Liedzeit

Abstract

Wittgenstein like Leibniz thought that for logical reasons there must be simple objects because it follows from the ideas of complexity and analysis. And like Leibniz’ monads these objects are the substance of the world. The exact nature of Wittgenstein’s objects however and their role in the “states of affairs” has remained unclear since the publication of the Tractatus. In an attempt to demystify the objects, Anthony Kenny once offered the game of chess as a model for the Tractatus world. I will slightly modify this model and introduce a different model based on Conway’s Game of Life.
In a second part I present a new “Causal Theory of Views” by physicist Lee Smolin who is heavily influenced by Leibniz. He postulates a new kind of objects that are actually events. These objects, although inspired by monads, might be candidates for the simple objects of Wittgenstein.

1. Facts and Monads

The world consists of birds and bees and trees and clouds. The world consists of elements. Of atoms. Of sub-atoms. The world consists of sense data. The world consists of bits and bytes (because it is a simulation). The world consists of facts. The world consists of Monads.
The last two claims are, of course, by Wittgenstein and Leibniz, respectively. And since a fact is the existence of atomic affairs and an atomic affair is a concatenation of objects it seems that at least in some sense both Leibniz and Wittgenstein are concerned with the ultimate constituents of being, the Elements of Things, as Leibniz calls them (M 3). And there are other similarities between the Monadology and the Tractatus.
“Both texts are composed as a sequence of numbered sentences; both lay out a picture of the world with little or no argument; both advance a pictorial conception of meaning; and both advance from a description of the structure of the world to reflections on ethics” (Sluga 2018).
And both admit the possibility of alternative worlds, and both deny that there is absolute space and time.
On the other hand, there are important differences.
For Leibniz (and this is one of the criticisms of Russell) every proposition is ultimately reducible to one which attributes a predicate to a subject. That is, relations are not real. (“I hold that paternity in David is one thing, and filiation in Solomon another, but the relation common to both is a merely mental thing” (quoted in Russell 1900: 206).) For Wittgenstein, it might be argued, only relations are real. The only properties of an object are the internal, that is the possible relations to other objects and the external (the actual relations).
The monads are simple only in respect to the compounds they enter. The monads must have qualities, Leibniz says, “otherwise they would not even be existing things” (M 8). This follows from his principle of the identity of indiscernibles. The qualities of a monad, it turns out, is its ability to mirror the qualities of all other monads (M 56).
As a consequence, this means that one cannot take away one monad without affecting all others. Wittgenstein completely disagrees. Not only are objects independent of one another but even the states of affairs are independent from one another. (TLP 2.061) This is the difference between a monistic and pluralistic view of the world.
Another major difference between a monad and an object is that the Wittgensteinian object is not really a constituent of the world. A state of affairs is the smallest ontological entity. Since all combinations of states of affairs can exist and the number of possible combinations of n states of affairs is 2n (TLP 4.27) it follows that a world with no states of affairs is possible. And even this world would have all objects in common with any other possible world. (TLP 2.022 and 2.023)

2. Life as a Model of the Tractatus World

Anthony Kenny once suggested a (modified) game of chess that would serve as a model “as near as we can get to [...] for the way the world is conceived in the Tractatus” (Kenny 1975: 74). In this model pieces and squares are the objects of the world and a state of affairs would be the relation between a piece and a square. This model would indeed honour some of the key concepts of the Tractatus. The world would be the totality of facts, not the pieces but its positions on the squares would be relevant. The rules for the positioning of the pieces give their logical form. The internal properties of for example a bishop would be its ability to move diagonally, its external property the actual position on the board. A piece not being on a particular square would be a negative fact. In Kenny’s modified chess pieces are not allowed to be taken, since objects are “indestructible”. But even with this modification, the analogy limps because as he admits “the atomic facts of chess are not independent on each other in the way that Tractatus states of affairs are” (Kenny 1975: 75).
Space and time (and colour) are forms of objects (TLP 2.051), so it seems wrong to regard the squares of the board as objects themselves. Somehow the position of a piece must be established in a way that avoids a reference to some (absolute) points in space.
Before I try to show how Kenny’s model could be enhanced, I want to introduce a different model that after all does go nearer to the Tractatus world, the Game of Life.
Wittgenstein illustrates the concept of truth by a black spot on a white paper. This spot, he says, could be described by “saying, for each point on the sheet whether it is white or black” (TLP 4.063).
This, in fact, is what one does in the case of cellular automatons. The most famous cellular automaton is the one invented (or “found”) by John Conway. “Life is a ‘game’ played on an infinite squared board. At any time, some of the cells will be live and others dead.” The initial state of the cells could be random or handpicked by the player. But once the game is started “the state of any later time follows inexorably from the previous one by the rules of the game” (Berlekamp, Conway, Guy 1982: 817).
The rules are that any dead cell with exactly three living neighbours becomes alive at the next tick in time and any living cell with exactly two or three neighbours stays alive. Which means that any cells with less than two or more than three neighbours die.
These simple rules are sufficient to generate rather complex patterns and behaviours. For our purpose what is interesting is that the cells are like objects. The dead cells (or off-cells) do not exist, but they “subsist” they form the substance of Life and are part of every possible Life world. (“The substance is what subsists independently of what is the case” (TLP 2.024).) What does exist, what is the case, are the on-cells, and more precisely, the patterns built by the concatenations of live cells.
The patterns are the states of affairs in Life. Now, since the board is infinite it makes no sense to describe a cell by its coordinates (in contrast to Wittgenstein’s illustration). The state of a cell is solely determined by its neighbouring cells.

