In logic and semantics , the term statement is variously understood to mean either:
118-478: An axiom , postulate , or assumption is a statement that is taken to be true , to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα ( axíōma ), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy , an axiom
236-457: A deductive system . This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms. Statement (logic) In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing
354-444: A first-order language . For each variable x {\displaystyle x} , the below formula is universally valid. x = x {\displaystyle x=x} This means that, for any variable symbol x {\displaystyle x} , the formula x = x {\displaystyle x=x} can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and
472-429: A metaproof . These examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have Existential Generalization : Axiom scheme for Existential Generalization. Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} ,
590-527: A will in which he asked to be buried next to his wife. Aristotle left his works to Theophrastus, his successor as the head of the Lyceum, who in turn passed them down to Neleus of Scepsis in Asia Minor. There, the papers remained hidden for protection until they were purchased by the collector Apellicon . In the meantime, many copies of Aristotle's major works had already begun to circulate and be used in
708-539: A "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry ). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that it
826-541: A "survival of the fittest" origin of living things and their organs, and ridiculed the idea that accidents could lead to orderly results. To put his views into modern terms, he nowhere says that different species can have a common ancestor , or that one kind can change into another , or that kinds can become extinct . Aristotle did not do experiments in the modern sense. He used the ancient Greek term pepeiramenoi to mean observations, or at most investigative procedures like dissection. In Generation of Animals , he finds
944-565: A branch of logic . Frege , Russell , Poincaré , Hilbert , and Gödel are some of the key figures in this development. Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions. In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent ; it should be impossible to derive
1062-547: A building known as the Lyceum (named after the sacred grove of Apollo Lykeios ), in which he established his own school. The building included a gymnasium and a colonnade ( peripatos ), from which the school acquired the name Peripatetic . Aristotle conducted courses and research at the school for the next twelve years. He often lectured small groups of distinguished students and, along with some of them, such as Theophrastus , Eudemus , and Aristoxenus , Aristotle built
1180-435: A contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom. It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry , and
1298-438: A fertilized hen's egg of a suitable stage and opens it to see the embryo's heart beating inside. Instead, he practiced a different style of science: systematically gathering data, discovering patterns common to whole groups of animals, and inferring possible causal explanations from these. This style is common in modern biology when large amounts of data become available in a new field, such as genomics . It does not result in
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#17327733283661416-574: A few salient points. Aristotle was born in 384 BC in Stagira , Chalcidice , about 55 km (34 miles) east of modern-day Thessaloniki . He was the son of Nicomachus , the personal physician of King Amyntas of Macedon , and Phaestis, a woman with origins from Chalcis , Euboea . Nicomachus was said to have belonged to the medical guild of Asclepiadae and was likely responsible for Aristotle's early interest in biology and medicine. Ancient tradition held that Aristotle's family descended from
1534-497: A fluid such as air. In this system, heavy bodies in steady fall indeed travel faster than light ones (whether friction is ignored, or not ), and they do fall more slowly in a denser medium. Newton's "forced" motion corresponds to Aristotle's "violent" motion with its external agent, but Aristotle's assumption that the agent's effect stops immediately it stops acting (e.g., the ball leaves the thrower's hand) has awkward consequences: he has to suppose that surrounding fluid helps to push
1652-406: A form of an apple. In this distinction, there is a particular apple and a universal form of an apple. Moreover, one can place an apple next to a book, so that one can speak of both the book and apple as being next to each other. Plato argued that there are some universal forms that are not a part of particular things. For example, it is possible that there is no particular good in existence, but "good"
1770-554: A large library which included manuscripts, maps, and museum objects. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira. They had a son whom Aristotle named after his father, Nicomachus . This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his philosophical works. He wrote many dialogues, of which only fragments have survived. Those works that have survived are in treatise form and were not, for
1888-409: A library in the Lyceum, which helped him to produce many of his hundreds of books on papyrus scrolls . Though Aristotle wrote many treatises and dialogues for publication, only around a third of his original output has survived , none of it intended for publication. Aristotle provided a complex synthesis of the various philosophies existing prior to him. His teachings and methods of inquiry have had
