Sunday, July 6, 2008

Everlasting Frontier - 1

After you cut the emphasis on personal spirituality (whatever it means), Science and Religion are two different ways of looking at the world. There are opposing qualities built into each of these two world views, which can never be reconciled.

Science always puts great value on observation. It has an evershifting point of view which always incorporates the latest experimental data. Just to illustrate science's viewpoint on cosmology it has changed from flat earth to spherical earth with sun and moon orbiting it to a rotating earth orbiting around sun in the solar system amidst fixed stars to expanding universe resulting from Big Bang to (currently the best known estimate of) increasingly expanding universe. There is no guarantee that this position will not change. In science theories are just mathamatical models to assist prediction and understanding (maybe by compactifying knowledge). The status of experimental observation is supreme.

In contrast, Religion (or at least organized religions) has a fixed orthodoxy. It often makes very strong claims about origin of universe, life, consciousness, knowledge , morality and just about everything. Actually just about every claim of religion inevitably clashes with those of science. Very often in the past its claims have not been borne out by observations. A majority of us discards the religious claims even while carrying on with religion with undiminished conviction. Religion is a force to reckon with chiefly because its proponents hugely outnumber the science proponents. So much so that survey after survey reveals the large number of religious practitioners among the community of scientists. I like to think that often this support has political causes (and shades of cowardice), the thinking being that otherwise it may alienate scientists from public. But the tenacity of religious belief among scientists has always surprised me and I must accept that their faith is not a deceptive facade in majority of them. It really comes from some corner of heart only.

Given the huge chasm between them it seems clear to me that reconciliation is simply not possible. One has to give to the other. The formidable convincing power that science has is pitted against even more formidable emotional attatchment to religion. Some of the very intelligent people are busy furthering the aims of either of these two with all the effort that they are capable of. It is no secret on whose side I am.

It is interesting to note how the hottest frontier of this battlefield has changed with time. Let me concern only with the organized religions here. I mean all those religions going under the collective name of Christianity or Islam. The other main group of religions are Indic ones (meaning Hinduism, Sikhism, Jainism, Buddhism etc). These ones are usually so amorphous as to cleverly dodge many skeptical enquiries. Any discussion/criticism of scientific position of these religions is easier and harder to do at the same time. I will discuss these religions later.

Since it made its debut during Ptolemic cosmology, Christianity has more or less plagiarized it as scriptural (how so unoriginal). At that time it must have sounded very high tech that God's book contains all the knowledge which is 'true'. Readers please note the political intentions and implications of this appropriation. The late comer Islam also did likewise. Too bad that cosmology was not much changed even then. During the late middle ages in europe there followed a turbulent and bloody war resulting from this orthodoxy with people who were intent on testing the claims themselves. We all know the burning of Copernicus and indictment of Gallileo. It seems clear that there must be many more who remained mute against this gangster like hold of theocracy. Islamic world was too steeped in Allah (or too scared to speak just like today) to question any dogma. We see that religion was cheifly fighting against the science of cosmology during those times. Since then the ground has shifted considerably. Now science holds unquestioned respect in this field. So much so that Pope (no less) thought it prudent to rescind the charges against Gallileo.

Since then a much more bloody battlefield has opened up after Darwin's theory of Natural Selection. This time other religions have also joined forces though not by any mutual collaboration. Islam and Hinduism are also now part of this war. I believe that this is probably the last frontier of this conflict. And no prizes for guessing who would eventually win.

We will talk about it in the next part.

Wednesday, January 9, 2008

The Omnipresence / Eternalness Riddle - B :
Dimension of Space Time.

Note : This is a follow up to this post. Please don't bother about the precise meaning of mathematical terms I use. Trust me, they have the nicest intuitive meaning ! Just read on.

A one dimensional world cannot be observed 'simultaneously' while being 'inside' the world. Only a being of 'higher' dimensions can have such a global view. A two dimensional being can observe one dimensional world without being part of it, and can also 'intervene' in it without being restrained by the topology of it. Similarly to be able to intervene in our usual (3,1) dimensional space-time, a being must necessarily inhabit a world which embeds our world as a part of it. Such a world must be at least of dimension 5. Hence an omnipresent / eternal God must inhabit this 5-D world which straddles our world. So for such a God to exist, this Embedding postulate (i,e our world being embedded inside a higher dimensional space) must hold. Please note that truth of embedding postulate does not imply existence of God, but its falsity forbids it.

Suppose such a God exists, then we can contemplate of a God of even higher dimensions. So let us denote and grade such Gods as G(m,n), where m denotes the spatial dimension and n the temporal dimension of the world it inhabits. We also assume that m and n are the maximal dimensions that G(m,n) inhabits e.g, G(4,1) can possibly 'rule' our world since it has the ability to 'intervene' in our (3,1)-D world, but would be 'powerless' against G(5,1) or G(5,2) and so on. As of now we have scant evidence of any higher dimension inside which we are embedded. But to allow this postulate we have these possibilities -

  1. If our world is exactly minkowskian of dim (3, -1) - it means it has 3 spatial and one temporal dimension and is geometrically flat throughout - then there doesn't seem to be any way to detect its embedding inside a higher dimensional world. In this case the embedding postulate can be possibly verified only when divine intervention occurs and gives us proof of His existence, which hasn't happened indubitably so far. Thankfully for God, our world is known to be non-minkowskian. So the God hypothesis makes a dashing comeback !
  2. It makes much more sense to pose this 'embedding' riddle for a non-minkowskian space, which holds for our natural world. However, mathematically, it is possible to embed it in a huge number of ways in some higher dimension. Let us demand the embedding to be a smooth topological embedding (I mean that it is not an immersion, plz don't bother about these terms, if it doesn't make sense to you). Further we distinguish two embeddings only upto isometries of the ambient space, which is a very reasonable assumption (in future, we may possibly have to distinguish between orientation preserving and orientation reversing cases). We also demand that the ambient space be minkowskian. This is the nicest possible case.
    So to illustrate : since a Klein bottle can be embedded in 4D space, so if our universe were a Klein bottle (which is 2 dimensional), its ambient space would need minimum 4 dimensions to properly embed it. I highlight this example merely to point out that the ambient space need not be of just one dimension higher.

