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	Commensurability (philosophy of science) - Wikipedia, the free encyclopedia
		The idea that scientific paradigms are incommensurable was popularized by the philosopher Thomas Kuhn in his book The Structure of Scientific Revolutions (1962).
		According to Feyerabend, the idea of incommensurability cannot be captured in formal logic, because it is a phenomenon outside of its domain.
		It follows then that two conceptual schemes would be incommensurable only in the case that it was not possible to translate the theory expressed in the language of one scheme into the ideas expressed in the language of another.
	en.wikipedia.org /wiki/Commensurability_(philosophy_of_science) (868 words)

	Commensurability (mathematics) - Wikipedia, the free encyclopedia
		That is, there is a common unit of length in terms of which a and b can both be measured; this is the origin of the term.
		Otherwise the pair a and b are incommensurable.
		In group theory, a generalisation to pairs of subgroups is obtained, by noticing that in the case given, the subgroups of the real line as additive group, generated respectively by a and by b, intersect in the subgroup generated by dc, where d is the LCM of m and n.
	en.wikipedia.org /wiki/Commensurability_(mathematics) (431 words)

	AS/SA Nº5 - Ronen: "Incommensurability and Representation"
		Incommensurability emerges at that specific point where the mechanism of signification in one theory cannot be accommodated with the mechanism implemented by another theory because these are brought about to make different claims about nature.
		Incommensurability pertains to those cases where representation aims to approximate an object (whether the object of representation is three-dimensional space, inner consciousness, social reality or any other dimension of the real).
		The incommensurability involved indicates that modes of representation cannot "contain" or exhaust an object, just as any given scientific concept is always short of catching a corresponding natural phenomenon because that object is always shaped by the concept in use, and because there are other, incompatible ways of conceptualizing the object.
	www.tau.ac.il /~rronen/documents/inc-rep.html (6045 words)

	Nicholson: Imaginary Paradigms (Site not responding. Last check: 2007-11-01)
		Incommensurability can be required as an issue of definition in order to distinguish one paradigm from another, though we can still debate the usefulness of such a distinction.
		They are incommensurable only in the trivial sense that they are asking different questions but they are not so in the sense of answering the same questions in different but incomparable ways.
		If the claims for incommensurability are to be taken seriously social science is going to be a badly disjointed affair with hardly anyone being able to talk to anyone else except to retreat into a corner with a plaintive cry that 'My paradigm's better than your paradigm'.
	www.kent.ac.uk /politics/research/kentpapers/nicholson.html (13066 words)

	Lecture 13
		Incommensurability of Standards or Cognitive Values - Scientists espousing different paradigms may agree on certain broadly defined desiderata for theories (that a theory should be simple, explanatory, consistent with the empirical data, of large scope and generality, and so on), but they typically disagree on their application.
		It is primarily in this sense that theories are "incommensurable" according to Kuhn.
		Thus, the reason Kuhn sees paradigms as incommensurable is simply that on his view there is no higher level to appeal to in deciding between the different values inherent in competing paradigms.
	www.soc.iastate.edu /sapp/phil_sci_lecture13.html (1299 words)

	Incommensurable Values- Local Government and Judicial Review
		Incommensurability is a contested notion and the theoretical basis of the paper depends upon the version of ´incommensurability´ propounded by Isaiah Berlin.
		Incommensurable values are permanently and inherently in competition and cannot be rationally prioritised or compared since there is no common measure in which they can be expressed.
		Incommensurability means however that this can never be conclusive as to the correctness of the ranking order but may be a convenient and arguably a fair device to resolve a dispute.
	www.ncl.ac.uk /nuls/research/wpapers/alder2.html (12440 words)

	crowder
		It follows from the notion of incommensurability in particular that value pluralism is opposed to value monism, the view that a single super-value or narrow range of such values overrides or serves as a common denominator for all others.
		When incommensurable values come into conflict, one of the senses in which choices among them are ‘hard’ is that there can be no complete compensation for whatever value or combination of values is chosen against.
		We have seen that particular decisions in cases of conflict among incommensurables must be determined by a particularist approach, one in which close attention is paid to the particulars of the concrete situation.
	berlin.wolf.ox.ac.uk /lists/onib/crowder.html (8790 words)

