Misplaced Pages

#424575

69-573: Brouwer also became a major figure in the philosophy of intuitionism , a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth . This position led to the Brouwer–Hilbert controversy , in which Brouwer sparred with his formalist colleague David Hilbert . Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl . In addition to his mathematical work, Brouwer also published

138-583: A "World of Ideas" (Greek: eidos (εἶδος)) described in Plato's allegory of the cave : the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers . A major question considered in mathematical Platonism is: Precisely where and how do

207-463: A conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive

276-497: A few years later by specific experiments. The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention. Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize

345-465: A pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportion to the arbitrary first "number" or "one". These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of

414-471: A regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid). He was combative as a young man. According to Mark van Atten, this pugnacity reflected his combination of independence, brilliance, high moral standards and extreme sensitivity to issues of justice. He

483-557: A role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy . Western philosophies of mathematics go as far back as Pythagoras , who described the theory "everything is mathematics" ( mathematicism ), Plato , who paraphrased Pythagoras, and studied the ontological status of mathematical objects, and Aristotle , who studied logic and issues related to infinity (actual versus potential). Greek philosophy on mathematics

552-492: A self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II "as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics" (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908: "After completing his dissertation, Brouwer made

621-480: A single consistent set of axioms. Set-theoretic realism (also set-theoretic Platonism ) a position defended by Penelope Maddy , is the view that set theory is about a single universe of sets. This position (which is also known as naturalized Platonism because it is a naturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of Paul Benacerraf 's epistemological problem . A similar view, termed Platonized naturalism ,

690-426: A special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant 's idea that mathematics is synthetic a priori .) Davis and Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend

759-471: A system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G , the extension of F equals the extension of G if and only if for all objects a , Fa equals Ga ), a principle that he took to be acceptable as part of logic. Frege's construction was flawed. Bertrand Russell discovered that Basic Law V is inconsistent (this is Russell's paradox ). Frege abandoned his logicist program soon after this, but it

SECTION 10

#1732884123425

828-424: Is Yoshikazu Giga ( University of Tokyo ). Volumes 1–80 (1869–1919) were published by Teubner . Since 1920 ( vol. 81), the journal has been published by Springer . In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy . Between 1945 and 1947,

897-406: Is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann . Subsequent managing editors were Felix Klein , David Hilbert , Otto Blumenthal , Erich Hecke , Heinrich Behnke , Hans Grauert , Heinz Bauer , Herbert Amann , Jean-Pierre Bourguignon , Wolfgang Lück , Nigel Hitchin , and Thomas Schick . Currently, the managing editor of Mathematische Annalen

966-590: Is a philosophy of the foundations of mathematics . It is sometimes (simplistically) characterized by saying that its adherents do not admit the law of excluded middle as a general axiom in mathematical reasoning, although it may be proven as a theorem in some special cases. Brouwer was a member of the Significs Group . It formed part of the early history of semiotics —the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably cannot be completely disentangled from

1035-609: Is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the foundations of mathematics program. At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology . Three schools, formalism , intuitionism , and logicism , emerged at this time, partly in response to

1104-528: Is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical. Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that

1173-472: Is much higher than elsewhere. For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. Circa the end of the 19th century, several paradoxes made questionable the logical foundation of mathematics, and consequently the validity of the whole mathematics. This has been called the foundational crisis of mathematics . Some of these paradoxes consist of results that seem to contradict

1242-400: Is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm . Where a special concept of rigor comes into play is in the socialized aspects of a proof. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered as trivial , easy , or straightforward , and therefore left to

1311-495: Is self contradictory. Several methods have been proposed to solve the problem by changing of logical framework, such as constructive mathematics and intuitionistic logic . Roughly speaking, the first one consists of requiring that every existence theorem must provide an explicit example, and the second one excludes from mathematical reasoning the law of excluded middle and double negation elimination . The problems of foundation of mathematics has been eventually resolved with

1380-477: Is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable , which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample . Similarly as in science, theories and results (theorems) are often obtained from experimentation . In mathematics,

1449-409: Is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view. Mathematische Annalen Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal. )

SECTION 20

#1732884123425

1518-431: Is thus analytic , not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths . Rudolf Carnap (1931) presents the logicist thesis in two parts: Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik ( Basic Laws of Arithmetic ) he built up arithmetic from