  3. The third case is when the ambient space is not minkowskian, but embedded in a higher dimensional space, which in turn is again embedded higher up, until we reach a minkowskian space. Let us assume that this embedding stops after a finite number of such embeddings.
  4. For the fourth case, we assume that embeddings never stops. By this we mean that never in the chain we get a minkowskian space. Please note from case 1 that a minkowskian space effectively neutralizes the embedding postulate and no need to further investigate. Getting a minkowskian space is tantamount to stopping of this chain.
  5. Fifth, we assume that the chain stops but the final space is still non-minkowskian. This final minkowskian space has a stand alone spooky existence not needing any ambient space to exist. I find this case the strangest.
Note : The reason I am discounting non trivial isometric immersions is that, then the topology of the space would not be the same as that of its ambient space (actually strictly finer), forbidding the God to intervene. I mean the intervention is also assumed to be topologically smooth.

We will continue ...

Tuesday, January 8, 2008

The riddle of Omnipresence / Eternalness - A

We would like to dissect the purported omnipresence and eternalness of God. Taken together it means that God is present at all places and at all times. How do we make sense of such a crazy idea ? How do we begin analyzing it ?

Let us ask some questions first. Can God view the future as well as past at the same time (I am not sure what it means) ? Maybe he can go to any moment of past or future at will ? If so can he visit his own beginning ? Suppose he does, then at that *moment* are there two copies of God ? How many copies is he allowed of himself ? Or maybe he is without any beginning. Even then suppose he visits any epoch twice, does that make two copies of him ? Maybe he can only *observe* any epoch, but not *enter* it bodily. If so, in what sense he is eternal ? Or maybe He is outside of time. Anyway, does He change with time ? If so how was a *young* God ? Maybe eternalness means unchanging. Then is he a fossilized being ? Maybe we should understand time's status in the Real System first.

What does going back in time means ? Let us say I was born 100 years back and living hale and hearty 50 years back. Now if my today's self visits the epoch 50 years back, am I there as two copies of myself ? Maybe the moment my current self enters that epoch, the earlier self vanishes. If so, then what happens if I want to leave that epoch after staying there 5 minutes ? Does the original self comes back to reoccupy that epoch ? Where it would come from ? Such a conundrum surely doesn't happen with space. If I revisit a point in space where I was 5 minutes back, that point doesn't get two copies of me at the same instant. Or rather my past self and current self are separated by 5 minutes in time even if they occupy the same point in space. We never observe anybody or anything occupying one instant of time at two different points of space. What about its dual : can somebody occupy one point of space at two different instants of time ? Surely it can. We have plenty of such experiences. We won't be surprised if a chimpanzee also understands this.

There appears a fundamental asymmetry in the nature of space and time. Unlike space, time is essentially unidirectional (at least as far as all our experience goes). Why this is so is the biggest of mysteries. To confound matters, many of our physical laws appear symmetric in time. Meaning they do not explicitly forbid backward time travel. As far as I know, only two laws are asymmetrical in time. They are the second law of thermodynamics and state vector collapse of a quantum state. The former is actually a statistical statement, which makes backward evolution ridiculously improbable. The latter imposes insurmountable barrier against backward evolution, as after a wave function has collapsed, it looses all information as to where it came from, hence cannot go back. This picture of real world (actually the real model) is certainly open to revision in future but as of now, there is no way it can be challenged. Some people do dispute this interpretation, but even the most serious dissenters would agree that quantum mechanics does impose a barrier against backward time evolution. And nobody can deny that there is no such restriction (either in the reality or in its model) for space.

What does this tell us about Omnipresence and Eternalness ? We will see ...

Monday, September 10, 2007

The Omnipotent God - B

This is a follow up to part A.

Recall that * version of Gods is the same as unstrarred one except it refers to knowledge as opposed to power. Also recall the definitions of more knowledgeable and more powerful, which unlike the colloquial sense are umambiguous. Nevertheless it is worth reiterating here. To be able to do some task (defined here) is a measure of Power of an entity and his ability to carry out a proof in a formal system is a measure of Knowledge of a k-entity, which as shown here is same as entity, which also shows that power and knowledge are equivalent concepts. Taking inspiration from the knowledge versions G2* and G3* here we define G2 and G3 Gods. After this we can equivalently talk about G2/G2* and G3/G3*. For the sake of completeness, we also reiterate the definition of G1 here.

  1. The *list of tasks which God (called as G1) can execute includes any conceivable task. i.e, if there exists any entity, then all the tasks from its *list are also present in the *list of G1. We have called this *list as L1. And we have shown that any entity with L1 as its *list cannot exist. The star-version of G1 is G1*, which is in possession of a formal system capable of telling him all the truth. This runs into same contradiction as G4. This point actually shows that *List concept itself is troublesome. Hence we change tack and work only with Lists.

