The Cosmic Chessboard, or, ONCE MORE FOR OLD TIMES SAKE!
A reader or two unwisely commented that my endless debate by Dr. Andreassen on radical materialism was worthwhile, perhaps as a case study on the pathology of a man like me who keeps arguing long after it is clear his opponent has left the room, but he continues to prance and mince about the stage, making grand orator’s gestures and barking rhetorical questions at an empty chair.
For the sake of those one or two unwisely flattering readers, I would like to offer one more argument along the same lines as the infinite number already spoken. Since this is not addressed to Dr. Andreassen, I will not try to simplify the argument to slow and childlike steps, but merely speak as if I were addressing someone learned in this discipline.
Let us start with two assumptions: first, all existing things whatsoever are made of matter.
Second, this or any other meaningful statement made about an existing thing is either true or false.
If a statement represents what it intends to represent, (namely, that if what a statement says is so indeed is so) that is what we call true; and if not, the statement is false. For the purposes of this argument, we need not bother with graduated variations of degrees of accuracy.
To this was must add a third assumption: that when a statement is true, a certain correspondence obtains between the statement in the brain, and the object the statement represents.
Hence, there is no possible condition or arrangement of the cosmos (including the parts of the cosmos where we human brains store statements both true and false about the cosmos) can be represented by a given number.
In other words, a true statement would be represented in some fashion by a particular constellation of brain atoms in the speaker’s mind, and in order for the same statement to be false, either the statement in the brain, or the object to which the statement corresponds, has to suffer a physical change. Either the sun has to go down or the statement ‘the sun is up’ has to have the word ‘not’ inserted, for the statement ‘the sun is up’ to change from true to false.
We do not for this argument need to say what this relationship is: only that the picture in a man’s brain of the universe around him if the picture is true cannot be the same as when the picture is false, and, under our axiom of materialism, the difference is a different physical arrangement of elements in the brain or the elements in the universe.
If everything is made of matter, then the cosmos is nothing more nor less than a collection of fundamental particles in motion, that is, the smallest possible bits of matter. And this includes the sentence in the second assumption, and the correspondence in the third assumption.
For the sake of ease of expression, let us call the fundamental particles ‘atoms’ as did the ancient Greeks, even though modern scientists use different words for fundamental particles and future scientists no doubt will use other words not yet coined.
Now let us introduce exhibit A, the cosmos.
In our imagination, let the cosmos be divided into a single four-dimensional hypercubical graph consisting of a sequential set of three dimensional cubical graphs. Let there be constructed any number of imaginary cubes as needed to form a three dimensional grid fine enough to hold only one and only one fundamental particle of matter such that no further meaningful subdivision of space is imaginable, and each cube covers a duration short enough that no further meaningful subdivision of time is possible.
So, are you imagining the four dimensional chessboard of ultrasubmicroscopic cubes extended throughout the several billion lightyear diameter of the visible universe, and back to the Big Bang and forward to the Heat Death of the Universe? Good.
For the sake of clarity of expression, I will call these ultrasubmicroscopic cubes of a given time ‘squares’ or ‘positions’ and merely trust that the reader will not be confused when I use a two dimensional word for a four dimensional expression of duration, height, breadth, and depth.
For ease of imagination, let us picture the cosmos at a given time to be like a chessboard where each square is so small that two fundamental particles cannot both fit into the same square. And at the next possible meaningful unit of time, the cosmos is another chessboard where the fundamental particles occupy each one its own square if it has not moved, and a new square, if it has.
Now, if since we have selected the smallest possible units of volume and duration for each cube of space and each instant of time, each square has only two possible states of being: full or empty. Therefore each square of the cosmos can be expressed by a simple binary number, one or zero.
(For the sake of completeness let me also mention that if the universe is continuous at a fine level, such that there is no fundamental particle individual into smaller particles, or no least events indivisible into smaller meaningful units of time, then this same argument can be made using a number line rather than discrete individual numbers. But since modern science assumes the universe is composed of quanta at a fine level, that alternate argument need not concern us now.)
