Website founded by
Milan Velimirović
in 2006
9:39 UTC
| |
MatPlus.Net  Forum General Dombrovskis paradox |
| |
|
|
|
|
You can only view this page!
| | Page: [Previous] [Next] 1 2 3 | | (21) Posted by Geoff Foster [Wednesday, Sep 29, 2010 10:06] |
The key move is an anticipatory closure of two black lines.
| | | (22) Posted by Jean-Marc Loustau [Wednesday, Sep 29, 2010 10:57] |
@Marjan : I agree with you, there is an extra theme: I don’t know if we both think to the same, but I suppose we do ! (I prefer not write it now to let some suspense and people propose themes…)
| | | (23) Posted by Paz Einat [Thursday, Sep 30, 2010 00:58] |
Well, there is a correction play by white, and as a more minor thing, also square vacation not only by the WSc2 tries and key, but also 1.b5? a5!
| | | (24) Posted by seetharaman kalyan [Thursday, Sep 30, 2010 21:23]; edited by seetharaman kalyan [10-09-30] |
I think the example by Alfred Pries is unique - this is only example I know where the thematic tries and the key are all by the same white piece. !!
| | | (25) Posted by Marjan Kovačević [Thursday, Sep 30, 2010 23:36] |
Seetharaman, having a single piece to begin all the phases is surely a valuable feature. Many other composers thought the same way, and composed such examples of the Dombrovskis theme, some of them with triple Dombrovskis, others with additional themes, too.
| | | (26) Posted by David Knezevic [Friday, Oct 1, 2010 03:33] |
For instance this magnicifent Tertiary Threat Correction:
Marjan Kovačević
1st Prize Sredba na solidarnosta 1984
 (= 10+7 )
#2 vvv | 1.Se2? ~ 2.Qxc6/Qf5#
1... Rxc5!
1.S~?(=Sa2?) ~ 2.Re1#
1... Bxd4 2.Qf5#
1... f3!
1.Sb3!? ~ 2.Qxc6#
1... f3 2.Qf5#
1... Rxc5 2.Sxc5#
1... Bxd4!
1.Sd3!! ~ 2.Sf2#
1... f3 2.Re1#
1... Bxd4 2.Qxc6#
1... Kf3 2.Qxc6# |
| | | (27) Posted by Jacques Rotenberg [Friday, Oct 1, 2010 03:47] |
Well...
I don't know what you mean, Marjan, but there are, I think, three other valuable features of this problem that may be pointed out :
1) Diagonal/Orthogonal Echo
You can see a good correspondence between
. the white line e4-e2 & the black line e2-e5
one one hand, and
. the white line h6-d2 & the black line d2-f4
on the other hand
e3 being a wonderful "pivot" of this echo
2) The change of functions of the black moves
. 1...Qe2 is used first to pin the wR (or say to attack the Ke5), and then to control c2
. 1...Bd2 is used first to control f4 and then to close c2-e2
3) The special feature of the key that closes 4 lines - two white & two black
| | | (28) Posted by Jean-Marc Loustau [Friday, Oct 1, 2010 12:07]; edited by Jean-Marc Loustau [10-10-01] |
@Marjan:
After some days, I give my own answer(s):
First I describe what happens in the problem itself: obviously the 2 mates Sf4#/Rd4# are prepared; so, in order to make them effective, White must organize an “extra-protection” of the “thematic” squares d2 or e2 (either directly by over-protecting e2, either indirectly by an anticritical move (Bc1)); these attempts fail; the key 1 Se3 paradoxically gives the 2 flights (thus under-protect) d2 and e2. Thus, this is a very good key, not only because it is a 2 flights-giving key, but also because it has a very paradoxical meaning. The fact the 2 waited mates appear in the solution is of course very important and underlines the paradox.
Now the theme could be expressed like this:
In each of 2 tries, white 1st move has a useful effect, in order to threat a mate (A or B). The key contains a double negative effect which has to be seen as the opposite of the useful effects of the tries. The mates A and B appear nevertheless in the solution.
For example, a different way to achieve this theme could be (I don’t know if it is actually possible, I have not seriously thought to this, it seems quite difficult, I say that just as example):
Mates A and B are respectively controlled by black units X and Y; 1st try pins X and threats A (resp. 2nd try pins Y and threats B); the key unpins a 3rd unit Z, over-controlling the mates A and B (or possibly unpins 2 black units…). Mates A and B appear in the solution. You can replace “pins” by interferes with” and/or unpins by “uninterferes”…
---------------------------------------------------
By the way, thinking to this problem, I have seen another possible theme, which has a minor importance here, but which is nevertheless there; you have 2 more tries:
The 1st one is trivial, but on the other hand will be seen by any solver or reader: 1 Re3+? Kd2!
