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Topic: Infinite descent

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Method of infinite descent

	Infinite descent - Wikipedia, the free encyclopedia
		In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction.
		More precisely, his method of infinite descent was an exploitation in particular of the possibility of halving rational points on an elliptic curve E by inversion of the doubling formulae.
		In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of algebraic number theory and the study of L-functions.
	en.wikipedia.org /wiki/Infinite_descent (564 words)

	Infinite Descent versus Induction
		It is often said that the principle of "infinite descent", developed by Fermat in his study of Diophantine analysis, is equivalent to induction.
		Those who argue that infinite descent is equivalent to induction typically refer to a proof such as the following for the fact that the sum of the first n integers is n(n+1)/2: If not, there is a least integer, say N, for which this is false.
		Thus, there is some validity in the claim that Fermat's infinite descent concept is one of the proto-types for reasoning about infintesimals and limiting sequences, which is perhaps not surprising in view of Fermat's own contributions to the foundation of the differential calculus.
	www.mathpages.com /home/kmath144.htm (1225 words)

	Infinite Descent (Site not responding. Last check: 2007-10-21)
		In an "Infinite Descent" proof, you first assume that a set of numbers with a certain property exists.
		That is, the fact that the descent is infinite affords you the luxury of not needing to assume minimality in the first place.
		Two Proofs that there are no four squares in arithmetic sequence, both by infinite descent.
	mcraefamily.com /MathHelp/PythagInfiniteDescent.htm (418 words)

	[No title]
		In this work, finite sums were calculated to confirm the convergence of infinite sums and measure the coverage of the integers by the predecessor tree.
		In this work, the infinite summations of the densities of integers in various disjoint subsets converge to finite values which measure various aspects of the coverage of the integers by the predecessor tree.
		left descent assembly -- The recurring structural unit, consisting of a left descent including both its header extension and its terminating leaf node, whose infinite instantiations provide a certain fraction of the odd integers present in the predecessor tree.
	www-personal.ksu.edu /~kconrow/glossary.html (1776 words)

	[No title]
		Inductively, we expect an infinite descent with terminated paths accumulating as the sum of 2i/3(i+1).
		Assertion: this formulation applies whether we are considering the infinite set of elements in the extension set of any integer or the totality of the elements of all the extension sets in the predecessor tree.
		The contribution of the density of the integers in each cell in the relevant table to the total number of integers is its entry (its frequency of occurrence) divided by 2^a*3^b (its density among the integers).
	www-personal.ksu.edu /~kconrow/counting.html (2300 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		With the benefit of a few centuries of hindsight we now know why infinite descent worked in his case: there's a lot of algebraic structure lurking in the solution set to x^4 = y^4 = z^2 (it's a mathematical group, as well as being topologically a donut) which just isn't there in the other cases.
		If he had tried infinite descent here, it would have failed, since (the necessary high-falutin' auxiliaries being absent) there is no way to take one such combination to give another one.
		Like I say, it's possible that in the more general case Fermat was able to _transform_ the problem into one where infinite descent would work, but as it stands, the Fermat problem is not amenable to infinite descent except (barely) in the case n=4.
	www.math.niu.edu /~rusin/known-math/96/flt.pop (838 words)

	PlanetMath: infinite descent (Site not responding. Last check: 2007-10-21)
		with certain properties implies that there exists a smaller one with these properties, then there are infinitely many of these, which is impossible.
		See here for a discussion of infinite descent vs. induction.
		This is version 10 of infinite descent, born on 2004-02-02, modified 2004-02-13.
	planetmath.org /encyclopedia/InfiniteDescent.html (134 words)

	Infinity (Site not responding. Last check: 2007-10-21)
		The very suggestion that certain objects, infinite in number, are "equal in magnitude" to others implies that not all such objects, infinite in number, are so equal.
		Galileo proposed adding an infinite number of infinitely small gaps to the smaller length to make it equal to the larger yet allow them to have the same number of points.
		the totality of all numbers is infinite, and that the number of squares is infinite.; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and, finally, the attributes "equal", "greater", and "less" are not applicable to the infinite, but only to finite quantities.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Infinity.html (4009 words)

	Fermat's Last Theorem: Infinite Descent
		He did provide one example of this method in his proof that the area of a right triangle cannot be equal to a square number.
		An elegant application of this proof is found in the case of FLT: n=4 where the proof rests on the method of infinite descent and the solution to Pythagorean Triples.
		In this case, infinite descent is impossible since it contradicts the Well Ordering Principle.
	fermatslasttheorem.blogspot.com /2005/05/infinite-descent.html (714 words)

