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For 2,000 years the foundations of mathematics seemed perfectly solid. Euclid's Elements (c. 300 BC BCE), which presented a set of formal logical arguments based on a few basic terms and axioms, provided a systematic method of rational exploration that guided mathematicians, philosophers, and scientists well into the 19th century. Even serious objections to the lack of rigour in Sir Isaac Newton's notion of fluxions (derivatives) in the calculus, raised by the Anglo-Irish empiricist George Berkeley (among others), did not call into question the basic foundations of mathematics. The discovery in the 19th century of consistent alternative geometries, however, precipitated a crisis, for it showed that Euclidean geometry, based on seemingly the most intuitively obvious axiomatic assumptions, did not correspond with reality as mathematicians had believed. This, together with the bold discoveries of the German mathematician Georg Cantor in set theory, made it clear that, to avoid further confusion and satisfactorily answer paradoxical results, a new and more rigorous foundation for mathematics was necessary.
Thus began the 20th-century quest to rebuild mathematics on a new basis independent of geometric intuitions. Early efforts included those of the logicist school of the British mathematicians Bertrand Russell and Alfred North Whitehead, the formalist school of the German mathematician David Hilbert, the intuitionist school of the Dutch mathematician L.E.J. Brouwer, and the French set theory school of mathematicians collectively writing under the pseudonym of Nicolas Bourbaki. Some of the most promising current research is based on the development of category theory by the American mathematician Saunders Mac Lane and the Polish-born American mathematician Samuel Eilenberg following World War II.
This article presents the historical background of foundational questions and 20th-century efforts to construct a new foundational basis for mathematics.
A remarkable amount of practical mathematics, some of it even fairly sophisticated, was already developed as early as 2000 BC BCE by the agricultural civilizations of Egypt and Mesopotamia , and perhaps even farther east. However, the first to exhibit an interest in the foundations of mathematics were the ancient Greeks.
Early Greek philosophy was dominated by a dispute as to which is more basic, arithmetic or geometry, and thus whether mathematics should be concerned primarily with the (positive) integers or the (positive) reals, the latter then being conceived as ratios of geometric quantities. (The Greeks confined themselves to positive numbers, as negative numbers were introduced only much later in India by Brahmagupta.) Underlying this dispute was a perceived basic dichotomy, not confined to mathematics but pervading all nature: is the universe made up of discrete atoms (as the philosopher Democritus believed) which hence can be counted, or does it consist of one or more continuous substances (as Thales of Miletus is reputed to have believed) and thus can only be measured? This dichotomy was presumably inspired by a linguistic distinction, analogous to that between English count nouns, such as "apple," and mass nouns, such as "water." As Aristotle later pointed out, in an effort to mediate between these divergent positions, water can be measured by counting cups.
The Pythagorean school of mathematics, founded on the doctrines of the Greek philosopher Pythagoras, originally insisted that only natural and rational numbers exist. Its members only reluctantly accepted the discovery that 2, the ratio of the diagonal of a square to its side, could not be expressed as the ratio of whole numbers. The remarkable proof of this fact has been preserved by Aristotle.
The contradiction between rationals and reals was finally resolved by Eudoxus of Cnidus, a disciple of Plato, who pointed out that two ratios of geometric quantities are equal if and only if they partition the set of (positive) rationals in the same way, thus anticipating the German mathematician Richard Dedekind (1831–1916), who defined real numbers as such partitions.
Another dispute among pre-Socratic philosophers was more concerned with the physical world. Parmenides claimed that in the real world there is no such thing as change and that the flow of time is an illusion, a view with parallels in the Einstein-Minkowski four-dimensional space-time model of the universe. Heracleitus, on the other hand, asserted that change is all-pervasive and is reputed to have said that one cannot step into the same river twice.
Zeno of Elea, a follower of Parmenides, claimed that change is actually impossible and produced four paradoxes to show this. The most famous of these describes a race between Achilles and a tortoise. Since Achilles can run much faster than the tortoise, let us say twice as fast, the latter is allowed a head start of one mile. When Achilles has run one mile, the tortoise will have run half as far again—that is, half a mile. When Achilles has covered that additional half-mile, the tortoise will have run a further quarter-mile. After n + 1 stages, Achilles has run
miles and the tortoise has run
miles, being still 1/2n + 1 miles ahead. So how can Achilles ever catch up with the tortoise (Figure 1see figure)?
Zeno's paradoxes may also be interpreted as showing that space and time are not made up of discrete atoms but are substances which are infinitely divisible. Mathematically speaking, his argument involves the sum of the infinite geometric progression
no finite partial sum of which adds up to 2. As Aristotle would later say, this progression is only potentially infinite. It is now understood that Zeno was trying to come to grips with the notion of limit, which was not formally explained until the 19th century, although a start in that direction had been made by the French encyclopaedist Jean Le Rond d'Alembert (1717–83).
The Athenian philosopher Plato believed that mathematical entities are not just human inventions but have a real existence. For instance, according to Plato, the number 2 is an ideal object. This is sometimes called an "idea," from the Greek eide, or "universal," from the Latin universalis, meaning "that which pertains to all." But Plato did not have in mind a "mental image," as "idea" is usually used. The number 2 is to be distinguished from a collection of two stones or two apples or, for that matter, two platinum balls in Paris.
What, then, are these Platonic ideas? Already in ancient Alexandria some people speculated that they are words. This is why the Greek word logos, originally meaning "word," later acquired a theological meaning as denoting the ultimate reality behind the "thing." An intense debate occurred in the Middle Ages over the ontological status of universals. Three dominant views prevailed: realism, from the Latin res ("thing"), which asserts that universals have an extra-mental reality—that is, they exist independently of perception; conceptualism, which asserts that universals exist as entities within the mind but have no extra-mental existence; and nominalism, from the Latin nomen ("name"), which asserts that universals exist neither in the mind nor in the extra-mental realm but are merely names that refer to collections of individual objects.
It would seem that Plato believed in a notion of truth independent of the human mind. In the Meno Plato's teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval. Thus, by clever questioning, Socrates managed to bring an uneducated person to "remember," or rather to reconstruct, the proof of a mathematical theorem.
Perhaps the most important contribution to the foundations of mathematics made by the ancient Greeks was the axiomatic method and the notion of proof. This was insisted upon in Plato's Academy and reached its high point in Alexandria about 300 BC BCE with Euclid's Elements. This notion survives today, except for some cosmetic changes.
The idea is this: there are a number of basic mathematical truths, called axioms or postulates, from which other true statements may be derived in a finite number of steps. It may take considerable ingenuity to discover a proof; but it is now held that it must be possible to check mechanically, step by step, whether a purported proof is indeed correct, and nowadays a computer should be able to do this. The mathematical statements that can be proved are called theorems, and it follows that, in principle, a mechanical device, such as a modern computer, can generate all theorems.
Two questions about the axiomatic method were left unanswered by the ancients: are all mathematical truths axioms or theorems (this is referred to as completeness), and can it be determined mechanically whether a given statement is a theorem (this is called decidability)? These questions were raised implicitly by David Hilbert (1862–1943) about 1900 and were resolved later in the negative, completeness by the Austrian-American logician Kurt Gödel (1906–78) and decidability by the American logician Alonzo Church (1903–95).
Euclid's work dealt with number theory and geometry, essentially all the mathematics then known. Since the middle of the 20th century a gradually changing group of mostly French mathematicians under the pseudonym Nicolas Bourbaki has tried to emulate Euclid in writing a new Elements of Mathematics based on their theory of structures. Unfortunately, they just missed out on the new ideas from category theory.
While the ancient Greeks were familiar with the positive integers, rationals, and reals, zero (used as an actual number instead of denoting a missing number) and the negative numbers were first used in India, as far as is known, by Brahmagupta in the 7th century AD CE. Complex numbers were introduced by the Italian Renaissance mathematician and physician Gerolamo Cardano (1501–76), not just to solve equations such as x2 + 1 = 0 but because they were needed to find real solutions of certain cubic equations with real coefficients. Much later, the German mathematician Carl Friedrich Gauss (1777–1855) proved the fundamental theorem of algebra, that all equations with complex coefficients have complex solutions, thus removing the principal motivation for introducing new numbers. Still, the Irish mathematician Sir William Rowan Hamilton (1805–65) and the French mathematician Olinde Rodrigues (1794–1851) invented quaternions in the mid-19th century, but these proved to be less popular in the scientific community until quite recently.
Currently, a logical presentation of the number system, as taught at the university level, would be as follows:B → B → B → B → B → B. Here the letters, introduced by Nicolas Bourbaki, refer to the natural numbers, integers, rationals, reals, complex numbers, and quaternions, respectively, and the arrows indicate inclusion of each number system into the next. However, as has been shown, the historical development proceeds differently:B+ → B+ → B+ → B → B → B, where the plus sign indicates restriction to positive elements. This is the development, up to B, which is often adhered to at the high-school level.
Although mathematics flourished after the end of the Classical Greek period for 800 years in Alexandria and, after an interlude in India and the Islāmic Islamic world, again in Renaissance Europe, philosophical questions concerning the foundations of mathematics were not raised until the invention of calculus and then not by mathematicians but by the philosopher George Berkeley (1685–1753).
Sir Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany had independently developed the calculus on a basis of heuristic rules and methods markedly deficient in logical justification. As is the case in many new developments, utility outweighed rigour, and, though Newton's fluxions (or derivatives) and Leibniz's infinitesimals (or differentials) lacked a coherent rational explanation, their power in answering heretofore unanswerable questions was undeniable. Unlike Newton, who made little effort to explain and justify fluxions, Leibniz, as an eminent and highly regarded philosopher, was influential in propagating the idea of infinitesimals, which he described as infinitely small actual numbers—that is, less than 1/n in absolute value for each positive integer n and yet not equal to zero. Berkeley, concerned over the deterministic and atheistic implications of philosophical mechanism, set out to reveal contradictions in the calculus in his influential book The Analyst; or, A Discourse Addressed to an Infidel Mathematician. There he scathingly wrote about these fluxions and infinitesimals, "They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" and further asked, "Whether mathematicians, who are so delicate in religious points, are strictly scrupulous in their own science? Whether they do not submit to authority, take things upon trust, and believe points inconceivable?"
Berkeley's criticism was not fully met until the 19th century, when it was realized that, in the expression dy/dx, dx and dy need not lead an independent existence. Rather, this expression could be defined as the limit of ordinary ratios Δy/Δx, as Δx approaches zero without ever being zero. Moreover, the notion of limit was then explained quite rigorously, in answer to such thinkers as Zeno and Berkeley.
It was not until the middle of the 20th century that the logician Abraham Robinson (1918–74) showed that the notion of infinitesimal was in fact logically consistent and that, therefore, infinitesimals could be introduced as new kinds of numbers. This led to a novel way of presenting the calculus, called nonstandard analysis, which has, however, not become as widespread and influential as it might have.
Robinson's argument was this: if the assumptions behind the existence of an infinitesimal ξ led to a contradiction, then this contradiction must already be obtainable from a finite set of these assumptions, say from:
But this finite set is consistent, as is seen by taking ξ = 1/(n + 1).
When Euclid presented his axiomatic treatment of geometry, one of his assumptions, his fifth postulate, appeared to be less obvious or fundamental than the others. As it is now conventionally formulated, it asserts that there is exactly one parallel to a given line through a given point. Attempts to derive this from Euclid's other axioms did not succeed, and, at the beginning of the 19th century, it was realized that Euclid's fifth postulate is, in fact, independent of the others. It was then seen that Euclid had described not the one true geometry but only one of a number of possible geometries.
Within the framework of Euclid's other four postulates (and a few that he omitted), there were also possible elliptic and hyperbolic geometries. In plane elliptic geometry there are no parallels to a given line through a given point; it may be viewed as the geometry of a spherical surface on which antipodal points have been identified and all lines are great circles. This was not viewed as revolutionary. More exciting was plane hyperbolic geometry, developed independently by the Hungarian mathematician János Bolyai (1802–60) and the Russian mathematician Nikolay Lobachevsky (1792–1856), in which there is more than one parallel to a given line through a given point. This geometry is more difficult to visualize, but a helpful model presents the hyperbolic plane as the interior of a circle, in which straight lines take the form of arcs of circles perpendicular to the circumference.
