How many different chess positions are there?

How does giving the conditions under which positions are regarded as the same for the purposes of art. 9.2 amount to a definition of "position"?

It does clearly show that what they mean by "position" has attributes that include at least the side to move and the possible moves of all the pieces (and the diagram).

And for the same reason I gave regarding the dead position rule, another attribute of a position in the competition rules game is the set of diagrams with the same material together with side to move that have previously occurred and how many times each of those has occurred. Otherwise you couldn't apply the rule you quote. 

#321

"How does giving the conditions under which positions are regarded as the same for the purposes of art. 9.2 amount to a definition of "position"?"
Position = placement of all pieces, player to move, en passant and castling rights.
or if you want: position = diagram + side to move + en passant and castling rights.

"another attribute of a position in the competition rules game is the set of diagrams with the same material together with side to move that have previously occurred and how many times each of those has occurred."
No, otherwise you cannot claim a draw by threefold repetition of the position: in your view no positions would ever repeat.
"9.2.1
The game is drawn, upon a correct claim by a player having the move, when the same position for at least the third time (not necessarily by a repetition of moves)"

Of course no positions ever repeat in a game played under FIDE competition rules. For each position counting toward a triple repetition the number of times the diagram has previously occurred and the ply count under the 50 or 75 move rules are necessarily different.

That is why art. 9.2.2

Positions are considered the same if and only if the same player has the move, pieces of the same kind and colour occupy the same squares and the possible moves of all the pieces of both players are the same.

is a necessary part of the triple repetition rule specification.

It's not a definition of the term position and applies only to art 9.2. For the remainder of the laws "position" means only "situation occurring during the game".

Although FIDE doesn't further define the term "position" the are several definitions that have been proposed that are more useful for analysing chess. These consider only a limited set of attributes of situations occurring in the game.

For a definition to be suitable for that purpose, the possible legal continuations from a defined position must usually correspond with those possible in any game in which it occurs and the result of the game for each continuation (and hence the theoretical result of the game) must be fully determined from the position alone.

There are exceptions - see my last paragraph here.

A diagram doesn't satisfy those requirements. For example this diagram

doesn't determine the result of the game under either FIDE basic rules or FIDE competition rules. It is apparent that the side to move at least must be one of the attributes considered in defining "position".

Why I posted earlier that @SciFiChess is not counting chess positions in his blog, merely a limited set of chess diagrams. 

For the purposes of defining "position" in chess under FIDE basic rules, the set of attributes you suggest in your definition above is sufficient for the purpose.

But it's not sufficient for defining "position" for chess under FIDE competition rules. Should a situation occur under competition rules where the diagram is as above, Black is to play and the ply count is 85 then White cannot force a win, but if the same applies but the ply count is less than 85 he can. Unlike chess under basic rules the position with a ply count greater than 135 is not a legal position under competition rules because the game would have terminated in a dead position earlier.

It is apparent that, under FIDE competition rules, the current ply count at least must also be one of the attributes considered in defining "position".

Similarly, under competition rules, whatever the ply count in the above position, if the diagram after ...Kf7 with White to play has already occurred twice then White cannot force a win, whereas a win is possible if the ply count is 0.

It is therefore apparent that a further attribute is necessary for the definition of "position" under FIDE competition rules, which could be "the set of diagrams with the same material together with side to move that have previously occurred and how many times each of those has occurred".

This obviously wouldn't prevent a triple repetition claim (or game termination under the quintuple repetition rule) because 9.2.2 specifies that for the purposes of those rules any attributes of the position other than those enumerated are to be ignored.

For some analysis purposes it is not necessary to consider all the attributes of a position that determine possible forward play. For example the Syzygy tablebases use only those attributes necessary under basic rules (neglecting castling rights) but still provide a weak solution of the positions they consider with ply count 0 under competition rules because the construction ensures no positions repeat the basic rules attributes. The Syzygy tablebases don't provide any solution for general positions under competition rules for that reason.

#323
For a triple or quintuple repetition of the position or for a 50 or 75 moves draw claim there are two other requirements: the score sheets and an arbiter to validate or reject the claim.
The arbiter will investigate the claim by inspecting the score sheets. He cannot do so by looking at the position.
The 3-fold repetition counter and the 50-moves counter are no attributes of a position (= FEN), they are attributes of the game (= PGN).

#325
The position = the board and the pieces on it = FEN
The game = the scoresheets = PGN
It is only logical that the 3-fold/5-fold repetition rules and the 50- and 75-moves rules belong to the same section as that about score sheets and the arbiter.
The 3-fold repetition counter and the 50-moves counter are not part of the position, otherwise you could not claim a draw by 3-fold repetition of the position, as the positions with the counters are not the same.