1587-434: Is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon Ω − . {\displaystyle \Omega ^{-}.} In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle , and the search for these particles. In both cases, these particles were discovered

1656-544: The Pythagorean theorem holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all. Another version of formalism is known as deductivism . In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one, if it follows deductively from

1725-475: The ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds . At this time, these concepts seemed totally disconnected from

1794-407: The consistency of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert to call such study metamathematics or proof theory . At the middle of the century, a new mathematical theory was created by Samuel Eilenberg and Saunders Mac Lane , known as category theory , and it became a new contender for

1863-456: The " axiom of reducibility ". Even Russell said that this axiom did not really belong to logic. Modern logicists (like Bob Hale , Crispin Wright , and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as Hume's principle (the number of objects falling under the concept F equals the number of objects falling under

1932-441: The 20th century led to new questions concerning what was traditionally called the foundations of mathematics . As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid around 300 BCE as the natural basis for mathematics. Notions of axiom , proposition and proof , as well as

2001-477: The Brouwer fixed point theorem. It is a corollary to the second, concerning the topological invariance of degree, which is the best known among algebraic topologists. The third theorem is perhaps the hardest. Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology , which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes , of

2070-434: The accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein 's general relativity , which replaced Newton's law of gravitation as a better mathematical model. There

2139-587: The actual infinitesimal —but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need. Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on

L. E. J. Brouwer - Misplaced Pages Continue

2208-407: The application of an inference rule. The Zermelo–Fraenkel set theory with the axiom of choice , generally called ZFC , is such a theory in which all mathematics have been restated; it is used implicitely in all mathematics texts that do not specify explicitly on which foundations they are based. Moreover, the other proposed foundations can be modeled and studied inside ZFC. It results that "rigor"

2277-424: The appropriate axioms. The same is held to be true for all other mathematical statements. Formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism .) But it does allow the working mathematician to continue in his or her work and leave such problems to

2346-489: The assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers , chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were seriously undermined by the second of Gödel's incompleteness theorems , which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain

2415-475: The common intuition, such as the possibility to construct valid non-Euclidean geometries in which the parallel postulate is wrong, the Weierstrass function that is continuous but nowhere differentiable , and the study by Georg Cantor of infinite sets , which led to consider several sizes of infinity (infinite cardinals ). Even more striking, Russell's paradox shows that the phrase "the set of all sets"

2484-403: The concept G if and only if the extension of F and the extension of G can be put into one-to-one correspondence ). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3

2553-409: The definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of syllogisms or inference rules , without any use of empirical evidence and intuition . The rules of rigorous reasoning have been established by the ancient Greek philosophers under the name of logic . Logic is not specific to mathematics, but, in mathematics, the standard of rigor

2622-437: The development of intuitionism at its source was taken up by his student Arend Heyting . Dutch mathematician and historian of mathematics Bartel Leendert van der Waerden attended lectures given by Brouwer in later years, and commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on — and only on — the foundations of his intuitionism. It seemed that he

2691-407: The experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from

2760-420: The finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than

2829-467: The increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge. Surprising and counter-intuitive developments in formal logic and set theory early in

L. E. J. Brouwer - Misplaced Pages Continue

2898-503: The intellectual milieu of that group. In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism , which has been described by the mathematician Martin Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Brouwer then "embarked on

2967-499: The investigation of formal axiom systems . Mathematical logicians study formal systems but are just as often realists as they are formalists. Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories, etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary. The main critique of formalism

3036-519: The mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble , a theory that postulates that all structures that exist mathematically also exist physically in their own universe. Kurt Gödel 's Platonism postulates

3105-469: The mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem . A second historical example is the theory of ellipses . They were studied by

3174-424: The minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ... Mathematical reasoning requires rigor . This means that

3243-474: The modern notion of science by not relying on empirical evidence. The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner . It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when

3312-498: The natural language of mathematical thinking. As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at the century's beginning. Hilary Putnam summed up one common view of the situation in the last third of the century by saying: When philosophy discovers something wrong with science, sometimes science has to be changed— Russell's paradox comes to mind, as does Berkeley 's attack on

3381-536: The notion of a proposition being true of a mathematical object (see Assignment ), were formalized, allowing them to be treated mathematically. The Zermelo–Fraenkel axioms for set theory were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics, as in physics, new and unexpected ideas had arisen and significant changes were coming. With Gödel numbering , propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into