Another way to distinguish the three geometries is to look at the sum of the angles of a triangle. It is 180° in Euclidean geometry, as first reputedly discovered by Thales of Miletus (fl. flourished 6th century BC BCE), whereas it is more than 180° in elliptic geometry and less than 180° in hyperbolic geometry. See Figure 2 figure.
The discovery that there is more than one geometry was of foundational significance and contradicted the German philosopher Immanuel Kant (1724–1804). Kant had argued that there is only one true geometry, Euclidean, which is known to be true a priori by an inner faculty (or intuition) of the mind. For Kant, and practically all other philosophers and mathematicians of his time, this belief in the unassailable truth of Euclidean geometry formed the foundation and justification for further explorations into the nature of reality. With the discovery of consistent non-Euclidean geometries, there was a subsequent loss of certainty and trust in this innate intuition, and this was fundamental in separating mathematics from a rigid adherence to an external sensory order (no longer vouchsafed as "true") and led to the growing abstraction of mathematics as a self-contained universe. This divorce from geometric intuition added impetus to later efforts to rebuild assurance of truth on the basis of logic. (See below The quest for rigour.)
What then is the correct geometry for describing the space (actually space-time) we live in? It turns out to be none of the above, but a more general kind of geometry, as was first discovered by the German mathematician Bernhard Riemann (1826–66). In the early 20th century, Albert Einstein showed, in the context of his general theory of relativity, that the true geometry of space is only approximately Euclidean. It is a form of Riemannian geometry in which space and time are linked in a four-dimensional manifold, and it is the curvature at each point that is responsible for the gravitational "force" at that point. Einstein spent the last part of his life trying to extend this idea to the electromagnetic force, hoping to reduce all physics to geometry, but a successful unified field theory eluded him.
In the 19th century, the German mathematician Georg Cantor (1845–1918) returned once more to the notion of infinity and showed that, surprisingly, there is not just one kind of infinity but many kinds. In particular, while the set B of natural numbers and the set of all subsets of B are both infinite, the latter collection is more numerous, in a way that Cantor made precise, than the former. He proved that B, B, and B all have the same size, since it is possible to put them into one-to-one correspondence with one another, but that B is bigger, having the same size as the set of all subsets of B.
However, Cantor was unable to prove the so-called continuum hypothesis, which asserts that there is no set that is larger than B yet smaller than the set of its subsets. It was shown only in the 20th century, by Gödel and the American logician Paul Cohen (b. 19341934–2007), that the continuum hypothesis can be neither proved nor disproved from the usual axioms of set theory. Cantor had his detractors, most notably the German mathematician Leopold Kronecker (1823–91), who felt that Cantor's theory was too metaphysical and that his methods were not sufficiently constructive (see below The quest for rigour: Formal foundations: Nonconstructive arguments).
While laying rigorous foundations for mathematics, 19th-century mathematicians discovered that the language of mathematics could be reduced to that of set theory (developed by Cantor), dealing with membership (∊) and equality (=), together with some rudimentary arithmetic, containing at least symbols for zero (0) and successor (S). Underlying all this were the basic logical concepts: conjunction (∧), disjunction (∨), implication (⊃), negation (¬), and the universal (∀) and existential (∃) quantifiers (formalized by the German mathematician Gottlob Frege [1848–1925]). (The modern notation owes more to the influence of the English logician Bertrand Russell [1872–1970] and the Italian mathematician Giuseppe Peano [1858–1932] than to that of Frege.) For an extensive discussion of logic symbols and operations, see formal logic: Logic systems: Formal logic.
For some time, logicians were obsessed with a principle of parsimony, called Ockham's razor, which justified them in reducing the number of these fundamental concepts, for example, by defining p ⊃ q (read p implies q) as ¬p ∨ q or even as ¬(p ∧ ¬q). While this definition, even if unnecessarily cumbersome, is legitimate classically, it is not permitted in intuitionistic logic (see below Intuitionistic logic). In the same spirit, many mathematicians adopted the Wiener-Kuratowski definition of the ordered pair XXltXX < a, bXXgtXX > as {{a}, {a, b}}, where {a} is the set whose sole element is a, which disguises its true significance.
Logic had been studied by the ancients, in particular by Aristotle and the Stoic philosophers. Philo of Megara (fl. flourished c. 250 BC BCE) had observed (or postulated) that p ⊃ q is false if and only if p is true and q is false. Yet the intimate connection between logic and mathematics had to await the insight of 19th-century thinkers, in particular Frege.
Frege was able to explain most mathematical notions with the help of his comprehension scheme, which asserts that, for every ϕ (formula or statement), there should exist a set X such that, for all x, x ∊ X if and only if ϕ(x) is true. Moreover, by the axiom of extensionality, this set X is uniquely determined by ϕ(x). A flaw in Frege's system was uncovered by Russell, who pointed out some obvious contradictions involving sets that contain themselves as elements—eelements—e.g., by taking ϕ(x) to be ¬(x ∊ x). Russell illustrated this by what has come to be known as the barber paradox: A barber states that he shaves all who do not shave themselves. Who shaves the barber? Any answer contradicts the barber's statement. To avoid these contradictions Russell introduced the concept of types, a hierarchy (not necessarily linear) of elements and sets such that definitions always proceed from more basic elements (sets) to more inclusive sets, hoping that self-referencing and circular definitions would then be excluded. With this type distinction, x ∊ X only if X is of an appropriate higher type than x.
The type theory proposed by Russell, later developed in collaboration with the English mathematician Alfred North Whitehead (1861–1947) in their monumental Principia Mathematica (1910–13), turned out to be too cumbersome to appeal to mathematicians and logicians, who managed to avoid Russell's paradox in other ways. Mathematicians made use of the Neumann-Gödel-Bernays set theory, which distinguishes between small sets and large classes, while logicians preferred an essentially equivalent first-order language, the Zermelo-Fraenkel axioms, which allow one to construct new sets only as subsets of given old sets. Mention should also be made of the system of the American philosopher Willard Van Orman Quine (b. 19081908–2000), which admits a universal set. (Cantor had not allowed such a "biggest" set, as the set of all its subsets would have to be still bigger.) Although type theory was greatly simplified by Alonzo Church and the American mathematician Leon Henkin (b. 19211921–2006), it came into its own only with the advent of category theory (see below Category theory).
The prominence of logic in foundations led some people, referred to as logicists, to suggest that mathematics is a branch of logic. The concepts of membership and equality could reasonably be incorporated into logic, but what about the natural numbers? Kronecker had suggested that, while everything else was made by man, the natural numbers were given by God. The logicists, however, believed that the natural numbers were also man-made, inasmuch as definitions may be said to be of human origin.
Russell proposed that the number 2 be defined as the set of all two-element sets, that is, X ∊ 2 if and only if X has distinct elements x and y and every element of X is either x or y. The Hungarian-born American mathematician John von Neumann (1903–57) suggested an even simpler definition, namely that X ∊ 2 if and only if X = 0 or X = 1, where 0 is the empty set and 1 is the set consisting of 0 alone. Both definitions require an extralogical axiom to make them work—the axiom of infinity, which postulates the existence of an infinite set. Since the simplest infinite set is the set of natural numbers, one cannot really say that arithmetic has been reduced to logic. Most mathematicians follow Peano, who preferred to introduce the natural numbers directly by postulating the crucial properties of 0 and the successor operation S, among which one finds the principle of mathematical induction.
The logicist program might conceivably be saved by a 20th-century construction usually ascribed to Church, though he had been anticipated by the Austrian philosopher Ludwig Wittgenstein (1889–1951). According to Church, the number 2 is the process of iteration; that is, 2 is the function which to every function f assigns its iterate 2(f) = f ○ f, where (f ○ f)(x) = f(f(x)). There are some type-theoretical difficulties with this construction, but these can be overcome if quantification over types is allowed; this is finding favour in theoretical computer science.
A number of 19th-century mathematicians found fault with the program of reducing mathematics to arithmetic and set theory as suggested by the work of Cantor and Frege. In particular, the French mathematician Henri Poincaré (1854–1912) objected to impredicative constructions, which construct an entity of a certain type in terms of entities of the same or higher type—itype—i.e., self-referencing constructions and definitions. For example, when proving that every bounded nonempty set X of real numbers has a least upper bound a, one proceeds as follows. (For this purpose, it will be convenient to think of a real number, following Dedekind, as a set of rationals that contains all the rationals less than any element of the set.) One lets x ∊ a if and only if x ∊ y for some y ∊ X; but here y is of the same type as a.
It would seem that to do ordinary analysis one requires impredicative constructions. Russell and Whitehead tried unsuccessfully to base mathematics on a predicative type theory; but, though reluctant, they had to introduce an additional axiom, the axiom of reducibility, which rendered their enterprise impredicative after all. More recently, the Swedish logician Per Martin-Löf presented a new predicative type theory, but no one claims that this is adequate for all of classical analysis. However, the German-American mathematician Hermann Weyl (1885–1955) and the American mathematician Solomon Feferman have shown that impredicative arguments such as the above can often be circumvented and are not needed for most, if not all, of analysis. On the other hand, as was pointed out by the Italian computer scientist Giuseppe Longo (b. born 1929), impredicative constructions are extremely useful in computer science—namely, for producing fixpoints (entities that remain unchanged under a given process).
Another criticism of the Cantor-Frege program was raised by Kronecker, who objected to nonconstructive arguments, such as the following proof that there exist irrational numbers a and b such that ab is rational. If is rational, then the proof is complete; otherwise take and b = 2, so that ab = 2. The argument is nonconstructive, because it does not tell us which alternative holds, even though more powerful mathematics will, as was shown by the Russian mathematician Aleksandr Osipovich Gelfond (1906–68). In the present case, the result can be proved constructively by taking a = 2 and b = 2log23. But there are other classical theorems for which no constructive proof exists.
Consider, for example, the statement∃x(∃yϕ(y) ⊃ ϕ(x)), which symbolizes the statement that there exists a person who is famous if there are any famous people. This can be proved with the help of De Morgan's laws, named after the English mathematician and logician Augustus De Morgan (1806–71). It asserts the equivalence of ∃yϕ(y) with ¬∀y¬ϕ(y), using classical logic, but there is no way one can construct such an x, for example, when ϕ(x) asserts the existence of a well-ordering of the reals, as was proved by Feferman. An ordered set is said to be well-ordered if every nonempty subset has a least element. It had been shown by the German mathematician Ernst Zermelo (1871–1951) that every set can be well-ordered, provided one adopts another axiom, the axiom of choice, which says that, for every nonempty family of nonempty sets, there is a set obtainable by picking out exactly one element from each of these sets. This axiom is a fertile source of nonconstructive arguments.
The Dutch mathematician L.E.J. Brouwer (1881–1966) in the early 20th century had the fundamental insight that such nonconstructive arguments will be avoided if one abandons a principle of classical logic which lies behind De Morgan's laws. This is the principle of the excluded third (or excluded middle), which asserts that, for every proposition p, either p or not p; and equivalently that, for every p, not not p implies p. This principle is basic to classical logic and had already been enunciated by Aristotle, though with some reservations, as he pointed out that the statement "there will be a sea battle tomorrow" is neither true nor false.
Brouwer did not claim that the principle of the excluded third always fails, only that it may fail in the presence of infinite sets. Of two natural numbers x and y one can always decide whether x = y or x ≠ y, but of two real numbers this may not be possible, as one might have to know an infinite number of digits of their decimal expansions. Similar objections apply to De Morgan's laws, a consequence of the principle of the excluded third. For a finite set A, if it has been shown that the assertion ∀x ∊ A¬ϕ(x) leads to a contradiction, ∃x ∊ Aϕ(x) can be verified by looking at each element of A in turn; i.e., the statement that no members of a given set have a certain property can be disproved by examining in turn each element of the set. For an infinite set A, there is no way in which such an inspection can be carried out.
Brouwer's philosophy of mathematics is called intuitionism. Although Brouwer himself felt that mathematics was language-independent, his disciple Arend Heyting (1898–1980) set up a formal language for first-order intuitionistic arithmetic. Some of Brouwer's later followers even studied intuitionistic type theory (see below Intuitionistic type theories), which differs from classical type theory only by the absence of a single axiom (double negation):∀x ∊ Ω(¬¬x ⊃ x), where Ω is the type of truth-values.
While it cannot be said that many practicing mathematicians have followed Brouwer in rejecting this principle on philosophical grounds, it came as a great surprise to people working in category theory that certain important categories called topoi (singular: topos; see below Topos theory) have associated with them a language that is intuitionistic in general. In consequence of this fact, a theorem about sets proved constructively was immediately seen to be valid not only for sets but also for sheaves, which, however, lie beyond the scope of this article.
