I think there was once an argument in the court of one King or another, when a philosopher claimed the non-existence of God. The King invited a mathematician (Euler) to argue that God did indeed exist, and, knowing that the philosopher had no knowledge of maths, he said something like:

(a+b)ⁿ/n tends to 1, hence God exists. And the philosopher couldn't counter that.

Another little anecdote. I think that Pascal once came up with a probabilistic proof that you should believe in god. He argued thus: the amount of effort it takes during your lifetime to believe in god (call it F) is finite, if god exists then your reward for believing in him is infinite, there is a nonzero probability p that god exists (fair enough in the absence of any evidence). Therefore, your expected gain if you believe in god is p x infinity -F=infinity.

I believe that Godel also came up with an ontological proof of the existence of god, if you're interested, I'll try and find a webpage with it on.

I think you will find a lot of pure mathemeticians who believe in a creator God (although not necessarily in the God of a particular religion).

I am constantly amazed that the axioms which we establish in mathematics have so much resonance in the world around us. Mathematics is essentially an abstract pursuit - we define axioms arbitrarily to create a consistent system which is powerful enough to work in. We usually choose axioms which are 'sensible' and which we have observed to be true, yet once we have defined our axioms the system in which we are working becomes entirely self contained. Yet somehow, using only these few 'common-sense' axioms, and simple rules of inference such as modus ponens, we deduce results which impact hugely on the world around us, and can be used to model and predict behaviour within it.

Why, also, does the mathematical beast continually defy attempts to tame it. One would think, probabilistically speaking, that any rigorous and well-defined intellectual system would be either trivial, or utterly impenetrable. Yet the various systems which exist within mathematics seem to offer accessibility, but still tantalise us with results and theorems that are just beyond our reach.

This is all just rhetoric, I know, but it is questions like these that lead me to believe in some unifying power in the universe. ''God created the natural numbers,'' said Leopold Kronecker. ''The rest,'' supposedly, ''is the work of man''. Yet if this is the case, why are there so many profound and beautiful results that rely on nothing but the natural numbers and their properties? Peano's axioms for the definition of the natural numbers are essentially additive, yet why do they lead to such a rich multiplicative structure? The primes are the building blocks of the natural numbers but they are notoriously difficult to find. And no matter how powerful and interesting our system of natural numbers is, we need to expand it to the rationals, and include limits of sequences to get the reals. Even then we find that most of the numbers in our system are transcendental and defy finite definition. And yet we still need to introduce another dimension before we have algebraic completeness.

Mathemeticians are in a unique position to see some of the true wonders of the world. I am certainly not a religious person, yet I believe that the opportunity to explore these wonders was given to us, not by the intellectual pursuits of man, but by the will of some creator God.

But surely it is not surprising that maths has a resonance with the world around us because, as you said, the axioms are choosen that have been ''OBSERVED to be true'', and are a product of ''common sense'', which is something that has been acquired. So the axioms themselves have not sprung from nowhere, but from the world itself.

As for our capacity to do maths at all, this is not surprising because there are clear evolutionary benefits in abstract reasoning.

Finally, it is not surprising that the universe is intelligible, even Einstein did say that the most unintelligile fact about the universe is its intelligibility. If the universe were so complicated that it was effectively impossible to find simple rules and patterns, it would be impossible for the reasoning present in just about all life forms aboe the amoeba: walking requires knowing how to move simply, but if, for example, gravity had come completely ridiculous force law, it wouldn't be possible to find the simple pattern.

Perhaps most importantly, I think it is a false argument to say that the existence of a God (or two or three!) somehow allows the universe to be intelligble, because in fact it places a limit on intelligibility, because in almost almost versions of God, God herself (itself?) is not intelligible. So if you allow for a creator God, you are saying the universe is ultimately unintelligible. And I think this contradicts at least my basic understanding of what science and in fact life in general is about, it is a proces of coming to grips with the world around you.

Yours,

Hi Sean, how's the basketball team doing? (Unless you're not the Sean Hartnoll I used to play Basketball with at St. John's).

I'll respond to your points paragraph by paragraph.

I think what I was trying to say about the wonder of the axiomatic structure of maths, is that once we have made these few, common-sense observations, the deep or profound results that we derive from them use only simple logical inference. So from these few axioms we end up with complex results which often model real-world behaviour very accurately. It is almost as if the abstract logical structure we apply to our maths is mirrored in the real world. It is this structure that leads me to believe in some cohesive force or influence.

I think it is dangerous to justify mankind's ability to reason abstractly in Darwinian terms. The development of mathematical (and other intellectual) reasoning has been simply too rapid to be due to natural selection. Darwinian evolution is a model which can be effectively applied in many many situations, and there are evolutionary arguments for consciousness and abstract reasoning (I think I read an argument which led from visio-spacial skills to mathematical skills). I'm not sure, however, of any evolutionary argument which leads to our ability to question the nature of our own intellectual thought, or the origins and purpose of our intellectual pursuits. We should certainly be wary of twisting existing scientific models until everything we observe follows them. That way we tend to ignore the discrepancies in our model which are the key to further scientific discovery.

I don't believe that the ability to reason that is present in man is reliant on the universe being governed by simple laws - particularly because we do not know what laws (if any) govern the behavioiur of our world. You cite gravity as an example of a simple force law, yet Newton's inverse square law does not describe gravitational attraction - it merely models it (very accurately). An even more accurate model of gravitation was provided by Einstein through general relativity. Though more complicated, this is essentially a simple model. The actual nature of the phenomenon of gravity is something we know nothing about. We simply have justifications for it within our current scientific model.

In my original posting, I was talking about mathematical complexity, which is much easier to quantify. In mathematics we are working within a known system, and results can be said to be true within that system. Since man did not define the system in which scientific results exist (the natural world), we cannot make statements about the truth or otherwise of results within that system.

Certain scientific field have their own mathematical axiom system (Quantum Mechanics, for example) and results within those fields can be said to be consistent with the axioms of that system. It is interesting, though, to think about how valid the original system was. In the case of quantum mechanics, and the increasingly ridiculous explanations which it is forcing, it is probably time for a swift slice to the jugular from Ocham's Razor.

I realise all of this is very abstract, and I am not in any way belittling the scientific study of evolution, or of gravity. But since we are thinking about the nature of the universe and the existence or otherwise of God, it seems to be the right level to talk on.

As for your final point, I rather hope that the universe is ultimately unintelligible. If not, what would we do once we had figured it out.