The 2nd one is a little bit more subtle even if the refutation is obvious: 1 Be3!? (2 Sf4#) Ke2! (Notice that this is a correction of the try 1 Bc1?: 1… Bd2 does not refute).
So you get a white Grimshaw (again a theme), in which the white weakness is obviously to give a flight (d2 or e2) to the black King. The key 1 Se3, again paradoxically, contains the 2 weaknesses, that is to say gives the 2 flights: this is the so-called “franco-russian” theme.
I don’t think this was intended by Dombrovskys, first because the try 1 Re3+ is trivial, and also because in the solution there is a dual after the flight 1… Kd2 (this is not a serious flaw because it does not prevent the threat, but in my opinion it would be a more serious flaw if the main theme were the franco-russian). Nevertheless probably he had seen this, and the first official achievement of the “franco-russian” theme has been published just some years later (1961 I think, but I cannot check now), also a wonderful problem by Dombrovskys, with Loshinsky as co-author.
| | | (29) Posted by Vladimir Tyapkin [Friday, Oct 1, 2010 18:26]; edited by Vladimir Tyapkin [10-10-01] |
Jean-Marc, your analysis follows pretty closely to what Yury Sushkov called the caprice paradox or caprice effect (тема каприза in Russian). The following definition is from his article "Choice of a key in the twomover problem" from Chess Composition 1977-82 (available at http://problem64.beda.cz/silo/tschepischnyi_kompozicija_1977_1982.djvu, pp. 5-31). His definition is fairly broad and a bunch of more strictly defined themes (like Franco-Russian theme you mentioned) fall into this category. There are more examples in the article starting from problem №34(check Yury's own problems since he has been working on this idea extensively).
The main feature of "caprice" and similar ideas is to understand the content of a problem one should consider tries and solution in the proper order. Together, they make up a logical chain of events. The key is usually a culmination, ether confirming tries or negating it. In a sense, tries and solution are separated in time and the latter is the direct consequence of tries even when it negates them (unlike a multi-phase problem with changed mates where all phases are logically equal and a try and solution could be interchanged). Sometimes, even tries should logically follow one another. p.23
Note, the meanings of 'confirming' and 'negating' are the same as yours 'useful' and 'negative' effects. In the article, he classified these effects and gave some examples in each category.
| | | (30) Posted by Marjan Kovačević [Friday, Oct 1, 2010 19:50] |
Vladimir, you mentioned one of my favorite articles. It discusses the general subjects of multiphase twomover, such as relations between phases and the logic behind it, and I would strongly recommend it to all chess composers.
Jean-Marc, this article really contains scenarios of the kind you discuss, but this wasn’t exactly what I had in mind as the possibly “main theme” of the #2 by A.Dombrovskis.
| | | (31) Posted by Darko Šaljić [Saturday, Oct 2, 2010 20:08] |
The problem contains a paradox that is rarely noticed by solvers.
The point is in lost relation between white line piece and square it controls.
After black self block mates the unit that had just lost control over that exact square.
Author achieved to show it in reciprocal manner, white piece that controls square A in mate takes control over B and vice versa.
| | | (32) Posted by Jacques Rotenberg [Sunday, Oct 3, 2010 13:10]; edited by Jacques Rotenberg [10-10-03] |
Again a very interesting analysis by JML.
All the paradox aspect underlined by JML was already evoked, when I recalled
"...When A. Dombrovskis defined his theme he also demanded that the changes will be done mainly by the key, and not by the keys of the tries !..."
this mainly brings to that, and, of course, the explicitation given :
"...In each of 2 tries, white 1st move has a useful effect, in order to threat a mate (A or B). The key contains a double negative effect which has to be seen as the opposite of the useful effects of the tries. The mates A and B appear nevertheless in the solution..." explains in details one of the ways this can be achieved.
The Grimshaw on e3 shown by JML is also, I think, a very important feature, because of the Franco-Letton (Franco-Russian).
Now, I may add still one more thing about the key :
The move Sc2-e3 is a withdrawal move ! in fact when a piece goes far from the bK it is a nice feature, because of a frequency argument : most of the attacking moves bring pieces closer to the bK.
