In analytic philosophy , anti-realism is the position that the truth of a statement rests on its demonstrability through internal logic mechanisms, such as the context principle or intuitionistic logic , in direct opposition to the realist notion that the truth of a statement rests on its correspondence to an external, independent reality. In anti-realism, this external reality is hypothetical and is not assumed.
140-424: There are many varieties of anti-realism, such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argument against a form of realism Dummett saw as 'colorless reductionism'. Anti-realism in its most general sense can be understood as being in contrast to a generic realism , which holds that distinctive objects of
280-425: A linguistic therapy. According to Ludwig Wittgenstein , for instance, philosophy aims at dispelling misunderstandings to which humans are susceptible due to the confusing structure of ordinary language . Phenomenologists , such as Edmund Husserl , characterize philosophy as a "rigorous science" investigating essences . They practice a radical suspension of theoretical assumptions about reality to get back to
420-587: 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
560-545: A doctor diagnoses a disease based on the observed symptoms. Logic also investigates incorrect forms of reasoning. They are called fallacies and are divided into formal and informal fallacies based on whether the source of the error lies only in the form of the argument or also in its content and context. Metaphysics is the study of the most general features of reality , such as existence, objects and their properties , wholes and their parts , space and time , events , and causation . There are disagreements about
700-497: 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
840-752: A given set of symbols in a theory can be mapped onto any number of sets of real-world objects—each set being a "model" of the theory—provided the relationship between the objects is the same (compare with symbol grounding .) In ancient Greek philosophy , nominalist (anti-realist) doctrines about universals were proposed by the Stoics , especially Chrysippus . In early modern philosophy , conceptualist anti-realist doctrines about universals were proposed by thinkers like René Descartes , John Locke , Baruch Spinoza , Gottfried Wilhelm Leibniz , George Berkeley , and David Hume . In late modern philosophy , anti-realist doctrines about knowledge were proposed by
980-444: A great variety of methods to arrive at philosophical knowledge. They include conceptual analysis , reliance on common sense and intuitions , use of thought experiments , analysis of ordinary language , description of experience , and critical questioning . Philosophy is related to many other fields, including the sciences, mathematics , business , law , and journalism . It provides an interdisciplinary perspective and studies
1120-442: A mathematician's knowledge of mathematics is one mathematical object making contact with another. The term " anti-realism " was introduced by Michael Dummett in his 1963 paper "Realism" in order to re-examine a number of classical philosophical disputes, involving such doctrines as nominalism , Platonic realism , idealism and phenomenalism . The novelty of Dummett's approach consisted in portraying these disputes as analogous to
1260-702: A metaphysical foundation of ethics. The modern period in Chinese philosophy began in the early 20th century and was shaped by the influence of and reactions to Western philosophy. The emergence of Chinese Marxism —which focused on class struggle , socialism , and communism —resulted in a significant transformation of the political landscape. Another development was the emergence of New Confucianism , which aims to modernize and rethink Confucian teachings to explore their compatibility with democratic ideals and modern science. Traditional Japanese philosophy assimilated and synthesized ideas from different traditions, including
1400-459: A moral claim may be either true or false. Specifically the moral anti-realist is committed to denying at least one of the following three statements: Different version of moral anti-realism deny different statements: specifically, non-cognitivism denies the first claim, arguing that moral statements have no meaning or truth content, error theory denies the second claim, arguing that all moral statements are false, and ethical subjectivism denies
1540-468: A noticeable growth in the number of female philosophers , but they still remained underrepresented. Arabic–Persian philosophy arose in the early 9th century CE as a response to discussions in the Islamic theological tradition . Its classical period lasted until the 12th century CE and was strongly influenced by ancient Greek philosophers. It employed their ideas to elaborate and interpret the teachings of
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#17327877408831680-469: 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
1820-460: A part of intellectual history , but it also investigates questions not covered by intellectual history such as whether the theories of past philosophers are true and have remained philosophically relevant. The history of philosophy is primarily concerned with theories based on rational inquiry and argumentation; some historians understand it in a looser sense that includes myths , religious teachings , and proverbial lore. Influential traditions in
