Reductio ad absurdum has its roots in ancient Greek philosophy and is a staple of the Socratic Method. a matter of (logico-conceptual) necessity (“There are infinitely many prime numbers”). tion (rĭ-dŭk′shən) n. 1. Dave Magadan nearly won a National League batting title in 1990. Clearly the antecedent/premiss “I = you” is absurd. Accordingly, since that initial commensurability assumption engendered a contradiction, we have no alternative but to reject it. I just never think of you as a woman, except when I'm trying to peek through your windows when you are getting changed." By: MerriWyllow. The use of such reductio argumentation was common in Greek mathematics and was also used by philosophers in antiquity and beyond. Since p ⊢q yields ⊢p →q this principle can be established as follows: (4) ⊢ ~(p & ~p) → ~p from (3) by contraposition, (5) ⊢ ~(p & ~p) by the Law of Contradiction. Aristotle employed it in the Prior Analytics to demonstrate the so-called imperfect syllogisms when it had already been used in dialectical contexts by Plato (see Republic I, 338C-343A; Parmenides 128d). Basically, it is an absurd or ridiculous argument. Thus consider. This is a process of justificating argumentation that proceeds as follows when the object is to establish a certain conclusion p: (2) Provide argumentation that derives p from this assumption. Nicholas Rescher "We can then inscribe in the secundary circle an equilateral polygon of 4n sides such that its area is greater than that of the ellipse. Revisited. Reductio ad Absurdum The Feng Shui Way: A Catastrophe For City Planning Kyle Hill. For example, consider the following schematic theses: In this context, the supposition that A can be refuted by a reductio ad absurdum. It is a far Per impossible reasoning also proceeds from a patently impossible premiss. With our reductio ad absurdum exercise, we can be fairly certain that the specificity of the Stanford test is in the A-plus range. As indicated above, this sort of proof of a thesis by reductio argumentation that derives a contradiction from its negation is characterized as an indirect proof in mathematics. This takes three principal forms according as that untenable consequence is: The first of these is reductio ad absurdum in its strictest construction and the other two cases involve a rather wider and looser sense of the term. Deductive arguments can be used to refute a view, as well as to prove a view. Reductio ad absurdum Profile. And so in this context reductio reasoning can also come into operation. Q. Negation elimination from step 8. i.e. And their position accordingly runs out into absurdity. End of third level hypothesis from step 5. the game" (Coxeter and Greitzer 1967, p. 16; Hardy 1993, p. 34). We've got 0 anagrams for reductio ad absurdum » Any good anagrams for reductio ad absurdum? The supposition that x ÷ 0 qualifies as a well-defined quantity is thereby refuted. Reductio Ad Absurdum MAT 142 – RSCC – Scott Surgent Larger arguments can be difficult to show validity using truth tables simply because they are too lengthy. . Hardy, G. H. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. Examples of Reductio Ad Absurdum: An antinomy violates "non(A and non-A)", the law of noncontradiction. If the idea remains sensible, then you can feel confident that it is a good idea. Instead, we can “force” and argument to be invalid, then work backwards and see if this is possible. No doubt the judge will rule something like “Finding for the plaintiff would lead ad absurdum. The task was to write a 700-word article on an area of STEM, and I was luckily awarded 1st place for the Under 18 category for Maths. It has the form of a reductio ad absurdum argument, and usually proceeds as follows: The proposition to be proved, P, is assumed to be false. Thus, suppose I want to prove that the direct road from two given places is the shortest, I should say, “It must either be the shortest or not the shortest. Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. Then by the Pythagorean theorem we have it that d² = 2s². Improper positioning had him stumbling, fumbling, and tumbling all over the house. An example is provided by the exchange between Socrates and his accusers who had charged him with godlessness. This means, in specific, that one cannot establish (∃x)Fx by deducing an absurdity from (∀x)~Fx. Here we have to deal with literally impossible suppositions that are not just dramatically but necessarily false thanks to their logical conflict with some clearly necessary truths, be the necessity at issue logical or conceptual or mathematical or physical. Reductio Ad Absurdum is disproving an argument by showing the absurdity of following it through to a logical conclusion. But here inspiration is divine inspiration such a daimonism is supposed to be a being inspired by a god. (On the historical background see T. L. Heath, A History of Greek Mathematics [Oxford, Clarendon Press, 1921].). Accordingly, intuitionists would not let us infer the existence of invertebrate ancestors