Aristotle discusses the definitions of numerous mathematical entities and properties, such as point, line, plane, solid, circle, commensurate, number, even and odd, three, etc., and uses others in interesting ways, such as prime and additively prime (not the sum of two numbers, i.e., 2 and 3, since 2 is the first number) in a definition of ‘three’.
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What contributions did Aristotle make to math?
Aristotle provides several examples of such triads of terms in mathematics, e.g., two right angles-angles about a point-triangle, or right angle-half two right angles-angle in a semicircle.
What did Aristotle say about mathematics?
Aristotle discussed two major concerns for the nature of mathematics. In one, he mentions that there must be some unprovable principles to avoid infinite regresses. And in the other, he mentions that the proofs should be explanatory. He mentioned two ways to start a proof, i.e., by axioms or by posits.
Is Aristotle a Greek mathematician?
Aristotle was a Greek philosopher and polymath born in 384-322 BC in Stageira, Chalcidice. He was well educated by the best teachers of the time namely Plato.
What is Aristotle best known for?
Aristotle was one of the greatest philosophers who ever lived and the first genuine scientist in history. He made pioneering contributions to all fields of philosophy and science, he invented the field of formal logic, and he identified the various scientific disciplines and explored their relationships to each other.
Who invented math?
Archimedes is known as the Father of Mathematics. Mathematics is one of the ancient sciences developed in time immemorial....Table of Contents.1.Who is the Father of Mathematics?2.Birth and Childhood3.Interesting facts4.Notable Inventions5.Death of the Father of Mathematics3 more rows
Who is the father of mathematics?
ArchimedesArchimedes is known as the Father Of Mathematics. He lived between 287 BC – 212 BC. Syracuse, the Greek island of Sicily was his birthplace. Archimedes was serving the King Hiero II of Syracuse by solving mathematical problems and by developing interesting innovations for the king and his army.
Who invented calculus?
Today it is generally believed that calculus was discovered independently in the late 17th century by two great mathematicians: Isaac Newton and Gottfried Leibniz.
What was Aristotle's theory?
In his metaphysics, he claims that there must be a separate and unchanging being that is the source of all other beings. In his ethics, he holds that it is only by becoming excellent that one could achieve eudaimonia, a sort of happiness or blessedness that constitutes the best kind of human life.
What did Aristotle teach?
Aristotle taught Alexander and his friends about medicine, philosophy, morals, religion, logic, and art.
What is Aristotle's most famous discovery?
Invented the Logic of the Categorical Syllogism This process of logical deduction was invented by Aristotle, and perhaps lies at the heart of all his famous achievements. He was the first person to come up with an authentic and logical procedure to conclude a statement based on the propositions that were at hand.
What is Aristotle's greatest work?
In 335, Aristotle founded his own school, the Lyceum, in Athens, where he spent most of the rest of his life studying, teaching and writing. Some of his most notable works include Nichomachean Ethics, Politics, Metaphysics, Poetics and Prior Analytics.
What are 3 facts about Aristotle?
Aristotle | 10 Facts On The Famous Ancient Greek Philosopher#1 He has been called the last person to know everything there was to know.#2 Most of his work which survives is in the form of lecture notes.#3 He might have had an intimate relationship with a man.#4 His works had a major impact on medieval Islamic thought.More items...•
What is Aristotle's definition of mathematics?
Aristotle defined mathematics as: The science of quantity. In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry.
What two sciences did Aristotle consider to be the most important?
Aristotle considers geometry and arithmetic, the two sciences which we might say constitute ancient mathematics, as merely the two most important mathematical sciences. His explanations of mathematics always include optics, mathematical astronomy, harmonics, etc. Click to see full answer.
3. Problems About Mathematical Objects
However, Aristotle rejected Plato’s mathematical cosmology, but he has gained the deepest knowledge of science from him only. On one hand, Aristotle treats mathematical sciences as a model of scientific knowledge, and on the other hand, he agrees with Plato’s assumption that authentic knowledge must have a real object.