1960-559: A realm outside of space-time"). Whilst our knowledge of physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. Field developed his views into fictionalism . Benacerraf also developed the philosophy of mathematical structuralism , according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism. Anti-realist arguments hinge on
2100-557: 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
2240-480: 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 ,
2380-426: 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
2520-709: A statement of the form "P or Q" as true only if we can prove P or if we can prove Q. In particular, we cannot in general claim that "P or not P" is true (the law of excluded middle ), since in some cases we may not be able to prove the statement "P" nor prove the statement "not P" . Similarly, intuitionists object to the existence property for classical logic, where one can prove ∃ x . ϕ ( x ) {\displaystyle \exists x.\phi (x)} , without being able to produce any term t {\displaystyle t} of which ϕ {\displaystyle \phi } holds. Dummett argues that this notion of truth lies at
2660-477: A subject-matter exist and have properties independent of one's beliefs and conceptual schemes. The ways in which anti-realism rejects these type of claims can vary dramatically. Because this encompasses statements containing abstract ideal objects (i.e. mathematical objects), anti-realism may apply to a wide range of philosophical topics, from material objects to the theoretical entities of science , mathematical statements , mental states, events and processes ,
2800-471: 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
2940-612: A variety of arguments. Deductive arguments are mainly studied by formal logic. An argument is deductively valid if the truth of its premises ensures the truth of its conclusion. Deductively valid arguments follow a rule of inference , like modus ponens , which has the following logical form : " p ; if p then q ; therefore q ". An example is the argument "today is Sunday; if today is Sunday then I don't have to go to work today; therefore I don't have to go to work today". The premises of non-deductive arguments also support their conclusion, although this support does not guarantee that
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#17327877408833080-405: A version of moral relativism, as different individuals would be required to follow different laws, but the moral facts are physical facts about the world, not mental facts, so they are metaphysically ordinary. Thus, different versions of moral relativism might be considered anti-realist or realist. Just as moral anti-realism asserts the nonexistence of normative facts, epistemic anti-realism asserts
3220-455: A way that is non causal, and not analogous to perception. This argument is developed by Jerrold Katz in his 2000 book Realistic Rationalism . In this book, he put forward a position called realistic rationalism , which combines metaphysical realism and rationalism . A more radical defense is to deny the separation of physical world and the platonic world, i.e. the mathematical universe hypothesis (a variety of mathematicism ). In that case,
3360-615: A wide area of philosophical subfields. A central topic in Arabic–Persian philosophy is the relation between reason and revelation . Indian philosophy combines the spiritual problem of how to reach enlightenment with the exploration of the nature of reality and the ways of arriving at knowledge. Chinese philosophy focuses principally on practical issues in relation to right social conduct, government, and self-cultivation . Major branches of philosophy are epistemology , ethics , logic , and metaphysics . Epistemology studies what knowledge
3500-494: Is a combination of different parts and attempts to identify what those parts are. An influential theory in this area claims that knowledge has three components: it is a belief that is justified and true . This theory is controversial and the difficulties associated with it are known as the Gettier problem . Alternative views state that knowledge requires additional components, like the absence of luck; different components, like
3640-510: Is a common issue in relation to the sources of knowledge and the justification they offer. It is based on the idea that beliefs require some kind of reason or evidence to be justified. The problem is that the source of justification may itself be in need of another source of justification. This leads to an infinite regress or circular reasoning . Foundationalists avoid this conclusion by arguing that some sources can provide justification without requiring justification themselves. Another solution
3780-609: 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
3920-541: Is a rational and critical inquiry that reflects on its own methods and assumptions. Historically, many of the individual sciences , such as physics and psychology , formed part of philosophy. However, they are considered separate academic disciplines in the modern sense of the term. Influential traditions in the history of philosophy include Western , Arabic–Persian , Indian , and Chinese philosophy . Western philosophy originated in Ancient Greece and covers