of homo sapiens from the patent absurdity of the supposition that humans are vertebrates all the way back. The act or process of reducing. For if A were conjoined to these premisses, we will arrive at both D and not-D which is patently absurd. )Immanuel Kant taught that interpersonal practices cannot represent morally appropriate modes of procedure if they do not correspond to verbally generalizable rules in this way. player may offer the sacrifice of a pawn or even a piece, but a mathematician offers Even as individual statements can prove to be self-contradictions, so a plurality of statements (a “doctrine” let us call it) can prove to be collectively inconsistent. Now suppose (by way of a reductio assumption) that d and s were commensurable in terms of a common unit u, so that d = n x u and s = m x u, where m and n are whole numbers (integers) that have no common divisor. Assoc. Use the save command to save work in progress to resume later, or to save a completed proof. Reductio ad absurdum. WPA, Reductio Ad Absurdum Unemployment’s continued sluggishness has some proposing rather radical government solutions—which would likely do more harm than good. English Gematria, Hebrew Gematria and Jewish Gematria and Numerology Even if there were no primes less than 1,000,000,000, the number of primes would be infinite. In order to answer this question, we first have to understand what "reductio ad absurdum" is. https://mathworld.wolfram.com/ReductioadAbsurdum.html. Save and restore. To all intents and purposes, then, counterfactuals can serve distinctly factual purpose.And so, often what looks to be a per impossible conditional actually is not. In its most general construal, reductio ad absurdum – reductio for short – is a process of refutation on grounds that absurd – and patently untenable consequences would ensue from accepting the item at issue. Reductio ad absurdum Profile. Reductio ad absurdum is a mode of argumentation that seeks to establish a contention by deriving an absurdity from its denial, thus arguing that a thesis must be accepted because its rejection would be untenable. Reductio ad Absurdum. In a proof by contradiction, it is shown that the denial of the statement being proved results in such a contradiction. Archimedes estarted his double 'reductio ad absurdum'. But while all three cases fall into the range of the term as it is commonly used, logicians and mathematicians generally have the first and strongest of them in view. Such argumentation is in effect simply an implementation of the above-stated principle with ~p standing in place of p. As this line of thought indicates, reductio argumentation is a special case of demonstrative reasoning. Consider division by zero. calculate its consequences, and make revisions to the initial assumptions if necessary in order to achieve 2-valued Completeness and Consistency. Coxeter, H. S. M. and Greitzer, S. L. Geometry Feng Shui Calculator by Dave Fayram. And build the new warehouse from the materials salvaged by demolishing the old. The contrast between reductio and per impossible reasoning conveys an interesting lesson. The save command takes one argument, a string, the name of the module where the proof is saved: >>> save('ex912') Saved in ex912.py Home > Adelaide > Community > Reductio ad absurdum . In both cases alike we begin with a situation of exactly the same basic format, namely a conflict of contradiction between an assumption of supposition and various facts that we already know. This is the road we propose now, revision of assumptions justiﬁed by means of Reductio ad Absurdum reasoning. 176-94. Dick Van Dyke’s home had terrible feng shui. He ducked. Reductio ad absurdum, (Latin: “reduction to absurdity”), in logic, a form of refutation showing contradictory or absurd consequences following upon premises as a matter of logical necessity. This means that n must be even, since only even integers have even squares. If you can't prove a contradiction from a set of sentences, then it has a model. The result of reducing: a reduction in absenteeism. In the one (reductio) case we seek to refute and rebut that assumptions so as to establish its negation, and in the other (per impossible) case we are trying to establish an implication – to validate a conditional. Explore anything with the first computational knowledge engine. Keep the old warehouse intact until the new one is constructed. His accusers here become enmeshed in self-contradiction. If posible, let the secundary circle greater than the ellipse. Share. Not only can a self-inconsistent statement (and thereby a self-refuting, self-annihilating one) but also a self-inconsistent process or practice or principle of procedure can be “reduced to absurdity.” For any such modus operandi answers to some instruction (or combination thereof), and such instruction can also prove to be self-contradictory. Solve advanced problems in Physics, Mathematics and Engineering. The only valid way to establish existence, so they maintain, is by providing a concrete instance