The moderate form of intuitionism considered here embraces Kronecker's constructivism but not the more extreme position of finitism. According to this view, which goes back to Aristotle, infinite sets do not exist, except potentially. In fact, it is precisely in the presence of infinite sets that intuitionists drop the classical principle of the excluded third.
An even more extreme position, called ultrafinitism, maintains that even very large numbers do not exist, say numbers greater than 10(1010). Of course, the vast majority of mathematicians reject this view by referring to 10(1010) + 1, but the true believers have subtle ways of getting around this objection, which, however, lie beyond the scope of this discussion.
While intuitionistic logic is obtained from classical logic by dropping the principle of the excluded third, other logics have also been proposed, though none has had a comparable impact on the foundations of mathematics. One may mention many-valued, or multivalued, logics, which admit a finite number of truth-values; fuzzy logic, with an imprecise membership relationship (though, paradoxically, a precise equality relation); and quantum logic, where conjunction may be only partially defined and implication may not be defined at all. Perhaps more important have been various so-called substructural logics in which the usual properties of the deduction symbol are weakened: relevance logic is studied by philosophers, linear logic by computer scientists, and a noncommutative version of the latter by linguists.
Russell's discovery of a hidden contradiction in Frege's attempt to formalize set theory, with the help of his simple comprehension scheme, caused some mathematicians to wonder how one could make sure that no other contradictions existed. Hilbert's program, called formalism, was to concentrate on the formal language of mathematics and to study its syntax. In particular, the consistency of mathematics, which may be taken, for instance, to be the metamathematical assertion that the mathematical statement 0 = 1 is not provable, ought to be a metatheorem—that is, provable within the syntax of mathematics. This formalization project made sense only if the syntax of mathematics was consistent, for otherwise every syntactical statement would be provable, including that which asserts the consistency of mathematics.
Unfortunately, a consequence of Gödel's incompleteness theorem is that the consistency of mathematics can be proved only in a language which is stronger than the language of mathematics itself. Yet, formalism is not dead—in fact, most pure mathematicians are tacit formalists—but the naive attempt to prove the consistency of mathematics in a weaker system had to be abandoned.
While no one, except an extremist intuitionist, will deny the importance of the language of mathematics, most mathematicians are also philosophical realists who believe that the words of this language denote entities in the real world. Following the Swiss mathematician Paul Bernays (1888–1977), this position is also called Platonism, since Plato believed that mathematical entities really exist.
Implicit in Hilbert's program had been the hope that the syntactic notion of provability would capture the semantic notion of truth. Gödel came up with the surprising discovery that this was not the case for type theory and related languages adequate for arithmetic, as long as the following assumptions are insisted upon:The set of theorems (provable statements) is effectively enumerable, by virtue of the notion of proof being decidable.The set of true statements of mathematics is ω-complete in the following sense: given any formula ϕ(x), containing a free variable x of type N, the universal statement ∀x ∊ Nϕ(x) will be true if ϕ(n) is true for each numeral n—that is, for n = 0, n = S0, n = SS0, and so on.The language is consistent.
Actually, Gödel also made a somewhat stronger assumption, which, as the American mathematician J. Barkley Rosser later showed, could be replaced by assuming consistency. Gödel's ingenious argument was based on the observation that syntactical statements about the language of mathematics can be translated into statements of arithmetic, hence into the language of mathematics. It was partly inspired by an argument that supposedly goes back to the ancient Greeks and which went something like this: Epimenides says that all Cretans are liars; Epimenides is a Cretan; hence Epimenides is a liar. Under the assumptions 1 and 2, Gödel constructed a mathematical statement g that is true but not provable. If it is assumed that all theorems are true, it follows that neither g nor ¬g is a theorem.
No mathematician doubts assumption 1; by looking at a purported proof of a theorem, suitably formalized, it is possible for a mathematician, or even a computer, to tell whether it is a proof. By listing all proofs in, say, alphabetic order, an effective enumeration of all theorems is obtained. Classical mathematicians also accept assumption 2 and therefore reluctantly agree with Gödel that, contrary to Hilbert's expectation, there are true mathematical statements which are not provable.
However, moderate intuitionists could draw a different conclusion, because they are not committed to assumption 2. To them, the truth of the universal statement ∀x ∊ Nϕ(x) can be known only if the truth of ϕ(n) is known, for each natural number n, in a uniform way. This would not be the case, for example, if the proof of ϕ(n) increases in difficulty, hence in length, with n. Moderate intuitionists might therefore identify truth with provability and not be bothered by the fact that neither g nor ¬g is true, as they would not believe in the principle of the excluded third in the first place.
Intuitionists have always believed that, for a statement to be true, its truth must be knowable. Moreover, moderate intuitionists might concede to formalists that to say that a statement is known to be true is to say that it has been proved. Still, some intuitionists do not accept the above argument. Claiming that mathematics is language-independent, intuitionists would state that in Gödel's metamathematical proof of his incompleteness theorem, citing ω-completeness to establish the truth of a universal statement yields a uniform proof of the latter after all.
Gödel considered himself to be a Platonist, inasmuch as he believed in a notion of absolute truth. He took it for granted, as do many mathematicians, that the set of true statements is ω-complete. Other logicians are more skeptical and want to replace the notion of truth by that of truth in a model. In fact, Gödel himself, in his completeness theorem, had shown that for a mathematical statement to be provable it is necessary and sufficient that it be true in every model. His incompleteness theorem now showed that truth in every ω-complete model is not sufficient for provability. This point will be returned to later, as the notion of model for type theory is most easily formulated with the help of category theory, although this is not the way Gödel himself proceeded. See below Gödel and category theory.
Peano had observed that addition of natural numbers can be defined recursively thus:x + 0 = x, x + Sy = S(x + y). Other numerical functions Bk → B that can be defined with the help of such a recursion scheme (and with the help of 0, S, and substitution) are called primitive recursive. Gödel used this concept to make precise what he meant by "effectively enumerable." A set of natural numbers is said to be recursively enumerable if it consists of all f(n) with n ∊ B, where f ∶ B → B is a primitive recursive function.
This notion can easily be extended to subsets of Bk and, by a simple trick called arithmetization, to sets of strings of words in a language. Thus Gödel was able to assert that the set of theorems of mathematics is recursively enumerable, and, more recently, the American linguist Noam Chomsky (b. born 1928) could say that the set of grammatical sentences of a natural language, such as English, is recursively enumerable.
It is not difficult to show that all primitive recursive functions can be calculated. For example, to calculate x + y when x = 3 and y = 2, making use of Peano's recursive definition of x + y and of the definitions 1 = S0, 2 = S1, and so on, one proceeds as follows:
But primitive recursive functions are not the only numerical functions that can be calculated. More general are the recursive functions, where f ∶ B → B is said to be recursive if its graph is recursively enumerable—that is, if there exist primitive recursive functions u, v ∶ B → B such that, for all natural numbers x and y, y = f(x) if and only if, for some z ∊ B, x = u(z) and y = v(z).
All recursive functions can be calculated with pencil and paper or, even more primitively, by moving pebbles (calculi in Latin) from one location to another, using some finite set of instructions, nowadays called a program. Conversely, only recursive functions can be so calculated, or computed by a theoretical machine introduced by the English mathematician Alan Turing (1912–54), or by a modern computer, for that matter. The Church-Turing thesis asserts that the informal notion of calculability is completely captured by the formal notion of recursive functions and hence, in theory, replicable by a machine.
Gödel's incompleteness theorem had proved that any useful formal mathematical system will contain undecidable propositions—propositions which can be neither proved nor disproved. Church and Turing, while seeking an algorithmic (mechanical) test for deciding theoremhood and thus potentially deleting nontheorems, proved independently, in 1936, that such an algorithmic method was impossible for the first-order predicate logic (see logic, history of: 20th-century logic). The Church-Turing theorem of undecidability, combined with the related result of the Polish-born American mathematician Alfred Tarski (1902–83) on undecidability of truth, eliminated the possibility of a purely mechanical device replacing mathematicians.
While many mathematicians use computers only as word processors and for the purpose of communication, computer-assisted computations can be useful for discovering potential theorems. For example, the prime number theorem was first suggested as the result of extensive hand calculations on the prime numbers up to 3,000,000 by the Swiss mathematician Leonhard Euler (1707–83), a process that would have been greatly facilitated by the availability of a modern computer. Computers may also be helpful in completing proofs when there are a large number of cases to be considered. The renowned computer-aided proof of the four-colour mapping theorem by the American mathematicians Kenneth Appel (b. born 1932) and Wolfgang Haken (b. born 1928) even goes beyond this, as the computer helped to determine which cases were to be considered in the next step of the proof. Yet, in principle, computers cannot be asked to discover proofs, except in very restricted areas of mathematics—such as elementary Euclidean geometry—where the set of theorems happens to be recursive, as was proved by Tarski.
As the result of earlier investigations by Turing, Church, the American mathematician Haskell Brooks Curry (1900–82), and others, computer science has itself become a branch of mathematics. Thus, in theoretical computer science, the objects of study are not just theorems but also their proofs, as well as calculations, programs, and algorithms. Theoretical computer science turns out to have a close connection to category theory. Although this lies beyond the scope of this article, an indication will be given below.
One recent tendency in the development of mathematics has been the gradual process of abstraction. The Norwegian mathematician Niels Henrik Abel (1802–29) proved that equations of the fifth degree cannot, in general, be solved by radicals. The French mathematician Évariste Galois (1811–32), motivated in part by Abel's work, introduced certain groups of permutations to determine the necessary conditions for a polynomial equation to be solvable. These concrete groups soon gave rise to abstract groups, which were described axiomatically. Then it was realized that to study groups it was necessary to look at the relation between different groups—in particular, at the homomorphisms which map one group into another while preserving the group operations. Thus people began to study what is now called the concrete category of groups, whose objects are groups and whose arrows are homomorphisms. It did not take long for concrete categories to be replaced by abstract categories, again described axiomatically.
The important notion of a category was introduced by Samuel Eilenberg and Saunders Mac Lane at the end of World War II. These modern categories must be distinguished from Aristotle's categories, which are better called types in the present context. A category has not only objects but also arrows (referred to also as morphisms, transformations, or mappings) between them.
Many categories have as objects sets endowed with some structure and arrows, which preserve this structure. Thus, there exist the categories of sets (with empty structure) and mappings, of groups and group-homomorphisms, of rings and ring-homomorphisms, of vector spaces and linear transformations, of topological spaces and continuous mappings, and so on. There even exists, at a still more abstract level, the category of (small) categories and functors, as the morphisms between categories are called, which preserve relationships among the objects and arrows.
Not all categories can be viewed in this concrete way. For example, the formulas of a deductive system may be seen as objects of a category whose arrows f : A → B are deductions of B from A. In fact, this point of view is important in theoretical computer science, where formulas are thought of as types and deductions as operations.
More formally, a category consists of (1) a collection of objects A, B, C, . . . , (2) for each ordered pair of objects in the collection an associated collection of transformations including the identity IA ∶ A → A, and (3) an associated law of composition for each ordered triple of objects in the category such that for f ∶ A → B and g ∶ B → C the composition gf (or g ○ f) is a transformation from A to C—i—i.e., gf ∶ A → C. Additionally, the associative law and the identities are required to hold (where the compositions are defined)—i—i.e., h(gf) = (hg)f and 1Bf = f = f1A.
In a sense, the objects of an abstract category have no windows, like the monads of Leibniz. To infer the interior of an object A one need only look at all the arrows from other objects to A. For example, in the category of sets, elements of a set A may be represented by arrows from a typical one-element set into A. Similarly, in the category of small categories, if 1 is the category with one object and no nonidentity arrows, the objects of a category A may be identified with the functors 1 → A. Moreover, if 2 is the category with two objects and one nonidentity arrow, the arrows of A may be identified with the functors 2 → A.
