Law of Continuity

Last updated

The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". [1] Kepler used The Law of Continuity to calculate the area of the circle by representing the latter as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely-many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations, from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. A mathematical implementation of the law of continuity is provided by the transfer principle in the context of the hyperreal numbers.

Gottfried Wilhelm Leibniz German mathematician and philosopher

Gottfried Wilhelm (von) Leibniz was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy. His most notable accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have generally favored Leibniz's notation as the conventional expression of calculus. It was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of all digital computers.

Nicholas of Cusa German philosopher, theologian, jurist, and astronomer

Nicholas of Cusa, also referred to as Nicholas of Kues and Nicolaus Cusanus, was a German philosopher, theologian, jurist, and astronomer. One of the first German proponents of Renaissance humanism, he made spiritual and political contributions in European history. A notable example of this is his mystical or spiritual writings on "learned ignorance," as well as his participation in power struggles between Rome and the German states of the Holy Roman Empire.

Johannes Kepler 17th-century German mathematician, astronomer and astrologer

Johannes Kepler was a German astronomer, mathematician, and astrologer. He is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonices Mundi, and Epitome Astronomiae Copernicanae. These works also provided one of the foundations for Newton's theory of universal gravitation.

Contents

A related law of continuity concerning intersection numbers in geometry was promoted by Jean-Victor Poncelet in his "Traité des propriétés projectives des figures". [2] [3]

In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem.

Jean-Victor Poncelet French engineer and mathematician

Jean-Victor Poncelet was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work Traité des propriétés projectives des figures is considered the first definitive text on the subject since Gérard Desargues' work on it in the 17th century. He later wrote an introduction to it: Applications d’analyse et de géométrie.

Leibniz's formulation

Leibniz expressed the law in the following terms in 1701:

In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included (Cum Prodiisset). [4]

In a 1702 letter to French mathematician Pierre Varignon subtitled “Justification of the Infinitesimal Calculus by that of Ordinary Algebra," Leibniz adequately summed up the true meaning of his law, stating that "the rules of the finite are found to succeed in the infinite." [5]

Pierre Varignon French mathematician

Pierre Varignon was a French mathematician. He was educated at the Jesuit College and the University of Caen, where he received his M.A. in 1682. He took Holy Orders the following year.

The Law of Continuity became important to Leibniz's justification and conceptualization of the infinitesimal calculus.

See also

Related Research Articles

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Non-standard analysis calculus using a logically rigorous notion of infinitesimal numbers

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.

Mathematical analysis Branch of mathematics

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

Hyperreal number

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

Infinitesimal extremely small quantity in calculus; thing so small that there is no way to measure them

In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.

<i>Method of Fluxions</i> book by Isaac Newton

Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxion is Newton's term for a derivative. He originally developed the method at Woolsthorpe Manor during the closing of Cambridge during the Great Plague of London from 1665 to 1667, but did not choose to make his findings known. Gottfried Leibniz developed his form of calculus independently around 1673, 7 years after Newton had developed the basis for differential calculus, as seen in surviving documents like “the method of fluxions and fluents..." from 1666. Leibniz however published his discovery of differential calculus in 1684, nine years before Newton formally published his fluxion notation form of calculus in part during 1693. The calculus notation in use today is mostly that of Leibniz, although Newton's dot notation for differentiation for denoting derivatives with respect to time is still in current use throughout mechanics and circuit analysis.

Leibnizs notation

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0.

In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently discovered calculus in the mid-17th century. However, both inventors claimed that the other had stolen his work, and the Leibniz-Newton calculus controversy continued until the end of their lives.

The term differential is used in calculus to refer to an infinitesimal change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx. The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.

Natura non facit saltus has been an important principle of natural philosophy. It appears as an axiom in the works of Gottfried Leibniz, one of the inventors of the infinitesimal calculus. It is also an essential element of Charles Darwin's treatment of natural selection in his Origin of Species. The Latin translation comes from Linnaeus' Philosophia Botanica.

In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal , the unique real infinitely close to it, i.e. is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.

Infinity mathematical concept of something without any limit

Infinity is a concept describing something without any bound, or something larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. This idea is also at the basis of infinitesimal calculus.

Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.

This is a glossary of terms that are or have been considered areas of study in mathematics.

In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded. Thus, if is finite and is infinitesimal, then one sets

References

  1. ↑ Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science. doi : 10.1007/s10699-011-9223-1 See arxiv
  2. ↑ Poncelet, Jean Victor. Traité des propriétés projectives des figures: T. 1. Ouvrage utile à ceux qui s' occupent des applications de la géométrie descriptive et d'opérations géométriques sur le terrain." (1865), pp. 13–14
  3. ↑ Fulton, William. Introduction to intersection theory in algebraic geometry. No. 54. American Mathematical Soc., 1984, p. 1
  4. ↑ Child, J. M. (ed.): The early mathematical manuscripts of Leibniz. Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child. Chicago-London: The Open Court Publishing Co., 1920.
  5. ↑ Leibniz, Gottfried Wilhelm, and Leroy E. Loemker. Philosophical Papers and Letters. 2d ed. Dordrecht: D. Reidel, 1970, p. 544
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.