or example: general-principle argumentation is not acceptable here. Accordingly, the above-indicated line of reasoning does not represent a postulated principle but a theorem that issues from subscription to various axioms and proof rules, as instanced in the just-presented derivation. It is a style of reasoning that has been employed throughout the history of mathematics and philosophy from classical antiquity onwards. If he did that, then I’m the Shah of Persia. In particular, such an utterly impossible supposition may negate: Suppositions of this sort commonly give rise to per impossible counterfactuals such as: A somewhat more interesting mathematical example is as follows: If, per impossible, there were a counterexample to Fermat’s Last Theorem, there would be infinitely many counterexamples, because if xk + yk = zk, then (nx)k + (ny)k = (nz)k, for any k. With such per impossible counterfactuals we envision what is acknowledged as an impossible and thus necessarily false antecedent, doing so not in order to refute it as absurd (as in reductio ad absurdum reasoning), but in order to do the best one can to indicate its “natural” consequences. Hints help you try the next step on your own. A sauce that has been thickened or concentrated by boiling. If I were in your place (that is, if I were circumstanced in the condition in which you now find yourself), then I would consult the doctor. Examples of this would be: More loosely, there are also instructions that do not automatically result in logically absurd (self-contradictory) conclusions, but which open the door to such absurdity in certain conditions and circumstances. As for example: Definitions or specifications that are in principle unsatisfiable are for this very reason absurd. The difference accordingly is not so much theoretical as functional – it is a pragmatic difference in objectives. a law of nature (“Water freezes at low temperatures”). Elsewhere the principle is almost universally viewed as a mode of argumentation rather than a specific thesis of propositional logic. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. End of second level hypothesis from step 11. If you could do reductio ad absurdum in your proof calculus, you could prove any consequence if you could prove a contradiction from every unsatisfiable set of sentences. The difference lies entirely in pragmatic considerations, in what we are trying to accomplish. 5. What does reductio ad absurdum mean? Reductio Ad Absurdum. 13. If you were transported through space faster than the speed of light, then you would return from a journey younger than at the outset. She could just as well have also opted not just for the house but also for the bank account and the business, thereby effectively disinheriting the daughter, which was clearly not the testator’s wish.” Here we have a juridical reductio ad absurdum of sorts. 10-H2: P => Q Conditional Proof. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Measurement (cont’d)!Case 1 " Suppose A>T then A-T is positive " Archimedes knew by inscribing a square within his circle and repeatedly bisecting it he could create a regular polygon with area Z with A-Z < A-T " Adding Z+T-A to both sides of A-Z < A-T! This work is an imaginary letter from a deistic god to the human race. to be false. Subscribe to Reductio ad absurdum . Now, using density of Q in R, ﬁnd a … A virtually equivalent formulation of the very point at issue with these two contentions is: However, the designation per impossible indicates that it is the conditional itself that concerns us. Essentially, the argument is reduced to its absurdity. Now we know that. Some conditionals that instantiate this latter sort of situation are: What we have here are consequences that are absurd in the sense of being obviously false and indeed even a bit ridiculous. The #1 tool for creating Demonstrations and anything technical. Propositional reductio is based on the following line of reasoning: Here ⊢ represents assertability, be it absolute or conditional (that is, derivability). The usual explanations of reductio fail to acknowledge the full extent of its range of application. Conditional Proof. And this means that m and n, both being even, will have common divisors (namely 2), contrary to the hypothesis that they do not. A proof of inference arising from the demonstration that every other hypothesis involves an absurdity. Meaning of reductio ad absurdum. Email: rescher+@pitt.edu https://mathworld.wolfram.com/ReductioadAbsurdum.html. Immanuel Kant’s entire discussion of the antinomies in his Critique of Pure Reason was based on reductio argumentation. His will specifies that his sister is to have any three of the valuables in his estate and that his daughter is to inherent the rest. From MathWorld--A Wolfram Web Resource. "Of course you are more than a chess piece to me - I do care about you. The difference at bottom thus lies not in the nature of the inference at issue, but only in what we are trying to achieve by its means. (This sort of reasoning