2006-597: A linear scale, and noted various exceptions, such as that sharks had a placenta like the tetrapods. To a modern biologist, the explanation, not available to Aristotle, is convergent evolution . Philosophers of science have generally concluded that Aristotle was not interested in taxonomy, but zoologists who studied this question in the early 21st century think otherwise. He believed that purposive final causes guided all natural processes; this teleological view justified his observed data as an expression of formal design. Aristotle's psychology , given in his treatise On
2124-502: A matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set
2242-495: A never-ending series of "primitive notions", either a precise notion of what we mean by x = x {\displaystyle x=x} (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol = {\displaystyle =} has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that. Another, more interesting example axiom scheme ,
2360-422: A particular object in our structure, then we should be able to claim P ( t ) {\displaystyle P(t)} . Again, we are claiming that the formula ∀ x ϕ → ϕ t x {\displaystyle \forall x\phi \to \phi _{t}^{x}} is valid , that is, we must be able to give a "proof" of this fact, or more properly speaking,
2478-427: A plant in the soil is potentially ( dynamei ) a plant, and if it is not prevented by something, it will become a plant. Potentially, beings can either 'act' ( poiein ) or 'be acted upon' ( paschein ), which can be either innate or learned. For example, the eyes possess the potentiality of sight (innate – being acted upon), while the capability of playing the flute can be possessed by learning (exercise – acting). Actuality
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#17327733283662596-596: A prediction that would lead to different experimental results ( Bell's inequalities ) in the Copenhagen and the Hidden variable case. The experiment was conducted first by Alain Aspect in the early 1980s, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that
2714-498: A priori principles. Aristotle's "natural philosophy" spans a wide range of natural phenomena including those now covered by physics, biology and other natural sciences. In Aristotle's terminology, "natural philosophy" is a branch of philosophy examining the phenomena of the natural world, and includes fields that would be regarded today as physics, biology and other natural sciences. Aristotle's work encompassed virtually all facets of intellectual inquiry. Aristotle makes philosophy in
2832-576: A researcher and lecturer, earning for himself the nickname "mind of the school" by his tutor Plato . In Athens, he probably experienced the Eleusinian Mysteries as he wrote when describing the sights one viewed at the Mysteries, "to experience is to learn" ( παθεĩν μαθεĩν ). Aristotle remained in Athens for nearly twenty years before leaving in 348/47 BC after Plato's death. The traditional story about his departure records that he
2950-403: A scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying ( falsified ) the theory that the postulates install. A theory is considered valid as long as it has not been falsified. Now, the transition between the mathematical axioms and scientific postulates
3068-409: A sentence is related to the statement it bears like a numeral to the number it refers to. Statements are abstract logical entities , while sentences are grammatical entities . Aristotle Aristotle ( Attic Greek : Ἀριστοτέλης , romanized: Aristotélēs ; 384–322 BC) was an Ancient Greek philosopher and polymath . His writings cover a broad range of subjects spanning
3186-503: A separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another ' hidden variables ' approach was developed for some time by Albert Einstein, Erwin Schrödinger , David Bohm . It
3304-646: A significant impact across the world, and remain a subject of contemporary philosophical discussion. Aristotle's views profoundly shaped medieval scholarship . The influence of his physical science extended from late antiquity and the Early Middle Ages into the Renaissance , and was not replaced systematically until the Enlightenment and theories such as classical mechanics were developed. He influenced Judeo-Islamic philosophies during
3422-416: A system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain. Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in
3540-562: A thrown stone, in the Physics (254b10), and "natural motion", such as of a falling object, in On the Heavens (300a20). In violent motion, as soon as the agent stops causing it, the motion stops also: in other words, the natural state of an object is to be at rest, since Aristotle does not address friction . With this understanding, it can be observed that, as Aristotle stated, heavy objects (on
3658-489: A variable x {\displaystyle x} and a term t {\displaystyle t} that is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid. ϕ t x → ∃ x ϕ {\displaystyle \phi _{t}^{x}\to \exists x\,\phi } Non-logical axioms are formulas that play
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3776-454: Is postulate . Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas. Non-logical axioms are often simply referred to as axioms in mathematical discourse . This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups,
3894-444: Is substitutable for x {\displaystyle x} in ϕ {\displaystyle \phi } , the below formula is universally valid. ∀ x ϕ → ϕ t x {\displaystyle \forall x\,\phi \to \phi _{t}^{x}} Where the symbol ϕ t x {\displaystyle \phi _{t}^{x}} stands for
