Space

The concept of space in chess is difficult. It "is not an easily definable or recognisable concept," according to Michael Stean. In Simple Chess (1978), Stean offers the capacity of a position as a guide. "Any given pawn structure has a certain capacity for accomodating pieces efficiently. Exceed this capacity and the pieces get in each other's way, and so reduce their mutual activity" (97).

Stean illustrates capacity with two positions that have an identical pawn structure. They are differentiated by the number of pieces on the board. In the first position, all of the pieces remain and White has a clear advantage with more space to manuever. In the second position, both players have traded off a knight and a bishop. White is overextended and Black has the advantage. With less space, Black's position lacks the capacity for the full complement of chess pieces. With two fewer pieces, "the size of Black's forces is here well within his position's 'capacity'" (98).

This diagram illustrates Stean's second position. In the first, White has an additional knight on c3 and bishop on e2. Black has a knight on f6 and bishop on c8. Black's position is congested.

In the old Modern School, space is one of the three elements, or "basic factors". Siegbert Tarrasch wrote, "the more space a player commands the freer his game and the easier his development" (The Game of Chess [1935], 225). Space triumphs the other factors.

The systematic utilisation of Space, or, to put it another way, the systematic disposition of the pieces, is the most important factor in a game of chess and, within certain limits, even more important than Force, that is to say than a superiority in material. Often a win is obtained because one player forces a decisively better position by a sacrifice of material--a triumph of mind over matter.
Tarrasch, The Game of Chess, 227.

Space is calculated by counting the number of squares that are controlled in the other player's side of the board. But matters are not so simple. In The Oxford Companion to Chess (1996), David Hooper and Kenneth Whyld present the game Tarrasch -- Marco, Vienna 1898. After 25...Bf6, the game reached this position.

White to move

Counting squares reveals, they note, "the players each control about the same number of squares" (380). However, Black is unable to make use of his greater space on the queenside. White's advantage on the kingside is of more account. White's pieces have greater mobility, and as a consequence, a combination is possible.

They observe at the beginning of their article on space that the terms space and mobility are sometimes indistinguishable.

To gain an advantage in space is to achieve the possibility of moving one's men to more squares than are available to the opponent's men. Not all the squares "gained" need to be controlled--they may simply be inaccessible to the opponent's men. Normally, to gain space is to gain mobility, and the terms are frequently synonymous.
Hooper & Whyld, The Oxford Companion to Chess, 379.

Stean does not use the terms interchangeably. Rather, mobility "is the essence of simple chess," and space is "the single most important factor in determining mobility" (97). Space, thus, appears to be a means to an end, and that end is mobility. The terms are not so much interchangeable as primary and secondary. Mobility is a principal factor in chess positions; space is important when it affects mobility.

Mobility, according to Dan Heisman, is an "element of positional evaluation," while space is a pseudo-element. In Elements of Positional Evaluation: How the Pieces Get Their Power, 4th edition (2010), Heisman asserts that, "space is not an end in itself, nor even a means to an end other than mobility/activity" (Loc 1528, emphasis in original). Heisman's examples are somewhat extreme, as he acknowledges, but his conclusion appears in harmony with Stean's discussion.

Mobility can be measured and quantified, just as can space according to some definitions. Activity is subjective, Heisman asserts. Activity is "how much the piece can do in an actual position" (Loc 744). Mobile pieces are more likely to be active.

Heisman offers an interesting position from the Deep Blue -- Kasparov rematch.

Black to move

Black has more pieces on White's half of the board, and contests more squares there. Black has more space. Heisman does not mention space in relation to this position, however. He employs it to assert the relation between mobility (quantifiable) and activity (subjective). White's pieces contest every entry point along the d-file, which at first glance appears to be controlled by a Black rook on d8. The other Black rook supports a passed pawn, but that pawn is blockaded by a White knight. Despite a quatifiable advantage in both material and mobility, Black lost this game. Heisman observes, "the rooks have good mobility but not all that much activity" (Loc 751). Heisman sees mobilty as the "basic tool," but activity is "the better 'real position' indicator" (Loc 759).

Might we conclude that space, the old "basic factor," can be understood better insofar as it contributes to mobility, and mobility is an element insofar as it correlates with activity? How do we distinguish activity from Stean's notion of capacity? Capacity is a static concept. Activity might be a consequence of adequate capacity. But for pieces to experience activity, they need targets. Hence the opponent's position must exhibit some vulnerability that can be attacked.