4. Ontological Status of Mathematical Objects
For the treatment of mathematical objects, Aristotle quoted five concepts in his discussions:-
6. Universal Mathematics
There is an analogy of mathematics suggested by Aristotle, which mentions that there is a super science of mathematics that covers all continuous magnitudes and discrete quantities such as numbers.
8. Unit (monas) and Number (arithmos)
Greek mathematicians believed that numbers are the plural of units. According to them, a number is made up of countable figures (unit). One is not a number, it is a unit that will construct a number. Later on, some questions arose on this concept such as overlapping problems.
9. Measure
Greek mathematicians used a system in which all fractions are proper parts, that we now call unit fractions. 2/3 is an exception, otherwise, all other fractions can be represented in the form 1/n. For example, 2/5 can be expressed as a sum of 1/3 and 1/15. Greeks used the same system of measurement as we do like 1 foot is 12 inches.
10. Time
Aristotle defined time as the number or count of change. He further distinguished two types of numbers, what is counted and by which we count. For example, 10 cows in a ground, here, the number counted is 10 and by which it is counted is 1 cow. In his view, time is a number where the change is measured by a unit of change.
11. Aristotle on the Infinite
Aristotle denied the existing theory of infinity in mathematics and physics and defined it in his terms. He mentioned,
Why does Aristotle not use uniform magnitudes?
On these two occasions ( Physics vi.7 and De caelo i.6) where Aristotle considers non-uniform magnitudes, he attempts to speak generally without a concrete example, so that his argument fails. However, one should also note that no one until the late Middle Ages seems to have noticed this. Greek mathematicians wisely avoided non-uniform magnitudes which could not be reduced to uniform magnitudes. The reason for this has partly to do with the difficulty of representing non-uniformity abstractly. Hence, Aristotle needs to consider non-uniform magnitudes for his proofs, but lacks the mathematics to deal with them.
What are some examples of Aristotle's problems?
Additionally, one of Aristotle's favorite examples is the problem of squaring a circle (finding a square equal to a given circle). The problem must be as old as Greek mathematics, given that the problem marks a transition from Egyptian to Greek style mathematics. Some have held that there was at least one solution, ...
What does Aristotle mean by spirals?
Although Aristotle is aware of curves generated by multiple motions and divides lines into straight, circular, and mixed, he only mentions only spirals, by which he may mean spherical sprials or any heavenly ‘twistings’ . (Some commentators have held that Aristotle means by mixed lines, lines that have straight and curved segments.) He does not mention, parabolas, ellipses, nor hyperboles, although these were a contemporary discovery. Nor does he mention two of the three major problems in contemporary mathematics, trisecting an angle and doubling the cube. Most today think that Aristotle's remark in Posterior Analytics i.7 about whether two cubes is a cube refers to their multiplication of two cube numbers (see above, 21), or less likely to the question of whether their sum is ever a cube, although it would be as natural a question in the 4th century BCE as it was to Fermat. As noted above, the third problem, squaring the circle, is a favorite example.
Which law of reflection was used by Aristotle in Meteorologica iii.5?
The law of reflection used by Aristotle in Meteorologica iii.5 is incorrect. Since the correct rule appears in pseudo-Aristotle, Problems xiv.4, 13 and in pseudo-Euclid, Catoptics , which may date from 3rd cent. BCE, it is reasonable to suppose that the correct law was unknown in Aristotle's time. Aristotle's rule is: Let M be the location of the mirror, G the object seen, and K the observer, then MG : GK is constant.
What does the number of propositions mean in Euclid's Elements?
Where a proposition occurs in Euclid's Elements , the number is given, * indicates that we can reconstruct from what Aristotle says a proof different from that found in Euclid). Where the attribution is in doubt, I cite the scholar who endorses it. In many cases, the theorem is inferred from the context.
Which philosophers have a solution to the problem of squaring a circle?
At least since Archimedes, we know that the problem of rectifying a circumference is equivalent to the problem of squaring a circle. Yet, Aristotle allows that the problem of squaring a circle may have a solution. It seems likely, then, that this equivalence was unknown in the 4th cent. BCE.
Who is the author of the locus theorem?