3450-411: The philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics. A major early proponent of formalism was David Hilbert , whose program was intended to be a complete and consistent axiomatization of all of mathematics. Hilbert aimed to show the consistency of mathematical systems from

3519-428: The philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th-century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic , set theory (both naive set theory and axiomatic set theory ), and foundational issues. It

SECTION 50

#1732884123425

3588-406: The physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. A striking aspect of the interaction between mathematics and physics

3657-459: The position carefully, they may retreat to formalism . Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of the excluded middle , and the axiom of choice ). It holds that all mathematical entities exist. They may be provable, even if they cannot all be derived from

3726-585: The reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years. Also, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof. Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that

3795-460: The rise of mathematical logic as a new area of mathematics. In this framework, a mathematical or logical theory consists of a formal language that defines the well-formed of assertions , a set of basic assertions called axioms and a set of inference rules that allow producing new assertions from one or several known assertions. A theorem of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by

3864-440: The sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world". Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. Logicists hold that mathematics can be known a priori , but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and

3933-434: The short philosophical tract Life, Art, and Mysticism (1905). Brouwer was born to Dutch Protestant parents. Early in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were his fixed point theorem , the topological invariance of degree, and the topological invariance of dimension . Among mathematicians generally, the best known is the first one, usually referred to now as

4002-551: The square root of two. Hippasus , a disciple of Pythagoras , showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. Simon Stevin

4071-450: The subject. The schools are addressed separately in the next section, and their assumptions explained. The view that claims that mathematics is the aesthetic combination of assumptions, and then also claims that mathematics is an art . A famous mathematician who claims that is the British G. H. Hardy . For Hardy, in his book, A Mathematician's Apology , the definition of mathematics

4140-483: The system to be proven consistent. Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation. Other formalists, such as Rudolf Carnap , Alfred Tarski , and Haskell Curry , considered mathematics to be

4209-842: The time of Pythagoras . The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism . Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects (see Mathematical object ). Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy , Charles Hermite , Henri Poincaré and Albert Einstein that support his views. Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in

SECTION 60

#1732884123425

4278-794: The treatment of general continuous mappings. In 1912, at age 31, he was elected a member of the Royal Netherlands Academy of Arts and Sciences . He was an Invited Speaker of the ICM in 1908 at Rome and in 1912 at Cambridge, UK. He was elected to the American Philosophical Society in 1943. Brouwer founded intuitionism , a philosophy of mathematics that challenged the then-prevailing formalism of David Hilbert and his collaborators, who included Paul Bernays , Wilhelm Ackermann , and John von Neumann (cf. Kleene (1952), p. 46–59). A variety of constructive mathematics , intuitionism

4347-488: Was continued by Russell and Whitehead . They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop much of mathematics, such as

4416-439: Was involved in a very public and eventually demeaning controversy with Hilbert in the late 1920s over editorial policy at Mathematische Annalen , at the time a leading journal. According to Abraham Fraenkel , Brouwer espoused Germanic Aryanness and Hilbert removed him from the editorial board of Mathematische Annalen after Brouwer objected to contributions from Ostjuden . In later years Brouwer became relatively isolated;

4485-647: Was later defended by the Stanford–Edmonton School : according to this view, a more traditional kind of Platonism is consistent with naturalism ; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence of abstract objects . Max Tegmark 's mathematical universe hypothesis (or mathematicism ) goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. Tegmark's sole postulate is: All structures that exist mathematically also exist physically . That is, in

4554-402: Was more like the aesthetic combination of concepts. Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato 's Theory of Forms and

4623-640: Was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy." About his last years, Davis (2002) remarks: Philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities. Major themes that are dealt with in philosophy of mathematics include: The connection between mathematics and material reality has led to philosophical debates since at least

4692-454: Was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with Leibniz , the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of Frege and of Russell , but was brought into question by developments in the late 19th and early 20th centuries. A perennial issue in

4761-463: Was strongly influenced by their study of geometry . For example, at one time, the Greeks held the opinion that 1 (one) was not a number , but rather a unit of arbitrary length. A number was defined as a multitude. Therefore, 3, for example, represented a certain multitude of units, and was thus "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of

#424575