  2. A G2 god is one whose list is union of lists of all the entities. This means that such a God has already executed all the conceivable tasks, maybe before the universe began.

  3. A G3 god by contrast has not performed all the tasks, but is in the process of finishing it and will do so eventually.

In the sequel it is sufficient if we talk only about either the starred or unstarred versions not both. So we use the terms interchangeably.

Also for the sake of completeness we mention that the corresponding notion of G4 in context of omnipotence is the *list L1. (Since G4 is a formal system not a k-entity so its omnipotence-counterpart must be a *list rather than an entity). We have already agreed that L1 is non-existent.

A point in Omnipotence vs Omniscience

In Dawkins' The God Delusion there is an argument showing why omniscience and omnipotence are mutually incompatible. Roughly it goes like this :
If God is omniscient he must be knowing what he is going to do at some point of time (say one year later) in the future. Then he is powerless against changing it. Hence He cannot be omnipotent
As we understand, defining omnipotence and omniscience is a troublesome task at best. And to give it a concrete unambiguous meaning seems impossible. We have tried various tacks of trying to define both the things, none of which works. Ultimately these concepts are ill defined and logically contradictory. Hence they cannot be used to form unambiguously meaningful sentences. So the above proof stands invalid. We are not saying that omnipotence and omniscience are mutually compatible. Just that it doesn't make sense to ask this question. In this respect it is like many english questions which are grammatically correct but meaningless to answer. like - what is the shape of red color ? flaw : shape is not defined for color.

In fact as we have seen here, if at all omniscience and omnipotence should be expected to go hand in hand. But again it is meaningless to explore this thing further.

Wednesday, September 5, 2007

The Hypothesis of Atheism

So far we have been trying to understand the Theism hypothesis. Upon some reflection, it turns out that it is not so easy to make this hypothesis precise, which is where most of the difficulty with God arises. Nevertheless we will try various approaches and see what we can do with it. For the sake of completeness, we mention a caricature of this hypothesis : There exists a (many ?) loving, caring, all powerful, all knowing, timeless, supremely perfect (and with limitless creative powers) being called God. The universe, life, humanity, ethics, morals, etc all are His creation. He rules the world with his infinite wisdom.

Now we come to the Hypothesis of Atheism. Again different people may have different interpretations of it, but let us have a initial working hypothesis, which we may refine as need comes (just like we are doing with theism hypothesis).
  • Life has got no spiritual meaning. We are all products of unconscious physical laws and our life is a result of choices/actions of all of us and the universe. There is no divine intervention in life. We are solely on our own.
  • Death has got no higher purpose. We have got just one precious life, which cannot be reincarnated and is lost irretrievably after death. In short, death is a full stop. No more outer journeys, no heaven, no hell and no after life.
  • There does not exist any soul, which is also one of the ambiguous concepts. We will try to understand it clearly in the future. Also we don't consider existence of ghosts, spirits, bhoots, demons, draculas, etc.
  • The universe was not made by God neither are we designed by any conscious being.
  • Nobody rules us from up there since there is no 'up there'. Our ethics, morals etc all are our creation and God/religion has no place in it. And nobody is 'up there' to punish for our wrongdoings, we have to do it on our own if at all.
  • Finally there is no God. So no divine intervention in the world too. No Allah, no Yahweh, no Ishwara, no Awesta, no trinity, no awatars, no prophets and no awaited messiah. Anybody who claims to be a prophet or awatar is either fraud, foolish, demented, demagogue, power hungry or all. Can't help but wonder how much our politicians (especially the dictators) fit this picture ? This we shall see is actually more true : such messiahs are usually very cunning politicians. In fact their aides are even more cunning and cruel.
  • Religion is a big eyewash. It serves no constructive purpose in our life. If something superficially seems nice about it, it is only our own inherent goodness and is not bestowed either by God or religion. Consequently prayers are meaningless. And so are all purported Books of revelations.

So let us steer clear of such prophets, messiahs and their revealed Books. Let us embrace the beautiful life that we have got with such a great luck. Let us pay our just tributes to mother earth by acknowledging its role in (figuratively speaking) creating life and not pander to imaginary / destructive Gods. We have already destroyed too much life in the name of God. Let us put a stop to all this madness.

Rationally speaking, the burden of proof should rest on the theists. We atheists are simply asserting the opposite of theists point of view. So technically theism is a hypothesis in need of proof not atheism. But to start with we consider the theism as well atheism on equal footing and start our investigation.

Tuesday, August 28, 2007

The Real World

Before we examine the ontological status of God further, we would like to have a look at the real world. We have to have a protocol on what we should accept as 'truth' in the real world. And this protocol better be an objective one. Any flimflam view such as spirituality or 'inner voice' wouldn't do unless it is exactly reproducible in the same circumstances by almost all observers. If possible we would like to have some recorded measurements (may be some numerical value) associated with any observation. Since we have already seen that 'truth' is always based upon a existing framework or formal system, we would like to know the formal system which is associated with 'reality'. To be precise we are elaborating this most important example, as promised.

Here we go :
  1. We accept the reality of the world out there. We take the facts that nature 'reports' us at face value. We do not entertain (at least for now) the notion of all of us being in a daze or hallucination. To be precise we distinguish between 'hypothesis' and 'reality'. Collective hallucination of humanity is a hypothesis, which we do not accept unless we have checked it. The 'truth' of data is determined by its reproducibility. Any impartial observer must be able to get the same data (within realistic errors), otherwise the 'true' status of the reported data is forfeited.