Hi Harry - thought it might be you, wasn't sure though. What are you doing these days?... I'm playing football instead of basketball this year, I hear the basketball team won the league though! (I don't know if these two events are connected!)

Anyway, some answers.

Firstly, I wasn't applying Darwinism as a model in this case but as an accurate description of evolution. I disagree that ''the developement of mathematical reasoning has been too rapid to be due to natural selection''. I would argue the ability has been there for ages, although it is only recently that material circumstances have been favourable enough to release the full potential. There are some fascinating experaments with monkeys and dolphins showing very high levels of abstract reasoning.

I wasn't initially speaking of consciousness per se, but as an aside it is worth noting some more interesting experiments by Lacan (French psycoanalistic bloke) which involved very young children looking in mirrors and which suggest that even the concept of Ï" is an acquired one (this is a more controversial statement, but I include it for completeness).

If you want to a call given theory of gravity a model of the actual phenomenon, that is fine by me. But the fact these models exist at all is a prerequisite for the evolution on "higher" life forms. This should be fairly clear: if nature were such that patterns were too extremely complicated and didn't have simple approximations that worked in low enery, low speed, low mass etc. limits, then the initial step of using these patterns to simple evolutionary advantage wouldn't be possible.

The final point is that introducing God(s) doesn't get you anywhere. You say you can't conceive how there could be so much intelligibility in the universe without God, but by introducing God you haven't ''explained'' anything, you've merely given what you don't know a different name. This seems to me the most important point.

I agree totally with Sean's final point. If you ask a religious person who created the universe, there answer, obviously, is God. If then asked who created God, a typical answer would be that 'God needs no creator as He is a necessary being whose cause is to be found within himself.' If one is to concede, however, that something (God) can exist without an external cause, why can't the universe exist without an external cause? I mean, why go so far back in the chain..... in other words what's the point in a God? If we are prepared to suggest that God doesn't need a creator, and that He in turn created the universe, it doesn't seem any more illogical (to me) to suggest that the universe doesn't need a creator to exist.

It seems I am outnumbered.

First of all Sean, I don't think I quite understand your argument that simple models of natural phenomena are essential to the evolution of higher life forms. Mankind, by necessity, developed higher reasoning skills before describing physical models. Simple models are not necessary for the visio-spacial predictions of physical behaviour which are evolutionarily important. Catching a ball, for example, is something that a small child can do - it would take a graduate mathematician to program a robotic arm to do the same thing. The child has no conception of Newton's equations of motion or gravitation, but can still perform the act. And it is the ability to perform the act which is relevant in evolutionary terms.

And even if the existence of simple models is necessary for evolution, what are you saying? Given that you claim that the existence of simple models is a prerequisite for the development of higher reasoning, are you claiming that in other parts of the universe (or other universes) where no such simple models exist there has been no evolution of higher creatures. Or are we simply very lucky that the phenomena of the universe can be modelled very simply. I, for one, don't believe in that level of coincidence.

Your final point, and one which seems to have been strongly supported by other members of the NRich team, was that introducing the concept of a God doesn't explain anything. I can appreciate this point, and I hate the facetious arguments offered by certain religious zealots as much as Carl. But as a scientific hypothesis, the existence of God is simple, elegant and irrefutable. It is also impossible to corroborate. It is irrelevant to ask the question ''who created God?'' Questioning the existence of God is something that is borne out of our attempts to describe the world we perceive. If we concede His (I use the masculine pronoun for convenience) existence we still cannot question the origins of God - in defining Him we have made him independent of such constraints as time and origin.

As far as Ockham's Razor goes, Dan, the existence of God is its ultimate application. Far from introducing greater complexity, introducing a creator God to our scientific model strips it away. If you listen to a committed Christian talking about the origins of the world and the nature of our existence, their explanations are simplicity itself. They merely conflict with our perception of the world, and our need to control our own destiny. The concept of a wave-nature of light similarly conflicted with Newton's perception of the world.

We shouldn't dismiss ideas because they offend our sensibilities. Though you clearly have a different perception of the world to me, will you at least admit 'Creator God' as a hypothesis to your scientific model?

I think that a scientific model has to disregard God, because, by definition, he is without cause. Consequently, he is irrelevant to the scientific model of creation because this is concerned with tracing a chain of cause and effect. If we look at the Big Bang theory, for example, we trace the model back to a certain point, before which we are unable to determine any data. We may then say that God is involved at this stage, or not, that is to say, he is unmeasurable and therefore does not exist in a scientific model.

The idea of an interference in the universe once established, according to some divine whim, is also irrelevant to a scientific model by means of its unrepeatability.

Harry, you say that God as a scientific hypothesis is irrefutable, which is a contradiction in terms. Any scientific hypothesis HAS to be falsifiable, God is not falsifiable, therefore the existence of God is not a scientific hypothesis.

I'm not being facetious. I simply mean, what is your basis for describing this hypothesis as nonsense. Surely it is that it disagrees with your current, observed and experienced, view of the world. It is certainly a scientific hypothesis, and it is entirely refutable. You could throw bricks under the door as it was opening. The bricks would knock the invisible men down if they were there. The door continues to open so we either have to dismiss this hypothesis, or refine it. We could, for example, say that the little men can jump over the bricks. Or that when one gets knocked over another one instantaneously appears in his place. All of these things are valid steps in a scientific process. We only dismiss them because they are not acceptable to us in our current world model.

Your comparisons with Pasteur and Einstein are fundamentally unfair. Pasteur and Einstein said things apparently ridiculous to their contempories, made PREDICTIONS, that were subsequently VERIFIED (to within a margin of experimental error). Now when someone makes a testable prediction based on the God hypothesis that is subsequently verified, I'll make room on my science bookshelf for a bible, but until then there is no way religion can be given anything remotely like a scientific status. Ethical, personal dimensions would be a whole different argument but God as a scientific hypothesis? Come on. Dan's point about irrefutablity is spot on.

The histoy of knowledge is a continuos process of explanations moving AWAY from the religious realm. Before Darwin and a fossil record, the existence of humans proved the existence of God, etc. etc.

A clarification about what I mean when I say the natural phenomena need to have simple approximations that work in order for life to evolve beyond random walks in primordial soup. To continue you example of a child catching a ball without doing calculations with differential equations. The point is he IS doing calculations, or rather his brain is. Abilities like this are easy to model with neural networks. Basically the way it works is that the brain is rapidly able to extrapolate from preexisting data. Not in a conscious way, but in the same way we learn to ride bicycles, or walk. The brain fits a curve to the data points, more or less. All I am saying, is that if the world was apparently unpredictable (i.e. the underlying patterns to phenomena were very complicated), you would not be able to fit the curve easily because your data points would be jumping all over the place.

