All entries for Monday 28 January 2008
January 28, 2008
Gödel’s incompleteness theorem is a mathematical formalism of a proposition on recursivity in knowledge that has been expressed by influential and, doubtless, lay thinkers throughout history.
It states that, within a complete set of true axioms that describe all observable phenomena, there must exist the statement, which is here arbitrarily called G, thus: “G = G is not true”. Thus for G to be true, it must not be true. Indeed, if G is not true, then G must be true. Here it can be seen that for the axioms to be complete, then G must be true. From this arises the inconsistency in that G is therefore not true. Likewise, if G is to be consistent with all other axioms, then one cannot include G in the set of true axioms, such that the set is incomplete.
The recursivity of G is quite difficult to comprehend and, as such, I here provide a thought experiment with interesting consequences.
A person, P, attempts to learn and know all things. Only this way can P ascertain that there is no axiom which lies outside of a complete set of true axioms. Clearly, there is a lot to learn not only for the time in which P exists but also for all other times in the universe such that P can acertain that the axioms are reliable in time. It is thus required of P to have lived for eternity and to have experienced all things. To encompass all things that have existed, or are to exist in the future, P must be omniscient, i.e. P must actually be the complete set of axioms, i.e. P completely encompasses and describes existence, i.e. P is existence itself. I, for one, know of no one who can claim to be existence in its entirity!
What this states, like in Gödel’s theorem, is that no individual entity can know the complete truth, i.e. it cannot know all things. By individual, I mean comparable to another entity, both individuals/entities being incorporated in existence. By existence, I mean a set of complete AND consistent axioms. Obviously, there must be a truth such that existence can exist! Gödel’s theorem can only apply to what an individual can know about the truth/existence. For something to know all things, i.e. the truth, it must itself be the truth and be existence itself. Existence is unity, i.e. there must not be any other form of existence. To prove this, let existence be divided into two, existence(A) and existence(B). From the previous thought experiment, existence(A) can only know all things in existence(B) if it incorporates/encompasses existence(B), and vice versa. If existence can be described by two entities that do not understand eachother such that there is an incomplete set of true axioms, then there is no absolute truth and it can be argued that we in fact don’t exist! This is somewhat problematic.
So, in order for existence to exist, as it must do for me to be writing this statement (analogous to Descarte’s “cognito ergo sum”: “I think therefore I am”) there can only be one truth, i.e. only one existence (unfortunately, the restriction of language makes this difficult to explain, as even the number “one” is comparable to other numbers. Such is the reason that I prefer to use the word “existence”, by which I not only mean “all encompassing” but “being all”). All things, including us as individual people, are encompassed by this unitary existence. In fact, all things are the existence! The only useful description of our individuality is as manifestations of existence.
I urge you here to not be put off by connotations with hippy culture or religious postulates: we are all of the same source, manifestations of the truth which is existence. We are all, for want of a better word, one.
One might ask how existence can incorporate that which does not yet exist physically. To answer this, I take an analogy from concepts in physics (although an analogy in its own right is no grounds for proof, the best we can hope for as individuals is an extensive set of analogies which explain existence to the best of our ability by coherence among the analogies). Energy, better described as potential energy, is a quantified measure of information stored for a system/body: it is very simply the potential to impart energy/information from one body to another by way of a force. Force is a description of the action of communication of information between the bodies (I might add that the commuication is mutual, i.e. every action has an equal and opposite reaction – Newton’s 3rd law. This states conservation of information. For this reason, I do not support the theory of the big bang, that something came from a singularity). A potential to impart information relates to the informatic existence of the two bodies at a later time. Thus potential, i.e. future information and knowledge, is contained within the current state of potential energy. This is in agreement with quantum mechanics as an observable state can be described as a linear combination of possible states. As such, existence encompasses all that is observable and knowable AND all that is to be observed and known.
In conclusion, hopes for a theory of everything are futile. Ultimately, any advancement in knowledge is futile. Of course, something must exist and the information contained within that existence is a sort of knowledge of it’s existence (again from Descarte’s famous conclusion). It is only natural for us to want to maintain existence and a good, proven way of doing this is to search for new knowledge. In fact, being of the same existence as those who are to live in the future, it seems only reasonable that we provide the best possible world in which they can live. Why bother thinking of them, especially if they do not yet exist? Because although they are a different manifestation to us, we are the same: of the same unitary existence.
So what is left for us, as manifestations of existence? We are all bound in unity by existence. We are of the same source. If the term consciousness could be coined to convey a means of communication of information, or the potential thereof, then consciousness permeates all things in existence. We ultimately share the same consciousness.
There are only two solutions for exiestence to be unitary and encompass all individuality in being. One is that in fact nothing exists. Non-existence is also unitary and encompasses all things that do not exist. However, as Descarte so rightly put, the fact that I am even thinking of this possibilty necessarily refutes it! The other solution is that we are all bound into a unitary existence. This binding must be a phenomena with a net attraction, such that existence is in a stable state. It must be advantageous for all individual bodies within existence to incorporate this binding phenomena. I would call this binding “love”.