  2. Consequently, any accepted 'truth' of reported data is conditional. We assign likelihood estimate of the 'truth' of the data based on the degree of its reproducibility. 100% reproducible data is considered highly likely to reflect the 'real world' extremely accurately and is considered a reliable guide to 'understanding' nature. Less that 100% reproducible data is considered that much unreliable in proportion.

  3. Based on the data, we make hypothesis concerning the observed phenomena. This is same as finding a model or a formal system which can be used as a computational machine to predict expected data under new conditions, which we promptly check to ascertain the correctness of the model. Any discrepancy is used to make (or tweak) the model better. We give a name to the model constructed so far i.e the Model of Reality (also referred to simply as the Model when no confusion arises).

  4. To decide upon whether to accept a given hypothesis, we conduct more and more experiments. In the accompanying figure we see two cases, the square one and the circular one. Each one is for testing a different hypothesis, say S for square and C for Circular. The size of the figure indicates the likelihood of the hypothesis being true. The darker color is the likelihood after more experiments (either similar or different) are conducted in both the cases. Clearly in the first case, the hypothesis S stands more strongly, whereas in the second case after more experiments are conducted, we find that the hypothesis C stands on shakier ground. Consequently we do not trust the truth of hypothesis C. This is the way we weigh evidences.

  5. We also accept Occam's razor. If two hypotheses perfectly well accounts for the observation, the complicated hypothesis is discarded in favor of a simpler one. The simplicity of a hypothesis is to be decided roughly by the consensus of people checking the phenomena. There is an objective criteria as well : the more computational resources a model requires and the more computationally complex it is, we judge it to be the more complicated hypothesis. For example, the hypothesis that all of us are in a hallucination is virtually uncomputable, whereas the alternative is to just accept the 'reality' of reality which requires nill computational resources. See note below.

For want of a better word, let us call this framework as the Real system. We distinguish between the 'Model of Reality' and 'reality'. Reality for us means information coming directly from the 'Real System', which are mostly experimental data. I urge the readers to accept the above mentioned framework of deciding 'reality' of real world. And then we will see what this system and model tells us about God or religion in particular. For the sake of completeness let me mention that we accept that some things are (may be at present) beyond the ambit of measurement, esp things which are highly subjective viz love, affection or anything connected with it. We do not deny the reality of it (how could we?), but then these are mostly the things for which there is no need to generate data, neither they can be generated by any conscious effort. We will have more to talk about 'love' in the future, both the personal kind (which we will never have any reason to object) as well as the 'transcendental' kind (the one 'felt' for God, which we will see as misguided at best and destructive at worst).

We usually have some hypothesis which we want to test viz, the existence, omnipotence, omniscience, etc of God. There are various degrees to which any hypothesis (could be any mundane one) may be investigated for its truth.
  • At the most obvious level, if it is directly supported by reproducible data it is designated highly likely (to be 'true') in the Real System. If it confirms with the already constructed model (as in point 3 above) well and good, else we change the model. The data from experiments is sacrosanct not the model.
    Ex - before discovery of high temperature superconductivity, it was considered unlikely based on the then accepted mechanism of superconductivity. After its discovery, we agree that earlier model was valid only for low temperatures and are still searching for a good model of high temperature superconductivity. Needless to say, nobody disputes that high temperature superconductivity exists.

  • Next, if it is not so far borne out in experiments (may be due to difficulty in doing experiments) but do not violate any of the experimental/scientific data already collected (in short, if it does not contradict the model obtained from the real system), it is given plausible status. In this case we do not rebuff the hypothesis but reserve our judgment till further information.
    Ex - some predictions of standard model are not yet confirmed by experiments but are not considered implausible as many other predictions are confirmed.

  • If on the other hand the hypothesis contradicts the model obtained from the real system and is not so far supported by data (again may be due to difficulty or expenses), we have every reason to distrust the hypothesis. Of course it may still be correct but we consider it implausible.
    Ex - cold fusion, based on the preliminary calculations is considered highly implausible (personally I consider it a waste of time). Needless to say, satisfactory data is awaited.

  • Next, if the hypothesis is logically contradictory, we have to reject it summarily. Such a hypothesis is considered impossible. It will be a waste of time to worry about such a hypothesis any more. Many hypotheses about the real world are unlikely to be labeled impossible (technically, we go only as far as implausible) as our model of truth is the Real system, which is a really kind hearted system. But as we have already seen our eminent friend God in many of his disguises tests our limits of patience. He alone seems capable of achieving the distinction of impossible. Oh God what hath you done !!

We call the above scale as the Plausibility scale. The tested hypothesis if qualifies as 'highly likely' is incorporated into the Model of Reality. 'Plausible' ones are only temporarily included. Rest all cases of hypotheses are discarded.

So far we have seen that G1, G1* and G4 types of Gods are impossible. We will examine G2, G2*, G3 & G3* in further articles. We will also try to understand God at various other levels and give many more definitions. Ultimately it turns out that any approach is either implausible or impossible. In the next article we will define atheism as well theism hypothesis precisely so as to be able to test it vis-a-vis the real system.

Note : This criteria might need refinement in the future when we encounter a well articulated and valid objection to it.

Monday, August 20, 2007

What is God : Omniscience - C

This article is a follow-up to part B.