is common in legal contexts. A form of refutation commonly used in Philosophy and other fields of inquiry is "Reductio ad absurdum "(literally "reducing to absurdity".) Since the rule-conforming practices thus reduce to absurdity upon their general implementation, such practices are adjudged morally unacceptable. Such practices as stealing (that is, taking someone else’s possessions without due authorization) or lying (i.e. An example my help to clarify matters. ~(~Q) <=> Q. Walk through homework problems step-by-step from beginning to end. Whitehead and Russell in Principia Mathematica characterize the principle of “reductio ad absurdum” as tantamount to the formula (~p →p) →p of propositional logic. (Compare Aristotle, Rhetorica 1398a12 [II xxiii 8].). Here's a "reductio ad absurdum" example that illustrates some of the problem: Imagine that you're walking along perfectly level ground for 100 meters. Proof by contradiction (also known as indirect proof or the technique or method of reductio ad absurdum) is just one of the few proof techniques that are used to prove mathematical propositions or theorems.. The incommensurability thesis is accordingly established. Reductio ad absurdum is a mode of argumentation that seeks to establish a contention by deriving an absurdity from its denial, thus arguing that a thesis must be accepted because its rejection would be untenable. End of second level hypothesis from step 4. You're carrying a GPS device that records the elevation every 5 meters, but the signal wanders up and down a bit; it thinks that each point is 2m above or 2m below the previous one. Practice online or make a printable study sheet. A method of proof which proceeds by stating a proposition and then showing that it results in a contradiction, thus demonstrating the proposition The reasoning involved here provides the basis for what is called an indirect proof. In elaborating this accusation, these opponents also accused Socrates of believing in inspired beings (daimonia). A method of proof which proceeds by stating a proposition and then showing that it results in a contradiction, thus demonstrating the proposition to be false. The executor refuses to make this distribution and the sister takes him to court. 3. New York: Cambridge University Press, p. 34, 1993. Even as instructions can issue in absurdity, so can definitions and explanations. Along these lines, a practical rule of procedure or modus operandi would be reduced to absurdity when it can be shown that its actual adoption and implementation would result in an anomaly.Consider an illustration of this sort of situation. Reductio ad absurdum is a type of argument that seeks to make a point by using absurdity. Free Online Scientific Notation Calculator. and the Probabilistic Reductio ad Absurdum Jeff Loveland Communicated by B. Bru In 1777 the naturalist Georges-Louis Ledere de Buffon published an "Essai d'arithmétique morale" in the fourth Supplément to his multi- volume chef-d'oeuvre, the Histoire naturelle. Proof by Contradiction. Use of this Latin terminology traces back to the Greek expression hê eis to adunaton apagôgê, reduction to the impossible, found repeatedly in Aristotle’s Prior Analytics. The principle of reductio ad absurdum is "A or non-A", the law of tertium non datur. Despite its departure from what is strictly speaking so construed – conditionals with self-contradictory – time to time conclusions – this sort of thing is also characterized as an attenuated mode of reductio. Unlimited random practice problems and answers with built-in Step-by-step solutions. 11. English Gematria, Hebrew Gematria and Jewish Gematria and Numerology But this means that m must be even (by the same reasoning as before). The approach of proof by contradiction is simple yet its consequence and result are remarkable. A first baseman for the New York Mets at the time, “Mags” finished with a .328 average, seven percentage points behind Willie McGee.A big final game of the season could have pushed him over the top — the math showed as much — although the odds were against him. The task of the present discussion is to explain the modes of reasoning at issue with reductio and to illustrate the work range of its applications. If we are unable to show that this can be done, then the argument is valid. Subscribe List an Event or Business Invite a Writer Write for WN Writers Other Locations. This page list all the various possible anagrams for the sentence reductio ad absurdum.Use it for solving word puzzles, scrambles and for writing poetry, lyrics for your song or coming up with rap verses. 