I finished exams today!

I don't think anyone was ganging up on Harry at all. I respect what Harry (and everyone else) has been saying, and I enjoy reading other people's comments, views and opinions. Inevitably, this topic is going to arouse big differences of opinion, but none of us can either prove or disprove other people's arguments (generally). However, I have to say that I was a little disappointed to read Harry's comment: 'I hate the facetious arguments offered by certain religious zealots as much as Carl.' I don't think that's fair, because I was only offering my opinion on the subject, and did not claim it to be correct in any way. I think the majority of people would actually agree more with Harry than with myself, Dan, Tom or Sean. Maybe I've been studying the scientific method for too long, but I'm going to need some sort of evidence before I could commit myself to believing in a God. I can't understand why people assume there is a God until proved otherwise, instead of vice-versa. Introducing a God, in my opinion, like I mentioned above, seems pointless if we are to suggest that some events need not a cause, and happen without prior action or rational reason (i.e. God); why not say that the material universe need no cause and happen without prior action? At least with the latter, there is a science that may falsify the claim that every event has a cause. That science is quantum cosmology. By it's very nature, this branch of physics is unpredictable: 'God' does play dice! However, to suggest that the whole universe is subject to quantum principles is debatable, but at least it's a start.

I know my views are less romantic than others, and I agree that it is incredible for beauty and elegance to appear in areas where you least expect them, such as mathematics. It is hard not to belief that there is a purpose to all this, and that we are just a small part of a much bigger picture. We may never know.

Sorry Carl, I didn't mean to imply any opinion on your part that you had not expressed. I actually thought you were referring to the phenomenon that I have encountered of an inability to listen to any points being made in an argument. One Christian friend responded to the old chestnut ''What about the Jews and the Muslims?'' with the comment ''...um...well I think Muslims are sad''.

On reflection, I don't think that I assume the existence of God unless proved otherwise. Rather, my conclusion that God exists is a probabilistic one. I look at the world around me and the existence of a creator God seems to be far more likely than any other explanation I have yet heard.

OK, I'm glad you're not feeling ganged up on, I mentioned it because this sort of argument can easily get quite antagonistic (as I expect you know). Getting back to the point, why does the existence of a creator god (who requires no cause) seem more likely than the existence of a universe (which requires no cause). One possible reply that I can see is that the creator god hypothesis explains the huge amount of structure of various sorts in the universe. I expect you know the anthropic principle, i.e. that if there weren't sufficient structure to create us, we wouldn't be here to observe it, so in some sense it isn't surprising that there is the sort of structures we observe. A more persuasive argument is the phenomenon of conscious experience, which I would agree poses a problem for a mechanistic world view (an impossible challenge for a mechanistic world view in my opinion), but doesn't require the existence of any sort of god as far as I can see.

I think what you refer to as the anthropic principle is what Sean was saying in some of the early discussion - the idea that the structure in the universe is what allows it to be observed. My original argument, though, was based on mathematical structure and not scientific structure, which is what we've drifted onto. There is no reason to believe that mathematical structure is a prerequisite for the phenomenon of consciousness.

I like the Sheldrake page. I do remember reading something about ''morphic fields'' by Sheldrake a while ago - it was kind of an 'alternative to Darwinism' proposition. It's worth noting though, how much respect Sheldrake has for Darwin. I think he is just keen to kickstart the scientific process.

However, no amount of invocation of Creator Gods and anthropic principles can explain the phenomenon that is the England back four. With the possible exception of Gary Neville, they were awful. Phil Neville should be shot. Anyway, as such I am recovering this morning, so may not be too coherent. I apologise.

I also think that we may have run the course with this one, so I'm signing off. I'm more than happy for somebody else to have the last word.

Harry, great to hear that there is no friction between any of us. Contrary to our differing views on the origin of the universe however, I wholeheartedly agree with your comments on England's performance:) To keep this conversation in the context of mathematics, if one was to model their performance as a function of time, one would observe two distinct stationary points: a maximum in the first quartile and a minimum extending from the midpoint (half-time) to t=93 minutes (full-time). That should satisfy NRICH.

Going back to the original question, which was do mathematicians, in general, believe in God, I also suspect that more pure than applied mathematicians do (exact proofs etc). Being an A level student, I can't exactly call myself a mathematician....which reminds me, when can you actually call yourself one....when you graduate, or if you do original research or teach?

Has anyone read Dawkin's book 'Unweaving the Rainbow'? It's a good attempt at trying to dismiss the claim that science has took the beauty and wonder out of who we are, and where we come from.

Just quickly, I think you are certainly qualified to call yourself a mathematician. Anyone who practises mathematics, and appreciates its intricacies and beauties can call themselves a mathematician - there is no prerequisite qualification or level of ability. The child who tries to find out why aeroplanes fly by making paper models is every bit the scientist.

The label professional mathematician is sometimes applied to those who have published research, or who are actively researching, but the boundaries are very slight. There are many amateur mathematicians who have made contributions to mathematics (e.g. Fermat) although it's less common now.

On this matter, I tend to side with Einstein, when he said ''Science without religion is lame, but religion without science is blind'' It is quite neccessary, in my mind to allow a God to exist indepentdent of the physical universe. He/she may perhaps interfere through quantum mechanics, but this is still the universe acting rather than phenomena. There will always be the question of ''why'' existing and the only plausible way to explain it is with religion.

On another note, evolution over time can be described to occur over a short period of time as time is not absolute. This seems to cohere with traditional religions (even though its a stretch).

Becuase the universe must be finite in time. Proof of this originates very simply from looking at the stars. If the universe was infinite in time, all we would see would be a flood of light or nothing at all. The fact that a god exists forever allows it to be exempt from an origin (start counting back from infinity and tell me when you get to 0). However, as the universe is finite, there will always be a why to something and the very fact that the physical universe does not provide an answer(or if it always does, it can't provide infinite answers) seems to prove that there must be some other reason.

I like that quote from Einstein. However, I'm sure he wasn't religous in the typical sense but rather appreciative of the beauty and elegance of nature. Rather like Carl Sagan and Richard Dawkins. Looks as if we've done full circle with Dan's final comments :) If there is a God, who thinks that it would have been a good idea to let any evolved intelligent species be aware of His/Her existence? I think I would have left some sort of non-random sequence/sign to be found within pi.

Perhaps we're not good enough at maths yet...watch this space.