Few today would credit Plato with original mathematics. More can be said for Aristotle. While it may be unlikely that Aristotle is the author of the locus theorem (13) from Meteorologica iii.5, it is interesting that in his commentary on Apollonius' Conics , the Byzantine mathematician Eutocius attributes the theorem to Apollonius, 150 years Aristotle 's junior.
What was Aristotle's contribution to mathematics?
As far as we know, his contributions were towards clarifying mathematical logic. His interest in universal quantification (and his sense that mathematics aspires to "barbara"-type universal assertions) is a very early cut at ideas which, after an enormously long period of dormancy, emerged in modern mathematical logic. How far he can be seen to have started this line of thought, rather than anticipated part of it, is doubtful. Aristotle has not been a "go-to" early authority in mathematics since the Archimedean tradition was resumed in the early modern period, and Aristotle (for mostly non-mathematical reasons) became the representative bogeyman of a superseded past.
What was Aristotle's greatest achievement?
A great achievement of Aristotle was laying the foundations for the first system of logic. His contributions to the field of logic are only a small amount of what we know today, but thanks to him later philosophers could build on what they learnt from Aristotle, and thus we now have many systems/forms of logic, e.g. propositional, modal, quantificational logic etc.
What did Aristotle show the world?
Aristotle showed the Western world that there is a way of thinking and speaking entirely as an individual. Before him, Socrates and Plato assumed that philosophy had to be done in discussion between thinkers, after Aristotle, the culture saw the possibility of the solitary thinker who then writes down his/her thoughts which other solitary thinkers could enrich their thinking.
How did Aristotle teach us to see logic?
Now how did he do it? First by teaching logic. Sentences he saw were propositions and propositions stood on their own, true or false but crucially they can be seen as standing independently of any person, even Aristotle. And logic, he taught us to see, logic, not the person, is the rule imposer, logic is the judge, and above all, not the speaker. By logic, one can produce a sound argument without anyone speaking it and no one was required to judge it.
How many papers did Aristotle write?
This a difficult question to answer, Aristotle was a philosopher, mathematician, botanist and more, he wrote more than 200 papers.
Which philosopher said that all four causes are necessary to explain the existence of any thing?
All four causes are necessary, according to Aristotle, to explain the existence of any thing.
Did Archimedes prove any new theorems?
For instance, the Stanford Encyclopedia of Philosophy commits an extraordinary howler, claiming that Archimedes had already proved that the rectification of the circumference of a circle is equivalent to the quadrature of its area. Well, for seriously disputable values of "proved", perhaps.
Why is Aristotelian realism important?
Because Aristotelian realism insists on the realisability of mathematical properties in the world, it can give a straightforward account of how basic mathematical facts are known: by perception, the same as other simple facts. Ratios of heights are visible (to a degree of approximation, of course). Infants and animals demonstrably do have the ability to recognise pattern and estimate number, shape and symmetry.
What is the philosophy of mathematics?
There is a name for a philosophy of mathematics that emphasises the way in which mathematical properties crop up in the actual world. It is called Aristotelian realism. It is based on Aristotle’s view, opposed to that of his teacher Plato, that the properties of things are real and in the things themselves, not in another world of abstracta. A version of it, holding that mathematics was the ‘science of quantity’, was actually the leading philosophy of mathematics up to the time of Newton, but the idea has been largely off the agenda since then.
What is the philosophy of Platonism?
Inspired by that thought, Platonism proposes a philosophy of mathematics opposite to nominalism. It says that mathematics is about a realm of non-physical objects such as numbers and sets, abstracta that exist in a mysterious realm of forms beyond space and time. If that sounds far-fetched, note that pure mathematicians certainly speak and often think that way about their subject. Platonism also fits well with the apparent success of mathematical proof, which seems to demonstrate how things must be in all possible worlds, irrespective of what the laws of nature might be in any particular world. The proof that the square root of 2 is an irrational number does not rely on any observationally established laws. It shows how things must be, suggesting that the square root of 2 is an entity beyond our changeable world of space and time.
Why do people care about mathematics?