4060-561: Is a set of essential features shared by all parts of philosophy. Others see only weaker family resemblances or contend that it is merely an empty blanket term. Precise definitions are often only accepted by theorists belonging to a certain philosophical movement and are revisionistic according to Søren Overgaard et al. in that many presumed parts of philosophy would not deserve the title "philosophy" if they were true. Some definitions characterize philosophy in relation to its method, like pure reasoning. Others focus on its topic, for example, as
4200-467: Is also concerned with the moral evaluation of character traits and institutions. It explores what the standards of morality are and how to live a good life. Philosophical ethics addresses such basic questions as "Are moral obligations relative?"; "Which has priority: well-being or obligation?"; and "What gives life meaning?" The main branches of ethics are meta-ethics , normative ethics , and applied ethics . Meta-ethics asks abstract questions about
4340-431: Is an expression of the philosopher's love of wisdom and has the aim of improving one's well-being by leading a reflective life. For example, the Stoics saw philosophy as an exercise to train the mind and thereby achieve eudaimonia and flourish in life. As a discipline, the history of philosophy aims to provide a systematic and chronological exposition of philosophical concepts and doctrines. Some theorists see it as
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4480-666: Is and how to acquire it. Ethics investigates moral principles and what constitutes right conduct. Logic is the study of correct reasoning and explores how good arguments can be distinguished from bad ones. Metaphysics examines the most general features of reality , existence, objects , and properties . Other subfields are aesthetics , philosophy of language , philosophy of mind , philosophy of religion , philosophy of science , philosophy of mathematics , philosophy of history , and political philosophy . Within each branch, there are competing schools of philosophy that promote different principles, theories, or methods. Philosophers use
4620-451: Is closely related to the disciplines of logic and linguistics. The philosophy of language rose to particular prominence in the early 20th century in analytic philosophy due to the works of Frege and Russell. One of its central topics is to understand how sentences get their meaning. There are two broad theoretical camps: those emphasizing the formal truth conditions of sentences and those investigating circumstances that determine when it
4760-408: Is concerned with meaning , understanding , or the clarification of language. According to one view, philosophy is conceptual analysis , which involves finding the necessary and sufficient conditions for the application of concepts. Another definition characterizes philosophy as thinking about thinking to emphasize its self-critical, reflective nature. A further approach presents philosophy as
4900-528: 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
5040-460: Is itself a borrowing from the Latin philosophia . The term philosophy acquired the meanings of "advanced study of the speculative subjects ( logic , ethics , physics , and metaphysics )", "deep wisdom consisting of love of truth and virtuous living", "profound learning as transmitted by the ancient writers", and "the study of the fundamental nature of knowledge , reality , and existence , and
5180-472: 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
5320-400: 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
5460-475: Is presented by coherentists , who state that a belief is justified if it coheres with other beliefs of the person. Many discussions in epistemology touch on the topic of philosophical skepticism , which raises doubts about some or all claims to knowledge. These doubts are often based on the idea that knowledge requires absolute certainty and that humans are unable to acquire it. Ethics, also known as moral philosophy, studies what constitutes right conduct . It
5600-495: 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
5740-418: Is sometimes understood as a proper science in its own right. According to some naturalistic philosophers , such as W. V. O. Quine , philosophy is an empirical yet abstract science that is concerned with wide-ranging empirical patterns instead of particular observations. Science-based definitions usually face the problem of explaining why philosophy in its long history has not progressed to the same extent or in
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5880-477: 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,
6020-460: Is suitable to use a sentence, the latter of which is associated with speech act theory . Mathematical empiricism 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
6160-439: Is that actions are in tune with those duties and not what consequences they have. Virtue theorists judge actions based on how the moral character of the agent is expressed. According to this view, actions should conform to what an ideally virtuous agent would do by manifesting virtues like generosity and honesty . Logic is the study of correct reasoning . It aims to understand how to distinguish good from bad arguments . It
6300-468: Is the 1687 book Philosophiæ Naturalis Principia Mathematica by Isaac Newton . This book referred to natural philosophy in its title, but it is today considered a book of physics. The meaning of philosophy changed toward the end of the modern period when it acquired the more narrow meaning common today. In this new sense, the term is mainly associated with philosophical disciplines like metaphysics, epistemology, and ethics. Among other topics, it covers