4012-454: Is a historical accident: his works on botany have been lost, but two books on plants by his pupil Theophrastus have survived. Aristotle reports on the sea-life visible from observation on Lesbos and the catches of fishermen. He describes the catfish , electric ray , and frogfish in detail, as well as cephalopods such as the octopus and paper nautilus . His description of the hectocotyl arm of cephalopods, used in sexual reproduction,
4130-474: Is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic , an axiom is a premise or starting point for reasoning. In mathematics , an axiom may be a " logical axiom " or a " non-logical axiom ". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., ( A and B ) implies A ), while non-logical axioms are substantive assertions about
4248-419: Is also a final cause or end. Then Aristotle proceeds and concludes that the actuality is prior to potentiality in formula, in time and in substantiality. With this definition of the particular substance (i.e., matter and form), Aristotle tries to solve the problem of the unity of the beings, for example, "what is it that makes a man one"? Since, according to Plato there are two Ideas: animal and biped, how then
4366-403: Is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply. It became more apparent when Albert Einstein first introduced special relativity where the invariant quantity
4484-541: Is man a unity? However, according to Aristotle, the potential being (matter) and the actual one (form) are one and the same. Aristotle's immanent realism means his epistemology is based on the study of things that exist or happen in the world, and rises to knowledge of the universal, whereas for Plato epistemology begins with knowledge of universal Forms (or ideas) and descends to knowledge of particular imitations of these. Aristotle uses induction from examples alongside deduction , whereas Plato relies on deduction from
4602-744: Is no more the Euclidean length l {\displaystyle l} (defined as l 2 = x 2 + y 2 + z 2 {\displaystyle l^{2}=x^{2}+y^{2}+z^{2}} ) > but the Minkowski spacetime interval s {\displaystyle s} (defined as s 2 = c 2 t 2 − x 2 − y 2 − z 2 {\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}} ), and then general relativity where flat Minkowskian geometry
4720-518: Is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system. Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself can be regarded as
4838-404: Is possible, for any sufficiently large set of axioms ( Peano's axioms , for example) to construct a statement whose truth is independent of that set of axioms. As a corollary , Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory. It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by
Axiom - Misplaced Pages Continue
4956-406: Is predicated. So, according to Aristotle, the form of apple exists within each apple, rather than in the world of the forms. Concerning the nature of change ( kinesis ) and its causes, as he outlines in his Physics and On Generation and Corruption ( 319b–320a), he distinguishes coming-to-be ( genesis , also translated as 'generation') from: Coming-to-be is a change where the substrate of
5074-502: Is probably not in its original form, because it was most likely edited by students and later lecturers. The logical works of Aristotle were compiled into a set of six books called the Organon around 40 BC by Andronicus of Rhodes or others among his followers. The books are: The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle's writings. It goes from
5192-415: Is replaced with pseudo-Riemannian geometry on curved manifolds . In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The ' Copenhagen school ' ( Niels Bohr , Werner Heisenberg , Max Born ) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in
5310-480: Is still a proper universal form. Aristotle disagreed with Plato on this point, arguing that all universals are instantiated at some period of time, and that there are no universals that are unattached to existing things. In addition, Aristotle disagreed with Plato about the location of universals. Where Plato spoke of the forms as existing separately from the things that participate in them, Aristotle maintained that universals exist within each thing on which each universal
5428-412: Is that which provides us with what is known as Universal Instantiation : Axiom scheme for Universal Instantiation. Given a formula ϕ {\displaystyle \phi } in a first-order language L {\displaystyle {\mathfrak {L}}} , a variable x {\displaystyle x} and a term t {\displaystyle t} that
5546-413: Is the fulfilment of the end of the potentiality. Because the end ( telos ) is the principle of every change, and potentiality exists for the sake of the end, actuality, accordingly, is the end. Referring then to the previous example, it can be said that an actuality is when a plant does one of the activities that plants do. For that for the sake of which ( to hou heneka ) a thing is, is its principle, and
5664-411: Is useful to regard postulates as purely formal statements, and not as facts based on experience. When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all. It
5782-423: Is what a statement means, it is the notion or idea that a statement expresses, i.e., what it represents. It could be said that "2 + 2 = 4" and "two plus two equals four" are two different statements that are expressing the same proposition in two different ways. Philosopher of language Peter Strawson (1919–2006) advocated the use of the term "statement" in sense (b) in preference to proposition . Strawson used