Perhaps the reason is that the certainty and objectivity of mathematics, its once-and-for-all establishment of rock-solid truths, stands as a challenge to many common philosophical positions. It is not just extreme sceptical views such as postmodernism that have a problem with it. So do all empiricist and naturalist views that hope for a fully ‘scientific’ explanation of reality and our knowledge of it. The problem is not so much that mathematics is true, but that its truths are absolutely necessary, and that the human mind can establish those necessities and understand why they must be so. It is very difficult to explain how a physical brain could do that.
What are some of the properties of mathematics?
Any digression into applied mathematics – rarely undertaken by philosophers of mathematics, who prefer the familiar ground of numbers and logic – will turn up, for the alert observer, many other quantitative and structural properties that are not themselves physical but can be realised in the physical world (and any other worlds there might be): flows, order relations, continuity and discreteness, alternation, linearity, feedback, network top ology, and many others.
Is mathematics about something?
T o the question: ‘Is mathematics about something?’ there are two answers: ‘Yes’ and ‘No’. Both are profoundly unsatisfying.
What did Aristotle think of physics?
Aristotle also had a theory of physics, which was of course wrong, but not as wrong as a lot of people seem to think. His equations for motion and gravitation, for example, are reasonably accurate approximations under normal conditions of friction and air resistance. Heavy things do fall faster in the real world, as long as you're not in outer space; and most things do stop on their own when you stop pushing them, because they are subject to friction. Aristotle noticed that there were exceptions (such as something launched from a catapult), and tried to come up with ad hoc theories to explain that---and these ad hoc explanations actually worked fairly well.
Why was Aristotle considered the first physicist?
Many consider Aristotle the first physicist (or at least the first we know of), because he was the first person in recorded history to make significant use of quantitative mathematics in understanding physical phenomena. His formalization of logic has largely been preserved in modern classical logic, though we now have...
Who was the most important thinker in history?
But overall, Aristotle was one of the most important thinkers who ever lived. Had he not existed, it's hard to say how long it would have been before someone else came along to think of empiricism, logic, and physics. Had it taken long enough, the course of human history could have been radically different---science and technology might have been held back hundreds of years.
What did Aristotle contribute to the world?
Aristotle was an ancient Greek philosopher and scientist who is widely considered to be one of the greatest thinkers in history. Moreover, along with Plato, he is considered the “Father of Western Philosophy”. During his lifetime, Aristotle wrote extensively making noteworthy contributions to numerous fields including physical sciences such as astronomy, anatomy, embryology, geology, geography, meteorology, zoology and physics. In the field of philosophy, Aristotle wrote about ethics, aesthetics government, politics, metaphysics, economics, rhetoric, psychology and theology. He also studied fine arts making significant contributions to subjects such as literature, poetry, drama and rhetoric. In many of these numerous fields, the works of Aristotle had an immense influence for almost two millennia making him one of the most influential people in the history of mankind. Moreover, Aristotle continues to influence some of these fields even in the modern era. Know more about the contributions of Aristotle through his 10 major accomplishments.
What did Aristotle write about?
In the field of philosophy, Aristotle wrote about ethics, aesthetics government, politics, metaphysics, economics, rhetoric, psychology and theology. He also studied fine arts making significant contributions to subjects such as literature, poetry, drama and rhetoric. In many of these numerous fields, the works of Aristotle had an immense influence ...
How did Aristotle use money?
He states that it came into use for the sake of convenience as people agreed to deal in something that is intrinsically useful and easily applicable, like iron or silver. Aristotle’s predecessor Plato believed in a communist social order where material things are held in common. Aristotle argued against this stating that this would lead to animosity among the citizens as people would feel that they didn’t receive what was rightly theirs; and reward would not be proportionate to work. Also, in contrast to Plato, Aristotle defended the right to private property. He saw property rights as an incentive mechanism where individuals would keep the fruits of his labor. Aristotle saw wealth creation as a mean to the ends of truth and virtue. He was against wealth becoming an end in itself rather than something subservient to a higher purpose. He was also against retail and interest. He believed that retail trade was done to make profit rather than procuring essential things; while interest was unnatural as it made a gain out of the money itself, and not from its use.
What is the soul of Aristotle?