6440-431: 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
6580-462: Is truth established?"; and "Can we prove causal relations?" Epistemology is primarily interested in declarative knowledge or knowledge of facts, like knowing that Princess Diana died in 1997. But it also investigates practical knowledge , such as knowing how to ride a bicycle, and knowledge by acquaintance , for example, knowing a celebrity personally. One area in epistemology is the analysis of knowledge . It assumes that declarative knowledge
6720-456: Is usually divided into formal and informal logic . Formal logic uses artificial languages with a precise symbolic representation to investigate arguments. In its search for exact criteria, it examines the structure of arguments to determine whether they are correct or incorrect. Informal logic uses non-formal criteria and standards to assess the correctness of arguments. It relies on additional factors such as content and context. Logic examines
6860-434: 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
7000-556: Is whether beauty is an objective feature of entities or a subjective aspect of experience. Aesthetic philosophers also investigate the nature of aesthetic experiences and judgments . Further topics include the essence of works of art and the processes involved in creating them. The philosophy of language studies the nature and function of language . It examines the concepts of meaning , reference , and truth. It aims to answer questions such as how words are related to things and how language affects human thought and understanding. It
7140-460: The German idealist Georg Wilhelm Friedrich Hegel . Hegel was a proponent of what is now called inferentialism : he believed that the ground for the axioms and the foundation for the validity of the inferences are the right consequences and that the axioms do not explain the consequence. Kant and Hegel held conceptualist views about universals. In contemporary philosophy , anti-realism was revived in
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#17327877408837280-556: 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
7420-527: The Quran . Al-Kindi (801–873 CE) is usually regarded as the first philosopher of this tradition. He translated and interpreted many works of Aristotle and Neoplatonists in his attempt to show that there is a harmony between reason and faith . Avicenna (980–1037 CE) also followed this goal and developed a comprehensive philosophical system to provide a rational understanding of reality encompassing science, religion, and mysticism. Al-Ghazali (1058–1111 CE)
7560-475: 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
7700-407: 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
7840-611: The past and the future . One kind of metaphysical anti-realism maintains a skepticism about the physical world, arguing either: 1) that nothing exists outside the mind, or 2) that we would have no access to a mind-independent reality, even if it exists. The latter case often takes the form of a denial of the idea that we can have 'unconceptualised' experiences (see Myth of the Given ). Conversely, most realists (specifically, indirect realists ) hold that perceptions or sense data are caused by mind-independent objects. But this introduces
7980-460: 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
8120-420: The "things themselves", that is, as originally given in experience. They contend that this base-level of experience provides the foundation for higher-order theoretical knowledge, and that one needs to understand the former to understand the latter. An early approach found in ancient Greek and Roman philosophy is that philosophy is the spiritual practice of developing one's rational capacities. This practice
8260-441: 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
8400-412: The 5th century CE. Its focus was on religious topics and many thinkers used ancient philosophy to explain and further elaborate Christian doctrines . The Renaissance period started in the 14th century and saw a renewed interest in schools of ancient philosophy, in particular Platonism . Humanism also emerged in this period. The modern period started in the 17th century. One of its central concerns
8540-434: 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
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#17327877408838680-587: 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
8820-407: 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"
8960-424: 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
9100-489: 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
9240-442: The basic limits of human understanding". Before the modern age, the term philosophy was used in a wide sense. It included most forms of rational inquiry, such as the individual sciences , as its subdisciplines. For instance, natural philosophy was a major branch of philosophy. This branch of philosophy encompassed a wide range of fields, including disciplines like physics, chemistry , and biology . An example of this usage
9380-441: The beginning of the 20th century. It discusses topics such as ethnophilosophy , négritude , pan-Africanism , Marxism, postcolonialism , the role of cultural identity, relativism , epistemological dualism , and the critique of Eurocentrism . Philosophical questions can be grouped into several branches. These groupings allow philosophers to focus on a set of similar topics and interact with other thinkers who are interested in