	The internal consistency of arithmetic with infinite descent, Yvon Gauthier (Site not responding. Last check: 2007-10-21)
		Arithmetic without an induction postulate, but with infinite descent, is Fermat Arithmetic coupled with Kronecker's "general arithmetic" of indeterminates.
		The main idea is to interpret a local (constructive) logic with a local "effinite" quantifier in a polynomial translation and show how logic is eliminated by infinite descent in the same way as the content is exhausted in the decomposition of polynomials (or forms) where the method of infinite descent is at work.
		Although not always effective, infinite descent provides a finite constructivist setting for an arithmetic that encompasses most of number theory and a large part of algebraic or arithmetic geometry.
	projecteuclid.org /getRecord?id=euclid.rml/1081878064 (196 words)

	Encyclopedia entries starting with INF
		Infinite Energy is a monthly magazine published in New Hampshire that details theories and experiments concerning alternative energy, usually based on concepts that are outside mainstream science.
		Infinitism is a theory in epistemology the branch of philosophy that treats of the possibility, nature, and means of knowledge.
		It is called the "infinitive" because the verb is usually not made "finite", or limited by inflection.
	encycl.opentopia.com /I/IN/INF (10299 words)

	ER Close Article Summer 2001 - SWC (Site not responding. Last check: 2007-10-21)
		But the separation of reality into the observer and the observed and the logic of infinite descent forces us to conclude that consciousness cannot be composed of quanta of matter or energy.
		The calculus of distinctions journey follows the logic of infinite descent (13), while the same journey experienced on a conscious spiritual level is a path of infinite ascent.
		The importance of combining the intellectual understanding of the infinite descent of distinctions with the direct experience of the infinite ascent of consciousness cannot be overstated.
	www.swcp.com /~hswift/swc/Summer01/close0101.htm (3489 words)

	Issues » Science and Culture » Mathematical Infinity and Kabbalah
		In fact, this is characteristic of infinite sets, namely, any infinite set has the same number of elements (via a one-to-one correspondence) as some of its subsets which are distinctly smaller than itself.
		The culmination of this process of infinite descent of spiritual worlds is the existence of the physical world.
		We needed the infinite gaps to represent the infinite transition (contraction) between Atzilus and B’riah, between Briah and Yitzira and between Yitzira and Asiya.
	www.algemeiner.com /generic.asp?ID=906 (5243 words)

	[No title]
		But it seems like the conclusion should be that the overdetermination argument needs addressing, for in a finite descent it strips causal potency away from higher-level entities and in an infinite descent it strips causal potency away from all entities.
		Schaffer states that an infinite descent is an egalitarian metaphysic wherein all levels are fully real and causally potent.
		H2 is the infinite descent, equivalent to a&b&c&I, where I is the infinite substructure of the quarks and leptons posited by Schaffer.
	philsci-archive.pitt.edu /archive/00000215/00/comments_on_schaffer.doc (2858 words)

	The method of descent (Site not responding. Last check: 2007-10-21)
		In the language of the ladder metaphor, if you know you can't reach any rung without first reaching a lower rung, and you also know you can't reach the bottom rung, then you can't reach any rungs.
		The above is often called finite descent, to distinguish it from the variant method known as infinite descent:
		The classic example of infinite descent is in Fermat's proof that there are no positive integer solutions to
	mathcircle.berkeley.edu /BMC4/Handouts/induct/node7.html (264 words)

	Set Theory and Logic - Numericana
		Although this common sense approach is perfectly sound, its seems the basic distinction between finite and infinite sets ought not to depend explicitely on the properties of something as complicated as the integers.
		At the most fundamental level (which may not be the most intuitive one) the existence of infinite sets is built into the axioms of modern set theory.
		On the other hand, infinite sets must be part of the theory, if it is to be of any use whatosever...
	home.att.net /~numericana/answer/sets.htm (3734 words)

	About "Fermat and His Method of Infinite Descent" (Site not responding. Last check: 2007-10-21)
		The basic method of the infinite descent is as follows: Assume one wants to prove no solution exists with a certain property.
		Repeat this argument an infinite number of times, thus infinitely descending through all integers.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/17554.html (120 words)

	Descent II from CD-ROM Access
		If you thought Descent went to the edge of 360 degree 3D power-blasting sensory insanity, then prepare yourself.
		"Descent II is also friendlier, although this characteristic has a mixed impact on game play.
		"Descent II is a solid extension of the Descent line.
	www.cdaccess.com /html/pc/descent2.htm (669 words)

	Fermat's Infinite Descent (Site not responding. Last check: 2007-10-21)
		Pierre de Fermat's method of infinite descent is beautifully illustrated by the proofs of the following two propositions in Number Theory.
		In a letter to Carcavi describing his methods he wrote "As ordinary methods, such as are found in books, are inadequate to proving such difficult propositions, I discovered at last a most singular method...which I called the infinite descent.
		At first I used it to prove only negative assertions, such as 'There is no right angled triangle in numbers whose area is a square'...
	www.mathpages.com /home/kmath288.htm (535 words)