5900-548: The Iliad , which reportedly became one of Alexander's most prized possessions. Scholars speculate that two of Aristotle's now lost works, On kingship and On behalf of the Colonies , were composed by the philosopher for the young prince. After Philip II's assassination in 336 BC, Aristotle returned to Athens for the second and final time a year later. As a metic , Aristotle could not own property in Athens and thus rented
6018-540: The Classical period . His father, Nicomachus , died when Aristotle was a child, and he was brought up by a guardian. At around eighteen years old, he joined Plato 's Academy in Athens and remained there until the age of thirty seven ( c. 347 BC ). Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon , tutored his son Alexander the Great beginning in 343 BC. He established
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#17327733283666136-652: The Early Modern period. John Philoponus (in Late antiquity ) and Galileo (in Early modern period ) are said to have shown by experiment that Aristotle's claim that a heavier object falls faster than a lighter object is incorrect. A contrary opinion is given by Carlo Rovelli , who argues that Aristotle's physics of motion is correct within its domain of validity, that of objects in the Earth 's gravitational field immersed in
6254-525: The Great Comet of 371 BC . Aristotle was one of the first people to record any geological observations. He stated that geological change was too slow to be observed in one person's lifetime. The geologist Charles Lyell noted that Aristotle described such change, including "lakes that had dried up" and "deserts that had become watered by rivers", giving as examples the growth of the Nile delta since
6372-476: The History of Animals in a graded scale of perfection, a nonreligious version of the scala naturae , with man at the top. His system had eleven grades of animal, from highest potential to lowest, expressed in their form at birth: the highest gave live birth to hot and wet creatures, the lowest laid cold, dry mineral-like eggs. Animals came above plants , and these in turn were above minerals. He grouped what
6490-456: The Milky Way was made up of "those stars which are shaded by the earth from the sun's rays," pointing out partly correctly that if "the size of the sun is greater than that of the earth and the distance of the stars from the earth many times greater than that of the sun, then... the sun shines on all the stars and the earth screens none of them." He also wrote descriptions of comets, including
6608-508: The Physics (215a25), Aristotle effectively states a quantitative law, that the speed, v, of a falling body is proportional (say, with constant c) to its weight, W, and inversely proportional to the density, ρ, of the fluid in which it is falling:; Aristotle implies that in a vacuum the speed of fall would become infinite, and concludes from this apparent absurdity that a vacuum is not possible. Opinions have varied on whether Aristotle intended to state quantitative laws. Henri Carteron held
6726-536: The natural sciences , philosophy , linguistics , economics , politics , psychology , and the arts . As the founder of the Peripatetic school of philosophy in the Lyceum in Athens , he began the wider Aristotelian tradition that followed, which set the groundwork for the development of modern science . Little is known about Aristotle's life. He was born in the city of Stagira in northern Greece during
6844-653: The philosophy of mathematics . The word axiom comes from the Greek word ἀξίωμα ( axíōma ), a verbal noun from the verb ἀξιόειν ( axioein ), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος ( áxios ), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers and mathematicians , axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof. The root meaning of
6962-519: The "extreme view" that Aristotle's concept of force was basically qualitative, but other authors reject this. Archimedes corrected Aristotle's theory that bodies move towards their natural resting places; metal boats can float if they displace enough water ; floating depends in Archimedes' scheme on the mass and volume of the object, not, as Aristotle thought, its elementary composition. Aristotle's writings on motion remained influential until
7080-566: The Lyceum of Athens, Alexandria , and later in Rome . With the Prior Analytics , Aristotle is credited with the earliest study of formal logic, and his conception of it was the dominant form of Western logic until 19th-century advances in mathematical logic . Kant stated in the Critique of Pure Reason that with Aristotle, logic reached its completion. Most of Aristotle's work
7198-646: The Middle Ages, as well as Christian theology , especially the Neoplatonism of the Early Church and the scholastic tradition of the Catholic Church . Aristotle was revered among medieval Muslim scholars as "The First Teacher", and among medieval Christians like Thomas Aquinas as simply "The Philosopher", while the poet Dante called him "the master of those who know". His works contain
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#17327733283667316-514: The ancient distinction between "axioms" and "postulates" respectively). These are certain formulas in a formal language that are universally valid , that is, formulas that are satisfied by every assignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in