The bSc2 is in position to give mate, as underlined by the try 1.b5?[2.Sb4#] a5! When it goes to e3 it leaves this for a weaker place.
| | | (33) Posted by Jean-Marc Loustau [Monday, Oct 4, 2010 14:27]; edited by Jean-Marc Loustau [10-10-04] |
Vladimir, thanks for quoting this wonderful article by Sushkov. I knew it because I have the book in which it has been published, but I’d better say “I know the problems” because I don’t understand russian language. I agree that the “scenario” described belongs to a “family of themes”, in that meaning they ask for a paradoxical relationship, based on effects, between some moves of the tries (keys, refutations, may be variations…) and the key of the solution, and this family can be called the Caprice family.
Nevertheless, in my opinion, all these themes are different and each of them deserves to be named. I think one of them has been named later Sushkov theme, the Franco-Russian (or Letton as Dombrovskys was from Latvia) is another one, and, to my knowledge, the Caprice theme asks for a relationship between refutations (black) and the key of the solution, what is quite different from the scenario described, which asks for a specific relationship between keys (white) of the tries and the key of the solution.
Indeed the article shows 2 problems with “almost” the scenario I described (I believe no more); just for my pleasure here is the less known and the first published (the other one is a well known masterpiece by Loshinsky, published in 1960):
Loshinsky and Ruchlis
Schach 1957 1st prize
 (= 9+10 )
2#
1 Qc6+? Kd4! 1 Qc5+? Ke6!
In order to make effective these moves, White over-controls d4 or e6:
1 Bc5? (2 Qc6#) Qe6! (2 Qc5??)
1 Rc6? (2 Qc5#) Bd4! (2 Qc6??)
The key gives the 2 thematic flights:
1 Qc7!! (Qxc4#) 1… Kd4 (Ke6) 2 Qe5 (Qxd7)#
I add the set play, which is not mentioned by Sushkov (may be he does and I have not understood because the article is in Russian): 1… Bd4 (Qe6) 2 Qc6 (Qc5)#
I am pretty sure it was intended by the authors because this set play is justified by the plausible try (underlining also the self-obstructions on c5 and c6): 1 Sd1? (Se3#) Sf5! and this is the only reason of the Pf3 (otherwise put the Pg3 on f3 or even much better on d3).
The only difference with the scenario I described is that none of the threat-mates of the tries appears in the actual play (it is the same in the other problem quoted by Suchkov). To my opinion it is an important thematic feature that the 2 threat-mates appear in the actual play, because the keys of the tries are played in the purpose to make these mates effective, and the key of solution seems to go in the opposite way: it “removes” the mates (it’s a reason why it is so paradoxical), so it is surprising when they reappear… In some way the problem by Dombrovskys shows (for the 1st time?) this feature!
Of course this is just a theoretical opinion, in practice the 2 problems quoted by Sushkov are absolutely excellent, and particularly the 2 new mates by the Queen on e5 and d7 in the problem above are really welcome!
About the post by JR I would like to say that I don’t fully agree with the 1st point when he writes “All the paradox aspect underlined by JML was already evoked, … explains in details one of the ways this can be achieved”.
Yes, as Dombrovskys wanted, here “the changes are done by the key, and not by the keys of the tries”. But these changes, allowing the achievement of the Dombrovskys theme, are the closing of the BLACK lines d2-f4 and e2-e5 (this is indeed a “positive” effect of the key). The fact the key closes also the WHITE lines h6-d2 and e4-e2 (giving thus 2 flights), is a specificity of the matrix (in our case these white lines are the same as the black ones), but not a specificity of the mechanism of the Dombrovskys itself; as evidence, you can replace the thematic Bh6 and Re4 by fictive fairy units, respectively combined B+Bhopper and R+Rhooper, and the solution works as well without flight giving key…
What I mean is that the 2 paradoxical themes (Dombrovkys theme, and still unnamed scenario described above) are of course very well integrated together but to my eyes clearly and completely differentiated from the point of view of the mechanism.
Another point interesting about the problem quoted above is the date: 1957; this shows that soviet composers were working in these years on this kind of paradoxes, and it seems to me very plausible that Dombrovskys wanted to achieve this combination of 2 themes. If you add the franco-letton/Russian theme (important or secondary here, I don’t know), you get an impressive combination of 3 beautiful paradoxical themes…
And finally: just sorry for this too long answer!
| | | (34) Posted by Oliver Petrov [Wednesday, Oct 6, 2010 08:31]; edited by Oliver Petrov [10-10-06] |
QUOTE
If you have a threat B you need to have some play around it as well.