Before going further, we show that a K-entity and a entity is one and the same.
Since a K-entity is already an entity by definition, it suffices to show that an entity is also a K-entity. Any entity is supposed to be able to finish a task. i.e, after a finite number of steps the task is over and the result is obtained. So we define a formal system whose axioms are the steps of the task and rules of derivation are the transition from one step to next. Since the task finishes (in technical jargon, terminates) this must constitute a formal system. We also need to include the elementary logical rules too. This system may not be minimal, so we make in into a minimal system. Under this formal system, the performing of the task is same as a proof of the initial input leading to the result of execution of the task. Consequently the above mentioned entity has completed a proof in a system and hence is a legitimate K-entity.
The above argument shows that there is no difference between an entity and a K-entity. This is the reason we defined types of God as G1, G1* and so on. The star versions are the same concepts but related to knowledge. Therefore an omnipotent God is same as an omniscient God and subject to the same contradictions. But we will spell out everything in detail and leave no room for doubt. We have already seen that G1 and G1* Gods are contradictory. Later we will see that G2, G2*, G3, G3* are also contradictory but that needs more work and we will revisit this issue after learning some more things.

For now let us try a different tack for defining God. Instead of defining God to be an entity, we define him to be knowledge itself. It is not too far off course as this is one of the glib interpretations of omniscience. This is tantamount to defining God to be a Formal system rather than somebody who works with the system. Let us call such a God as G4. The question is which system is G4? Since we have many (far too many) systems, all of them cannot be God. Think about the elementary examples we considered in the earlier part. This is not what we intuitively understand by God. So we consider some stronger systems. Any God worthy of his name must be at least capable of doing arithmetic and be able to prove arithmetical statements (or at least understand the proofs if presented). This simple requirement actually poses the most formidable difficulty.

As proved by
Gödel, any system which includes ordinary arithmetic cannot be both consistent and complete. This is referred to as the Incompleteness theorem. Meaning, however many axioms we include in our system, there will necessarily be left out statements which will neither be false nor true in the system. So the system can always be enlarged by adding one more axiom and there is no canonical choice of any largest system. That makes the choice of G4 very difficult. On the one hand we want to include all the arithmetical theorems in our system and on the other hand we want G4 to be as large as possible (so that God comes closer and closer to 'omniscience'). But whatever system we choose as G4, there are competing systems which are as large and also many (far too many) which are strictly larger. This dilemma is akin to choosing the largest natural number. Whatever number you choose, there exist a (and infinitely many of them) larger number. Hence, G4 also becomes non-existent. In hindsight we can see that G1* was a defined to be a k-entity in possession of the formal system G4. Since G4 itself is problematic, G1* too cannot exist. So we see that all the grandoise talks of God being 'Pure Knowledge' is just vacuous.

Actually even if G4 existed, it makes little sense to worship a list of axioms. And in any case, this is not the popular understanding of God. As we have seen till now, the ontological status of God seems problematic and is getting mired into more and more contradictions. It leads us to suspect whether God is just a psychological construct. Or worse : an illusion constructed by the way our brain works. Not unlike the illusion we have of sun moving round the earth or the desert mirage. See this for a good example of an optical illusion :
If you move your head towards the screen while looking at the center black dot, the two outer circles seem to rotate.

If the above is true, then to be an atheist is to be free of precisely such illusions. In the future we will also discuss the emotional / psychological apologies for God / religion and examine whether religiosity is really needed for our life. Whatever we conclude, our motto is not to take any thing on faith, not even atheism. Whatever stands up to rigorous scrutiny will be our stand.

In the next article we will take a break from this grind of God and take a look at the real world.

Thursday, August 16, 2007

What is God : Omniscience - B

This article is a follow-up to part A.

To 'know' something is to know it to be 'true'. Being 'true' means true in some Formal system. That means there is a formal system to which the statement refers to and is derivable from its axioms by a sequence of steps. Which means, the given statement is a proved theorem in some formal system. Which formal system is immaterial (it may be implicitly assumed, but once at least we should do the exercise of concretely identifying the axioms and rules) and it also doesn't matter whether the supposed statement is elementary or involved. We only want it to be a theorem to say that it contributes to our knowledge. Also a related point is the likelihood of 'truth'. For this we incorporate the usual rules of probability in our formal system and demand the probability of the said statement being true given all the data.

Motivated by the above discussion, let us define an K-Entity to be an entity who can prove a theorem of some formal system (K stands for knowledge). Recall (from the omnipotence article) that an entity is supposed to be capable of carrying out a task. Also recall that a task is a sequence of well defined steps explicitly spelt out to the entity. By our definition any computational program such as Matlab or Mathematica is a K-entity, so are all the humans and (supposedly) God too. Knowledge of a k-entity is the collection of all the statements he has proved till date. The true statements that one 'knows' must be a part of her/his knowledge indeed. A is said to More knowledgeable than B if A's knowledge subsumes that of B and is strictly larger. As is clear we are allowing one's knowledge to increase with time. Before going further we will see some examples of 'proof' required of somebody to be a K-entity. We will also see later that a K-entity is actually no different than a entity.
  1. Our formal system consists of 1, 0, + and =. The axioms are 1+1=0, 1+0=1, 0+1=1. So let us try to see some of the theorems of this system. 1+1+1+1=0 is a theorem as ca be clearly seen by repeatedly applying either axioms. Similarly 1+1+1=1 is also a trivial theorem. And anybody who can perform this task is a K-entity according to our understanding.
  2. We include some more symbols now. Let us have 0,1,2,+ and =. The axioms are 1+1=2, 1+2=2+1=0, 0+1=1+0=1, 0+2=2+0=2. It is easy to see that 2+2=1 is a theorem.