Brad, I hope we can ask why forever, otherwise things would get dull eventually. If there exists a God, will it be simply an expression of a higher mathematical system? Is God constrained by determinism and/or probability? It seems difficult to imagine that he is not, but if he is, then he is merely another physical/mathematical concept?

I am not sure, however, I think that Einstein was somewhat religious(I have heard otherwise in books though). I believe that he also said something along the lines of 'I believe in a god that influences no part of it after creation ' and 'I want to know if God had a choice in creating the universe'. He also said other things that contradict this.

Anyway, I think that it would have been wise to let all know of your existance- not just tell a mortal to tell about you. This has created the problem of conflicting religions.

Well, there has to be someway to explain the end of what we can see. We have certain limits to the amount of things that we can examine. perhaps I should be clearer. Eventually, we will break matter down into so small of things that they can no longer be examined. These will be atoms in the truest sense of the word. These will most probably influence all aspects of physics. I am not saying that the way these behave will be limited, I am just saying that these will the smallest thing possible. There must be some reason for these to exist- outlined above, as they are a product of finite time. Nothing else can form them, so what is there reason for existance. I think most people can tell where I am going here.

Tom- you state that if God is mathematically desribable, then he is part of this universe. By some logic(somewhat distorted I might add), this leads to a paradox, but I'll ignore that for the time being. In plato's Republic, he mentions two independent realms, the ideal realm and physical realm. In the ideal realm exists mathematics, music, art, etc. He believes that we are simply using this ideal realm to desribe our physical world. It however does not necessarily only describe things in the physical world. An example of this is points(according to string theory). Therefore, I think that a god could be described by mathematics and not exist in our realm.

Brad - if we regress to some indivisible and then explain this through the existence of God, we are merely regressing to a further indivisible. By the way, is there any reason why we can't have infinite regression - that is, is it certain that there exist indivisible particles?

That is in fact what I am saying. In my mind God is portrayed in our gravity and other forces. He has created the universe and thus created all things irreducable by science. These things act in certain ways creating the rest of the universe. He simply after creating the universe(whether he had a ''choice'' or not) did not choose to interfere while disrupting science. This may seem as though I am denouncing religions where god appears to some; but I am not. It seems that he could 'control' quantum mechanics and use this to his good.

As far as your first statement, it seems to me that science a study of finding a reason for everything to occur. If God is the reason and God has existed forever, then there is a reason for all things.

Brad, I don't think that you have proved either that the universe is finite in time or the existence of an origin in time of the universe, invalidating your proof of the existence of god.

Firstly, your proof of the finiteness in time of the universe is flawed because you need additional axioms to use some sort of variant of Olber's paradox to get your result, but this is the less important point as the universe is believed to be finite in time for other reasons.

Assuming for the moment that the universe is finite, the second part of your argument also breaks down. As far as I can tell, there are only two types of ''why'' question that can be asked here: Why is the universe in the particular state it is in at time t? Why is there an object we call ''the universe'' at all? I think that what you're trying to say is that as long as t> 0, the state of the universe at time t can be explained in terms of the state of the universe at times 0< =s< t, so there is no need for divine intervention here. However, at time t=0, there is no s with 0< =s< t, so this cannot be part of the explanation of the state of the universe at time t=0, hence the need for an external explanation. The flaws in this argument are (1) you've assumed that the external explanation is god without further proof, one way to solve this is to define god as the cause of the state of the universe at t=0, but you have yet to prove that this entity exists and so this definition is not well defined (a mathematical analogy, I can define N as the integer such that N=N+1, but it isn't well defined because there is no such integer, similarly I could define N as the integer such that (N+1)-1=N, but this is not well defined (every integer satisfies this relation)). Even if such an entity exists (the entity which has the property of being the cause of the state of the universe at time t=0), you've yet to demonstrate that this being corresponds to any notion we have of god. (2) You've assumed the existence of the point t=0 in the history of the universe, however it is perfectly conceivable that there is no point t=0, only points t> 0. This would mean that causation alone could account for the state of the universe at any particular time. The second type of "why" question is, why the existence of the object we call "the universe" at all? One explanation is that of a creator god, but this is in fact not an explanation as has been repeatedly pointed out earlier in this conversation. Summing up, you haven't proved that a creator god is necessary to prove the existence of the universe.

I'd agree with the point you made earlier that there is no proof or disproof of the existence of god. However, for sceptics like me it is always worthwile demonstrating that (a) all known ''proofs'' of the existence of god are invalid, (b) all known ''plausibility arguments'' for the existence of god are invalid.

I think it would be pertinent to the discussion if anyone believing in the existence of a creator god of some sort would try to describe some aspect of what they believe him to be like. It seems that the best any of these ''proofs'' of the existence of god can do is to invalidate our current physical model of the universe, and there are many considerably better ways to do this than those presented above.

I realise that I'm repeating myself, but nobody has yet challenged my basic arguments against the necessity of the existence of a creator god. I'm perfectly happy to allow people to have faith in the existence of a creator god, I think it's a bit strange, but it's honest. However, I feel that all arguments that ''prove'' the existence of god have been created because of a prior belief in god, which is in some sense dishonest.

But still I am being biased and going to question your argument. I suppose that I should define God as something that creates the Universe at t=0. This would mean that as the universe's time is finite, there would have to be something that created the 'laws' at t=0. This seems to be the only plausible way to complete the scientific model. Perhaps though, the ''God'' I am speaking of is not the traditional god. It is merely something external creating the universe. You can take this where you want from here. I don't think that your first counter is very valid.

I am not sure what is meant by the universe's origin is possibly not at t=0. Any origin at all would serve as an origin, perhaps you could explain this better. If it is meant that from an external perspective, the same one we would view ''God'' from, relativity would cause time to dilate and dilate and dilate, perhaps you are right. But note that at the point where time dilation is infinite, this is the begginning of time as we know it. So, I don't think that a finite amount of finite time periods added up, no matter how big, will add up to infinity. This is probably not explained well though.

Simply to destroy most of what I have just written, I will say that taking in mind Hawking and Guth baby universes, there is a possibilty of a universe to become it's own grandfather, so to speak. However, it is also important to remember that for a universe to start as a baby universe, it needs relatively high energy density. The current belief is that our universe doesn't contain that high of overall energy density.

I don't think that there is any way to prove the existance of a god (but it is fun trying) and it is certainly not possible to desribe this god without some sort of divine intervention. But, I do believe that the goal of science actually supports the existance of a god, despite many other's beliefs on the subject.