If there is a second threat #B in the try, does this compromise the pattern?
| | | (35) Posted by Jacques Rotenberg [Wednesday, Oct 6, 2010 21:41] |
@ Oliver :
see post (9) in this file
@ Marjan :
Let's summarize the answers obtained to your question till now :
1)The key move is an anticipatory closure of two black lines.
2)There is a correction play by white
3)Square vacation not only by the WSc2 tries and key, but also 1.b5? a5!
4)Diagonal/Orthogonal Echo
5)The change of functions of the black moves
6)The key that closes 4 lines - two white & two black
7)In each of 2 tries, white 1st move has a useful effect, in order to threat a mate (A or B). The key contains a double negative effect which has to be seen as the opposite of the useful effects of the tries. The mates A and B appear nevertheless in the solution.
8)White Grimshaw
9)Franco-Letton (Franco-Russian)
10)The point is in lost relation between white line piece and square it controls.
After black self block mates the unit that had just lost control over that exact square.
11)The move Sc2-e3 is a withdrawal move
12)The 2 threat-mates of the tries appear in the actual play, when the keys of the tries are played in the purpose to make these mates effective, and the key of solution seems to go in the opposite way.
...
so ?
what's yours ?
| | | (36) Posted by Marjan Kovačević [Wednesday, Oct 6, 2010 23:57] |
I believe it is not a waste of time to try to analyze the problem that was once nominated to be “the twomover of millennium”. Although it wouldn't be my favorite, there must be a good reason for several composers to vote for this one.
Let me now add some more facts and comments. The author himself, Alfred Dombrovskis, wrote that he had composed this twomover in 1954, but he didn’t dare to publish it next four long years. He didn’t believe it would be well accepted, and he couldn’t dream at all how popular it would become.
This may be an explanation for adding 3 black pieces (Se8, Pa6, Pb6) instead of WPb5. He probably wanted to add an illusion of more active WQ, including the mentioned try (1. b5? a5!) that had no relation to the main content. Another curious point is the mentioned minor dual (1… Kd2 2.Qc2,Qd1) that could have been easily avoided with the same total number of pieces:
 (= 8+7 )
So, the author wasn’t convinced, and the construction wasn’t convincing, but the reaction of judges was absolutely great, right from the start. The problem won 1st places in a row: the Probleemblad tourney, very strong at that time; then the Latvian Championship; and finally the USSR Championship, an incredibly strong competition at that time. The great Lev Loshinsky was among these judges, too.
Jacques, thanks for the help with summing up the answers! This gave an opportunity to use only the numbers you mentioned, instead of some more words:
Out of this 12 points, only 9) and 10) don’t seem correct. For the point 9) each try should gave given one of the thematic flights, when the key offers both. The point 10) is correct only for the WR. The other variation would probably need another WQ instead of WB, if not another WK to be pinned, too.
From the remaining 10 points, 2) and 8) seem welcome but incidental features; while 3) and 11) seem irrelevant. Other 6 points describe some elements and features of the whole combination that I see as the main theme.
Jean-Marc, the alternative scenario you mentioned seems to be on the same lines of thought as the combination by Dombrovskis, but much more difficult to find and to present in such a clear and impressive way. This was probably the main reason that majority of composers followed only the formal relation of tries and solution in this famous problem, until they reduced it to the most banal methods. Such a change of direction, from tactical content of moves to their exact notation described by letters, changed the history of twomover, too.
And Jacques, let me postpone my answer one more day, in order not to spend too much space at once.
| | | (37) Posted by Jacques Rotenberg [Thursday, Oct 7, 2010 01:57]; edited by Jacques Rotenberg [10-10-07] |
Marjan,
Thank you very much for openning the discussion about this very important problem. That's right, I am a bit impatient of your answer.
Before you give your point of vue, some questions :
- the point 9) seems perfectly shown :
1.Re3+? but 1…Kd2! due to the flight giving key (1…Kc4 2.Sa3‡)
1.Be3? [2.Sf4‡] but 1…Ke2! due to the flight giving key
1.Se3!! gives these two flights, so :
Franco-Letton + w-Grimshaw
- The construction you propose is an evident question. Due to the huge success of this problem 'as is' I did not dare to ask about the way the pb is built :
To say it sharper : I am not sure the construction you propose is better.
