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Topic: Analogies and scientific knowledge

The human reason is very restricted. It seems that the Lobachevski geometry is an objective reality (it works in the physicist on super-large-scale distances), but its human reason is not capable to imagine. It is and is ready more hard cases. Here me at some forums consider as the freak; I trust in a reality paranormal and  the phenomena, orthodox and Buddhist miracles etc. it seems To me, all it basically is explainable. But here recently I saw on the Internet such  that days badly felt. Fortunately, such sufferings (from  a dissonance) quickly enough come to an end. I heard such interpretation: the human reason is similar to the car; doing its more and more powerful, it is possible to increase its speed, but it all the same never reaches to the moon. In spite of the fact that we cannot imagine the same Lobachevski geometry, we can successfully study its properties by means of a method of analogies. Here there is a question - that such generally analogies? I cannot give the good answer, I offer two variants: 1) If we intuitively feel that it is possible to name something a word "analogy", "metaphor" is and there is an analogy. 2) as analogies allow to do a prediction, it can serve and criterion - if any analogy gave true predictions, it means has the right to be called as analogy. Further I will sound important idea: the amount of analogies can compensate quality. I already wrote that many feeble arguments can turn in the total to strong just as indications of many people on court turn to the proof (if to eliminate arrangement possibility). The addition theorem of involuntary events here works. I will give an example. Those who play computer games (strategy), start to understand some laws of real wars a few. We tell, in the modern strategy the rule "a stone, scissors and a paper" which can be transferred and on real situations often works. A classical example:  are effective against  as while those approach,  will long throw their arrows; the cavalry is effective against  as quickly them reaches; and  are effective against a cavalry. It too it is possible to name all analogies (in a broad sense). I Will give an example, as set of analogies allow to do more authentic prediction. In the majority of strategy to defend easier, than to attack; same works and in real wars (exceptions - for example are possible, in nuclear war by the first can attack to be easier). And so, in different strategy this rule is implemented differently: in one to defend easier for the effect account "fire walls"; in others - the defending side has the tactical initiative and can counterattack there where it is the most favourable; in the third - the attacking side is more vulnerable for different bombardments since it is moved; in the fourth - in the territory to supply armies it is easier. Etc. Now I will give an example, as by means of analogies it is possible to study properties of the same Lobachevski geometry. I know four suitable analogies: 1) the Internal surface of sphere. We assume, there is a sphere surface in three-dimensional space. Between two points of this surface it is possible to draw a straight line (in three-dimensional space). Also between two points it is possible to lead a line on surfaces, as though a shade from the first straight line if at sphere center there was a light source (and further to continue this line in both sides that the circle on a sphere surface turned out). We name these abstractions accordingly "a line in the first sense" and "a line in the second sense". Now we invent a homonym - "straight line" so-called which means both that and that. Now we receive "the theorem" corresponding to Lobachevski geometry: Through set by a point it is possible to lead any number of "straight lines", parallel given "straight line". 2) "straight line" Poincare Nazovyom's Model so-called a circle arc, perpendicular an exterior surface of a circle (i.e. under what corner this arc quits a circle wall, on the same corner and enters into this wall; the line from center of a circle to a wall divides this arc on two symmetric half). We name these "straight lines" "parallel" so-called if they have no general points. We receive "theorem": Through one point it is possible to lead any number of "straight lines", "parallel" given "straight line" 3) upper halfplane Model we Name "straight lines" of a circle which are halved by a x axis. 4) model on a hyperboloid In this model "lines" are called intersection areas between a hyperboloid (a surface formed by rotation of a hyperbola round axis Z in a picture) and the planes transiting through center of coordinates. Now we take the second and third model, and we define, what in them should be the total of corners of a triangle: From pictures it is visible that the total of corners of a triangle turns out less than 180 degrees. Really, in Lobachevski geometry there is such theorem: https://ru.wikipedia.org/wiki/_ the Total of corners of any triangle is less  also can be as much as close to zero

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: K> Now I will give an example, as by means of analogies it is possible to study properties...  from the cradle learn all using analogy...

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Re: Analogies and scientific knowledge

...Everyone  ... K> a question - that such generally analogies? You for what purpose are interested? Can it is necessary take the appropriate professional literature and to read? https://www.google.com/search?q=analogi … rogramming

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: K> From pictures it is visible that the total of corners of a triangle turns out less than 180 degrees. Really, in Lobachevski geometry there is such theorem: And you in which of geometries considered this total?

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Re: Analogies and scientific knowledge

K>> From pictures it is visible that the total of corners of a triangle turns out less than 180 degrees. Really, in Lobachevski geometry there is such theorem: S> And you in which of geometries considered this total? The second and the third (Poincare's model and upper halfplane model). There in pictures it is visible that it is less 180.