	value.html
		That is, incommensurability - of particular rights with social welfare - and qualitative distinctions among values - between freedom of contract, for example, and other public policy interests - lay at the heart of the Lochner judicial framework.
		By focusing on incommensurable values and the social relations that different conceptions of value construct, I have tried to develop the stakes in this choice between competing conceptions of value and to suggest the considerations in terms of which this choice ought to be made in every context the conflict arises.
		This is the conception of incommensurability that, presented in an extreme form, is the heart of Alisdair MacIntyre's nostalgic critique of modernity.
	netlaw.samford.edu /olson/value.html (16811 words)

	20th WCP: Lakatos and MacIntyre on Incommensurability and the Rationality of Theory-change
		I show that Lakatos' rejection of the incommensurability thesis and dismissal of actual history are motivated by the belief that neither are compatible with the rationality of theory-change.
		Incommensurability of meaning is present in many of the key terms that comprise rival theories, at IT1 holds.
		MacIntyre, however, affirms incommensurability both as a semantic thesis, and as a thesis about the absence of criteria that are at once common and decisive.
	www.bu.edu /wcp/Papers/Scie/ScieMine.htm (3250 words)

	Euclid's Elements, Book X, Definitions I
		In particular X.5 and X.6 state that two magnitudes are commensurable if and only if their ratio is the ratio of a number to a number.
		The proof referred to at the beginning of this definition is that of X.10 which finds lines commensurable in square only, and lines incommensurable in square.
		Thus, the diagonal on the square on the standard line is rational, even though it's incommensurable with the standard line, since it's commensurable in square with it.
	aleph0.clarku.edu /~djoyce/java/elements/bookX/defX.I.html (504 words)

	Two Faces of Liberalism
		Our author does not grasp how difficult it is to show that values are incommensurable; but unless he can show clear and important instances of this phenomenon, his argument for value pluralism collapses.
		He states: "Two goods that are incommensurable as a pair may be commensurate when either or both are compared with other goods" (p.
		He does not see that one of the two incommensurable values must be commensurable with a third, and that the other member of the pair cannot be.
	www.mises.org /misesreview_detail.asp?control=182 (1564 words)

	Golden Ratio is made from irrational & rational numbers. Applications & construction. No Phi reduction
		If you think incommensurables (irrationals and transcendentals) are real numbers then you have some work to do in moving these numbers from your left brain and into the right [no need to have Dedekind kind of cut-cut lobotomy].
		It may not be difficult to accept the fact that a circle, by virtue of being incommensurable, has an infinite mantissa of nonrepeating numbers in the measure of its circumference.
		Incommensurable numbers' applications are also found in the Great Pyramid, and the book reveals the critical component in the realization of incommensurables' infinite (mantissa) length.
	www.hyperflight.com /golden_numbers-proportion.htm (4414 words)

	Henri Poincare: Science and Hypothesis: Chapter 2: Mathematical Magnitude and Experiment (Site not responding. Last check: 2007-11-01)
		If we did not remember it, we could hardly understand that Kronecker gives the name of incommensurable number to a simple symbol — that is to say, something very different from the idea we think we ought to have of a quantity which should be measurable and almost tangible.
		From Kronecker's point of view, the incommensurable number 2 is nothing but the symbol of this particular method of division of commensurable numbers; and to each mode of repartition corresponds in this way a number, commensurable or not, which serves as a symbol.
		The common part will appear to us to be a point which will still remain as we imagine the bands to become thinner and thinner, so that we admit as an intuitive truth that if a line be divided into two half-rays the common frontier of these half-rays is a pint.
	spartan.ac.brocku.ca /~lward/Poincare/Poincare_1905_03.html (3684 words)