Building upon the works of earlier philosophers, Aristotle wrote one of the earliest comprehensive texts in psychology. Titled De Anima (On The Soul), the focus of the text is not spiritual but bio-psychological, the study of psychology within a biological framework. The term soul in the text may be better translated as life-force. According to Aristotle, animals and plants also have souls like humans. Plants have a vegetative soul, which comprises the powers of growth, nutrition and reproduction. In addition to this, animals have a sensitive soul, which comprises the powers of perception and locomotion. In addition to the above two, humans also possess a rational soul, which comprises of the powers of reason and thought. Aristotle was one of the first to examine the impulses that drove life stating that the urge to reproduce (‘Libido’) was the overriding impulse of all living things and this was derived from the vegetative or plant soul. Alongside libido, human actions were also determined by ‘Id’ and ‘Ego’. Aristotle’s work in psychology was unchallenged for many years and it determined the history of the subject. Moreover, many of his proposals continue to influence modern psychologists.
How many species of animals did Aristotle identify?
In his works, Aristotle names around 500 species of bird, mammal and fish; he distinguishes dozens of insects and other invertebrates; and describes the internal anatomy of over a hundred animals, of which he dissected around 35.
What is Aristotle's main focus in metaphysics?
The primary focus of Aristotle in Metaphysics is the nature of existence; how things exist while undergoing change in the natural world; and how this world can be understood. The Metaphysics is regarded as one of the greatest philosophical works.
Which philosopher wrote the first literary theory?
Poetics by Aristotle. Poetics is a work by Aristotle that is the earliest surviving work of dramatic theory and also the first surviving western philosophical treatise to focus on literary theory, the systematic study of the nature of literature.
What did Aristotle contribute to the world?
He made pioneering contributions to all fields of philosophy and science, he invented the field of formal logic, and he identified the various scientific disciplines and explored their relationships to each other. Aristotle was also a teacher and founded his own school in Athens, known as the Lyceum.
How many works did Aristotle write?
Aristotle wrote as many as 200 treatises and other works covering all areas of philosophy and science. Of those, none survives in finished form. The approximately 30 works through which his thought was conveyed to later centuries consist of lecture notes (by Aristotle or his students) and draft manuscripts edited by ancient scholars, notably Andronicus of Rhodes, the last head of the Lyceum, who arranged, edited, and published Aristotle’s extant works in Rome about 60 BCE. The naturally abbreviated style of these writings makes them difficult to read, even for philosophers.
Where did Aristotle live?
After his father died about 367 BCE, Aristotle journeyed to Athens, where he joined the Academy of Plato. He left the Academy upon Plato’s death about 348, traveling to the northwestern coast of present-day Turkey. He lived there and on the island of Lésbos until 343 or 342, when King Philip II of Macedonia summoned him to the Macedonian capital, Pella, to act as tutor to Philip’s young teenage son, Alexander, which he did for two or three years. Aristotle presumably lived somewhere in Macedonia until his (second) arrival in Athens in 335. In 323 hostility toward Macedonians in Athens prompted Aristotle to flee to the island of Euboea, where he died the following year.
How did Aristotle influence subsequent philosophy and science?
Aristotle’s thought was original, profound, wide-ranging, and systematic. It eventually became the intellectual framework of Western Scholasticism, the system of philosophical assumptions and problems characteristic of philosophy in western Europe during the Middle Ages. In the 13th century St. Thomas Aquinas undertook to reconcile Aristotelian philosophy and science with Christian dogma, and through him the theology and intellectual worldview of the Roman Catholic Church became Aristotelian. Since the mid-20th century, Aristotle’s ethics has inspired the field of virtue theory, an approach to ethics that emphasizes human well-being and the development of character. Aristotle’s thought also constitutes an important current in other fields of contemporary philosophy, especially metaphysics, political philosophy, and the philosophy of science.
Where did Aristotle go after his father died?
After his father’s death in 367, Aristotle migrated to Athens, where he joined the Academy of Plato (c. 428–c. 348 bce ). He remained there for 20 years as Plato’s pupil and colleague. Get a Britannica Premium subscription and gain access to exclusive content. Subscribe Now.
What are Aristotle's two surviving works on logic and disputation?
