GöDEL's THEOREMS- "the argument against Theory of Everything"

Gödel's incompleteness theorems are two theorems of mathematical logic that establish limitations against completeness or consistency of any physical or arithmetic theory. They were devised by Kurt Gödel in 1931.

Kurt Gödel 

1st INCOMPLETENESS THEOREM-

“For any consistent axiomatic system in arithmetic there always exists a set of arithmetic propositions which can’t be proved by it.”

It means that, a consistent (free from any sort of contradiction) mathematical theory-X that completely covers up certain mathematical topic-x cannot be stated as a complete theory because it is always possible to formulate a statement belonging to topic-X whose validity or invalidity cannot be proved by theory-X.

So, basically, Gödel’s first theorem states that a theory can’t be consistent as well as complete at the same time and therefore there would always exist unprovable statements.

2nd INCOMPLETENESS THEOREM-

“Any consistent axiomatic system cannot prove its own consistency and if it does so, it is inconsistent.”

So, the 2nd theorem clearly states that any consistent mathematical theory cannot be used to prove itself.

Now, what has Gödel’s theorems got to do with Theory of Everything?

First we must remember that a physical theory is always in form of a mathematical model. So, the Gödel’s theorems hold as good for physics as they do for mathematics. Now, suppose we get a theory that could be claimed as a theory of everything (call it ToE v1.0). For it to be accepted as valid, it is bound to be consistent. So, as it is consistent, by Gödel’s 1st theorem, it will be incomplete or there will always be something that couldn't be explained by it (thus contradicting it from a theory of “EVERYTHING”). Now, even if we construct a even higher theory (call it ToE v2.0) which could explain the thing that was unexplainable by ToE v1.0, Gödel’s 1st theorem would still apply, due to which something else will get formulated which would be unexplainable by ToE v2.0. So, this process of refinement of theories will go on forever but still none of them will succeed to be claimed as a true Theory of Everything.

Suppose we are able to create a satisfying theory of everything disregarding Gödel’s first theorem. Now, this theory by definition would be able to explain everything. And everything includes itself. This is where Gödel’s 2nd theorem will come to play. By Gödel’s 2nd theorem, we know that no theory can prove itself. So, if this theory of everything could explain itself, it will be inconsistent and eventually be invalid.

Therefore Gödel’s incompleteness theorems will deviate any proposed theory of everything from its very definition of being “of EVERYTHING” and will disprove its validity. So, this is how Gödel’s theorems prove to be strong arguments against a theory of everything.

A number of scholars claim that Gödel's incompleteness theorem suggests that any attempt to construct a ToE is bound to fail. Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable. 

He says In his lecture ‘Gödel and the End of Physics’ -

“Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Godel’s theorem ensured there would always be a job for mathematicians. I think M theory will do the same for physicists.”

Still many physicists believe that a ToE is possible to formulate. Most physicists believe that Gödel's Theorem does not mean that a ToE cannot exist. They argue that the theory of everything may not refer to a set of underlying rules but to the understanding of how and why the universe exists and how it works. Surely they will leave no stone unturned to get to it.

(To tell the truth I as a lover of physics am quite scared and startled of the consequences of these theorems)

Click Here for Part-3 (Pillars for the Toe)

Click Here for Part-5 (Reasons and Consequences of the ToE)

Dimensions (in Physics)

Well, dimension broadly means measurement. In mathematics a dimension of a space or object is understood as the minimum number of coordinates required to specify the position of a point within it. The term, dimension of course has mathematical origins. It is quite fundamental in mathematics. But, in physics dimension is a much wider concept. In physics, dimension is perceived as the means by which we can precisely describe a body’s position and structure in the space it occupies (although the natural definition of dimension is the same as provided by mathematics). It is of great importance to physicists, especially cosmologists.

Conventionally, we deal with three dimensions in our daily lives. Those are length, breadth and height. These are what required to describe the 3-dimensional structure of our beautiful surroundings. These are known as the spatial dimensions. Almost all study related to dimensions in maths is confined to these three dimensions. But, these three spatial dimensions can only define the structure of the 3-dimensional bodies. What about its duration? Thus, physics has attached an additional dimension to the conventional 3-dimensional objects. It is the temporal dimension or the time dimension. Time is often referred to as the fourth dimension for this reason, but that is not to imply that it is a spatial dimension. It defines the time duration of a body or its position in the passage of time. It is perceived differently from the three spatial dimensions as there is only one of it, and we cannot move freely in time but subjectively move in one direction. Thus, these four dimensions are required to describe the structure of spacetime. The term spacetime has originated from the special theory of relativity formulated by Albert Einstein.  It is the accepted physical theory regarding the relationship between space and time. It proved that space and time are not two different things. But, they are combined into a single interwoven continuum.

To describe ‘dimensionless’ there is even a zero dimension. It is represented by a point which has no dimensions. This dimensionless object is the base of the other higher spatial dimensions (length, breadth & height). 

Now, let’s see how we can visualize the spatial first four dimensions. 

Well, the 1^st dimension can be represented by a line or a line segment which just has a length (infinite in case of a line and finite in case of a line segment). A line can be seen as a collection of parallel points.

The 2^nd dimension which consists of both length and breadth can be represented in the simplest way as a square (where length and breadth are finite) or in a broad sense, a plane (where length and breadth are infinite). Two parallel lines when connected form a square.

The 3^rd dimension which consists of length, breadth and height can be represented in the simplest way as a cube. Two parallel squares when connected by their corresponding vertices form a cube.

Then comes the 4^th dimension. It is represented by a tesseract which is theoretically formed when two parallel cubes are connected by their corresponding vertices. According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book,”A New Era of Thought”. A tesseract, also known as a hypercube, is a hypothetical object which cannot be shown on a plane surface or even in form of a 3-D model as it is a higher dimensional object. The tesseract and even higher dimensional hypercubes can only be shown through computer simulations.

A Tesseract

                                                        (I got these nice pics from Wikipedia)

Higher Dimensions

In physics 3-dimensional space and one time dimension is the accepted norm. And our visible universe consists of this 3+1 dimensional subspace. But there are attempts to unify the four fundamental forces in physics (or to formulate a theory of everything) by introducing higher dimensions. The string theory or the M-theory is a modern higher dimensional theory proposed by physicists which predicts the existence of up to 11 dimensions! This 11-dimensional M-theory is considered as a valid and the most reliable candidate for “the theory of everything” although till date there is no experimental evidence confirming existence of these additional higher dimensions.

Where do these dimensions exist? The observable universe is 3+1 dimensional but the mutiverses (theory of existence of multiple universes) are expected to be higher dimensional (although there is no experimental evidence for multiverses as well). It is also predicted that the maximum number of dimensions that can exist in any universe other than ours is 11, because beyond that the universe becomes unstable and immediately collapses into a 11-dimensional form.

Now, if higher dimensions are predicted to exist, why can’t we see them or gather evidence about them? Physicists till now can’t predict what exactly the higher dimensions would look like. Simply speaking, we don’t exactly know what we are looking for. The world around us including us is 3-dimensional and thus we can calibrate our scientific instruments up to 3 dimensions only. The dark matter and dark energy that is predicted to consume most of our universe is expected to be higher dimensional stuff as we can see their effects but we can’t see them. It is also a possibility that the higher dimensional multiverses may be hovering right below our noses, but we can’t detect them or get a slightest hint of them because our world is functioning is a 3-dimensional system.

Scary! Isn’t it? But don’t worry it is just a speculation.