9520-480: The bottom of various classical forms of anti-realism, and uses it to re-interpret phenomenalism , claiming that it need not take the form of reductionism . Dummett's writings on anti-realism draw heavily on the later writings of Ludwig Wittgenstein , concerning meaning and rule following, and can be seen as an attempt to integrate central ideas from the Philosophical Investigations into
9660-560: The color red, can exist at different locations at the same time. This is not the case for particulars including individual persons or specific objects. Other metaphysical questions are whether the past fully determines the present and what implications this would have for the existence of free will . There are many other subfields of philosophy besides its core branches. Some of the most prominent are aesthetics, philosophy of language, philosophy of mind, philosophy of religion, philosophy of science, and political philosophy. Aesthetics in
9800-481: 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"
9940-403: 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
10080-461: The conclusion is true. One form is inductive reasoning . It starts from a set of individual cases and uses generalization to arrive at a universal law governing all cases. An example is the inference that "all ravens are black" based on observations of many individual black ravens. Another form is abductive reasoning . It starts from an observation and concludes that the best explanation of this observation must be true. This happens, for example, when
10220-614: The consequences of the general theories developed by normative ethics in specific situations, for example, in the workplace or for medical treatments. Within contemporary normative ethics, consequentialism, deontology , and virtue ethics are influential schools of thought. Consequentialists judge actions based on their consequences. One such view is utilitarianism , which argues that actions should increase overall happiness while minimizing suffering. Deontologists judge actions based on whether they follow moral duties, such as abstaining from lying or killing. According to them, what matters
10360-443: The constructive tradition of analytic philosophy deriving from Gottlob Frege . In philosophy of science , anti-realism applies chiefly to claims about the non-reality of "unobservable" entities such as electrons or genes , which are not detectable with human senses. One prominent variety of scientific anti-realism is instrumentalism , which takes a purely agnostic view towards the existence of unobservable entities, in which
10500-409: 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
10640-399: The dispute between intuitionism and Platonism in the philosophy of mathematics . According to intuitionists (anti-realists with respect to mathematical objects), the truth of a mathematical statement consists in our ability to prove it. According to Platonic realists, the truth of a statement is proven in its correspondence to objective reality. Thus, intuitionists are ready to accept
10780-400: The emergence and application of formal logic , the focus on the role of language as well as pragmatism , and movements in continental philosophy like phenomenology, existentialism , and post-structuralism . The 20th century saw a rapid expansion of academic philosophy in terms of the number of philosophical publications and philosophers working at academic institutions . There was also
10920-482: The emergence of non-Vedic teachings, like Buddhism and Jainism . Buddhism was founded by Gautama Siddhartha (563–483 BCE), who challenged the Vedic idea of a permanent self and proposed a path to liberate oneself from suffering . Jainism was founded by Mahavira (599–527 BCE), who emphasized non-violence as well as respect toward all forms of life. The subsequent classical period started roughly 200 BCE and
11060-407: 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
11200-407: The features that all entities have in common. Specific metaphysics is interested in different kinds of being, the features they have, and how they differ from one another. An important area in metaphysics is ontology . Some theorists identify it with general metaphysics. Ontology investigates concepts like being , becoming , and reality. It studies the categories of being and asks what exists on
11340-420: 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
11480-413: The form of empirio-criticism , logical positivism , semantic anti-realism and scientific instrumentalism (see below). In the philosophy of mathematics , realism is the claim that mathematical entities such as 'number' have an observer-independent existence. Empiricism , which associates numbers with concrete physical objects, and Platonism , in which numbers are abstract, non-physical entities, are
11620-476: The history of philosophy include Western , Arabic–Persian , Indian , and Chinese philosophy . Other philosophical traditions are Japanese philosophy , Latin American philosophy , and African philosophy . Western philosophy originated in Ancient Greece in the 6th century BCE with the pre-Socratics . They attempted to provide rational explanations of the cosmos as a whole. The philosophy following them
11760-488: The idea that a satisfactory, naturalistic account of thought processes can be given for mathematical reasoning. One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm , as in the argument given by Sir Roger Penrose . Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in