An arrow f ∶ A → B is called an isomorphism if there is an arrow g ∶ B → A inverse to f—that is, such that g ○ f = 1A and f ○ g = 1B. This is written A ≅ B, and A and B are called isomorphic, meaning that they have essentially the same structure and that there is no need to distinguish between them. Inasmuch as mathematical entities are objects of categories, they are given only up to isomorphism. Their traditional set-theoretical constructions, aside from serving a useful purpose in showing consistency, are really irrelevant.
For example, in the usual construction of the ring of integers, an integer is defined as an equivalence class of pairs (m,n) of natural numbers, where (m,n) is equivalent to (m′,n′) if and only if m + n′ = m′ + n. The idea is that the equivalence class of (m,n) is to be viewed as m − n. What is important to a categorist, however, is that the ring B of integers is an initial object in the category of rings and homomorphisms—that is, that for every ring B there is a unique homomorphism B → B. Seen in this way, B is given only up to isomorphism. In the same spirit, it should be said not that B is contained in the field B of rational numbers but only that the homomorphism B → B is one-to-one. Likewise, it makes no sense to speak of the set-theoretical intersection of π and -1, if both are expressed as sets of sets of sets (ad infinitum).
Of special interest in foundations and elsewhere are adjoint functors (F,G). These are pairs of functors between two categories and ℬ, which go in opposite directions such that a one-to-one correspondence exists between the set of arrows F(A) → B in ℬ and the set of arrows A → G(B) in —that is, such that the sets are isomorphic.
The original purpose of category theory had been to make precise certain technical notions of algebra and topology and to present crucial results of divergent mathematical fields in an elegant and uniform way, but it soon became clear that categories had an important role to play in the foundations of mathematics. This observation was largely the contribution of the American mathematician F.W. Lawvere (b. born 1937), who elaborated on the seminal work of the German-born French mathematician Alexandre Grothendieck (b. born 1928) in algebraic geometry. At one time he considered using the category of (small) categories (and functors) itself for the foundations of mathematics. Though he did not abandon this idea, later he proposed a generalization of the category of sets (and mappings) instead.
Among the properties of the category of sets, Lawvere singled out certain crucial ones, only two of which are mentioned here:There is a one-to-one correspondence between subsets B of A and their characteristic functions χ ∶ A → {true, false}, where, for each element a of A, χ(a) = true if and only if a is in B.Given an element a of A and a function h ∶ A → A, there is a unique function f ∶ B → A such that f(n) = hn(a).
Suitably axiomatized, a category with these properties is called an (elementary) topos. However, in general, the two-element set {true, false} must be replaced by an object Ω with more than two truth-values, though a distinguished arrow into Ω is still labeled as true.
Topoi are closely related to intuitionistic type theories. Such a theory is equipped with certain types, terms, and theorems.
Among the types there should be a type Ω for truth-values, a type N for natural numbers, and, for each type A, a type ℘(A) for all sets of entities of type A.
Among the terms there should be in particular:The formulas a = a′ and a ∊ α of type Ω, if a and a′ are of type A and α is of type ℘(A)The numerals 0 and Sn of type N, if the numeral n is of type NThe comprehension term {x ∊ A|ϕ(x)} of type ℘(A), if ϕ(x) is a formula of type Ω containing a free variable x of type A
The set of theorems should contain certain obvious axioms and be closed under certain obvious rules of inference, neither of which will be spelled out here.
At this point the reader may wonder what happened to the usual logical symbols. These can all be defined—for example, universal quantification∀x ∊ Aϕ(x) as {x ∊ A|ϕ(x)} = {x ∊ A|x = x} and disjunctionp ∨ q as ∀t ∊ Ω((p ⊃ t) ⊃ ((q ⊃ t) ⊃ t)). For a formal definition of implication, see formal logic.
In general, the set of theorems will not be recursively enumerable. However, this will be the case for pure intuitionistic type theory ℒ0, in which types, terms, and theorems are all defined inductively. In ℒ0 there are no types, terms, or theorems other than those that follow from the definition of type theory. ℒ0 is adequate for the constructive part of the usual elementary mathematics—arithmetic and analysis—but not for metamathematics, if this is to include a proof of Gödel's completeness theorem, and not for category theory, if this is to include the Yoneda embedding of a small category into a set-valued functor category.
It turns out that each topos has an internal language L(), an intuitionistic type theory whose types are objects and whose terms are arrows of . Conversely, every type theory ℒ generates a topos T(ℒ), by the device of turning (equivalence classes of) terms into objects, which may be thought of as denoting sets.
Nominalists may be pleased to note that every topos is equivalent (in the sense of category theory) to the topos generated by a language—namely, the internal language of . On the other hand, Platonists may observe that every type theory ℒ has a conservative extension to the internal language of a topos—namely, the topos generated by ℒ, assuming that this topos exists in the real (ideal) world. Here, the phrase "conservative extension" means that ℒ can be extended to LT(ℒ) without creating new theorems. The types of LT(ℒ) are names of sets in ℒ and the terms of LT(ℒ) may be identified with names of sets in ℒ for which it can be proved that they have exactly one element. This last observation provides a categorical version of Russell's theory of descriptions: if one can prove the unique existence of an x of type A in ℒ such that ϕ(x), then this unique x has a name in LT(ℒ).
The interpretation of a type theory ℒ in a topos means an arrow ℒ → L() in the category of type theories or, equivalently, an arrow T(ℒ) → in the category of topoi. Indeed, T and L constitute a pair of adjoint functors.
It is now possible to reexamine Gödel's theorems from a categorical point of view. In a sense, every interpretation of ℒ in a topos may be considered as a model of ℒ, but this notion of model is too general, for example, when compared with the models of classical type theories studied by Henkin. Therefore, it is preferable to restrict to being a special kind of topos called local. Given an arrow p into Ω in , then, p is true in if p coincides with the arrow true in , or, equivalently, if p is a theorem in the internal language of . is called a local topos provided that (1) 0 = 1 is not true in , (2) p ∨ q is true in only if p is true in or q is true in , and (3) ∃x ∊ Aϕ(x) is true in only if ϕ(a) is true in for some arrow a ∶ 1 → A in . Here the statement 0 = 1 in provision 1 can be replaced by any other contradiction—econtradiction—e.g., by ∀t ∊ Ωt, which says that every proposition is true.
A model of ℒ is an interpretation of ℒ in a local topos . Gödel's completeness theorem, generalized to intuitionistic type theory, may now be stated as follows: A closed formula of ℒ is a theorem if and only if it is true in every model of ℒ.
Gödel's incompleteness theorem, generalized likewise, says that, in the usual language of arithmetic, it is not enough to look only at ω-complete models: Assuming that ℒ is consistent and that the theorems of ℒ are recursively enumerable, with the help of a decidable notion of proof, there is a closed formula g in ℒ, which is true in every ω-complete model, yet g is not a theorem in ℒ.
A Platonist might still ask whether, among all the models of the language of mathematics, there is a distinguished model, which may be considered to be the world of mathematics. Take as the language ℒ0 pure intuitionistic type theory (see above Intuitionistic type theories). It turns out, somewhat surprisingly, that the topos generated by ℒ0 is a local topos; hence, the unique interpretation of ℒ0 in the topos generated by it may serve as a distinguished model.
This so-called free topos has been constructed linguistically to satisfy any formalist, but it should also satisfy a moderate Platonist, one who is willing to abandon the principle of the excluded third, inasmuch as the free topos is the initial object in the category of all topoi. Hence, the free topos may be viewed, in the words of Leibniz, as the best of all possible worlds. More modestly speaking, the free topos is to an arbitrary topos like the ring of integers is to an arbitrary ring.
The language ℒ0 should also satisfy any constructivist: if an existential statement ∃x ∊ Aϕ(x) can be proved in ℒ0, then ϕ(a) can be proved for some term a of type A; moreover, if p ∨ q can be proved, then either p can be proved or q can be proved.
The above argument would seem to make a strong case for the acceptance of pure intuitionistic type theory as the language of elementary mathematics—that is, of arithmetic and analysis—and hence for the acceptance of the free topos as the world of mathematics. Nonetheless, most practicing mathematicians prefer to stick to classical mathematics. In fact, classical arguments seem to be necessary for metamathematics—for example, in the usual proof of Gödel's completeness theorem—even for intuitionistic type theory.
In this connection, one celebrated consequence of Gödel's incompleteness theorem may be recalled, to wit: the consistency of ℒ cannot be proved (via arithmetization) within ℒ. This is not to say that it cannot be proved in a stronger metalanguage. Indeed, to exhibit a single model of ℒ would constitute such a proof.
It is more difficult to make a case for the classical world of mathematics, although this is what most mathematicians believe in. This ought to be a distinguished model of pure classical type theory ℒ1. Unfortunately, Gödel's argument shows that the interpretation of ℒ1 in the topos generated by it is not a model in this sense.
A topos is said to be Boolean if its internal language is classical. It is named after the English mathematician George Boole (1815–64), who was the first to give an algebraic presentation of the classical calculus of propositions. A Boolean topos is local under the following circumstances. The disjunction property (2) holds in a Boolean topos if and only if, for every closed formula p, either p is true or ¬p is true. Moreover, with the help of De Morgan's laws, the existence property (3) may then be rephrased thus: if ϕ(a) is true for all closed terms a of type A, then ∀x ∊ Aϕ(x) is true. As it turns out, a Boolean local topos may be described more simply, without referring to the internal language, as a topos with the following property: if f, g : A → B are arrows such that fa = ga for all a : 1 → A, then f = g. (Here 1 is the so-called terminal object, with the property that, for each object C, there is a unique arrow C → 1.) For the Boolean topos to be ω-complete requires furthermore that all numerals—that is, closed terms of type N in its internal language—be standard—that is, have the form 0, S0, SS0, and so on.
Of course, Gödel's completeness theorem shows that there are plenty of Boolean local topoi to model pure classical type theory in, but the usual proof of their existence requires nonconstructive arguments. It would be interesting to exhibit at least one such model constructively.
As a first step toward constructing a distinguished ω-complete Boolean model of ℒ1 one might wish to define the notion of truth in ℒ1, as induced by this model. Tarski had shown how truth can be defined for classical first-order arithmetic, a language that admits, aside from formulas, only terms of type N. Tarski achieved this essentially by incorporating ω-completeness into the definition of truth. It is not obvious whether his method can be extended to classical higher-order arithmetic—that is, to classical type theory. In fact, Tarski himself showed that the notion of truth is not definable (in a technical sense) in such a system. If his notion of definability corresponds to what is here meant by constructibility, then it is possible to conclude that, indeed, no Boolean model can be constructed.
One may be tempted to consider as a candidate for the distinguished Boolean local topos the so-called von Neumann universe. This is defined as the union of a class of sets containing the empty set (the initial object in the category of sets) and closed under the power-set operation and under transfinite unions—thus, as a subcategory of the category of sets. But what is the category of sets if not the distinguished Boolean local topos being sought?
A better candidate may be Gödel's constructible universe, whose original purpose was to serve as a model of Zermelo-Fraenkel set theory in which the continuum hypothesis holds. It is formed like the von Neumann universe, except that the notion of subset, implicit in the power-set operation, is replaced by that of definable subset. Is it possible that this universe can be constructed syntactically, like the free topos, without reference to any previously given category of sets, or by a universal property?
In the internal language of a Boolean local topos, the logical connectives and quantifiers have their natural meanings. In particular, quantifiers admit a substitutional interpretation, a desirable property that has been discussed by philosophers (among them, Russell and the American logician Saul Kripke [b. 1941born 1940])—to wit: if an existential statement is true, then it can be witnessed by a term of appropriate type in the language; and a universal statement is true if it is witnessed by all terms of the appropriate type.
Note that, in the internal language of the free topos, and therefore in pure intuitionistic type theory, the substitutional interpretation is valid for existential quantifiers, by virtue of the free topos being local, but that it fails for universal quantifiers, in view of the absence of ω-completeness and the fact that in the free topos all numerals are standard. For a Boolean local topos, ω-completeness will also ensure that all numerals are standard, so that numerals mean exactly what they are intended to mean.
Some mathematicians do not believe that a distinguished world of mathematics should be sought at all, but rather that the multiplicity of such worlds should be looked at simultaneously. A major result in algebraic geometry, due to Alexandre Grothendieck, was the observation that every commutative ring may be viewed as a continuously variable local ring, as Lawvere would put it. In the same spirit, an amplified version of Gödel's completeness theorem would say that every topos may be viewed as a continuously variable local topos, provided sufficiently many variables (Henkin constants) are adjoined to its internal language. Put in more technical language, this makes the possible worlds of mathematics stalks of a sheaf. However, the question still remains as to where this sheaf lives if not in a distinguished world of mathematics or—perhaps better to say—metamathematics.
These observations suggest that the foundations of mathematics have not achieved a definitive shape but are still evolving; they form the subject of a lively debate among a small group of interested mathematicians, logicians, and philosophers.