7434-494: The ball along to make it continue to rise even though the hand is no longer acting on it, resulting in the Medieval theory of impetus . Aristotle suggested that the reason for anything coming about can be attributed to four different types of simultaneously active factors. His term aitia is traditionally translated as "cause", but it does not always refer to temporal sequence; it might be better translated as "explanation", but
7552-640: The basics, the analysis of simple terms in the Categories, the analysis of propositions and their elementary relations in On Interpretation , to the study of more complex forms, namely, syllogisms (in the Analytics ) and dialectics (in the Topics and Sophistical Refutations ). The first three treatises form the core of the logical theory stricto sensu : the grammar of the language of logic and
7670-399: The becoming is for the sake of the end; and the actuality is the end, and it is for the sake of this that the potentiality is acquired. For animals do not see in order that they may have sight, but they have sight that they may see. In summary, the matter used to make a house has potentiality to be a house and both the activity of building and the form of the final house are actualities, which
7788-574: The broad sense coextensive with reasoning, which he also would describe as "science". However, his use of the term science carries a different meaning than that covered by the term "scientific method". For Aristotle, "all science ( dianoia ) is either practical, poetical or theoretical" ( Metaphysics 1025b25). His practical science includes ethics and politics; his poetical science means the study of fine arts including poetry; his theoretical science covers physics, mathematics and metaphysics. In his On Generation and Corruption , Aristotle related each of
7906-433: The cause of earthquakes was a gas or vapor ( anathymiaseis ) that was trapped inside the earth and trying to escape, following other Greek authors Anaxagoras , Empedocles and Democritus . Aristotle also made many observations about the hydrologic cycle. For example, he made some of the earliest observations about desalination: he observed early – and correctly – that when seawater is heated, freshwater evaporates and that
8024-447: The conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc.). In the field of mathematical logic , a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to
8142-487: The correct rules of reasoning. The Rhetoric is not conventionally included, but it states that it relies on the Topics . What is today called Aristotelian logic with its types of syllogism (methods of logical argument), Aristotle himself would have labelled "analytics". The term "logic" he reserved to mean dialectics . The word "metaphysics" appears to have been coined by the first century AD editor who assembled various small selections of Aristotle's works to create
8260-399: The data to come to the theory of evolution . Aristotle's writings can seem to modern readers close to implying evolution, but while Aristotle was aware that new mutations or hybridizations could occur, he saw these as rare accidents. For Aristotle, accidents, like heat waves in winter, must be considered distinct from natural causes. He was thus critical of Empedocles's materialist theory of
8378-808: The death. Following Alexander's death, anti-Macedonian sentiment in Athens was rekindled. In 322 BC, Demophilus and Eurymedon the Hierophant reportedly denounced Aristotle for impiety, prompting him to flee to his mother's family estate in Chalcis, Euboea , at which occasion he was said to have stated "I will not allow the Athenians to sin twice against philosophy" – a reference to Athens's trial and execution of Socrates . He died in Chalcis, Euboea of natural causes later that same year, having named his student Antipater as his chief executor and leaving
8496-676: The definitive foundation for mathematics. Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance, Newton's laws in classical mechanics, Maxwell's equations in classical electromagnetism, Einstein's equation in general relativity, Mendel's laws of genetics, Darwin's Natural selection law, etc. These founding assertions are usually called principles or postulates so as to distinguish from mathematical axioms . As
8614-479: The earliest known formal study of logic, and were studied by medieval scholars such as Peter Abelard and Jean Buridan . Aristotle's influence on logic continued well into the 19th century. In addition, his ethics , although always influential, gained renewed interest with the modern advent of virtue ethics . In general, the details of Aristotle's life are not well-established. The biographies written in ancient times are often speculative and historians only agree on
8732-518: The elements of the domain of a specific mathematical theory, for example a + 0 = a in integer arithmetic. Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry ). To axiomatize
8850-553: The end of his life, the two men became estranged having diverging opinions over issues, like the optimal administration of city-states, the treatment of conquered populations, such as the Persians, and philosophical questions, like the definition of braveness. A widespread speculation in antiquity suggested that Aristotle played a role in Alexander's death, but the only evidence of this is an unlikely claim made some six years after
8968-448: The first three Postulates, assert the possibility of some construction but expresses an essential property." Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept. The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments ( syllogisms , rules of inference )