11900-420: The importance of a strong state and strict laws. Buddhism was introduced to China in the 1st century CE and diversified into new forms of Buddhism . Starting in the 3rd century CE, the school of Xuanxue emerged. It interpreted earlier Daoist works with a specific emphasis on metaphysical explanations. Neo-Confucianism developed in the 11th century CE. It systematized previous Confucian teachings and sought
12040-467: 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
12180-419: The indigenous Shinto religion and Chinese and Indian thought in the forms of Confucianism and Buddhism, both of which entered Japan in the 6th and 7th centuries. Its practice is characterized by active interaction with reality rather than disengaged examination. Neo-Confucianism became an influential school of thought in the 16th century and the following Edo period and prompted a greater focus on language and
12320-483: The interconnectedness of all things. Latin American philosophy during the colonial period , starting around 1550, was dominated by religious philosophy in the form of scholasticism . Influential topics in the post-colonial period were positivism , the philosophy of liberation , and the exploration of identity and culture. Early African philosophy, like Ubuntu philosophy , was focused on community, morality, and ancestral ideas. Systematic African philosophy emerged at
12460-499: 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
12600-551: The manifestation of cognitive virtues instead of justification; or they deny that knowledge can be analyzed in terms of other phenomena. Another area in epistemology asks how people acquire knowledge. Often-discussed sources of knowledge are perception , introspection , memory , inference , and testimony . According to empiricists , all knowledge is based on some form of experience. Rationalists reject this view and hold that some forms of knowledge, like innate knowledge , are not acquired through experience. The regress problem
12740-527: 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
12880-472: 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
13020-424: 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
13160-474: 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
13300-596: The most fundamental level. Another subfield of metaphysics is philosophical cosmology . It is interested in the essence of the world as a whole. It asks questions including whether the universe has a beginning and an end and whether it was created by something else. A key topic in metaphysics concerns the question of whether reality only consists of physical things like matter and energy. Alternative suggestions are that mental entities (such as souls and experiences ) and abstract entities (such as numbers) exist apart from physical things. Another topic in metaphysics concerns
13440-404: The most influential philosophers of the subsequent period. The increasing influence of Western thought and institutions in the 19th and 20th centuries gave rise to the intellectual movement of Islamic modernism , which aims to understand the relation between traditional Islamic beliefs and modernity. One of the distinguishing features of Indian philosophy is that it integrates the exploration of
13580-472: The natural and social sciences, religion, and mathematics. Epistemology is the branch of philosophy that studies knowledge. It is also known as theory of knowledge and aims to understand what knowledge is, how it arises, what its limits are, and what value it has. It further examines the nature of truth , belief , justification , and rationality . Some of the questions addressed by epistemologists include "By what method(s) can one acquire knowledge?"; "How
13720-498: 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
13860-504: The natural world. The Kyoto School emerged in the 20th century and integrated Eastern spirituality with Western philosophy in its exploration of concepts like absolute nothingness ( zettai-mu ), place ( basho ), and the self . Latin American philosophy in the pre-colonial period was practiced by indigenous civilizations and explored questions concerning the nature of reality and the role of humans. It has similarities to indigenous North American philosophy , which covered themes such as
14000-455: The nature and sources of morality. It analyzes the meaning of ethical concepts, like right action and obligation . It also investigates whether ethical theories can be true in an absolute sense and how to acquire knowledge of them. Normative ethics encompasses general theories of how to distinguish between right and wrong conduct. It helps guide moral decisions by examining what moral obligations and rights people have. Applied ethics studies
14140-471: The nature of reality, the ways of arriving at knowledge, and the spiritual question of how to reach enlightenment . It started around 900 BCE when the Vedas were written. They are the foundational scriptures of Hinduism and contemplate issues concerning the relation between the self and ultimate reality as well as the question of how souls are reborn based on their past actions . This period also saw
14280-576: The nonexistence of facts in the domain of epistemology . Thus, the two are now sometimes grouped together as "metanormative anti-realism". Prominent defenders of epistemic anti-realism include Hartry Field , Simon Blackburn , Matthew Chrisman, and Allan Gibbard , among others. Philosopher Philosophy ('love of wisdom' in Ancient Greek ) is a systematic study of general and fundamental questions concerning topics like existence , reason , knowledge , value , mind , and language . It