text:
Verna Lee "Happy" Freeman passed away Thursday, May 30, 2013, at her home of 70 years north of Cortez.
Happy was born in Amarillo on April 4, 1917, the second child of William I. Vance and Mable U. Thomas Vance.
Happy was raised on a homestead in eastern New Mexico just out of Portales. She met Steve Freeman, who lived on a nearby farm.
They were married March 6, 1940, and moved to the Cortez area in 1943. They had one son, Gary Freeman.
Happy loved working in the yard.
From early spring till late fall, she could always be found digging somewhere in a flowerbed.
In the winter, she would work on her quilts. She started quilting when she was just big enough to sit at a quilt frame and sewing machine.
When she grew tired of quilting, she would crochet or work on crossword puzzles.
Every year her three grandchildren would spend the summer with her and Steve, and they had high adventures camping and fishing.
She was preceded in death by her parents, by three brothers. Dan Vance, Bill Vance, and Paul Freeman, and by her husband of 57 years, Steve Freeman.
Happy is survived by her son Gary and his wife Charlotte of Dolores, three grandchildren, Kevin and his wife Karen of Sun Prairie, Wisc., Shelley of Oklahoma City, Okla., and Kelley and her husband Steve of Oklahoma City, Okla.; six great-grandchildren, Kaili, Kaylyn, Kiara and Kellen Freeman of Sun Prairie, Wisc., Jessica Freeman of Dallas, Texas, and Sean Freeman of Oklahoma City, as well as several nieces and nephews, great nieces and nephews, and great-great nieces and nephews, and Betsy Freeman, her sister-in-law, of Portales, N.M.
Happy was a member of Cortez First Assembly of God since her move here in the 1940's.
There will be a celebration of her life July 5, at 14149 Rd. 25, the family home.
This will be a potluck dinner, and those who attend should bring favorite story of Happy.