9086-533: The following. Brood size decreases with (adult) body mass, so that an elephant has fewer young (usually just one) per brood than a mouse. Lifespan increases with gestation period , and also with body mass, so that elephants live longer than mice, have a longer period of gestation, and are heavier. As a final example, fecundity decreases with lifespan, so long-lived kinds like elephants have fewer young in total than short-lived kinds like mice. Aristotle distinguished about 500 species of animals , arranging these in
9204-403: The form of the substance is the actual house, namely 'covering for bodies and chattels' or any other differentia that let us define something as a house. The formula that gives the components is the account of the matter, and the formula that gives the differentia is the account of the form. Like his teacher Plato, Aristotle's philosophy aims at the universal . Aristotle's ontology places
9322-493: The formula ϕ {\displaystyle \phi } with the term t {\displaystyle t} substituted for x {\displaystyle x} . (See Substitution of variables .) In informal terms, this example allows us to state that, if we know that a certain property P {\displaystyle P} holds for every x {\displaystyle x} and that t {\displaystyle t} stands for
9440-414: The foundation of the various sciences lay certain additional hypotheses that were accepted without proof. Such a hypothesis was termed a postulate . While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if
9558-515: The four elements proposed earlier by Empedocles , earth , water , air , and fire , to two of the four sensible qualities, hot, cold, wet, and dry. In the Empedoclean scheme, all matter was made of the four elements, in differing proportions. Aristotle's scheme added the heavenly aether , the divine substance of the heavenly spheres , stars and planets. Aristotle describes two kinds of motion: "violent" or "unnatural motion", such as that of
9676-428: The ground, say) require more force to make them move; and objects pushed with greater force move faster. This would imply the equation incorrect in modern physics. Natural motion depends on the element concerned: the aether naturally moves in a circle around the heavens, while the 4 Empedoclean elements move vertically up (like fire, as is observed) or down (like earth) towards their natural resting places. In
9794-405: The group operation is commutative , and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups. Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define
9912-440: The image. According to Aristotle, spontaneity and chance are causes of some things, distinguishable from other types of cause such as simple necessity. Chance as an incidental cause lies in the realm of accidental things , "from what is spontaneous". There is also more a specific kind of chance, which Aristotle names "luck", that only applies to people's moral choices. In astronomy , Aristotle refuted Democritus 's claim that
10030-954: The immediately following proposition and " → {\displaystyle \to } " for implication from antecedent to consequent propositions: Each of these patterns is an axiom schema , a rule for generating an infinite number of axioms. For example, if A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are propositional variables , then A → ( B → A ) {\displaystyle A\to (B\to A)} and ( A → ¬ B ) → ( C → ( A → ¬ B ) ) {\displaystyle (A\to \lnot B)\to (C\to (A\to \lnot B))} are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens , one can prove all tautologies of
10148-429: The interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid . The ancient Greeks considered geometry as just one of several sciences , and held the theorems of geometry on par with scientific facts. As such, they developed and used
10266-412: The learner is in doubt about the truth of the postulates. The classical approach is well-illustrated by Euclid's Elements , where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions). A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from
10384-564: The legendary physician Asclepius and his son Machaon . Both of Aristotle's parents died when he was still at a young age and Proxenus of Atarneus became his guardian. Although little information about Aristotle's childhood has survived, he probably spent some time in the Macedonian capital, making his first connections with the Macedonian monarchy . At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Plato's Academy . He became distinguished as
10502-424: The logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view. An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that: When an equal amount is taken from equals, an equal amount results. At
10620-663: The mathematical assertions (axioms, postulates, propositions , theorems) and definitions. One must concede the need for primitive notions , or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa , Mario Pieri , and Giuseppe Peano were pioneers in this movement. Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory , group theory , topology , vector spaces ) without any particular application in mind. The distinction between an "axiom" and
10738-468: The modern zoologist would call vertebrates as the hotter "animals with blood", and below them the colder invertebrates as "animals without blood". Those with blood were divided into the live-bearing ( mammals ), and the egg-laying ( birds , reptiles , fish ). Those without blood were insects, crustacea (non-shelled – cephalopods, and shelled ) and the hard-shelled molluscs ( bivalves and gastropods ). He recognised that animals did not exactly fit into