14420-536: 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
14560-411: 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
14700-436: The philosophical sense is the field that studies the nature and appreciation of beauty and other aesthetic properties, like the sublime . Although it is often treated together with the philosophy of art , aesthetics is a broader category that encompasses other aspects of experience, such as natural beauty. In a more general sense, aesthetics is "critical reflection on art, culture, and nature ". A key question in aesthetics
14840-428: 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
14980-406: 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
15120-459: 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
15260-421: The possibility of another kind of skepticism: since our understanding of causality is that the same effect can be produced by multiple causes, there is a lack of determinacy about what one is really perceiving, as in the brain in a vat scenario. The main alternative to this sort of metaphysical anti-realism is metaphysical realism . On a more abstract level, model-theoretic anti-realist arguments hold that
15400-417: The precise definition of the term and its meaning has changed throughout the ages. Metaphysicians attempt to answer basic questions including " Why is there something rather than nothing? "; "Of what does reality ultimately consist?"; and "Are humans free?" Metaphysics is sometimes divided into general metaphysics and specific or special metaphysics. General metaphysics investigates being as such. It examines
15540-416: The preeminent forms of mathematical realism. The " epistemic argument" against Platonism has been made by Paul Benacerraf and Hartry Field . Platonism posits that mathematical objects are abstract entities. By general agreement, abstract entities cannot interact causally with physical entities ("the truth-values of our mathematical assertions depend on facts involving platonic entities that reside in
15680-400: The problem of identity . One question is how much an entity can change while still remaining the same entity. According to one view, entities have essential and accidental features . They can change their accidental features but they cease to be the same entity if they lose an essential feature. A central distinction in metaphysics is between particulars and universals . Universals, like
15820-428: The provocative, vexing, and enduring problems central to the human condition. The philosophical pursuit of wisdom involves asking general and fundamental questions. It often does not result in straightforward answers but may help a person to better understand the topic, examine their life, dispel confusion, and overcome prejudices and self-deceptive ideas associated with common sense. For example, Socrates stated that "
15960-436: The rational study of reality, knowledge, and values. It is distinguished from other disciplines of rational inquiry such as the empirical sciences and mathematics . The practice of philosophy is characterized by several general features: it is a form of rational inquiry, it aims to be systematic, and it tends to critically reflect on its own methods and presuppositions. It requires attentively thinking long and carefully about
16100-585: 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
16240-460: 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
16380-488: The same questions. Epistemology, ethics, logic, and metaphysics are sometimes listed as the main branches. There are many other subfields besides them and the different divisions are neither exhaustive nor mutually exclusive. For example, political philosophy, ethics, and aesthetics are sometimes linked under the general heading of value theory as they investigate normative or evaluative aspects. Furthermore, philosophical inquiry sometimes overlaps with other disciplines in
16520-410: The same way as the sciences. This problem is avoided by seeing philosophy as an immature or provisional science whose subdisciplines cease to be philosophy once they have fully developed. In this sense, philosophy is sometimes described as "the midwife of the sciences". Other definitions focus on the contrast between science and philosophy. A common theme among many such conceptions is that philosophy
16660-563: The scope and fundamental concepts of these fields. It also investigates their methods and ethical implications. The word philosophy comes from the Ancient Greek words φίλος ( philos ) ' love ' and σοφία ( sophia ) ' wisdom ' . Some sources say that the term was coined by the pre-Socratic philosopher Pythagoras , but this is not certain. The word entered the English language primarily from Old French and Anglo-Norman starting around 1175 CE. The French philosophie
16800-440: 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
16940-551: 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
17080-598: The study of the biggest patterns of the world as a whole or as the attempt to answer the big questions. Such an approach is pursued by Immanuel Kant , who holds that the task of philosophy is united by four questions: "What can I know?"; "What should I do?"; "What may I hope?"; and "What is the human being?" Both approaches have the problem that they are usually either too wide, by including non-philosophical disciplines, or too narrow, by excluding some philosophical sub-disciplines. Many definitions of philosophy emphasize its intimate relation to science. In this sense, philosophy