Brad, you haven't really addressed Dan's very clear points. Perhaps you could try and answer explicity the following quesitons:

(1) What does it mean for God to be ''external'' to the universe? (i.e. in what sense does God exist?)

(2) Why does God exist?

(3) Why did God create the universe?

(4) Why did God create the universe in one way and not another?

More technically:

(5) What does ''creation'' mean in an atemporal sense of the word (note that the definition of creation as the act of generating being form nothingness, has a causal structure, i.e. there is a cause and effect, but I don't think we can use these concepts when the universe as such doesn't exist)?

In particular, while questions (2)-(4) remain unanswered, the concept of God has provided zero explanatory power to understanding the universe. Added to the fact that God is beyond experiment as well as understanding there is really no sense in which its existence can have anything to do with science.

I will qualify this again, as I have done before, with the observational that ethical-type arguments are quite a different kettle of fish. Ultimately I think the other arguments are also wrong, but they are slightly more stable than the God-as-explanation arguments.

I am simply saying that there must be a cause for everything. If the universe was born without a cause, that seems to throw off the notion that there is a cause for every effect. Maybe quantum mechanics created the universe. But, I hardly see, with the plank constant being so small, how quantum mechanics could effect something with infinite or very high density.

As said before, something that has existed forever could perhaps be exempt from a cause and anyway our time only exists in our universe from its start. Some people use this as an excuse, but I think it's a very bad one. There would have to be a flow of what is happening outside of our space. But once again, I am simply asking what wrote the rules. If the rules have been around forever then they are also exempt from this cause though. Instead of invoking the use of the word God, I will just call what wrote the rules X, and we know that this must exist. I believe that this is God, however you can believe what you wish it to be.

Substituting God for X, your questions are easier to answer.

(1)external means not in this universe. Existing perhaps in hyperspace or some other space.

(2)because X always has existed- X has no origin. Perhaps X exists to create this universe, but I think that that's a little vain and wrong.

(3)I really can't answer that

(4)I really can't answer that either

(5)creation means that something, X, would have caused the universe to start. From point t=0 on there are causes, as creation happened at t=0, then there must be a cause.

Once again, all I can prove is that there must be an X so long as the universe hasn't existed for infinite time. I cannot prove that it is God, has a personality, or what form it takes. It could maybe be another universe, although I don't think according to current theory that this is correct. I'll perhaps ask a theologian about some of these questions, although I doubt he could give that many answers either.

Any origin, it would seem, would serve as an origin anyway. It seems to me that you are saying that time has started, yet it didn't start. Any starting point would be sufficient as a place to ''write the rules''. Maybe you could explain a little better, I don't think I see what you are saying.

You might also have an infinite time period over which the universe has existed, if we have a cyclic universe; big bang, big crunch, big bang, big crunch..., then we don't have an initial cause.

Origin or no origin, if there is something ''external'' to the universe which sets it in motion at t=0 or whenever, however, the main point to have in mind, I think, is that you haven't explained anything by saying God created the universe.

It is interesting though that there exist theories which do not depend on an origin, in fact it's more than interesting, it's vital because it hints at an alternative explanation that could genuinely tell us something.

If what Tom said is true, or if another universe could ''give birth to our universe'', then there certainly isn't any need for what I have labeled as X. But, as Tom also said, according to current observations, this isn't very possible. I also don't think that what you are saying is very possible either,Dan; however, my notion of a God probably doesn't seem very likely to you either. With current theory and observation being where they are, I don't think that it is possible to rule out any sort of suggestion dealing with what we are talking of.

I do however still think that as long as what Tom said is false, then the universe must have had something to ''write the rules'' and to form one or several things to control it all. Whether these things are strings, branes, or an undiscovered particle, there must have been something to create the first one of them and the laws that they obey. But, that definitely doesn't rule out what Tom or Dan have said.

By the way, what theories do not depend on an origin?

Where does a circle begin? I think this is the essence of Hawking's No Boundary theory. In general, the idea of an origin comes from looking at how time passes for us, and extrapolating form this by running the clock back. However, general relativity being as it is, there is no reason to assume this ''nice'' behaviour of time is true everywhere in the universe. For exmple, falling into a blakc hole, I would see the whole universe pass before my eyes as I fall, does this mean the black hole is the ''end'' of time? Similarly, suppose I was born right on the very very edge of the event horizon, this would effectively be time 0. Does this mean the black hole is the ''origin'' of time?

The black hole you mentioned doesn't seem that it would slow down the growth of the universe as a whole. It must be kept in mind that we are discussing the origin of time as seen by an external observer. Also, I suppose that even if we were to consider the event horizon of black hole formed at the start of time to be the start of time, this is practically irrevalent as the laws ''written'' here (or indeed ''being written here'') cannot escape from the event horizon to effect our universe at all. I am fairly certain that if these laws are to be ''written'' at the same 'time' as other laws, they would be the same anyways(in case the black hole would evaporate).

This does also bring up the interesting question of what laws exist inside of a black hole. Perhaps a finished theory of quantum gravity would aid in this discussion, but I don't believe that we are anywhere close to there yet.

What exactly are we taking the universe to be?

The term can be used in different senses

I mean is it everything that exists?

in which case it would be incorrect to say it must have an external cause.

But, in actuality, it is not meaningless as a thought expiriment. Perhaps, you also believe that Einstein's visioning of the topology of the universe being that of Tensor Analysis to be meaningless as well. By your logic, because no one can vision the universe from hyperspace, it cannot be visioned from that perspective. We are simply invisioning the universe from hyperspace for simplicity.

My personal interpretation of the word universe (having never looked it up) is that it includes (possibly not exclusively) everything that is physical (stars, planets etc). This is only natural as if there was anything else, some sort of anti-physical stuff, then we wouldn't have any method of detecting it.

One point I would make here is that, according to Einstein equations, the speed of lightis impossible to reach because of matter/energy restrictions. (Incidentally, that seems to imply light isn't matter). The point is that if you were accelerated towards light speed, would you perhaps cease to be matter, but NOT cease to be entirely. This sort of spirit thing that would be left would perhaps remain until you slow down a bit again. and you become matter. So if that was the case, the spirit thing might be the anti-physical stuff I was talking about. And of course, God is the ultimate spirit.

I hope that makes a bit of sense

Light is matter, it just doesn't have any mass in the conventional sense (i.e. the constant m is equal to zero). See some other discussions on this board that explain why this means it can still have inertia.