subset: commoncrawl


text: I know nothing about cars, but have some questions for those who do: which is faster, a 1967 XKE convertible in excellent condition or a 2012 Camaro? Are they similar in performance, or vastly different? Assume typical engines in each, so whatever they come with (nothing souped up). Which handles better? What would happen if a 67 Jaguar was trying to chase down a Camaro through suburban streets? And how much might a 1967 Jaguar in excellent condition cost? Bonus question: which is faster, new Dodge Challenger or new Camaro? Please refrain from too much car mumbo-jumbo in your answers. Remember, I am not a car guy.


text:
I was just over at Hypable reading the reviews of the first of those who have managed to gain entry into Pottermore beta. An example of what they say we will find out is backstory to Professor McGonagall.
That bit is just the beginning of it all. The idea of the amazing batches of information we will receive over the the following years, I expect it will revive fanfiction plots and ideas exponentially.
As I understand it, only information that is contained in the books constitute canon (according to mnff Help section). I know fanfiction writers constantly refer to Rowling's interviews previously for new, fill-in-the-blanks information on new characters. But now that will be mechanical and new information will keep on coming.
My question (more than suggestion, unless it triggers a debate) is whether we still ought to stick to the way canon is currently defined at mnff. Writers here are going to use that information to be inspired anyway, but will it have any effect on warnings placed on the story that utilises information from Pottermore? Not canon or canon?
I understand that this can be wholly sorted when we know the nature of information Pottermore will reveal, but it's starting now and the information will leak already before October, and meanwhile we could give this some thought.


text:
This Puzzle is dedicated to test the students in a more creative thinking which gives a pleasure in learning in the topic object oriented programming(OOP) in Visual Basic.NET Programming (Visual Studio 2010)
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text:
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text:
The pain experienced during a heart attack and during a severe heartburn episode can be difficult to distinguish. It is not unusual for people to mistake symptoms of heart disease (such as angina and even a heart attack) for heartburn. Similarly, many people go to the emergency room each year out of fear that they are having a heart attack, only to find out they have severe heartburn. In fact, it often takes medical testing to make the determination.
Pain May Be Difficult to Distinguish
Here are some possible differences between heartburn and heart attacks.
- A sharp, burning sensation below the breastbone or ribs
- Burning sensation may move up toward the throat
- Pain often occurs after eating, particularly when lying down
- Pain increases when bending over, lying down, exercising, or lifting heavy objects
- Bitter or sour taste at the back of the throat
- Symptoms respond quickly to antacids
Note: Call for emergency medical services right away if you have any chest pain, even if you think it may be heartburn.
- A feeling of uncomfortable fullness, pressure, squeezing, tightness, or pain in the center of the chest that lasts for more than a few minutes or goes away and comes back
- Symptoms brought on by physical exertion or emotional stress
- Pain or discomfort that spreads to one or both arms, the back, stomach, neck, or jaw
- Shortness of breath
Other symptoms, such as:
- Breaking out in a cold sweat
- Nausea
- Lightheadedness
- Fainting
- Rapid heartbeat
Other Causes of Chest Pain
Heartburn and heart attacks are not the only conditions that can cause chest pain. Other problems that can cause chest pain include:
Other heart conditions, such as:
- Coronary spasm—arteries supplying blood to the heart go into spasm, temporarily limiting blood flow to the heart muscle
- Pericarditis—inflammation of the sac surrounding the heart
- Aortic dissection—rare, but dangerous condition in which the inner layers of the aorta separate
- Panic attack—periods of intense fear accompanied by anxiety, chest tightness, rapid heartbeat, rapid breathing, profuse sweating, and shortness of breath
- Costochondritis—inflammation of the rib cage cartilage
- Muscle-related chest pain—often accompanies fibromyalgia and other chronic pain syndromes
- Injured ribs, pinched nerves—can cause localized chest pain
- Shingles —infection of a nerve root, caused by reactivation of the chickenpox virus
- Pleurisy—inflammation of the lining of the chest and lungs, which causes chest pain that increases with coughing, inhalation, or deep breathing
- Pulmonary embolism —a blood clot lodged in the artery of the lung
Other lung conditions, such as:
- Atelectasis—collapsed lung
- Severe asthma
- Pneumonia—lung infection
- Gallbladder or pancreas problems—gallstones or inflammation of the gallbladder or pancreas can cause abdominal pain, which can radiate to the chest
- Disorders of the esophagus—swallowing disorders such as esophageal spasms and achalasia (failure of esophageal muscle to relax)
- Cancer —cancer involving the chest or that has spread from another part of the body
Seeking Medical Attention for Chest Pain
Chest pain can be difficult to interpret. Call for emergency medical services right away if you have any chest pain, especially if you have other signs and symptoms of a heart attack. Do not have someone drive you to the emergency room. Emergency medical crews can provide treatment on the way to the hospital.
- Reviewer: Michael Woods, MD
- Review Date: 05/2016 -
- Update Date: 06/11/2014 -


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The agency said Wednesday that the animals caught at the spill site at Mayflower have been taken to a wildlife rehabilitation center. Some of the animals have been returned to the wild.
Volunteers have helped scrub the oil off birds, mammals and turtles rescued from near Lake Conway after the March 29 spill.
Game and Fish officials say water samples are still being taken by state environmental officials and ExxonMobil. The agency says there's no sign of oil having seeped into Lake Conway, though the oil contaminated a cove south of the lake. Sampling also shows no oil in Palarm Creek.