17220-455: 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
17360-483: 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
17500-469: The teachings of Advaita Vedanta to argue that all the different religions are valid paths toward the one divine. Chinese philosophy is particularly interested in practical questions associated with right social conduct, government, and self-cultivation . Many schools of thought emerged in the 6th century BCE in competing attempts to resolve the political turbulence of that period. The most prominent among them were Confucianism and Daoism . Confucianism
17640-410: The third claim, arguing that the truth of moral statements is mind dependent. Examples of anti-realist moral theories might be: There is a debate as to whether moral relativism is actually an anti-realist position. While many versions deny the metaphysical thesis, some do not, as one could imagine a system of morality which requires you to obey the written laws in your country. Such a system would be
17780-842: 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
17920-601: The unexamined life is not worth living " to highlight the role of philosophical inquiry in understanding one's own existence. And according to Bertrand Russell , "the man who has no tincture of philosophy goes through life imprisoned in the prejudices derived from common sense, from the habitual beliefs of his age or his nation, and from convictions which have grown up in his mind without the cooperation or consent of his deliberate reason." Attempts to provide more precise definitions of philosophy are controversial and are studied in metaphilosophy . Some approaches argue that there
18060-433: The unobservable entity X serves as an instrument to aid in the success of theory Y and does not require proof for the existence or non-existence of X. In the philosophy of ethics , moral anti-realism (or moral irrealism) is a meta-ethical doctrine that there are no objective moral values or normative facts. It is usually defined in opposition to moral realism , which holds that there are objective moral values, such that
18200-472: Was a strong critic of the idea that reason can arrive at a true understanding of reality and God. He formulated a detailed critique of philosophy and tried to assign philosophy a more limited place besides the teachings of the Quran and mystical insight. Following Al-Ghazali and the end of the classical period, the influence of philosophical inquiry waned. Mulla Sadra (1571–1636 CE) is often regarded as one of
18340-437: Was characterized by the emergence of the six orthodox schools of Hinduism : Nyāyá , Vaiśeṣika , Sāṃkhya , Yoga , Mīmāṃsā , and Vedanta . The school of Advaita Vedanta developed later in this period. It was systematized by Adi Shankara ( c. 700 –750 CE), who held that everything is one and that the impression of a universe consisting of many distinct entities is an illusion . A slightly different perspective
18480-488: 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
18620-660: Was defended by Ramanuja (1017–1137 CE), who founded the school of Vishishtadvaita Vedanta and argued that individual entities are real as aspects or parts of the underlying unity. He also helped to popularize the Bhakti movement , which taught devotion toward the divine as a spiritual path and lasted until the 17th to 18th centuries CE. The modern period began roughly 1800 CE and was shaped by encounters with Western thought. Philosophers tried to formulate comprehensive systems to harmonize diverse philosophical and religious teachings. For example, Swami Vivekananda (1863–1902 CE) used
18760-511: Was founded by Confucius (551–479 BCE). It focused on different forms of moral virtues and explored how they lead to harmony in society. Daoism was founded by Laozi (6th century BCE) and examined how humans can live in harmony with nature by following the Dao or the natural order of the universe. Other influential early schools of thought were Mohism , which developed an early form of altruistic consequentialism , and Legalism , which emphasized
18900-512: Was how philosophical and scientific knowledge are created. Specific importance was given to the role of reason and sensory experience . Many of these innovations were used in the Enlightenment movement to challenge traditional authorities. Several attempts to develop comprehensive systems of philosophy were made in the 19th century, for instance, by German idealism and Marxism . Influential developments in 20th-century philosophy were
19040-647: 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
19180-402: 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
19320-458: 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
19460-449: Was shaped by Socrates (469–399 BCE), Plato (427–347 BCE), and Aristotle (384–322 BCE). They expanded the range of topics to questions like how people should act , how to arrive at knowledge , and what the nature of reality and mind is. The later part of the ancient period was marked by the emergence of philosophical movements, for example, Epicureanism , Stoicism , Skepticism , and Neoplatonism . The medieval period started in
19600-467: 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
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