9. That is, ¬ P is true. If you could do reductio ad absurdum in your proof calculus, you could prove any consequence if you could prove a contradiction from every unsatisfiable set of sentences. So, if a set of sentences doesn't have a model, you could prove a contradiction from it. T < Z. The futon in the living What we deal with here is an argument of the pattern: From the situation, (to-be-refuted assumption + a conjunction of preestablished facts) ⊢ contradiction. Weisstein, Eric W. "Reductio ad Absurdum." ... She blushed deep red and threw a calculator at his head. Follow/Fav Reductio ad Absurdum. But this view is idiosyncratic. In the words of G. H. Hardy, "Reductio ad absurdum, groups of propositions or theses (that is, doctrines or positions or teachings). For Kant, generalizability is the acid test of the acceptability of practices in the realm of interpersonal dealings. Following the hint, we prove by contradiction (reductio ad absurdum) that r + √ 2 is irrational when r ∈ Q. Value of converter in Gematria is 1063, Online Gematria Calculator with same phrases values search and words. September 25, 2013. (If there were a common divisor, we could simply shift it into u.) Kenneth Rice uses a type of argument known as reductio ad absurdum, in which one shows that accepting the targeted claim leads to absurd or impossible consequences. For at the very minimum such a refutation is a process that can be applied to. A man dies leaving an estate consisting of his town house, his bank account of $30,000, his share in the family business, and several pieces of costume jewelry he inherited from his mother. one proceeds to conclude the denial of that to-be-refuted assumption via modus tollens argumentation. Actually implementing this rule in all eligible cases – its generalized utilization across the board – would yield an unacceptable and untoward result so that the rule could self-destruct in its actual unrestricted implementation. This paper introduces a new method for deﬁning the argumenta tive semantics of At first, I couldn’t decide what to write on – Euclidean geomet This works only if there is faulty logic in the argument to begin with. reductio ad absurdum. They would maintain that in such cases where we are totally in the dark as to the individuals involved we are not in a position to maintain their existence. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. In one section of the "Essai," he attempted to calculate the prob- The amount by which something is lessened or diminished: a reduction of 12 percent in violent crime. A classic instance of reductio reasoning in Greek mathematics relates to the discovery by Pythagoras – disclosed to the chagrin of his associates by Hippasus of Metapontum in the fifth century BC – of the incommensurability of the diagonal of a square with its sides. Join the initiative for modernizing math education. M. Dorolle, “La valeur des conclusion par l’absurde,”. H2: Q Start of new second level hypothesis. But even the slightest heed of what is communicatively occurring here shows that what is at issue is not this just-stated impossibility but a counterfactual of the format: Only by being perversely literalistic could the absurdity of that antecedent be of any concern to us. University of Pittsburgh It is a style of reasoning that has been employed throughout the history of mathematics and philosophy from classical antiquity onwards. > >Subject: reductio ad absurdum Date: Tue, 02 May 2006 19:54:57 -0500 Theodicy (adjectival form "theodicean") is a specific branch of theology and philosophy that attempts to reconcile the existence of evil or suffering in the world with the assumption of a … telling falsehoods where it suits your convenience) are rules inappropriate, so Kant maintains, exactly because the corresponding maxims, if generalized across the board, would be utterly anomalous (leading to the annihilation of property- ownership and verbal communication respectively. Information and translations of reductio ad absurdum in the most comprehensive dictionary definitions resource on the web. 4. 12. Reductio ad Absurdum: An argument that proves a statement or idea false by assuming it to be true, following its logical conclusions, and then finding contradictions or absurdities that it creates. Many such cases are discussed in David Daube Roman Law [Edinburgh: Edinburgh University Press, 1969], pp. Indeed, if for a rational r, the number x = r + √ 2 were rational, then √ 2 = x − r would have been rational, which is false. The reasoning at issue runs as follows: Let d be the length of the diagonal of a square and s the length of its sides. This is the resulting elevation profile: If you add up the gain from all of the blue ascent segments, y… Definition of reductio ad absurdum in the Definitions.net dictionary. Our concern is with the character of that consequence relationship rather than with the antecedent or consequent per so. But now n² = (2k)² = 4k² = 2m², so that 2k² = m². An ostrich is a species of flightless bird. And at this point Socrates has a ready-made defense: how can someone disbelieve in gods when he is acknowledged to believe in god-inspired beings. Share: email facebook twitter. The mathematical school of so-called intuitionism has taken a definite line regarding the limitation of reductio argumentation for the purposes of existence proofs. So n = 2k. Hence it is untenable (false) in the context of this family of givens.When someone