The fact you remind that Alfreds Dombrovskis kept it some years before publication speaks more for a complete choice from him, I would say : it seems that A. D. had rather have a dual on the flight in balance with a more active Queen and Knight.
So it might be that the "illusion of activity" was not considered by him as "having no relation" with the main content.
| | | (38) Posted by Marjan Kovačević [Friday, Oct 8, 2010 00:21]; edited by Marjan Kovačević [10-10-08] |
Here is the little story I wanted to tell:
In 1995, the thematic condition for the #3 round of Liga Problemista, suggested by Milan Velimirovic, asked for an anticipatory interference of the black line in the second white move. I wanted to find a suitable matrix for two variations, orthogonal-diagonal. It was logical to have two tries in the W2 moves, refuted by opening of the thematic black line, or by arrival of black line-piece. Then, it would be best if thematic refutations appear as variations in the solution, producing the thematic mates. The easiest way to use the harmful effects of black thematic moves seemed to be self-block when the thematic black piece arrives at a flight square, and this dictated the matrix…
To make it shorter: step by step, the best and only mechanism I found was exactly the one by Dombrovskis 1958 (extended to #3, and with two different WSs coming to the interference point e3 in W2 moves).
The shocking discoveries were: 1) “my” idea could have been presented in #2, 2) this was already done by Dombrovskis decades ago, 3) My perception of this (and not only this) #2 radically changed.
From then on, I see the AD58 #2 as a great example of the double anticipatory interference THEME, in its most impressive – paradoxical FORM. This is also a rarely successful example of what I would call “logical twomover” (the French-Russian theme would belong to the same family, still open for serious investigations).
The tries in AD58 are only attempts, without a developed thematic play, and the threats serve as the main plan (Probespiel), reminding us of the Logical school. With the use of the thematic mating moves the combination gets longer then usually in #2, and could be compared with a logical #3.
However, the most important element, the flavor, comes from the sharp tactical play, and the paradoxical relation (as with the paradoxical flavor of the Keller theme, or the Lepuschitz theme). And, as Hans Peter Rehm explained, the quality of the Lepuschitz paradox is not in its pattern, but in the quality, beauty, surprise … of the play itself.
The same could be said about AD58. The most impressive element isn’t the naked pattern (and AD wasn’t the first to present it), it is the whole tactical content, and the whole spectacular turnover around the lines and flights. The separate elements of this content were described in this Post at points 1), 4), 6), and 7), as summarized by Jacques.
We got used to say the Dombrovskis theme is paradoxical. In order to find some paradox in the #2 by Everts (1921, but not the oldest incidental presentation) from this Post, we need to THINK or READ the letters, while in AD58 all of us FEEL the paradox directly, from its sparkling play.
Unfortunately, many formal examples of the Everts type entered FIDE Album, with small differences: they came 60 years later, and they didn’t have such an interesting key as Everts. On the other hand, the Dombrovskis pattern by Pries (1951) seems to be intentional, and it also presents one (more complex) anticipatory interference, but a single variation doesn’t make a theme, and the whole impression is hard to compare.
We also got used to see the Dombrovskis pattern as a theme, but in the case of AD58, it could be perceived as a form, too. Many great problems mix two different ideas (themes or forms) in a similar way, so that you could perceive the content as whole. Or you could perceive one of these two ideas as the form, another as the real content. As in the Rorschach test, or in the optical illusions, one of them becomes invisible, while we are focused on another.
Such a natural crossing of two elements makes the composition much stronger, it cements the content, even more when we are not aware of the elements themselves, as in the case of the relation between drawing and color. The banal presentations of the Dombrovskis pattern appear as a sketch, or rather as a frame without a picture in it.
AD58 could be perceived as a whole, and its scenario is so impressive that it could be described with words (not letters), as a story. For instance: “in order to close his lines, you have to close yours”, or: “if you can not beat the enemy, make him beat himself”, or “instead of taking his flights, give him double, and you will get it all back (with a high interest – if you want to be pragmatic)”, etc. If a composition gives you so funny ideas, what more could you ask?
P.S. Jacques, about the theme explained at point 9), I wouldn't consider a checking try as a serious part of the French-Russian theme.
| | | (39) Posted by seetharaman kalyan [Friday, Oct 8, 2010 07:04]; edited by seetharaman kalyan [10-10-08] |
QUOTE
For instance this magnicifent Tertiary Threat Correction.