Figure 1: A glider approaching a block

Some of the configurations in Life are either so common or so extraordinary that they get names. In Figure 1 there is a pattern called glider in the lower left. Pic 2 shows the situation at the next generation. The pattern in the top left has not changed. It is one of Life’s stable patterns, called a Block. But the glider has moved upwards and to the left and has a different shape. Three of its initial cells have survived and two new cells are now part of the pattern. In the next generation the shape is the same as in the first one but rotated. Like the ship of Theseus the glider in the third pictures consists of entirely different cells but is still identical (in a sense) with the glider of the first generation.
A state of affairs that constitutes a glider is completely independent of other states of affairs. The position of the glider can only be given in respect to its former or subsequent incarnations. So “there is a glider” would qualify as an elementary propositions because it depicts five cells within a 3x3 grid in one of the glider shapes. But to say “there is a glider 3 cells south and 2 cells east of a block” would describe a complex fact. A cell of a glider has the external properties of the state of its eight neighbouring cells. The internal properties are defined entirely by the rules. The internal properties of a Life cell object are exactly the external properties it will have in the next generation. The reason is that Life’s rules are simple and deterministic and all objects are equal. One could imagine different kind of cells though with different rules for survival. Or it could be that the external properties of generation n do not deterministically define the state of the cell in of generation n+1 but leave room for randomness.

3. A new Chess Model

And this brings me back to Kenny’s game of chess. Instead of having pieces and squares as inhabitants of the Tractarian chess world we will allow only one kind of object, the squares. Chess would be a kind of cellular automaton like Life. Only this time a square is not a space marker for the piece object but stays on its own. And instead of having only two states, on and off, as properties the square has multiple possible states. A square could be, for example, in a “bishop” state. Normally one would say there is a bishop on the square and this can be moved diagonally in every direction. But one can get rid of the actual pieces and say that the square possesses the ability to transpose its internal properties to any square in a diagonal direction.
The external properties limit this ability to all diagonal squares with an empty state or a non-empty square with a different parity (meaning it is of the opposite colour). So instead of describing a chess piece with either an absolute position on the board or relative to other pieces we just describe the current state of a square by the sum of its possible moves. Only that there is nothing that actually moves. A “move” means that the source square switches to an empty state and the target square switches to the state of the source square. The advantage of this model is that it allows ‘pieces to be taken’. Only there are no actual pieces, a piece is just a (human readable) marker for the state of the field, no more essential to the square than the colour. What happens with a move is just that two cells change its state.
This chess game without pieces would still be chess because it shares the logical form with standard chess. It has the same multiplicity. A single red ball, Kenny says, cannot represent a game of chess, but if we allow to bounce the ball, we could create a system where every possible chess situation is represented by us bouncing the ball a specified number of times (Kenny 1975: 75).
The bouncing system would not be very useful but one very interesting way of transferring the game of chess to a different medium is to have a chess game represented by a piece of music. A piece moving to e4 could for example be mapped by playing E natural in the 4th register.
The note defines the piece (or its state), e.g., Pawn = 1/16th note, Knight = 1/8th note (Stokes 2011). With some clever musical transformations, one could actually make the musical game quite pleasing.
And the advantage of this chess incarnation is that we not only got rid of the pieces but also of the space frame.
One of the games Stoke transformed is the famous one played by Adolf Anderssen and Lionel Kieseritzky on 21 June 1851, the so-called Immortal Game.