Light certainly is matter in the sense that Brad and Neil require. What they need to group into this category is everything that in everyday life has effects on the dynamics of our universe. In other words, everything that we regard as normal, going on without an intervention from a potential God. Light must be in this category. For examples of the dynamical effects of light, consider the:

1. Photoelectric effect - energy can be transferred from light to electrons

2. Compton effect - photons carry momentum

Also, Neil, your deductions from the Einstein equations (which you don't really need - these are GR - you only need SR) seem to imply that God is travelling at the speed of light. Also, you are comparing God to a photon. I'm not sure most people who believe in God will find this very convincing.

Simon -

I wasn't really saying God was light as such, just that the 'other place' [as opposed to the universe] that God inhabits is perhpas the same place in which matter goes whilst above or at the speed of light. If God is light, it would explain why he is omnipresent of course. But I was really meaning something Asimov had in mind... I can't remember the name of the story but it was the robot field testers Donovan and Powell testing a spaceship which the robot brain had designed to beat light. Of course, they died for a while and came back to life when it slowed down, but they existed in 'another place' in between. And because Asimov's robot laws mean you can't hurt humans, the brain became a bit of a joker and filled the ship with baked beans. I hope someone else knows that story!

Sean-

I'd like to have a bash at your questions:

I should point out that I am taking God to mean the usual Christian/Jewish/Islamic One, rather than trying to define a God from the questions.

(1) What does it mean for God to be ''external'' to the universe? (i.e. in what sense does God exist?)

Well, I think the answer is that he's not. He is omnipresent apparently, so he IS everything. Thus the universe is part of the God, not necessarily the other way round.

(2) Why does God exist?

He created everything, and he is everything, so I suppose we can say he exists because the universe exists.

(3) Why did God create the universe?

Why do people grow beards, or have tattoes! Seriously, I'm not sure about the standard religious answer to this. It just goes ''In the beginning He created..''. There is no reason. Its probably a hobby. You are exhibit 4981/B!

(4) Why did God create the universe in one way and not another?

We are not in a position to consider possible answers to this question, as we do not know why he did it, and we do not know if there was another way. Do you mean, ''why is the universe just so?''

More technically:

(5) What does ''creation'' mean in an atemporal sense of the word (note that the definition of creation as the act of generating being form nothingness, has a causal structure, i.e. there is a cause and effect, but I don't think we can use these concepts when the universe as such doesn't exist)?

But it wasn't nothingness was it. If God is the universe, then he creates it from himself.

Well, my point in asking these particular questions was to get answers, as indeed happened, showing that in (1) and (2) God is for all practical purposes (FAPP) equated with the universe itself and (3) and (4) are not answerable. I was hoping this would bring out the initial point I was trying to make that the concept of ''God'' doesn't add anything to our ''deep'' understand of why things are as they are, which by contrast, I think science is all about. What is happening is that we give things a different name and shove the real uncertanties somewhere where by definition we're not allowed to look, i.e. effectively saying we don't know and never will.

Thus it seems to me that the only real statement that the existence of God is presenting is that there are questions we will never know the answer to. Now, this may well be the case but I think the uncertainty is great! And I intend to do my best to dig a little further and discover some more about the universe and I would be really dissapointed if at some point the digging hit a concrete wall with the word ''God'' written on it.

Can anyone come up with something that the concept of God actually really contributes to our understanding?

We might say that the existence of God implies a purpose to the universe, which makes the whole affair more aesthetically pleasing, although I and others would disagree, because of Sean's concrete wall.

Neil - the Asimov story is called Escape! and doesn't really propose the existence of an afterlife in hyperspace; that was just the Brain's practical joke. Relativity states that you cannot accelerate to light speed, however, objects already travelling at light speed are fine, which is why Asimov and other sci-fi writers have to use hyperspace.

I do particularly hope that we as a human race would never run into that concrete wall; then what would we do. I am just saying that there has to be a cause for everything and the cause for our universe is either God or another universe or perhaps some agent in quantum gravity. I am also saying that most probably we eventually will run into the building block of all other things. This, to me, implies a divine choice having been made.

I think that I'm going to agree to disagree on this one. I don't believe that you can willy-nilly extend the concept of causation or existence beyond the physical universe, and so none of the above arguments for the existence of god have any meaning for me whatsoever, since they all rely on intuitive, everyday notions of both causation and existence. However, if anyone would like to present a plausible argument as to why these two notions might extend beyond the physical universe, I'd be very interested to hear them.

I suppose you are right that we will have to agree to disagree on this one. While I certainly can't provide anything close to a proof of why causation would extend outside of our universe, it just seems altogether fitting and proper that all things follow reason. Perhaps Voltaire mixed with Plato is a bad combination, and I have been reading too much of both recently.

Brad - A good cure to Plato+Voltaire is probably Aristotle+Rousseau! Aristotle attempts to eliminate Plato's mysticism by placing the essence of things in the things themselves as opposed to in some mysterious world of forms. Rousseau similarly tries to place Voltaire's rational optimism in the social context of the time and is generally more down-to-Earth than the other Enlightenment thinkers. Mind you, I like both Plato and Voltaire too.

Dan - I agree entirely. I would add to the agreeing to disagree bit that I don't know anyone who has ever changed their view of God due to rational conversation (in one direction or another). I have only seen it change as people's world views have changed either in dramatic events or (more commonly) just through time passing and their perspective on the world changing.

Tom - not entirely sure. I'm not sure new physics is required throughout the black hole, only near the singularity itself (which is in fact shown ot exist using the usual laws of physics). This is because near the singularity it is not possible to ignore gravity when dealing with particle physics. I think one would hope the laws would not depend on the black hoel in question...

I have, as I think many of us do, a bizarre position in this discussion of defending both sides even though I don't particularly side one way or the other.

I agree with Dan about the reasons of religion, but these are NOT the same as reasons of God, as God was the vague name given to the source of anything beyond understanding at the time (and is now therefore an outdated concept). People who have strange dreams had a lot more influence around 1 BC than now.

Having said that, I can't really see why anything exists.

One more philosophical note, another one worth reading is (probably) Kant's Critique of Pure Reason. I say probably, because I'm finding it extremely difficult to understand. What I'm about to write may not be what he actually said, but it's my understanding of it, feel free to correct me if I've made any blunders. He makes the distinction between the phenomenal realm, which is the realm of things which we can directly observe, like tables and chairs, and the noumenal realm of things we can't, like ''the soul'' and things like that. He also argues that Reason cannot be used to prove anything in the noumenal realm (I think), thus effectively rendering a proof of the existence or nonexistence of god impossible, as god surely resides in the noumenal realm if in any at all.