Thank you Milan for quoting this really magnificent problem. The transferred mate 1..f3 2.Qf5 in another phase makes this memorable.
| | | (40) Posted by Jean-Marc Loustau [Friday, Oct 8, 2010 12:42] |
Marjan, you write : “about the theme explained at point 9), I wouldn't consider a checking try as a serious part of the French-Russian theme”.
Well, first I knew that it would be your opinion, and notice that I told that the checking try is probably the reason why this theme was not intended by AD. I would like to add also that I agree with you from a general (generic) point of view or even an “artistic” point of view: a checking try is not a satisfactory way to achieve a theme (except if the theme asks for such checking moves). But general principles must always be reviewed on each particular case.
I have thought somewhat to this matter, and here is another possible point of view (yours, of course is valuable too, even if opposite). Curiously, I will use same words as you (feel, logical)…
What makes the beauty, the artistic dimension of a paradox, it is to be actually be seen as a paradox, I should say to be felt as a paradox. For this reason, it is of the highest importance to make the tries very plausible. So let us see an example of a kind of process a solver could follow in our case (I write it intentionally in a very naïve way):
- Well well… Why 1 Re3 is not a mate??? OK if 1… Kc4 2 Sa3#, but obviously if 1… Kd2 there is no mate; thus I can’t give the flight d2 (of course! c1, d1 are not controlled…)
- Sh5 is clearly there to mate on f4, but again if 1 Sf4+?? Kd2, so I will try 1 Be3 (may be 1 Bc1 first, but after seeing 1… Bd2! it becomes more interesting to close the line d2-f4)… But… Nothing after 1… Ke2! Damn! I can’t give the flight e2!
- So, let us forget the possibility to give these flights, and let us try something else: 1 b5? Or 1 Sg3? etc.
Well, after a while, he will find 1 Se3!! But, what is absolutely sure in this case, he will be really surprised, and he will really feel how this key is paradoxical, may be he doesn’t know at all what is the franco-Russian/letton theme, but he will feel the spirit of the theme itself, and I believe it is very important! I even think there are serious chances that this solver will see the franco-Russian/letton paradox and not the Dombrovskys paradox (which probably remains the main theme)!
Suppose now a problem with franco-Russian/letton theme, with 2 very subtle tries, with for example very hidden refutations, nice or difficult variations, clever 1st moves… As an experienced problemist I will like a lot the subtlety, the effects, and so on, I will probably agree that it is very artistic… But it is then also highly probable that the paradox by itself will be very formal, I mean depending of the way the problem is explained or shown, and may be even a little artificial. If there is actually a paradox there, it would be a “cultural” paradox (but, of course, this can be good nevertheless)…
So I would say that AD58 shows in its deep frame a real Franco-Letton paradox, even, of course, if I agree that the tries 1 Re3+? 1 Be3? are not very artistic, from a classical point of view (I say that, because “what is artistic?”… One could say also that things are artistic when they can be felt!... Well I don’t want to debate here about such matters)
You mention the logical school… As you know, it is very common in logical moremovers to see first a short thematic checking try, immediately refuted… It’s then a way to introduce the introductory manoeuvre (and the most surprising it is, the better) and to underline the “logical aspect” of this manoeuvre. The so-called “thematic” try is not artistic at all, it can be even trivial, it is just a technical way to emphasis the logical structure. The same way can be (and probably should be) used for the Franco-Russian aspect of the AD58: we have 2 technical tries emphasizing the paradoxical key 1 Se3! In a logical problem, if the thematic try is unsound (or absent), then the problem is less clear, is not so good… In AD58 I think the key is much better because there are the tries 1 Re3+? 1 Be3? (So you can say the Franco-Russian is not well achieved, but then, how do you call this actual paradox?)
Finally, I have to say that I feel (I do want this word, “feel”) a great difference between logical school and let us call “paradoxical 2#” school: in the logical school the try is there to explain why things works (why should I play this introductory manoeuvre? See the try!), and in the other school the tries (or try) are there to show why the solution should not work (Why should I not play this key? See the tries!): the explanation comes after! The “thematic” tries are there for opposite purposes: in logical school they explain why the solution is not so paradoxical, in “anti-logical” school (Damn! Excuse me, I can’t prevent myself…) they explain why the solution is so paradoxical… This is the kind of things which makes that, personally I am much more attracted by the second one than by the first one!
I know, I am to verbose… Sorry again…
| |
Read more... | Page: [Previous] [Next] 1 2 3
MatPlus.Net Forum General Dombrovskis paradox |
|
|
|
ISC 2024