	Irrational numbers. Evolution of the real numbers. (Site not responding. Last check: 2007-11-01)
		Incommensurability has been and is the historical problem in the relationship of arithmetic to geometry.
		The student should know that this discovery -- The diagonal and side of a square are incommensurable -- was a landmark in the history of mathematics.
		It drove home the distinction between geometry and arithmetic; between what is continuous and what is discrete; and, as we are about to see, it led to the invention of irrational numbers.
	www.themathpage.com /aReal/irrational-numbers.htm (866 words)

	Merriam-Webster Online
		The two theories are incommensurable, making any attempt at comparison across disciplines ridiculous.
		"Commensurable" means "having a common measure" or "corresponding in size, extent, amount, or degree." Its antonym "incommensurable" generally refers to things that are unlike and incompatible, sharing no common ground (as in the "incommensurable theories" of the example sentence), or to things that are very disproportionate, often to the point of defying comparison ("incommensurable crimes").
		They came to English by way of Middle French and Late Latin, ultimately deriving from the Latin noun "mensura," meaning "measure." "Mensura" is also an ancestor of "commensurate" (meaning "coextensive" or "proportionate") and "incommensurate" ("disproportionate" or "insufficient"), which overlap in meaning with "commensurable" and "incommensurable" but are not exact synonyms.
	leopard.eb.com /cgi-bin/mwwodarch.pl?Jun.01 (172 words)

	Real numbers 1
		Theodorus was writing out for us something about roots, such as the sides of squares whose area was 3 or 5 units, showing that the sides are incommensurable with the unit: he took the examples up to 17, but there for some reason he stopped.
		We suppose that the discovery that √2 was not commensurable with 1 came earlier which is why Theodorus started with √3.
		, √17 are incommensurable with a segment of unit length.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Real_numbers_1.html (2733 words)

	SSRN-Value Pluralism and Rational Choice by David Luban
		Theorists skeptical of rational choice theories often base their doubts on the claim that values are plural and incommensurable.
		I deny both the IUT and the IIT, and argue that because these theses fail, the issue of value incommensurability is largely beside the point in debates over the usefulness of rational choice techniques.
		Provided that each of the incommensurable values may be quantitatively measured (although of course the measures cannot be compared between values), a small set of plausible assumptions permits rational choice among value packages, and also induces a transitive ordering of value packages.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=264335 (549 words)

	Jerrold R. Coombs - Rorty, Critical Thought, and Philosophy of Education
		Although Rorty claims there can be no argument for adopting one incommensurable language game over another, he clearly does argue that his pragmatic view of knowledge, mind and language are preferable to foundationalist alternatives.
		We might conclude from this that he is either mistaken about these two vocabularies' being incommensurable or he is wrong about it's being impossible to genuinely argue about which of two incommensurable vocabularies should be adopted.
		The only way I know to give concrete application to the term "incommensurable" is by interpreting it in such a way that two vocabularies are incommensurable if and only if neither can be translated into the other to any substantial degree.
	www.ed.uiuc.edu /EPS/PES-Yearbook/97_docs/coombs.html (1232 words)

	Intentional Dynamics and the Development of Space-Time: Reconciling the Two Incommensurable 'Rivers' (the a posteriori ...
		A schematic of the conjunction capturing the 'minimal ontology' of the epistemic act.
		Everything else with apparent intentional content, to the extent that it occurs (e.g., thoughts, representations, symbols, rules), in order that it has intentional content, ultimately serves, is contextualized by, parasitic upon, or differentiates, or develops (phylogenetically and ontogenetically) out of these relations (Swenson, in press a, b, c; see also Matsuno, in press).
		The previous section asserted that there are entailments which make a decontextualized view of inentionality or the epistemic act (of cognition, perception-action, and so on) untenable.
	members.tripod.com /spacetimenow/section3.html (688 words)

	Reordering Networks of Incommensurable Concepts in Phased Cycles
		Whilst mapping is a major step beyond currently available information tools, it does not address the question of how adequately to interrelate incommensurable perspectives, especially in those cases where wisdom indicates that they are complementary.
		Such complementarity is usually signalled by periodic conceptual shifts from one perspective or framework to another, whether it be in the case of wave/particle theories in physics, or alternation in government between control by the "left" or by the "right".
		It has been argued earlier that seemingly incommensurable theoretical positions or social policies could be fruitfully explored as "frozen" portions of social learning cycles.
	www.laetusinpraesens.org /docs/cycles.php (9946 words)