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Padres recall three; Smith to start Friday
By Corey Brock / MLB.com
SAN DIEGO -- The Padres added three relief pitchers to their roster Tuesday when the team recalled right-handers Brad Brach, Brad Boxberger and Anthony Bass to their roster.
San Diego manager Bud Black said he and the organization will use the final 25 games of the season to evaluate each as well as other young players on the roster.
One of those young players is 23-year-old right-hander Burch Smith, who will get the start Friday at Petco Park against the Rockies.
Smith, ranked by MLB.com as the Padres' No. 9 prospect, began the season with Double-A San Antonio and then was added to the big league roster in time for a start -- his Major League debut -- May 11 against the Rays. He allowed six earned runs in just one inning, part of a six-game stint that saw him post a 11.37 ERA.
Smith returned to the Minor Leagues where he went 5-1 with a 3.39 ERA in 12 starts with the Triple-A Tucson Padres. On Tuesday, he was actually named the Pacific Coast League's Pitcher of the Week. He tossed seven scoreless innings in a no-decision against Reno with 10 strikeouts.
Smith said Tuesday that he's a different pitcher than the one who struggled earlier in the season with the Padres.
"I'll get back out there and go at it differently this time," Smith said. "I'm going to try and execute my pitches and not just try to blow the ball by guys."
Black is excited to see Smith pitch again.
"The last number of games he's pitched in the Minor Leagues have been good," Black said.
Alonso won't rush back from hand injury
SAN DIEGO -- First baseman Yonder Alonso had an MRI on his sore right hand Tuesday that revealed fluid in and between the joints in the hand.
"That's what's causing it to hurt," Alonso said.
Alonso was scheduled to have a cortisone shot in the hand later Tuesday, but at this point he's been told the only thing he can do for the hand is rest it.
"We're working through a couple of scenarios where we'll give it rest with the hope that I will be able to come back sooner than later," he said.
The good news was that no surgery is expected on the hand, which caused him to miss 34 games earlier this season. He said that even after returning from the disabled list, the strength in his hand wasn't the same as it was earlier in the season.
The initial diagnosis was that Alonso would miss between seven and 10 days, but it now appears that he will miss more time than that.
"He could be back in a couple of weeks," said Padres manager Bud Black. "He wants to get back and play again, which is great."
Alonso said there's no sense in him rushing back, especially since there's just 24 games remaining in the regular season after Tuesday.
"I'm not even 90 percent now, so what's the point in coming back before I'm healthy?" he said.
Alonso said his hand hasn't felt fully healthy since before his previous injury. He hit .284 with six home runs, 29 RBIs and seven doubles in 190 at-bats before that first injury. Since then, he's hitting .278 with four doubles and 16 RBIs.
Short hops
• Third baseman Chase Headley, who has missed the last six games with a sore back, took ground balls and batting practice before Tuesday's game against the Giants. Headley, who had back spasms in a game Aug. 27 against the D-backs, said his back was feeling better. If he doesn't start Wednesday then he'll certainly do so Friday when the team opens a three-game series against the Rockies.
• Outfielder Carlos Quentin had surgery on his right knee Tuesday in San Diego. The surgery was performed by team physician Heinz Hoenecke. As expected, the procedure was done on the patella area of the knee and included the removal of loose fragments, removing scar tissue and cleaning up loose cartilage. Quentin is expected to be ready for the start of Spring Training in mid-February.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.


text: S200 Second-Year Spanish I (3 credits) P: S105 or S135 or S150 or equivalent. This course reviews some of the basic structures studied in the first year, and examines them in more detail. Emphasis remains on the four skills and on critical thinking skills. Readings are both journalistic and literary. Grades are based on exams, oral tests, homework, compositions, and a cumulative final exam. Homework load is substantial. The next course in this sequence is S250. See Schedule of Classes for section numbers, times, days, and rooms.


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Sefko: Behind Mayo, suddenly 'shallow' Mavericks romp past Raptors
It's a fact of life in the NBA that when you put together back-to-back 30-point games, opponents tend to notice.
The next time out, you're guaranteed to see double teams early and often.
O.J. Mayo saw it, and proved that he's not just a one-trick pony.
With the Mavericks shorthanded and the Toronto defense loading up against him, Mayo went from scorer to complete player in a hurry, filling up the stat sheet as the Mavericks knocked off the Raptors 109-104 Wednesday night.
Mayo had a team-best six assists — many coming when the Raptors threw a trapping defense at him — in addition to 22 points and five rebounds.
"O.J. can play," said point guard Darren Collison, who was the recipient of one of Mayo's fourth-quarter dishes that led to an easy layup. "He's not just a scorer. He can make plays coming off the pick and roll. He's not only looking for his shot, he's looking to get everybody else involved.
"They're going to trap him. He's the hot scorer right now and he's got it going, so they're going to throw a couple things at him. So we've got to make them pay."
Mayo said the idea is to take advantage of whatever the defense is giving him. For two games, they chased him around with one defender and he scored at will. Against the Raptors, he had to slice through double teams with the dribble and find open teammates.
"Just be poised," he said of his game plan. "If you've got two people on you, somebody has to be open. Try to find them. Sometimes, you might not, but the hockey assist is always fine, too. I got great teammates."
The Mavericks, who led by double figures much of the way but had to play solid defense late to protect the victory, improved to 4-1 for the first time since 2007-08. They were 1-4 last year after five games.
It took the perspective of a coach to frame the quick start to the season.
"It points to the fact that, to this point, we've taken advantage of a favorable schedule," Mavs coach Rick Carlisle said. "All these wins are very good wins. These are quality teams. Portland's going to be a factor in the West. Toronto's going to be a factor in the East. The Lakers are going to be a top team in the West before it's done. We caught them at a good time. We're on the road now and it gets tougher."
It starts Friday in New York, against the Knicks and Jason Kidd. But the key Wednesday was that the Mavericks finished off a 3-0 homestand with a quality effort despite some tough circumstances.
The Mavericks were down to 11 healthy players as Dirk Nowitzki and Shawn Marion were out with knee trouble, Elton Brand was in New York with his wife for the birth of their baby girl, and Roddy Beaubois remained out with a sore ankle.
"As usual," Toronto coach Dwane Casey said, "Rick's doing a hell of a job with this group with bailing wire."
You know things are going good when the Mavericks even win the rebounding battle, something they had only done once this season.
They pounded Toronto on the boards, leading the Raptors 42-26 through three quarters. They ended up plus-nine.
It helped that they played bigger, with Chris Kaman getting his first start of the season and responding with 22 points and eight rebounds.
Rookie Bernard James came off the bench with nine boards in 18 minutes.
And the Mavericks played defense, too.
Rookie Jae Crowder made a terrific defensive play on high-scoring DeMar DeRozan. Crowder forced a tough, 18-foot jumper with a hand in DeRozan's face. Moments later, Vince Carter also snuffed DeRozan on a jump shot that protected the Mavericks' lead.


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SARASOTA — Introduced with a car chase, overturning police cruiser and exploding helicopter, the producers of the biggest suspense thriller in Sarasota County revealed their final act this week.
Officials for the top-secret movie-studio project intended for Lakewood Ranch, known only as Project Waterboy since it was first discussed in June, finally unveiled their plans to the public Sept. 20.
Sanborn Studios hopes to begin producing TV shows and independent films in Lakewood Ranch as soon as November.
"This is a win-win for the community," said Rex Jensen, president and CEO of Schroeder-Manatee Ranch. "We've had a lot of near misses, but this is one of the biggest transactions I've seen."
Running his own studio has been Ken Sanborn's longtime dream. The Longboat Key resident's father was a filmmaker, and Sanborn started his own career as a cameraman for WLFA-TV in Tampa, before he began shooting and directing documentaries and TV commercials.
"I've always had a passion for film," he said.
Sanborn got away from his passion when he built a company called Gyrocam, which made cameras for military vehicles and law-enforcement vehicles and helicopters.
"I was not doing what I really loved," he said.
Sanborn sold the business last year and then had the time to help his son, Harrison, make a film called "Paradise Lost" in Bradenton.
"I realized I missed filmmaking," Sanborn said. "So I told my son I wanted to start a studio."
The plan is to retrofit the existing TVC building at 7321 Trade Court in Lakewood Ranch so it can house two sound stages and the studio's production offices.
The company also will begin to move into the old Gyrocam headquarters at the Sarasota-Bradenton International Airport.
Phase two of the project includes four stage buildings on vacant land at Lakewood Ranch Boulevard and Satellite Court. Construction is expected to begin in six months and be complete a year later in 2012. The total project is estimated at $30 million and is expected to have an economic impact of $164 million.
Sanborn has committed to hiring 117 people with an average salary of $72,000, which according to Kathy Baylis, president of the Economic Development Corporation of Sarasota County, is nearly double the current average salary in Sarasota and Manatee counties.
"As an economic developer, you can imagine what I lie awake in bed and think about … that big one," she said. "The one that's going to be a game-changer. This is one of the biggest ones I've worked on in my career."
Sanborn promised, though, he would not be shipping in all of those employees from California.
"We want to mentor young people to work in production," he said. "These are jobs people will have for a long time."
That aspect of the project was encouraging to Dr. Larry Thompson, president of Ringling College of Art and Design.
"It's an amazing opportunity for our students," he said.
The school has digital-animation and digital-filmmaking programs, and Thompson said Sanborn Studios could keep his graduates in the area instead of fleeing to California to seek careers in film.
"I've been looking forward to having opportunities for our students and to now be able to interact with a real studio is amazing," Thompson said.
Sanborn Studios first production is a 22-episode television show called "Miami 24/7," which will begin shooting in November. Each episode will cost about $1 million to produce, Sanborn said.
There should be plenty of opportunities for Ranch residents to watch the filmmaking take place.
"Lakewood Ranch is essentially our back lot," Sanborn said. "It's not just a place to put (our buildings)."
The new studio head said Sarasota County has a big advantage over Miami and even Southern California because it's easy to get around and within a 30-mile radius there are a variety of backdrops, such as beaches and country towns.
The announcement was treated as a major event, with three of Sarasota County's state legislators, four Sarasota city commissioners and all five Sarasota County commissioners in attendance.
Said Baylis: "This truly is a dream come true for Sarasota County."
Contact Robin Roy at [email protected].
Longboat Key resident Sanborn vacationed in Anna Maria Island throughout his childhood, but it wasn't until around 1999 that he returned to the area. At the time, he owned an aerial-photography business (which became Gyrocam Systems in 2000), and he realized that to expand the company, he needed a facility with a private hangar. Because large Fortune 500 companies surrounded the airports in New Jersey, he thought that he would have a better shot at getting such a facility in Florida.
Sanborn met with officials in his native Lakeland, who weren't interested. Finally, he met with Nancy Engel, executive director of the Manatee County Economic Development Council, who helped him get permission to build his facility at the Sarasota-Bradenton International Airport.
SARASOTA — Introduced with a car chase, overturning police cruiser and exploding helicopter, the producers of the biggest suspense thriller in Sarasota County revealed their final act this week.