  3. In real world, we deal with physical evidences. Let us say a piece of skull is found at a archaeological site with cut marks on its surface. Also let us say that by radio carbon dating, its age is estimated to be contemporaneous to stone age. This is a probabilistic situation, and all we infer from this is the likelihood of that person being butchered by its fellow human beings and possibly cannibalized. But alternate interpretations are also possible and wherever possible the likelihood estimates can be done too. All this may be done by a computer or a human being, both of which are K-entities. Here it is cumbersome to pinpoint the exact formal system, but we assume the truth of physical laws, and also some earlier estimates of similar sites. This formal system is very special and we will need to explicitly elaborate it. I will come to it in a future article.
The above two examples illustrate the essential common-sensical nature of our definition. Also in the above examples the symbols 1,2,0,+,= are used just because of ease of typing and familiarity. You may use any symbols which you may as well have designed yourself, it is totally superfluous, and doesn't carry any meaning. The theorems are arrived at just mechanically. in practice for a complicated formal system such as Algebraic Topology, it may require considerable ingenuity to prove theorems of interest. But just to be a K-entity, it doesn't take much effort.

Having done all this, now we ask the following question : since God is supposed to be omniscient, exactly what is meant by omniscience in the above context ? Intuitively, God's knowledge must include all the true statements possible. This implies one of the three things (in cases I and II God is supposed to be infinitely knowledgeable) -

  1. God is in possession of a 'divine' system which tells him proofs of theorems of all the possible formal systems. We will define such a God as type G1*. If we demonstrate the existence of two formal systems which are mutually incompatible, then these two systems can't be part of a larger single formal system. Hence, God's ultimate weapon (the 'divine' system) cannot exist. It is a routine task to come up with such examples. But historically such a thing has seen much acrimonious debate, most notably between Hilbert vs Brower. Category theory provides the usual framework to settle such issues. This is a issue of completeness of a formal system, which we discuss more fully while considering G4 in the next article. Suffice it to say that the idea of a unique formal system giving all the true statements is logically absurd hence impossible hence G1* is non-existent.

  2. He picks one by one every possible formal system and diligently proves all its theorems and consequently 'knows' it, moreover he must already have done all this analysis (maybe much before the universe began, else what use as a God He is) as his knowledge includes all the true theorems. We define such a God as of type G2*. This gives God a quality of timelessness or eternalness. We will have occasion to deal with eternalness and a related concept omnipresence in future. For the moment we are leaving this issue and will come back to it.

  3. He is in the process of proving all the true statements and will eventually complete the task. We define such a God as of type G3*. This tells us that God is essentially no different from us. We are also in the process of finding all the true theorems and 'potentially' can find it. However we may not uncover all the truth in any conceivable time but God may do it. So we will also deal with God 'eventually' completing the task (of knowing all true theorems) when we deal with 'eternalness' of God and see how this leads to contradictions.

So is God really not omniscient or are we missing something ? None of the above three cases are really tenable. These cases actually rule out omniscience as a property of any entity, God or otherwise. However there is still one way for 'something' to be omniscient. But here we run into Godel's incompleteness and will see this in the next article.

Note : Observe that G1* is analogous to G1 (defined here). G2 and G3 types are defined here.

Tuesday, August 7, 2007

What is God : Omniscience - A

Apart from Omnipotence, the other putative property of God is Omniscience. Let us try to see how valid this is. To be omniscient is to know everything. We would require any knowledgeble being (entity ?) to know of *true* things and only true things. It doesn't make much sense to know of false things and certainly not from God. That brings us to meaning of *truth*. But here is a problem.

Strictly speaking there is no such thing as absolute logical truth. Truth only has a conditional status. Every statement can be analyzed for its truth value, which rests on *accepted* truth of some earlier statements, which has similar status themselves. It boils down to *truth* of some irreducible statements, which are accepted as true. These statements are called Axioms. Axioms are defined to be true beyond question, they need not be self-evident, realistic, intuitive or even simple but most of them are. Based on these axioms, certain statements are analyzed for truth value and thereby declared (true) Theorems if found to be derivable from the axioms. A Proof (or Derivation or Deduction) is a sequence of logical steps (which themselves have axiomatic status but are self evident) by which a set of axioms lead to the given statement whose truth is being analyzed, and then that statement is considered to be Proved. The axioms themselves cannot be arbitrary, but a strong condition of Consistency is being imposed, i.e a self contradictory statement should not be a proved theorem. We also require the set of axioms to be mutually Independent, i.e not derivable from each other. A classic and typical self contradictory statement is "The umbrella is black and not black". A consistent set of axioms and all of its (or some) theorems is said to constitute a Formal System. The word 'formal' should warn readers that *truth* is decreed formally and has no other justification. As I clarified in my earlier post, I am not being pedantic in definitions, just careful enough without compromising readability. Let me illustrate Formal systems with some examples. The examples are from Physics so that things seem familiar to many readers.