10856-532: The most part, intended for widespread publication; they are generally thought to be lecture aids for his students. His most important treatises include Physics , Metaphysics , Nicomachean Ethics , Politics , On the Soul and Poetics . Aristotle studied and made significant contributions to "logic, metaphysics, mathematics, physics, biology, botany, ethics, politics, agriculture, medicine, dance, and theatre." While Alexander deeply admired Aristotle, near
10974-504: The near-by island of Lesbos . During this time, Aristotle married Pythias , Hermias's adoptive daughter and niece, and had a daughter whom they also named Pythias. In 343/42 BC, Aristotle was invited to Pella by Philip II of Macedon in order to become the tutor to his thirteen-year-old son Alexander ; a choice perhaps influenced by the relationship of Aristotle's family with the Macedonian dynasty. Aristotle taught Alexander at
11092-404: The oceans are then replenished by the cycle of rainfall and river runoff ("I have proved by experiment that salt water evaporated forms fresh and the vapor does not when it condenses condense into sea water again.") Aristotle was the first person to study biology systematically, and biology forms a large part of his writings. He spent two years observing and describing the zoology of Lesbos and
11210-437: The primary kind of knowledge; but if there is some motionless independent thing, the knowledge of this precedes it and is first philosophy, and it is universal in just this way , because it is first. And it belongs to this sort of philosophy to study being as being, both what it is and what belongs to it just by virtue of being. Aristotle examines the concepts of substance ( ousia ) and essence ( to ti ên einai , "the what it
11328-571: The private school of Mieza , in the gardens of the Nymphs , the royal estate near Pella. Alexander's education probably included a number of subjects, such as ethics and politics , as well as standard literary texts, like Euripides and Homer . It is likely that during Aristotle's time in the Macedonian court, other prominent nobles, like Ptolemy and Cassander , would have occasionally attended his lectures. Aristotle encouraged Alexander toward eastern conquest, and his own attitude towards Persia
11446-510: The propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens . Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed. These axiom schemata are also used in the predicate calculus , but additional logical axioms are needed to include a quantifier in the calculus. Axiom of Equality. Let L {\displaystyle {\mathfrak {L}}} be
11564-412: The related demonstration of the consistency of those axioms. In a wider context, there was an attempt to base all of mathematics on Cantor's set theory . Here, the emergence of Russell's paradox and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent. The formalist project suffered a setback a century ago, when Gödel showed that it
11682-409: The role of theory-specific assumptions. Reasoning about two different structures, for example, the natural numbers and the integers , may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups ). Thus non-logical axioms, unlike logical axioms, are not tautologies . Another name for a non-logical axiom
11800-433: The same certainty as experimental science, but it sets out testable hypotheses and constructs a narrative explanation of what is observed. In this sense, Aristotle's biology is scientific. From the data he collected and documented, Aristotle inferred quite a number of rules relating the life-history features of the live-bearing tetrapods (terrestrial placental mammals) that he studied. Among these correct predictions are
11918-418: The same statement. By a statement, it is meant "that which one states", not one's stating of it. There are many interpretations of what the term statement means, but generally, it indicates either: a meaningful declarative sentence that is either true or false ( bivalence ), or: a proposition. A proposition is an assertion that is made by (i.e., the meaning of) a true or false declarative sentence. A proposition
12036-400: The sentence "Pegasus exists." is a statement is a subject of debate among philosophers. Bertrand Russell held that it is a (false) statement. Strawson held it is not a statement at all. In some treatments, "statement" is introduced in order to distinguish a sentence from its informational content. A statement is regarded as the information content of an information-bearing sentence. Thus,
12154-476: The strict sense. In propositional logic it is common to take as logical axioms all formulae of the following forms, where ϕ {\displaystyle \phi } , χ {\displaystyle \chi } , and ψ {\displaystyle \psi } can be any formulae of the language and where the included primitive connectives are only " ¬ {\displaystyle \neg } " for negation of
12272-558: The surrounding seas, including in particular the Pyrrha lagoon in the centre of Lesbos. His data in History of Animals , Generation of Animals , Movement of Animals , and Parts of Animals are assembled from his own observations, statements given by people with specialized knowledge, such as beekeepers and fishermen, and less accurate accounts provided by travellers from overseas. His apparent emphasis on animals rather than plants
12390-498: The system of natural numbers , an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. Furthermore, using techniques of forcing ( Cohen ) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as