	[No title]
		Secondly, he observes, because conceptual schemes would always be embodied in languages, one conceptual scheme would be incommensurable with another insofar as, and only insofar as, the languages in which these conceptual schemes were embodied were non-intertranslatable (p.67,68).
		However, the incommensurability criterion, on the other hand, requires that if there are incommensurable conceptual schemes, they must be embodied in non-intertanslatable languages.
		But, Davidson argues, if there is something in common between even apparently incommensurable conceptual schemes, they cannot be genuinely incommensurable, since the languages which embody them should be intertranslatable by reference to this very thing they have in common.
	www.unc.edu /~plmiller/writing/What_Metaphors_Really_Mean_long.doc (8000 words)

	incommensurables (Site not responding. Last check: 2007-11-01)
		In our example, both the nominator (12) and the denominator (36) can be divided by 12, thus obtaining 1/3, which expresses the ratio between a foot and a yard in the smallest possible numbers.
		Two quantities which are not commensurable are incommensurable.
		Suppose AC and AB are commensurable; let a/b be their ratio expressed in the smallest possible numbers.
	www.siue.edu /~evailat/incommensurables.html (559 words)

	The conclusion in the previous section was that, when confronted with the inevitable choice between ic options, it ...
		Although we agree that one book is better than the other with respect to some of these criteria, we are unable to come to a conclusion about their overall quality.
		Both options are supported by incommensurable values: the first one by the value of friendship, the second one by the values of sincerity and honesty.
		However, when choosing between two incommensurable options, your reasons for a particular choice cannot be the reasons that justify that choice.
	www.bezinningscentrum.nl /teksten/bert/reasons.htm (5181 words)

	Dissents in the Final Appellate Tribunal: Tragic Choices
		The nature of incommensurability means that there may be ineradicable disagreement as to what counts as least sacrifice so that social engineering is unlikely to avoid the need to make tragic choices.
		A third kind of incommensurable disagreement is relatively unusual and cuts across the two kinds of disagreement that I have already mentioned.
		[119] Apart from the central claim that uncertainty between incommensurables is a positive good this argument faces the difficulty that the relative importance of certainty in relation to other values is itself a matter of incommensurable disagreement and therefore cannot be a reason for excluding dissent.
	www.ncl.ac.uk /nuls/research/wpapers/alder1.html (12278 words)

	[No title]
		Barsky demonstrated that all triangles resulting from the drawing of all the diagonals in a regular pentagon are isosceles triangles with base angles measuring either 72 degrees or 36 degrees.
		A geometric demonstration showing that the side of a pentagon and a diagonal connecting two vertices of a pentagon are incommensurable was presented as a possible basis for this problem in early Greek mathematics because of the Pythagorean's relationship with the pentagon.
		A geometric proof was given showing the side and diagonal of a pentagon are incommensurable was given, but I have to admit that I did not fully understand it.
	public.csusm.edu /DJBarskyWebs/330CollageSep05.html (1615 words)

	incommensurable - OneLook Dictionary Search
		We found 17 dictionaries with English definitions that include the word incommensurable:
		Tip: Click on the first link on a line below to go directly to a page where "incommensurable" is defined.
		incommensurable : The American Heritage® Dictionary of the English Language [home, info]
	www.onelook.com /?w=incommensurable (154 words)

	Draw star of pentagram, pentacle, pentagon: instruction & illustration
		The segment of a circle you created geometrically, however, is and (possibly) remains a transcendental number and then you could divide one incommensurable (transcendental) number by another incommensurable (irrational) number for a perfect result.
		Irrationals and transcendentals are in the family of incommensurables, but transcendentals are not constructible through the Pythagorean theorem (from 2D of the curve to 1D of the hypotenuse), while additional differences between irrationals is based on applications.
		incommensurable numbers when put in the ratio (or are proportioned) do result in a rational number.
	www.hyperflight.com /pentagon-construct.htm (5468 words)

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