The board and the pieces on it ≠ FEN. I just pointed that out already.

There is no requirement for scoresheets under basic rules, so for a game played under those rules definitely

The game ≠ the scoresheets.

As far as basic rules chess is concerned "position" in the FIDE laws has its normal English meaning, which would not be "the board and the pieces on it".

The competition rules include art. 9.2 which specifies conditions under which positions are considered equal. I would say that is not a definition of "position" and in any case applies only to that article and 9.6.1.

You are free to make up your own definition of course, so long as you don't assert that it corresponds with the FIDE meaning of the term. Since there is no definition in the FIDE laws I would say that meaning is simply "situation occurring in the game".

The word is certainly used in several incompatible ways in chess literature.

The following board layout could be (and often is) an attribute of a position legitimately occurring in a chess game, with the English meaning of position.

however it could only be an attribute of an illegal position as defined by FIDE:

3.10.3 A position is illegal when it cannot have been reached by any series of legal moves.
since pieces occupy squares in the starting position and legal moves never result in a piece not occupying a square.  

For the purposes of analysis only positions that are not illegal need be considered and most definitions are restricted only to legal positions as defined by FIDE (ergo not all legitimate positions).

For the purposes of analysing chess your definition of "position" in post #326 is quite useless and for the purposes of analysing chess under competition rules you different definition in post #322 is also quite useless.

In particular if you're planning to solve chess by a partial forward search, as you do in this thread, then you cannot, under competition rules, identify the nodes in the search space with positions under either of those definitions (as you also do in that thread). Estimating the size of the search space using either of those definitions, as you do, will simply result in a vast underestimate.

I think that there is certainly a way to give a value in a given chess position, so it is interesting to know how much different chess position are possible first

Yes; but it's not possible to give a value in a given chess position in a game played under competition rules if "position" is understood to have either of the meanings advanced by @tygxc in post #322 or post #326. That is because he doesn't include enough attributes to uniquely determine a value in a given chess position.

(See my example and comments in this post.)

doesn't matter, just knowing how many positions is already interesting, to have an idea of the number of possibilities

Before you can decide how many positions there are you need to decide what a position is. 

What do you mean by the term?

There is a vast difference in the number of positions under competition rules according to @tygxc's proposed meanings (neither even a FEN) and mine (a FEN plus a legal list of positions with the same material that have occurred and how many times they have occurred).

My definition would give a value in a given chess position under competition rules; @tygxc's would not. If a definition of "position" doesn't do that, what would be the interest in knowing how many "positions" it results in? (I would agree with his first definition (post #322) for basic rules chess, which in that case would give a value in a given chess position.)

OK, @tygxc's second definition

The position = the board and the pieces on it 

would appear to define a position to mean the same thing as an endgame classification and it might be of interest to know how many of those there are. Not what most people would understand as the number of positions though.

#331
Per Tromp there are (4.59 +- 0.38) * 10^44 legal positions.
The vast majority of these contain multiple underpromotions to pieces not previously captured.
https://github.com/tromp/ChessPositionRanking

Per Gourion an upper bound for the total number of legal chess diagrams without promotion to pieces not previously captured is equal to 3.8521 × 10^37.
https://arxiv.org/pdf/2112.09386.pdf 

The Laws of Chess state when positions are the same and hence also when they are different:
"9.2.2
Positions are considered the same if and only if the same player has the move, pieces of the same kind and colour occupy the same squares and the possible moves of all the pieces of both players are the same."
https://handbook.fide.com/chapter/E012018 

second definition

If you take the 32 pieces and make all different positions, naturally the pawns doesn't count they are the 8 the same, same with the rook, night and, bishop, how many different  positions on the board . (call it a picture) and take then 1 figure out one by one

Do you mean "all different diagrams" when you say, "all different positions"?

If so, does that mean you make no distinction between diagrams and positions. Would you for instance say that this

is the same position regardless of who has the move, and, if it's under competition rules, whether the ply count under the 50 move rule is 0 or 99?

#336
"Incidentally you keep stating that he counts positions with adjacent kings, but looking at this page that would seem to have never been the case"

Tromp generated and counted exactly 8726713169886222032347729969256422370854716254 possible positions, including positions with adjacent kings.
Then he sampled 1 million of these and found 55958 legal and thus he arrived at his number of legal positions.

yes, exactly no need to think about who moves, just  all different PICTURES

oh, I find the best word : The same DISPOSITION

Yes, my apologies.

Indeed, he did effectively cut out adjacent kings only in the illegality tests.

Though I doubt if he could have generated 

8726713169886222032347729969256422370854716254 positions.

In his earlier calculations of an upper bound adjacent kings were specifically excluded.