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: K> 1) the Internal surface of sphere. K> we assume, there is a sphere surface in three-dimensional space. Between two points of this surface it is possible to draw a straight line (in three-dimensional space). Also between two points it is possible to lead a line on surfaces, as though a shade from the first straight line if at sphere center there was a light source (and further to continue this line in both sides that the circle on a sphere surface turned out). K> we name these abstractions accordingly "a line in the first sense" and "a line in the second sense". K> Now we invent a homonym - "straight line" so-called which means both that and that. K> Now we receive "the theorem" corresponding to Lobachevski geometry: K> Through set by a point it is possible to lead any number of "straight lines", parallel given "straight line". I here did not understand Something. How though one parallel line to lead? They all are intersected on sphere. The sphere surface has the positive curvature, there the total of corners is more 180, it not that at Lobachevsky.

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: K> In spite of the fact that we cannot imagine the same Lobachevski geometry, we can successfully study its properties by means of a method of analogies. K> Here there is a question - that such generally analogies? The mathematician would tell that the analogy is a forgetting functor. https://ru.wikipedia.org/wiki/%D0%97%D0 … 0%BE%D1%80 actually, all essence of the category theory just that we study things through their relations. The more we know about relations, the we know about object though inside we can and not look more. Just as learning something through analogies. (And it too analogy)

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Re: Analogies and scientific knowledge

Hello, D. Mon, you wrote: DM> I here did not understand Something. How though one parallel line to lead? They all are intersected on sphere. Why are intersected? Unless parallels of the Earth (a line, parallel to equator) are intersected? DM> the sphere Surface has the positive curvature, there the total of corners is more 180, it not that at Lobachevsky. Truth is more 180? It is sad...

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: DM>> I here did not understand Something. How though one parallel line to lead? They all are intersected on sphere. K> why are intersected? Unless parallels of the Earth (a line, parallel to equator) are intersected? Parallels of the Earth - not straight lines in this sense, them not to receive the method described by you. The immediate analog of a straight line on sphere is geodesic, "the big circle". Such, as meridians, as equator. And all of them are intersected. DM>> the sphere surface has the positive curvature, there the total of corners is more 180, it not that at Lobachevsky. K> truth is more 180? It is sad... Truth. As a classical example - a triangle made of two perpendicular meridians and equator. At it all corners straight lines, the total 270. At other triangles the total can be less, and at small will come nearer to 180, but all the same always hardly more than 180 will be.

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: K>>> From pictures it is visible that the total of corners of a triangle turns out less than 180 degrees. Really, in Lobachevski geometry there is such theorem: S>> And you in which of geometries considered this total? K> the second and the third (Poincare's model and upper halfplane model). There in pictures it is visible that it is less 180. In pictures of it it is not visible, ....

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Re: Analogies and scientific knowledge

Hello, D. Mon, you wrote: K>> Truth is more 180? It is sad... DM> Truth. As a classical example - a triangle made of two perpendicular meridians and equator. At it all corners straight lines, the total 270. At other triangles the total can be less, and at small will come nearer to 180, but all the same always hardly more than 180 will be. Any analogy gives any correct predictions, and any wrong. The "more strongly" (more close) the analogy, the is more percent of the correct predictions. It is possible to take as an example analogy between atom and solar system. She allows to foretell that if kinetic energy of an electron will be enough high, it departs from a kernel on infinity. On the other hand, from this analogy the wrong prediction follows that the electron should radiate energy and gradually fall on a kernel. As I already wrote, the amount of analogies can compensate quality. In an initial post 4 analogies are resulted, from them two or most likely three foretell that the total of corners is less 180, and one (first) - that is more than all 180. Therefore as a whole these analogies predict properties of Lobachevski geometry.

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Re: Analogies and scientific knowledge

K>>>> From pictures it is visible that the total of corners of a triangle turns out less than 180 degrees. Really, in Lobachevski geometry there is such theorem: S>>> And you in which of geometries considered this total? K>> the second and the third (Poincare's model and upper halfplane model). There in pictures it is visible that it is less 180. S> In pictures of it it is not visible, .... You look inattentively (the last pictures with places of honor).

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Re: Analogies and scientific knowledge

Hello, Khimik, you wrote: K>>>>> From pictures it is visible that the total of corners of a triangle turns out less than 180 degrees. Really, in Lobachevski geometry there is such theorem: S>>>> And you in which of geometries considered this total? K>>> the second and the third (Poincare's model and upper halfplane model). There in pictures it is visible that it is less 180. S>> In pictures of it it is not visible, .... K> you look inattentively (the last pictures with places of honor). And you a case do not operate with Euclidian geometry, including corners which are drawn in a projection non-Euclidean on ?

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Re: Analogies and scientific knowledge

Hello, Sheridan, you wrote: K>> you look inattentively (the last pictures with places of honor). S> And you a case do not operate with Euclidian geometry, including corners which are drawn in a projection non-Euclidean on ? Did not understand a question. All four models naturally imply Euclidean geometry.