	Diophantus' 20th Problem and Fermat's Last Theorem for n = 4 (Site not responding. Last check: 2007-10-21)
		The proofs are completed using Fermat's "wonderful" method of infinite descent.
		This problem was negatively solved by Fermat in the 17th century, who used the "wonderful" method ("ipse dixit" Fermat) of infinite descent.
		This method, which is, historically, the first use of induction, consists in producing smaller and smaller non-negative integer solutions assuming that one exists; this naturally leads to a "reductio ad absurdum" reasoning because we are bounded by zero.
	pauillac.inria.fr /coq/contribs/Fermat4.html (255 words)

	Related Works
		Infinite Ascent and Descent: Proof of the Primacy of Consciousness
		6, No. 1, Summer, 2001, focuses on the similarity between the mathematical logic of infinite descent used by the famous French mathematician Pierre de Fermat, and the infinite ascent spiritual inquiry of Ramana Marishi.
		Infinite Continuity, a Theory Unifying Relativity and Quantum Physics, 1990, E. Close, Currently out of print.
	www.transcendentalphysics.com /relatedworks.html (192 words)

	Descent II: The Infinite Abyss from CD-ROM Access
		The megahit Descent II sent you over the edge into 360 degree pounding sensory insanity.
		Now Descent II: The Infinite Abyss intensifies the insanity, launching you on a headlong plung into mind-bending 3D destruciton with the all-new Vertigo Series.
		Bend your mind around the infinite possibilities for destruction as you blast your way through the ultimate Descent.
	www.cdaccess.com /html/pc/desc2ia.htm (469 words)

	Brainware Home Page (Site not responding. Last check: 2007-10-21)
		The Descent Mission Builder II demo is the same as the released version of the editor except that it limits the size of the levels that can be saved.
		Descent / Descent 2 / Levels of the World / Vertigo Series / Mission Builder II Contains 3 Disks and an Instruction Manual:
		Descent Mission Builder II because it is property of Interplay.
	www.brainware.net /index.php?page=533289.txt (251 words)

	Open to man are the two ways
		Always open to man are the two ways to infinite ascent or to infinite descent.
		For the All-Wise Creator has implanted in the nature of man an infinite impotence and poverty so that he may be a comprehensive mirror reflecting the infinite manifestations of God’s Compassion and Power, and of His Richness and Generosity.
		He is the guest of a Generous One Who has put the treasuries of His infinite Compassion at his disposal, and subjugated to him His peerless works of creative power and even His special servants.
	www.islamanswers.net /happiness/twoWays.htm (4918 words)

	IMO 1988/B3 solution (Site not responding. Last check: 2007-10-21)
		But we cannot have an infinite sequence of decreasing numbers all greater than zero, so we must eventually get A = ka - b = 0.
		k was unchanged during the descent, so k is a perfect square.
		Note that jumping straight to the minimal without the infinite descent avoids some of the verification needed in the infinite descent.
	www.kalva.demon.co.uk /imo/isoln/isoln886.html (453 words)

	Infinite Descent - but not into Hell! - Resonance - February 2003
		Everyone knows the principle of mathematical induction (PMI for short); it is now standard fare even at the high-school level.
		Curiously, very few seem to know its close relative -- the principle of descent (PD for short); curious, because the two principles are complementary to one another.
		In this article we study some typical applications of this principle.
	www.ias.ac.in /resonance/Feb2003/Feb2003p42-55.html (96 words)

	The Twenty Third Word
		In regard to destruction and evil, the evil-commanding soul may commit infinite crimes, but concerning creativity and good, its power is extremely little and partial.
		For the All-Wise Creator has included in man’s nature an infinitely vast impotence and boundlessly huge poverty, so that he can be an extensive mirror containing the innumerable manifestations of an All-Powerful and Compassionate One Whose power is infinite, an All-Generous All-Rich One Whose wealth is boundless.
		He is the guest of One so generous that infinite treasuries of mercy have been opened to him and innumerable unique beings and servants subjugated to him.
	www.nur.org /newfiles/risale-inurcollection/words/word23a.htm (4738 words)

	Mudd Math Fun Facts: Irrationality by Infinite Descent
		The reasoning about an infinite sequence of decreasing positive integers is another form of mathematical induction (both depend on the fact that any non-empty subset of the positive integers has a least element).
		This form of reasoning was invented by Fermat and is called the method of infinite descent.
		Suggestions or references can be made using our suggestion form.
	www.math.hmc.edu /funfacts/ffiles/30005.5.shtml (424 words)

	Problem Sets from Mathcamp (Site not responding. Last check: 2007-10-21)
		While I'm quite certain that all of my students did their problem sets promptly, I will still archive them here for those students or any other interested viewers (and, indeed, for myself).
		The class covered proof by contradiction, the pigeon hole principle, proof by induction, infinite descent, invariants, and whatever else could be thrown into that wonderful stew.
		It served as an essential toolbox for students to draw on throughout the rest of the camp.
	www.math.uiuc.edu /~dzaharo2/teaching/mathcampsets (388 words)

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