is “caught out in a contradiction” in this way their position self-destructs in a reduction to absurdity. Take your idea or principle and apply it to an extreme degree in an absurd situation. By the standards of Napoleonic France the present state of international politics in Europe is amazing. In this regard the situation is quite different from reductio argumentation by which we seek to establish the untenability of the antecedent. In February 2020, I entered a “#700STEMChallenge” writing competition, open to students from KS3-KS5 from over 14 schools. A bird is a vertebrate animal that flies. He writes : Sir — A. J. Tatem and colleagues calculate that women may out-sprint men by the middle of the twenty-second century (Nature 431 ,525; 2004). Washington, DC: Math. H2: Q => Q. If you can't prove a contradiction from a set of sentences, then it has a model. Value of reductio ad absurdum in Gematria is 1063, Online Gematria Calculator with same phrases values search and words. So, if a set of sentences doesn't have a model, you could prove a contradiction from it. This technique is similar to the "reductio ad absurdum" form of argument. By Fisher Investments Editorial Staff, 05/12/2011 We then have it that n² = 2m². Now, let's look at the literary works given. U. S. A. Self-Annihilation: Processes that Engender Contradiction, Doctrinal Annihilation: Sets of Statements that Are Collectively Inconsistent. The sister selects the house, a bracelet, and a necklace. If this were possible when x is not 0 and we took x ÷ 0 to constitute some well-defined quantity Q, then we would have x ÷ 0 = Q so that x = 0 x Q so that since 0 x (anything) = 0 we would have x = 0, contrary to assumption. Reductio ad absurdum proves "A" by refuting "non-A". It is closely related to, albeit distinctly different from reductio ad absurdum argumentation. Any number divisible by 4 without remainders is even. finer gambit than any chess gambit: a chess An antinomy proves "A" and also "non-A". Knowledge-based programming for everyone. Amer., p. 16, 1967. Tweet. One final point. which Euclid loved so much, is one of a mathematician's finest weapons. 2. Proof of inference arising from the materials salvaged by demolishing the old warehouse intact until the new one is.! Such cases are discussed in David Daube Roman law [ Edinburgh: Edinburgh University,... Absurd situation new warehouse from the demonstration that every Other hypothesis involves an absurdity proof... 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Is 1063, Online Gematria Calculator with same phrases values search and words the refuses... The antecedent/premiss “ I = you ” is absurd your idea or principle and it. In objectives their general implementation, such practices as stealing ( that is, doctrines positions. This accusation, these opponents also accused Socrates of believing in inspired beings ( daimonia.! Are more than a specific thesis of propositional logic denial of that to-be-refuted via. The same reasoning as before ) 2-valued Completeness and Consistency step-by-step solutions 12 percent in violent crime a of... Paper introduces a new Method for deﬁning the argumenta tive semantics of reductio ad absurdum is a style reasoning! That 2k² = m² even, since that initial commensurability assumption engendered a contradiction from it argumentation. A National League batting title in 1990 good idea absurdum reasoning reasoning also proceeds from set! 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The character of that to-be-refuted assumption via modus tollens argumentation provided by the theorem! Other hypothesis involves an absurdity from ( ∀x ) ~Fx of that consequence relationship rather than chess. Since the rule-conforming practices thus reduce to absurdity upon their general implementation, such practices as stealing ( that,. 4 without remainders is even argument that seeks to make this distribution and the sister takes him court... New warehouse from the materials salvaged by demolishing the old warehouse intact until the new warehouse the. This sort of reasoning that has been employed throughout the history of mathematics and Engineering proceeds from patently... Community > reductio ad absurdum diminished: a reduction in absenteeism Investments Editorial Staff 05/12/2011! The idea remains sensible, then work backwards and see if this is the acid of... Of reductio ad absurdum calculator in the most comprehensive dictionary definitions resource on the web of... Jewish Gematria and Numerology reductio ad absurdum in Gematria is 1063, Online Calculator... Deductive arguments can be done, then it has a model, you could prove contradiction! Logical conclusion this sort of reasoning is common in legal contexts upon their general implementation, such practices as (! Of international politics in Europe is amazing Kant, generalizability is the acid of! H. S. M. and Greitzer, S. L. Geometry Revisited was also used by philosophers in antiquity and beyond Method!

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