  1. Classical mechanics (with the Newtonian Gravitation) as propounded by Newton, and formalized by Lagrange is a typical Formal System. Force, potential energy, kinetic energy, mass, velocity, momentum etc are the elements of the System either defined or axiomatically decreed. The laws of motions (aptly called laws, as they must be accepted without question) and Gravitation are the axioms and rules of derivation are elementary logical rules plus the mathematical formalism of calculus. The theorems of this system are the physical predictions viz, keplers planetary laws of motion, orbits of asteroids, etc. We distinguish this formal system from the physical world it is supposed to model. The formal system is no more true or false if it fails to describe the physical world, it is just a model after all. If some real phenomena is not described by this model, we try to construct a new Formal System (or model) which would be appropriate for our job.
  2. If we impose a limit on velocity (in this case that of speed of light) that any body may achieve, it gives us the Special Theory of Relativity. Among many of its astonishing predictions are time dilation, length contraction, etc. Again this is just a model and is not required to correspond to physical reality. But if not, we would abandon studying this model and find a better a model instead. This is the usual paradigm of Science. Experimentation / observation lead to more data, which helps in refining (or renewing) the model which suggests experiments in a particular direction to refine the model even more. This is how science progresses.
  3. Special relativity along with the principle of equivalence and some more laws (essentially imposing speed limit in accelerated frames as well as inertial frames) gives us the General theory of Relativity.
  4. All the above examples are conventionally considered classical physics. Quantum mechanics is a system which is obtained by changing much of classical physics. The postulates are counter-intuitive as well as its predictions (theorems). Nevertheless till date no experiment has been found to contradict these predictions. It is beyond the scope of this article to describe the axioms. I can do no better than to suggest you an expert, who is also a highly skilled expositor. Please refer Mathematical foundations of Quantum Mechanics, K.R. Parthasarathy (Hindustan publishing, TRIM-35) to study the formal system.
Actually in the above examples, to clearly identify the axioms and rules of derivation is a tough task and requires extreme originality. But then who can say Newton was not a genius of the highest rank ?

As such to decide consistency / independence of any Formal System is also a highly non trivial task and Category theory is the usual playground of such considerations. In mathematics, one usually plays in a universe called ZFC, the usual formalism of set theory. Many professional mathematicians leave such task to professional logicians and are content that consistency is guaranteed. As we will soon see God can have no such luxury. Historically this type of understanding came much later. We already had a System (in mathematics) and nobody seriously doubted it. Only after Cantor's seminal work on infinite sets, the need to explicitly spell out the axioms arose. Let me give you a quick example of a toy non-system.
The undefinables are umbrella, black, cloth, cover of an unbrella, color of a cloth. First axiom : cover of an umbrella is made of cloth. We define black to be color. By color of an umbrella we mean color of the cloth of its cover. Second axiom : the color of all umbrellas is black. Third axiom : the color of all cloth is not black. Clearly it implies that all umbrellas are black as well as not black. This contradiction violates consistency requirement and so cannot be a formal system.
The above discussion makes it very clear that there is no unique Formal system. We shall see many more examples too. Changing an axiom leads us to a new system and and any omniscient being is supposed to *know* all the Formal systems. Just what could this preposterous idea might mean ? We also need to undertand to the concept of *knowing*. We will see this in the next article.

Wednesday, August 1, 2007

What is God : Omnipotence - A

To really understand the good/bad consequences of religion, we must first understand God on whose name all this is done. Almost every pious / theologian I have talked to evades this question. In fact they feel proud of the elusiveness of God. We as scientific rationalists would try to have a thorough understanding of God and decide His likely existence and influence on the material world. After thinking through it, I have several different possible definitions of God. In this article I will explore one possible approach. First let us have some preliminaries. The bold entries are definitions to be taken in a literal sense, quite often self-evident. I am not being mathematically precise (in spite of being a mathematician) here just to maintain the flow.

In order that God be able to influence something, he must be capable of executing some well defined actions. Let me define an Entity as 'something' which can execute a task to its end. A Task is a sequence of clearly well defined steps, which if stops gives us Result of the Execution. I don't want to call this an algorithm as I want to consider non-computational tasks too. Technically a Turing machine is synonymous with a algorithm. But we will differentiate between the two. In our scheme Turing machine is also an 'entity'. Our tasks / algorithms are not restricted to computations. One example each of the non-computational and computational type -

  • To transfer all the apples kept on the table to a basket.
    1. If there are still apples left on the table pick one and transfer it to the basket.
    2. Check whether there are apples left on the table, if yes go to step 1, else go to step 3.
    3. Stop. The Result is basket full of apples.

  • To find remainder and quotient of N when divided by M -
    1. Put Q = 0, R = N.
    2. If R less than M, go to step 4, else step 3
    3. Subtract M from R, put it as new value of R, increment Q by 1. Go to step 2.
    4. Stop. Result: Q is the quotient and R is the remainder.

Existence of an entity for us means it must be capable of executing at least one task and in fact must have actually executed at least one task, either Spontaneously (i.e without any external switches as all of us do) or Summoned (as we kick start a computer, or implore God to do something for us and he 'obliges'). I think this is a reasonable definition of existence of a being (which I am calling as entity). A drawback of this wide definition is that all computers are entities as well as several other machines, which is precisely why I shy from calling them beings. Some other entities according to our definition are -
  • Cellular machinery for reproduction, protein production, etc. These presumably act spontaneously.
  • Automated machines / robots employed in a industry for manufacturing. Summoned execution.
  • Computers which automatically allot us airplane / train seats, etc.
  • Don't forget that all human beings are also entities.
We need this wide definition of entity so as not to leave any possible entities out. An entity may be summoned at some time or spontaneous at others. It is clear that our God also must be an 'Entity', else He is inconsequential and thereby non-existent for all purposes. It is also clear that any entity has a corresponding List of tasks it has already executed. So according to our definition, empty list means non-existence. We also postulate that for any entity, its list must exist. Otherwise what possible meaning can the existence of an entity might mean? We will use this postulate as a test for existence. Two entities may have overlapping list or non intersecting ones. An entity X is said to be More Powerful than Y if X's corresponding list is strictly larger and subsumes that of Y. This I believe is a very reasonable definition of 'power'. We also consider a different type of list called *List which is just the collection of tasks the entity can execute. Let us take the colloquial meaning of 'can'. An entity's Action is the sum total of his executing any or all tasks from his *list.