It is possible that two of Aristotle’s surviving works on logic and disputation, the Topics and the Sophistical Refutations, belong to this early period. The former demonstrates how to construct arguments for a position one has already decided to adopt; the latter shows how to detect weaknesses in the arguments of others. Although neither work amounts to a systematic treatise on formal logic, Aristotle can justly say, at the end of the Sophistical Refutations, that he has invented the discipline of logic—nothing at all existed when he started.
Why do we need to do philosophy?
Everyone must do philosophy, Aristotle claims, because even arguing against the practice of philosophy is itself a form of philosophizing. The best form of philosophy is the contemplation of the universe of nature; it is for this purpose that God made human beings and gave them a godlike intellect.
Aristotle’s Deductive Approach in Mathematics
- Aristotle discussed two major concerns for the nature of mathematics. In one, he mentions that there must be some unprovable principles to avoid infinite regresses. And in the other, he mentions that the proofs should be explanatory. He mentioned two ways to start a proof, i.e., by axioms or by posits. Axioms are the statements that are accepted as...
Aristotle’s Criticism of Plato’s Mathematical Cosmology
- Aristotle criticizes Plato’s mathematical cosmology by claiming that the study of the motion of the sublunary body is Physics rather than mathematics. Therefore, he accepted Eudoxean astronomy as a helping hand in the study of the motion of heavenly bodies. As a part of the theory of natural places, Aristotle mentions that all heavenly bodies fall towards the centre of the universe, and th…
Problems About Mathematical Objects
- However, Aristotle rejected Plato’s mathematical cosmology, but he has gained the deepest knowledge of science from him only. On one hand, Aristotle treats mathematical sciences as a model of scientific knowledge, and on the other hand, he agrees with Plato’s assumption that authentic knowledge must have a real object. These two statements raise questions about the e…
Aristotle’s Dialectical Method
- Aristotle claims that dialects provide a path to the first principles of philosophical sciences. There is a list of aporia given in ‘Metaphysics’ out of which three aporias are related to the problem of mathematical objects. The way he had treated every problem in his first philosophy is aporetic as he didn’t try to break the resulting deadlock. We cannot say that ‘Analytics’ serves as a true guid…
Universal Mathematics
- There is an analogy of mathematics suggested by Aristotle, which mentions that there is a super science of mathematics that covers all continuous magnitudes and discrete quantities such as numbers. He claims that although mathematicians had proved theorems such as a:b=c:d⇒a:c=b:d, separately for numbers, lines, planes, and solids, now we have one universal pr…
Aristotle’s View of Infinite Divisibility and Continuity
- Some philosophers in Plato’s academy reported that lines constitute indivisible magnitude, whether it is a finite number or infinite number. Aristotle denied this hypothesis by building a theory of continuity and infinite divisibility. He claimed that a line comprising of an infinite number of potential points is equivalent to saying that a line can be divided anywhere on it, bringing pote…
Unit
- Greek mathematicians believed that numbers are the plural of units. According to them, a number is made up of countable figures (unit). One is not a number, it is a unit that will construct a number. Later on, some questions arose on this concept such as overlapping problems. This problem says that what is the guarantee that when 3 is added to 5 then the correct result is not …
Measure
- Greek mathematicians used a system in which all fractions are proper parts, that we now call unit fractions. 2/3 is an exception, otherwise, all other fractions can be represented in the form 1/n. For example, 2/5 can be expressed as a sum of 1/3 and 1/15. Greeks used the same system of measurement as we do like 1 foot is 12 inches. However, Aristotle shared a different view in whi…
Time
- Aristotle defined time as the number or count of change. He further distinguished two types of numbers, what is counted and by which we count. For example, 10 cows in a ground, here, the number counted is 10 and by which it is counted is 1 cow. In his view, time is a number where the change is measured by a unit of change.
Aristotle on The Infinite
- Aristotle denied the existing theory of infinity in mathematics and physics and defined it in his terms. He mentioned, According to him, in the case of magnitudes, infinitely small or large magnitudes do not exist. He also supported Anaxagoras thought, that from any given magnitude, it is always possible to take smaller. Hence, his remark about infinite magnitudes was in a differ…