 256/81$. This is our reconstruction of the rule's accuracy, not a claim that its author used our symbol $\pi$ or formulated an abstract theory of that constant. A worked numerical procedure is mathematical knowledge even when it is not presented as a Greek-style proof.^[ref]

The case for Greece therefore needs a more specific claim. In Physics by Aristotle, nature is investigated through principles of motion and change, and explanation requires identifying different kinds of cause. His distinctions between material, formal, moving and final causes constitute an explicit programme of explanation, not merely a collection of useful recipes. Greek mathematics, exemplified by Euclid's Elements, made the organization of results into chains of proof unusually prominent. These traditions help explain why later European scholars gave Greek texts such authority. They do not establish that earlier or non-Greek practitioners lacked reasoning, nor that all Greek explanations were correct.^[ref][ref]

The European sixteenth and seventeenth centuries nevertheless deserve attention as more than the recovery of ancient truths. Galileo Galilei's Two New Sciences by Galileo Galilei joins precise definitions of motion to geometrical demonstrations and reported tests. An inclined plane lengthens a descent sufficiently to compare its distance and time; a mathematical dependence becomes something an apparatus can discriminate. Francis Bacon's Novum Organum by Francis Bacon instead urges a disciplined reconstruction of inquiry through ordered experience and the correction of intellectual prejudices. Their programmes differ, but both show participants deliberately contesting how knowledge should be made. Neither text authorizes the claim that observation or reasoning had only just been invented.^[ref][ref]

Institutional criteria also change the answer. If science means a modern profession with specialized laboratories, credentials and journals, even sixteenth-century Europe is too early for much of it. If it means any successful craft technique, the category becomes so broad that the first surviving written text cannot identify its beginning. If it means public, revisable claims about nature supported by disciplined observation and reasoning, several ancient traditions qualify, with different strengths and social settings. Each definition selects different evidence; none should be smuggled into a geographical verdict.

The strongest conclusion is consequently plural but discriminating. Greece was a major centre of explanatory natural philosophy and demonstrative mathematics; early modern Europe was a major site of their reorganization; neither was the unique beginning of systematic knowledge of nature. This conclusion recognizes real differences in standards of explanation without using proximity to present-day science as the sole measure of past achievement. It also leaves open the historical work of tracing transmission and transformation rather than imagining isolated civilizations independently possessing, or wholly lacking, a timeless scientific essence.

Scientific knowledge is deeply shaped by historical context in the questions asked, evidence available and standards of credibility, but context does not make results arbitrary or determine their truth by itself. The strongest argument identifies mechanisms through which context enters inquiry and then examines what experimental or mathematical constraints can resist it. This avoids both the claim that facts emerge untouched by human activity and the opposite claim that a social explanation is a substitute for testing them.

First, inherited conceptual resources determine what an investigator initially thinks an observation might be. Joseph Priestley's experiments on oxygen illustrate this directly. Heating mercury(II) oxide produced an unusually combustion-supporting gas, but the experiment did not arrive labelled with a modern chemical formula. Phlogiston supplied an available vocabulary: the gas could be interpreted as exceptionally dephlogisticated air rather than a newly isolated chemical element. His own account acknowledges the difficulty of revising expectations even when striking observations appeared. This is specific evidence that interpretation has a history, rather than a general accusation that past investigators were irrational.^[ref]

Antoine Lavoisier's material balances reorganized the same field by treating the air consumed during calcination as a constituent incorporated into the product. The retort, enclosed air supply and quantitative examination of products made some interpretations more difficult to maintain. Yet his identification of oxygen as the universal acidifying principle was itself wrong: hydrogen chloride gives an chemical acid without containing oxygen. The example establishes two distinct conclusions. An interpretive framework helps make observations intelligible; further observations and successful alternative explanations can expose its limits. Historical situatedness is compatible with correction.^[ref]

Second, instruments and techniques shape what can become an observable object. In nineteenth-century embryology, an optical microscope did not simply magnify a transparent miniature adult. Investigators needed suitable organisms, controlled access to eggs and sperm, ways of preparing thin or fixed material, and conventions for ordering developmental stages. Transparent sea-urchin eggs and external fertilization made processes accessible that were much harder to observe in opaque mammalian material. Such a choice was productive but selective: knowledge gained from one organism needed further work before it could be generalized to others.^[ref]