The question that immediately comes to mind is : what is the corresponding list of God. We may have never seen Him, but consequences of his action must be evident in our world. Also a God worthy of His name must be more powerful than any other entity. Let us examine one possible list as worthy of God's, as well as analyze possible problems with it. First we try to ascribe one hypothetical *list to Him and see the consequences.

Consider all possible entities in the universe and sum total (logically, the union) of all possible corresponding *lists. God's *list is supposed to be strictly larger than this list. Let us call it the *list L1 and such a God as G1, which would be omnipotent. We want God's *list to be this collection as we want God to be able to do all such tasks that is why we have taken this definition of omnipotence. This very soon runs into logical contradictions. For example can God make a pile of stones heavy enough so that he cannot lift it up 1 meter from the ground ? It is straight forward to convert this crude instruction into algorithmic format (say the task T1) and it is also very clear that I am (you can put yourself too here) capable of executing this task, hence T1 is in my list. But if God can execute this task then he cannot lift that very pile. So we convert lifting of that pile into an task (say T2) and this task is clearly not in God's list. Otherwise if He cannot execute T1 then too He doesn't have at least one task (T1) in his list. Either way He is not omnipotent. We shall have more occasions to talk about Omnipotence in the future, where we will meet successively better Gods. And we shall also address objections to this reasoning.

We have actually demonstrated non-existence of the *list L1. Hence G1 also cannot exist, as we hypothesized. The above example also demonstrates that G1 cannot be strictly more powerful than all human beings. What a letdown God !!

PS : Here we see that talking about *lists leads to troublesome concepts. In the sequel we will only talk about Lists in the context of omnipotence. Also we had defined omnipotence using *lists, so we will redefine omnipotence using lists in the future and see the consequences.

Tuesday, July 31, 2007

Why God ?

The universality of religion/God indicates at least two things -
  1. Religion is a consequence of our evolutionary history. It perhaps helped in our wild past. The reasons are not difficult to find out. It is no brainer that having a aggressive religion helps tremendously in wars. Increased cohesion and harmony also helps in ensuring food security. It also promotes xenophobia against other tribes, which is necessary for wars and aggressive expansion of tribe. Those societies which are hostile and aggressive against other friendly tribes will have much more incentive to massacre those other tribes and expand their folk/religion. No wonder, most of the wars all over the world are either fought over religion or escalated to unsolvable levels due to religion.
  2. There is also probably an innate aptitude for religion in humans. Humans are unique in having a unmatched and limitless power of thinking and imagination, which is greatly amplified by language. With a power of thinking also comes amplification of fears, anxiety and stress. And also proclivity for generalization / gullibility of all sorts. If a weaker man can break a thin branch, a stronger person can break a thicker one, and an elephant can tear off an entire trunk it follows that the tornado which can uproot an entire forest must be the handiwork of a super powerful being. The obvious generalization is to postulate existence of an omnipotent force much more powerful than the tornado. This is the innate aptitude for God I am talking about. This is perhaps akin to the innate aptitude for language we have. If true, it is a worthwhile experiment to raise a child without teaching her any religious ideas and only inculcating a scientific rationality to see whether she later acquires any taste for God.

Sunday, July 22, 2007

Preface / Aim / Program of this Blog

Throughout human history, religion is one of the universal features common to almost all the societies. Along with religion comes a baggage of complex rituals, belief and many times an elaborate theology also. A naive argument can run along the lines - since it is universal, it must be intrinsic to us. Some may extend this to conclude *truth* of it. While universality does indicate some kind of intrinsic-ness, it does not go beyond it. And we certainly cannot allow popularity of a belief to be a test of validity. At one point of time all of us believed in the *obviously true* Geocentric theory and were willing to burn those who question it. Another school of thought about religion talks about its utility to the society, which also has large grains of truth. But as we shall see, this *utility* has also promoted vicious hatred among humans and a prima-facie case can be made against futility (and dangerousness) of religion.

As we shall see, often religion and theology are intertwined, but they should not be. Ones *relationship* to God must be personal as by definition God should not need a mediator. But largely religion acts as a mediator and almost all human societies are only too willing to concede their personal ground to religion. Thus theology takes a backseat to religion, often with disastrous results. We shall explore on this more as we go along.

A major theme of this project will be study of theology, which we will do in a systematic manner taking up every major religion one by one. Any question about God must confront with His ontological status. We shall have much to talk about all such issues. We will also examine God's role as a creator of universe, life and humans. Further we will examine His role as a ruler of the world. As we go along, we will separate God with religion and make a plausible case for atheism. Even this does not confront the societal impact of religion. So we will make a case against religion also. A related issue is the morality and its supposed derivation from religion. This is one of the two most vacuous apologies for religion. It is the aim of this project to convince the reader to 'convert' to atheism because this turns out to be a more humane approach in all respects than any religious system. When we study the major religions of the modern/ancient world, we will contrast it with atheism and see which is a superior prospect for humanity, logically as well as practically. The other patronizing apology for religion is 'it is for the masses' as if all the masses will turn into marauding hordes the moment they become atheists. We will see that it is incorrect to equate atheism with immoral wantonness.

I hope I can do justice to this project and find enough time for it.

Glossary :
Geocentric theory : a theory of sun going around the earth
Theology : study of God and/or religion
Ontology : study of existence of object under investigation