This implies, as said above, that there is no possible condition or arrangement of the cosmos cannot be represented by a given number.
An example will make this clear: If our universe were much smaller and simpler than the one we find ourselves in, let us say a place called Lineland, which is line of no duration with only three possible positions for atoms to occupy, we could visualize each possible universe thusly:
- 001
- 010
- 011
- 100
- 101
- 110
- 111
The alert reader will note that this is merely the first six numbers in base two. Hence every possible state of the universe of Lineland can be expressed as a unique number.
(I am not considering the possibility of an entirely empty universe, 000, because then nothing would exist whatsoever, and the second axiom assumed above, ‘This sentence exists’ would be untrue. So for the purposes of this argument, let us, like the ancient Greek, not consider zero to be a number.)
There is nothing about this simple universe which makes it differ in kind from our more complex universe, which we shall call Spacetimeland.
Our universe is represented by a four dimensional hypercube of three dimensional cubes rather than a line of three positions, but the number of possible positions in our four dimensional chessboard, while it is very high, is nonetheless a finite number, and there is one unique number for each possible arrangement of atoms in it.
I draw the readers’ attention to the third position in the simple universe of Lineland: the sequence is 1,0,1,0,1,1. This is once again, merely a number in base two: 43. As above, if any part of the simple universe of Lineland can be expressed in terms of a number, and likewise for the more complex universe of Spacetimeland we inhabit.
Therefore, if materialism is true, and everything is made of matter, then the whole cosmos can expressed as a unique number, and every part of the cosmos expressed as a number.
Again, if materialism is true, then the human mind and all its contents, including every symbol, idea, statement made about the universe, whether true or false or meaningless, including all the statements in this argument and all other arguments, are nothing but a part of the cosmos, and therefore correspond to an arrangement of atoms and therefore can be expressed as a number.
This means the human mind is the comprised of the physical characteristics of the brain and nothing but the physical characteristics of the brain, and, as already been established, the brain is just one part of the chessboard of the universe, and all its contents and all their motions can be expressed as a large by finite number.
So that, for example, in my brain, my disbelief in radical materialism could be expressed as 111000 if, for example, the argument here on this page fills the first three positions in my brain and leave empty the next three. This is equal to 56. Again, a believer in materialism might have his first and third position filled: 101000 equals 40.
And likewise for every belief in the mind whatsoever: materialism is 40, skepticism about materialism is 56, and so on.
Now let us hypothesize a certain statement G. This is a statement which, when I think it, the atoms of my brain line up into an arrangement represented by a number, let us say 07.
This statement G is “The statement R is true.”
And statement R is “There is no number that represents Statement G.”
It cannot be doubted that I am capable of writing this sentence, for I have done so in the paragraph above, and I cannot write it without thinking it.
A paradox arises: Statement G is either false or true. In other words, either it corresponds to the physical conditions of the universe (the entire universe including all human brains within it), or it does not.
If Statement G is false, then it does not correspond to the conditions of the universe, in which case the conditions of the universe are some number other than 07.
But if Statement G is false then statement R is false, and therefore there is a number that represents Statement G, that is, a condition of atoms whose positions on the cosmic chessboard, including those lined up in my brain, which represent Statement G. We have already said this number is 07. Since the universe cannot both be and not be in condition 07, it is impossible that Statement G be false.
But if statement G is true, then the universe corresponds to the condition that obtains when statement R is true. And if Statement R is true then the statement there is no number that represents Statement G is also true. If there is no number that represents Statement G, or my brain thinking Statement G, r even even one of any possible arrangements of atoms in the cosmos, then materialism is false, because then something exists aside from what exists in the arrangement of atoms on our cosmic chessboard.
(G for Goedel, if you saw where I was going with this.)
P.S. I promise never to talk about this topic again. Unless someone actually comes up with an argument I have never heard before.
About John C Wright
John C. Wright is a practicing philosopher, a retired attorney, newspaperman, and newspaper editor, and a published author of science fiction. Once a Houyhnhnm, he was expelled from the august ranks of purely rational beings when he fell in love; but retains an honorary title.