And this serves as a good example for the difference between Leibniz’s monistic and Wittgenstein’s pluralistic system.
With the seventh move White puts a pawn on d3. This move has been criticised by German grandmaster Robert Hübner who suggested Nc3 instead. This would have been a legal alternative. And in one sense it would be natural to say that with the alternative move the positions of a pawn and a knight has changed, but “everything else remains the same” (TLP 2.21). On the other hand, Leibniz would say, that everything has changed, since every chess piece is defined by its relation to all other pieces (more precisely by its perceptions of the other pieces). Nc3 would not be compossible with the other pieces that make up the Immortal Game world. It belongs to a different world. And of course, with the Hübner move we would not be in the Immortal Game anymore. Wittgenstein would say that there is no mythical connection between one chess piece and all the others. Any legal move leads to a perfectly normal new position and the game can non-deterministically continue to the end. The fact that in this particular position at one time the pawn was moved to d3 is just that a historical, contingent fact.

4. A Causal Theory of Views

Are there Wittgensteinian objects? Wittgenstein was not sure. “It always looks as if there were complex objects functioning as simples, and then also really simple ones like the material points of physics...” (NB June 21, 1915). He doubted that questions like the one about the nature of the object could be answered. “It looks as if I could say definitively that these questions could never be settled at all” (NB September 3, 1914). And it is probably safe to assume that there is no group of scientists at CERN in Geneva searching for Wittgensteinian objects – or monads.
But the idea is not as ludicrous as one might think. Because theoretical physicist Lee Smolin recently advanced a new theory that explicitly refers to Leibniz. He lets himself be guided by five principles all of them being aspects of a single principle: Leibniz’s principle of sufficient reason (Smolin 2020: 233). From quantum theoretical assumptions Smolin holds that “space and time cannot both be fundamental. Only one can be present at the deepest level of understanding: the other must be emergent and contingent” (Smolin 2020: 235).
Smolin was inspired by the Monadology. He even calls the elements of his relational model of the universe “nads”. Nads, of course, are not really monads. For one thing, they do not have a soul. “Nads have two kinds of properties: intrinsic properties, which belong to each individual nad, and relational properties, which depend on several of the nads” (Smolin 2020: 242).
This actually sounds more like a description of a Wittgensteinian object than a monad. Especially as monads do not really have relations. (Remember, relations are only a mental thing.) On the other hand, objects certainly do not obey the principle of the identity of indiscernibles (TLP 2.0233). Next, Smolin adopts the Leibnizian idea that nads have a “view of the universe” and he even thinks that the actual universe differs from other possible ones in that it has as much perfection as possible. Only the “quantity that is maximized, which Leibniz called ‘perfection’, we call an action” (Smolin 2020: 243).
A nad, Smolin goes on, is an event and the relation between events is causation (Smolin 2020: 254).
“I then would propose that each event has a certain quantity of energy, and that energy is transmitted from past events to future events along the causal relations” (Smolin 2020: 261).
In our musical chess world, a note would cause the next note.
In his summary he says that “the universe consists of nothing but views of itself, each from an event in its history, and the laws act to make the views as diverse as possible” (Smolin 2020: 271).
Out of context this sounds as mysterious and esoteric as any philosopher’s speculation. It is a new theory, and “as is the case with any new theory”, Smolin says “it is most likely wrong”. But, and here there is a difference to the systems of Wittgenstein and Leibniz, “one good thing about it is that it will be most likely be possible to test it against experiment” (Smolin 2020: 248). And I think that both Leibniz and Wittgenstein would agree that this in itself is an improvement on their own ideas.

References

Berlekamp, E.R.; Conway, J.; Guy, R.K. (1982) Winning Ways for Your Mathematical Plays, Vol. 2 Games in Particular, London: Academic Press.
Kenny, Anthony (1975) Wittgenstein, Harmondsworth: Penguin.
Leibniz, Gottfried Wilhelm (1714) Monadologie, https://www.plato-philosophy.org/wp-content/uploads/2016/07/The-Monadology-1714-by-Gottfried-Wilhelm-LEIBNIZ-1646-1716.pdf.
Russell, Bertrand (1900), A Critical Exposition Of The Philosophy Of Leibniz, Cambridge: Cambridge University Press.
Sluga, Hans (2018), “Simple Objects, Complex Questions”, https://www.truthandpower.com/?p=32.
Smolin, Lee (2020), Einstein’s Unfinished Revolution, London: Penguin Books.
Stokes, Jonathan, 2011, Chess Music.