Well, firstly I wouldn't really try and read read Kant just like that, it is a famously hard book!

The good bit in Kant on god is that he first ''proves'' the existence of god and then ''disproves'' it. It is one of the four so-called antinomias of Kant.

I'm not sure I like his distinction between noumena and phenomena, for example he makes no reference to how the two interact, which is surely an essential part of science (i.e. how does what we see depend on what the underlying cause is).

As for existence... An unhelpful definition is that being is what is. This is so unhelpful that many philosophers in the analytic (i.e. Britain and US) traditions treat the word ''being'' with contempt. However, something useful may be found in existentialist philosophy...

... draw a pigeon, now draw a red pigeon, now draw an existing pigeon. This sequence is meant to bring out the fact that existence is not a property of some object, such as being red or whatever (these are called essences sometimes). Rather being is what makes us able to encounter something in the world, in a more or less direct sense. So an existing pigeon is one that we can encounter in the world. ''encountering'' does not have to necessarily be through senses, a unicorn which we encounter in a vivid dream or in our imagination exists in a very real sense, although we would not say it exists as matter.

I tried reading it straight off, gave up, this time I'm reading a book called ''Kant - Selections'', which has an extensive introduction, still very difficult. I can only read it for about 20 minutes before I get a headache, so I'm making slow progress. On the phenomena / noumena point, I think that it is a good distinction. As far as I can see, he says that since you cannot ever make any reasonable statements about the noumenal world (because you can't observe it, and you can't apply phenomenal concepts to it), you have to limit yourself to what you can make statements about. I think this was a useful thing to do in the light of things like Descartes discourse on method. Although we may be living in a fantasy world projected by some other being (something a bit like the Matrix perhaps), we'll never be able to argue about it, so in a sense it's pointless worrying about it. Instead, limit yourself to the potentially observable realm (phenomenal), and worry about that only. I like the way he thinks, but I think he's missed a few points which it is easier to see now than it would have been then, thanks to modern physics, etc.

Dan -

If you happen to have read Sartre's Nausea, there is a famous section in which the main character describes how he has suddenly discovered what it means for a tree to exist.

I think part of the problem of determining existence in terms of its manifestations is that there are infinitely many, i.e. an apple can appear in infinitely many ways, depending on lightling etc. I seem remember this has a solution of some sort but I can't remember (it's terrible, for years I kept dragging a copy of Being and Nothingness between London and Cambridge on the basis that I might want to look at bits every so often, and now when I want to refer to it, I don't have it with me!).

My problem with noumena is that it seems unnecessary. Firstly, there must be a relation between a noumenon and a corresponding phenomenon, otherwise the noumena do not interact with anything and so are completely irrelevant. But if there is an interaction the noumena are in some sense ''chained'' to the phenomena. This is a sort of one-to-one correspondence. But as any mathematician knows, if two things are isomorphic, they are pretty much the same. Basically, I think the noumenal world IS accessible, and what isn't accessible is irrelevant and possibly a result of the misuse of language (see below). I agree with not being able to make statements about things (such as god), but I would interpret this as the things in question not existing at all, rather I think they result from extending conepts we have in an invalid way (so for example god takes the idea of a human, takes the idea of power or perfection in a person and then makes it infinite. This is invalid in a similar way to how certain limits don't exist).

Brad -

How can god be vengeful? God is presented as something abstract, living out of the universe, outside time, distinctly non-human, yet is endowed with a load of human attributes such as vengeful, forgiving, etc. Freud once said that god was a manifestation of a collective desire for a father figure (noting the appearance of god as an old man in many religions). I wouldn't quite go that far, but it is worth taking the point. Even in the ancient world, gods appear as an attempt to humanise nature, so for example you have a rain god, this takes a feature of nature (rain) and gives it a human face, making it in some sense understandable and, more importantly, accessible (you can pray to him and if you are lucky it will rain on your crops). Similarly modern gods of the christian type appear as a human face on creation ("man was created in gods image", jesus as the human manifestation of god etc., etc.). For the record, I think there many other reasons why people in god, but I think the humanisation of the unknown is certainly one of them. And I don't see any reason why this should be valid. To call something living in hyperspace vengeful is a grammatically correct sentence, but is like asking what colour an idea is, it doesn't make sense.

Exist is too general a word. ''Be'' is perhaps even more general. It could be argued that everything we know of and much that we don't know of 'is'. Then you go down the dubious road of separating out everything that 'is' into various categories, depending on their perception to man. Suppose our method of vision was not by reception of wavelengths in a small part of the EM spectrum. Suppose we could 'see' by mass-detection, as in one of Asimov's books (here we go again...).

We might have a different way of dividing what 'is' into the categories.

Dan, I think the point about extending concepts beyond the area w here they are obviously meaningful is the critical one for the whole discussion about existence.

'Existence' has a variety of different meanings when used in relation to different things. The following are some:

1. In a limited range of cases, its meaning can pretty much be reduced to a set of statements of the form: 'if you do x then you will see y'. For example, the statement 'there exists a chair in my living room' (assuming I know what all the other words mean) really just amounts to 'I strongly expect that if I were to go into the living room, I would see a chair'. Obviously that needs amplification - what it means to 'see' a chair is rather contentious. Anyhow, this is one area in which the word 'exist' is well defined.

2. 'There exists a prime greater than any given integer'. 'There exists oscillating solutions to this equation'. These are slightly different - referring to mathematical objects, but their meaning can basically be reduced to statements of the form: 'if you go and do x, you will find y'. For example: 'if you substitute in an oscillating solution into the equation then you will get an identity'.

In both the above cases, 'existence' is well defined in terms of other concepts. However, it sort of feels as though all the different ways in which 'existence' can be used have something in common, and so from this we abstract the general concept of existence, which is the thing that is objectionably applied in cases that appear to be beyond its realm of significance.

I think this process is intended to be analogous to the abstraction from the familiar ways in which 'length' can be used to the concept of a norm in a vector space. Now, the mathematical abstraction is rigorous, but the philosohpical one is dodgy. The difference is in the fact that a norm is defined rigorously to have certain properties, e.g. the triangle inequality. In the philosophical case there is just a sort of vague handwaving argument - 'all these different things exist so there is something that they all share'.

I think the section of Sartre that you are talking about is saying that modern philosophy has come to the conclusion that this philosophical abstraction is dodgy. I'm not really sure though what he means by modern philosophy. It is likely though that he means the positivists and particularly those who influenced them: Frege, Russell and Wittgenstein. Since that time though, their take on existence has come under vast attack. Russell's claim that all existence can be represented by the existential quantifier is believed by very few.