12508-960: The term "statement" to make the point that two declarative sentences can make the same statement if they say the same thing in different ways. Thus, in the usage advocated by Strawson, "All men are mortal." and "Every man is mortal." are two different sentences that make the same statement. In either case, a statement is viewed as a truth bearer . Examples of sentences that are (or make) true statements: Examples of sentences that are also statements, even though they aren't true: Examples of sentences that are not (or do not make) statements: The first two examples are not declarative sentences and therefore are not (or do not make) statements. The third and fourth are declarative sentences but, lacking meaning, are neither true nor false and therefore are not (or do not make) statements. The fifth and sixth examples are meaningful declarative sentences, but are not statements but rather matters of opinion or taste. Whether or not
12626-402: The thing that has undergone the change has itself changed. In that particular change he introduces the concept of potentiality ( dynamis ) and actuality ( entelecheia ) in association with the matter and the form. Referring to potentiality, this is what a thing is capable of doing or being acted upon if the conditions are right and it is not prevented by something else. For example, the seed of
12744-475: The time of Homer , and "the upheaving of one of the Aeolian islands , previous to a volcanic eruption ."' Meteorologica lends its name to the modern study of meteorology, but its modern usage diverges from the content of Aristotle's ancient treatise on meteors . The ancient Greeks did use the term for a range of atmospheric phenomena, but also for earthquakes and volcanic eruptions. Aristotle proposed that
12862-478: The traditional rendering will be employed here. Aristotle describes experiments in optics using a camera obscura in Problems , book 15. The apparatus consisted of a dark chamber with a small aperture that let light in. With it, he saw that whatever shape he made the hole, the sun's image always remained circular. He also noted that increasing the distance between the aperture and the image surface magnified
12980-401: The treatise we know by the name Metaphysics . Aristotle called it "first philosophy", and distinguished it from mathematics and natural science (physics) as the contemplative ( theoretikē ) philosophy which is "theological" and studies the divine. He wrote in his Metaphysics (1026a16): If there were no other independent things besides the composite natural ones, the study of nature would be
13098-492: The universal ( katholou ) in particulars ( kath' hekaston ), things in the world, whereas for Plato the universal is a separately existing form which actual things imitate. For Aristotle, "form" is still what phenomena are based on, but is "instantiated" in a particular substance. Plato argued that all things have a universal form , which could be either a property or a relation to other things. When one looks at an apple, for example, one sees an apple, and one can also analyse
13216-426: The word postulate is to "demand"; for instance, Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line). Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, Proclus remarks that " Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like
13334-498: Was created so as to try to give deterministic explanation to phenomena such as entanglement . This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the EPR paradox in 1935). Taking this idea seriously, John Bell derived in 1964
13452-432: Was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions ( theorems , in the case of mathematics) must be proven with the aid of these basic assumptions. However,
13570-681: Was disappointed with the Academy's direction after control passed to Plato's nephew Speusippus , although it is possible that the anti-Macedonian sentiments in Athens could have also influenced his decision. Aristotle left with Xenocrates to Assos in Asia Minor , where he was invited by his former fellow student Hermias of Atarneus ; he stayed there for a few years and left around the time of Hermias' death. While at Assos, Aristotle and his colleague Theophrastus did extensive research in botany and marine biology , which they later continued at
13688-478: Was strongly ethnocentric . In one famous example, he counsels Alexander to be "a leader to the Greeks and a despot to the barbarians". Alexander's education under the guardianship of Aristotle likely lasted for only a few years, as at around the age of sixteen he returned to Pella and was appointed regent of Macedon by his father Philip. During this time, Aristotle is said to have gifted Alexander an annotated copy of
13806-421: Was to be") in his Metaphysics (Book VII), and he concludes that a particular substance is a combination of both matter and form, a philosophical theory called hylomorphism . In Book VIII, he distinguishes the matter of the substance as the substratum , or the stuff of which it is composed. For example, the matter of a house is the bricks, stones, timbers, etc., or whatever constitutes the potential house, while
13924-588: Was widely disbelieved until the 19th century. He gives accurate descriptions of the four-chambered fore-stomachs of ruminants , and of the ovoviviparous embryological development of the hound shark . He notes that an animal's structure is well matched to function so birds like the heron (which live in marshes with soft mud and live by catching fish) have a long neck, long legs, and a sharp spear-like beak, whereas ducks that swim have short legs and webbed feet. Darwin , too, noted these sorts of differences between similar kinds of animal, but unlike Aristotle used
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