The Wilhelm Roux–Hans Driesch comparison shows why the details of an intervention matter. Killing an attached frog blastomere and separating a living sea-urchin blastomere differ both in material and operation. Partial development in one setting and a complete small larva in another cannot honestly be flattened into an experiment with one obvious answer. Nevertheless, the second outcome constrains the universality of an exclusively fixed mosaic theory of development. A capacity for regulation is empirically established in the tested conditions even if disagreement remains about its explanation. Technical and theoretical contexts shape the result's significance without allowing the observer simply to choose its outcome.^[ref]

Third, social interests can supply a problem or analogy without fully fixing the resulting theory. Charles Darwin's adaptation of Thomas Robert Malthus's population argument is a clear case. A discussion of human subsistence helped him formulate a mechanism for changes among organisms. However, his broad struggle for existence includes dependence and physical conditions, not only direct competition. The resulting natural selection explanation must be evaluated through genetic variation, heredity and differential reproduction, not accepted or rejected merely because a stimulus came from political economy. The route by which a hypothesis is formed is a historical question; the evidence sustaining its application is a connected but distinct question.^[ref][ref]

Fourth, conventions of demonstration affect what a community can reliably criticize. Galileo Galilei's definitions of accelerated motion let readers derive a squared time relation and compare it with repeated descents on an inclined plane. A reader can dispute whether the apparatus sufficiently approximates the model, distinguish a rolling ball from a freely falling body, or check whether the mathematical conclusion follows. The idealization is historically chosen, but its consequences are not whatever the author wishes. For a constant acceleration from rest, the distance ratio for elapsed times in the ratio $1:2$ must be $1:4$; changing that conclusion requires changing the assumptions or the mathematics.^[ref]

These cases support degrees rather than a single all-purpose answer. Historical context strongly affects research priorities, available specimens, instruments, vocabularies and access to credibility. It also affects which uncertainties become urgent and which applications receive resources. Its influence on a particular measured relation or mathematical entailment must be demonstrated rather than presumed. The same social setting can sustain competing interpretations, and a mathematical framework can travel beyond the setting in which it was first developed. Conversely, the ability to reproduce a result elsewhere does not erase the historical work needed to make that reproduction possible.

Scientific knowledge is historically produced and materially constrained. Context helps explain why a claim was thinkable, testable and persuasive at a particular time; it does not alone tell us whether the claim survives comparison, intervention and criticism. A convincing history therefore traces both the circumstances that make knowledge possible and the resistant evidence through which those circumstances can themselves be challenged.

For each student, the probability of both falling asleep and dreaming about mathematics is $(1-p)q$. These indicators are independent across students, so the same Poisson thinning calculation gives, for the requested count $D$,Explicitly $G_D(s)=\exp (\lambda (1-p)q(s-1))$, the probability generating function of that Poisson distribution.

Project Euler presents problems combining mathematics with the design of efficient algorithms.^[ref]

Set theory studies sets, their set elements, and the constructions and sizes possible for them. It supplies a shared vocabulary for mathematics: a function connects two sets, a binary relation specifies which pairs of objects are related, and a probability space starts with a set of possible outcomes. Giving these objects a common language lets a construction from one subject be used in another without repeating its underlying bookkeeping.

A particularly beautiful question is how to compare sizes without first assigning a number. Two sets have the same cardinality when a bijection pairs every member of one with exactly one member of the other. For finite sets, this agrees with ordinary counting. For infinite sets, it produces surprises: $n\mapsto 2n$ bijects the natural numbers with the nonnegative even integers, and the Cantor pairing function bijects the Cartesian product ${\mathbb{N}}^2$ with $\mathbb{N}$. A proper subset can have the same cardinality as its containing set.

This does not make all infinite sets equally large. Cantor's diagonal argument shows that all infinite binary sequences cannot be listed by the natural numbers. Cantor's theorem gives the wider principle: the power set of any set has strictly greater cardinality. The contrast between an explicit enumeration and an impossibility argument makes the meaning of infinity much sharper than a picture of something merely continuing forever.

Set formation itself requires care. Russell's paradox shows why not every condition defines a set of all objects satisfying it. Zermelo–Fraenkel set theory instead uses axioms specifying permitted constructions, including set comprehension within a previously given set. The restrictions are useful: they let us form the collections ordinary arguments need while blocking the unrestricted construction that produces the paradox.

The course catalogue below covers undergraduate degrees, integrated master's degrees, taught master's programmes and research master's programmes in mathematics, physics, chemistry, biology, computer science, engineering and medicine. Course names and examination stages follow the University's published directories and departmental teaching information.^[ref] Graduate programmes and historical examination holdings are distinguished from one another: a current course listing does not establish which past papers are held.