Sean,

I am not sure that we can assure that a God has non-human characteristics any more than we can assure that he has human characteristics. We really can't rely on any sort of description of a God at all. So saying that a God could be vengeful is really no more an anomoly than saying any other undetermined statement could occur.

Sartre readers - I'm amazed to find several mathematical Sartre readers, anyway, I think it's great. I can answer the question as to what Sartre meant by ''modern philosophy'', he was referring to the phenomenology school initiated by Husserl around the start of the century and adapted by Heiddeger. The essence of phenomenology is, surprisingly enough, to start with the phenomenon, without assuming anything about existence. What Sartre does is to start with the phenomenon as the act of perceiving the world and from there attempts to discuss the existence of the world (being-in-itself) and of a perceiver (being-for-itself). This is the first part of Being and Nothingness. Once he has set up the world and people in the world he then explores these concepts further. His approach is much more powerful than many other previous attempts because from the very outset man is IN the world. The whole philosophy is constructed from the outset in a way in which consciousness is always consciousness of something, and consciousness is concrete (compare this with Descartes who says ''I think therefore I am'', but the I in this sentence is ill-defined, it is some abstract concept, Sartre in contrast has the I as existing within a context). This is important also because Sartre is frequently presented as naively believing in freedom before all else, but this is misleading because his freedom is always freedom within a situation.

How exactly does Kant define phenomena and noumena?

There are certainly problems defining existence. But then again there are ultimately problems defining anything. Consider a dictionary. If I don't already know the meaning of at least a few words it is useless as it will ultimately be circular. Wittgenstein talks about how we learn to use words through language-games, which are certainly not riguorous.

So what I think Sartre is doing, especially in his novels/plays, is to give a ''feeling'' for what existence means, rather than a ''definition''. This could be realted (although I don't think it generally is) to Wittgenstein's phrase ''what we cannot speak about we must pass over in silence''.

Brad - precisely what I was trying to say was that ''we can't really rely on any sort of description of God at all'' like you say. But the corrollary of this is that God is a fairly useless concept. It's like saying saying ''there is something out there that I don't know about, but I don't know what it is''. I am perfectly prepared to agree with this, but I don't feel the need to call it God.

I don't think the questions will ever be answered philosophically, I don't even know if they are the right questions. It seems to me that the most significant cause for shifts in philosophical world view over the past two centuries (perhaps even more) have come from science. Of course one could dispute the boundary between science and philosophy, but after toying (briefly) with the idea of studying philosophy it seemed obvious that theoretical physics actually has a lot more to say about the world.

Yes, I definitely agree with Dan that physics tells us nothing with certainty. There is some debate over whether that is really the point. Some, such as Feyerabend for example, say that it tells us nothing at all. Others like Popper thought that completely true theories were the goal, but that they were unattainable. I think I agree with Popper on this point, although not on anything else.

Dan, also completely right is that modern scientific revolutions played an important part in this. Pre 20th century physics was regarded by many as being necessarily true. Newton regarded his laws as self-evident dictates of God no less. The classic example, which I think you will find interesting involves Kant, who claimed (I think in the Critique of Pure Reason) that the euclidean structure of space was true a priori. Of course the revolution of general relativity made that point of view obsolete since that theory deals precisely with the non-euclidean structure of space in the region of massive matter.

I'm not sure if ethics and morality are as important as the other things you mention. I mean it's never really philsophical discussions about ethics that change people's minds about things. You often hear people say I always though X was OK, but then it happened to me and I realised it was wrong', but you don't often hear: 'I always thought X was OK and then I had a long philsophical discussion about it and now I think X is wrong'. Of course you can ask questions like: 'if you kill someone but you didn't mean to, is it wrong?', but I think the philosophical interest lies not in trying to answer these, but in analysing what they mean, and what is going on in the head of the person saying them.

In my opinion, it is very true that physics can tell us nothing with certainty. However, as long as the current model desribing our universe seems to agree with every expiriment and is mathematically consistent, it seems to be ''true''. That being said, there really is no way that we can rule out the possibilty that all things happen because of some divine winds. But it certainly seems more logical that these divine winds have an underlying reason behind them, and that it isn't just coincidence that certain events seem to follow a set of rules for action occuring in them. But, I do think that eventually, physics will run into these ''divine winds'', although the 'divine winds' will follow a rigid set of rules and thus not really be devoid of reason like Aristotle's Science. Disagree with me if you like, but those winds are what I'm after.

Anyways, most words do create somewhat of a problem when we attempt to define them only with other words. For this reason, most words that can be defined are easily represented by a picture. A picture to represent existence seems to be very hard to find though.

I find physics to be more rewarding than philosophy as well. You at least get some sense of closure with physics, but any sense of closure in philosophy wil be torn apart by relativism and other doctrines of the like.

Whilst I agree (almost) completely with the two previous e-mails, two qualifications:

1) I think philosophy of ethics and politics are important. Not because they will change anyone's ideas - they won't - but because once you have a position they help you understand the consequencies of the position and in general give a more sophisticated structure to beliefs that most people hold in a fairly unrigurous sense.

2) I think Brad's concept of ''closure'' is a very useful one. It is not unlike the concept of a metanarrative in postmodernism, the idea that some theory can (or can't, as the postmodernists would hold) justify itself without reference to soething else. So it makes sense to ask whether physics is closed. It seems obvious that in a strict sense it isn't, there are a load of things which are assumed, such as the existence of the an objective world and the reliability of data to represent the world etc. etc. However, I think that as _method_, physics can be called closed and in fact is more closed han what is normally called philosophy. When has a philosophical concept helped physics? rarely, in fact normally it has caused damage, such as positivism which demanded everything have immediate observables and might or might not have been a factor in Boltzmann's suicide (his kinteic theory of gases didn't meet the required observability criteria). When has physics influenced philosophy? All the time. It's impossible to do metaphysics now without reference to quantum mechanics and relativity, and many previous philosophical ideas, like the a priori structure of space mentioned by Simon are just wrong.

And to be more controversial, what says physics needs to jutify itself anyway? It is only a philosophical position of requiring everything to be well-defined that required a justification of the scientific method. Does physics care at the moment whether there is in fact and objective reality? When it becomes relevant physics, physics will have found its own way of dealing with the problem, the same way it found a way of dealing with relative time when it needed to. And it seems that whenever physics, through necessity, brings its power to bear on a problem that until that moment had been considered ''philosophical'', it does so with much more effect than yeras of philosophical pondering have done.

Sean