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Just turned the tv on... 30-7? WTF KNICKS?
Just turned the tv on... 30-7? WTF KNICKS?
I understand why Stat, Meta, Prigioni dont want to play for our Knicks organization .. WE R NBA JOKE
How can u be down by 20 points in the first at HOME .... 34 -11 guess who got 34
This is gonna end in a brawl. Mark my words.
Prigioni and Udrih need to play the point the rest of the game.
Sig made by Sons of Thunder
Get JR out the game
I like Stat, Meta, Shump, and Prig lineup .. they dont need JR or Melo as their 5th man .. Woodson always want these two idiots in the game
Our Starters are EMBARRASSING that includes 6th man JR .. those 6 players played all the first qtr. for 34-11 qtr.
Prigioni dont want to be on the court with 4 poor playing Starters
I don't get this: We are going onto a run without Melo and JR and once Boston took a timeout, Woodson put both of them back into the game, adding Bargs, going back to the line up which didn't work in the first place. Is he tanking? Somebody should tell him that we haven't got a draft pick.
Last edited by pat; Dec 08, 2013 at 13:07.
"Those who believe in telekinetics, raise my hand."
Kurt Vonnegut
Naaaah it's not the uniform nor the crowd….maybe just maybe it could be the team with the 5-14 record, those who are playing the game. How many yrs are we going to use the….."Matinee Trap Game' excuse.
I just woke up about 15min ago and didn't expect this, but when I saw it I wasn't surprised.
Last edited by Broadway; Dec 08, 2013 at 13:35.
Entire starting lineup is -23 on the floor or worse, with Melo -31 & Felton -32. Woody will continue to ride these guys until they're down 50.
JR, Felton, Shump and more with no pts and uts the 3rd qtr...come on, the Celts aren't this good


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Old Raliegh Pursuit racer, all original and complete. Bought with the intention of turning it into a fixie, but too many other interests have got in the way. Had very little use, most marks are from storage. No dents or cracks. Size 21"/53cm
£50 collected from Mold N. Wales (near Chester)


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Bella Donna: Cleopatra
I thought you might enjoy a trip back in time for this week's Bella Donna. So we're going waaaay back to look at Cleopatra, the queen of Egypt. It's hard to think of another woman whose beauty has left such a lasting impression. Her intense, kohl-lined eyes continue to inspire modern-day designers such as Alexander McQueen, who used a strong Egyptian eye on models for his fall 2007 collection. The jury's out on whether Cleopatra was physically attractive, but an officially commissioned portrait of her (it's the coin in the gallery here) suggests that she wasn't necessarily the Angelina Jolie of her day.
Cleopatra had a short life, but it was a tumultuous one. Trust me, even if you yawn when you think about history, you'll love this stuff—it is filled with all sorts of sordid drama and is too juicy to pass up. As a child, Cleopatra's older sister tried to poison her (but wound up being killed herself). Miss Cleo's reign didn't begin so well: At 18, she was queen... and was also married to her 12-year-old brother, Ptolemy III. Bossy Cleopatra wanted to rule alone, so she started to leave her brother out of official documents and events. Surprise! This didn't go over so well, and Cleopatra was forced into exile. But then her brother made the kind of mistake that only 15-year-old boys can make. To get the whole story,
So, some background: The Roman Empire was embroiled in a civil war at the time. Ptolemy, Cleopatra's pimply teenage brother, had taken over as the king of Egypt. Now, he owed some money to Rome, so to please Julius Caesar, he thought, "Hey, I know! I'll have Caesar's opponent Pompey beheaded, and then I'll hand him the head as a gift, and he'll totally think I'm an ally!" (Like I said, only a teenage boy would think of this.) So Ptolemy was all, "Look what I have for you, Caesar! Ta-da! A severed head!"
Bad idea. Caesar flipped out because Pompey was the widower of his late daughter. Personalities before politics, that sort of thing. So Caesar took over Egypt, and when Cleopatra heard about this, she decided that it was time to return. She's said to have returned to the palace rolled up inside a Persian carpet, only to tumble out of it in front of Caesar. He loved it, they became lovers, and Cleopatra gave birth to a son, Caesarion, nine months later. Apparently this warmed Caesar's heart, because he decided that instead of taking Egypt for Rome, he'd acknowledge Cleopatra as the rightful ruler.
Ah, but another love affair was on its way. After Caesar died, Mark Antony summoned Cleopatra to talk about where her loyalty lay. So she shows up in a grand, over-the-top way—think Pimp My Ride, but for ancient times—and impresses him. Cue the love affair. Cue the heirs. Cue yet another war, during which Antony mistakenly heard that Cleopatra had committed suicide. So then he committed suicide. A few days later, Cleopatra heard the news of his death, and then she really did commit suicide. See? Lots of drama.


text: A wet practice can offer some unusual images, even from the pitlane area. Here Kimi Raikkonen is in his Ferrai, holding nothing back as he heads for the first corner. I have reduced the shutter speed to increase the movement on the image but kept car in focus. Japanese Grand Prix, Suzuka, October 4, 2009


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For conference room reservations please call 212-869-1212 or email email@example.com
The hotel also offers a luxurious conference room for up to 12 people equipped with state-of-the-art technology for your meeting in NYC (a 50" computer monitor/TV screen).
Business travelers will have no problem unwinding, as the Casablanca Hotel is situated just steps from Times Square in the heart of New York City's theater district and in the middle of Manhattan's magical sightseeing attractions, boutiques, restaurants, NY events and nightlife.
The Casablanca Hotel Times Square is a long time favorite Manhattan hotel choice for business travelers in the Fashion and Entertainment industries who feel they have found a hidden gem, a small oasis of hospitality amidst the action and excitement of the fashion and theater districts.
Free high speed Internet access including wireless throughout the hotel
Computer station available free of charge in Rick's Cafe
Elegant state of the art boardroom for business meetings for a small hourly fee
Rick's Cafe is available 24 hours a day for small informal meetings with clients throughout the day
Complimentary selection of coffees, teas, cookies, and fruits throughout the day
Complimentary selection of newspapers
Access to fax and copier
Club member discount available to frequently returning travelers
Located on second floor. Rick's Cafe offers a complimentary computer station with printer and high speed Internet.
Only $100 per hour room rental plus taxes for up to 12 people maximum.Overnight guests of the Casablanca Hotel receive 25% discount.
Includes complimentary refreshments breaks in Rick's Café .
This price is available for meetings during the hours of 9AM-6PM only *Tax and gratuities are extra. A two hour minimum applies.
After 6PM or on weekends, the room rental is $150 per hour.
Enhance your meeting with added refreshments set up in the meeting room
Add Breakfast for $10 per person++
Add Lunch for $50 Per person++
Add Sodas for $5 Per person++
Add afternoon break for $10 Per person++
Add Wine and Cheese for $20 per person++
++ Taxes and service fees are additional
From our own beautiful Rick's Café, adjacent to the Conference room, enjoy pastries, cookies, coffee, cappuccino, espresso, tea, soda, juice and water for your breaks all through the day.
Rick's Café also offers a computer station for checking emails during breaks and there is an outdoor courtyard for those who would like to enjoy a bit of fresh air during the break.
Email email@example.com to check availability for your requested dates.


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ISS032-E-020576 (20 Aug. 2012) --- Russian cosmonaut Gennady Padalka, Expedition 32 commander, participates in a session of extravehicular activity (EVA) to continue outfitting the International Space Station. During the five-hour, 51-minute spacewalk, Padalka and Russian cosmonaut Yuri Malenchenko (out of frame), flight engineer, moved the Strela-2 cargo boom from the Pirs docking compartment to the Zarya module to prepare Pirs for its eventual replacement with a new Russian multipurpose laboratory module. The two spacewalking cosmonauts also installed micrometeoroid debris shields on the exterior of the Zvezda service module and deployed a small science satellite.