– The basic modal logic K is sufficient for proving T1, C and T2.
– Modal logic S5 is not needed for proving T3; the logic KB is sufficient.
– Without the first conjunct φ(x) in D2 the set of axioms and definitions
would be inconsistent.
– For proving theorem T1, only the left to right direction of axiom A1 is needed.
However, the backward direction of A1 is required for proving T2.
So, does god exist or not? Can anyone proficient in symbolic logic translate this for us?
As someone who's read the first 2/3 of GEB at least twice, I'm planning to reorganise my entire worldview around this.
Like you, I've read the first 2/3 of GEB at least twice. I've also read the novels Rymdväktaren (The Space Guardian) and Nyaga by Swedish author Peter Nilson, which centers around the idea that the main protagonists, mathematical mastermind Diana Emerson and her companion Peter Lorentzen, successfully implements the proof in a supercomputer, giving rise to lots of adventures and mysteries.
Great books, unfortunately never translated from Swedish. The story is also about whether information can be destroyed, which of course was a hot question around the time of Nilson writing the books (they were released in the first half of the 90s, I think).
A little deviation, but I just wanted to underline how strongly I feel, again like you, to bring this into my worldview! I need to study the scriptures! :-D
>> Does an omnipotent being exist in a heaven above? Not literally, no.
Can you prove that an omnipotent being doesn't exist? You make a statement of fact without any backup evidence.
I wouldn't be surprised if it is impossible to prove or disprove, using the material world we're a member of, the existence of an omnipotent being who lives outside of this material world.
IMHO what Godel's 'proof' boils down to is that if it is possible for God to exist, then God must exist. The discussion of possible worlds is an attempt to strictly formalize and make consistent the argument that it's possible for God to exist.
The proof's axioms deal with what kinds of things have the property of being 'positive' (A1-A5). But lacking a separate definition of what positive means, we really have no choice but to view these 5 axioms as an implicit description of 'positivity' and wonder what mathematical models realize these axioms.
One thing to note from the axioms is that for every property either that property or the negation of that property (but not both) has the label of 'positive'. This is already pretty strong, because it's making claims the about the 'positivity' of properties that are completely unrelated to the rest of the argument. It's almost just a labeling game: given p and -p, you must label exactly one as 'positive'. Is the property of being red positive? Saying no is equivalent (according to these axioms) to saying that the property of not being red is positive.
You can pick one model for the axioms where being red is 'positive', and then the God that's shown to exist will be red. You can pick another where being not red is 'positive', and then the proof implies that God will not be red. It's not a logical contradiction because they're different models, but it certainly makes one wonder what the proof is talking about when it mentions 'positivity'. I have a suspicion that the axioms themselves generate an inconsistency and hence they could generate proofs for any statement.
Anyways, this paper is about formal theorem provers, not theology, so hats off to the researchers implementing higher-order logic.
The problem is that there may be a god such that your belief and your attitude in life may affect ultimate fate (life after death, etc.). If you worry about going hungry tonight, as you have established, then you should worry all the more about going hungry forever. Therefore, the existence and the nature of God is actually of the utmost importance to you (objectively, if not subjectively).
The exceptions are:
- if you have confidently established (to your personal satisfaction, even if you don't discuss it with anyone else) that there is no god, or at least no god whose existence would affect you. But you seem to imply that that is not the case.
- if you are in such a state of urgent need that you have no time to worry about anything else but your day-to-day survival (which is necessary to be able to consider theological issues, and thus can take precedence). But the fact that you are on Hacker News eliminates that possibility.
In practice I mostly act the same as you do, ignoring the issue. But I am aware that it is a completely irrational choice, and that I in fact do not have better things to do.
Pardon my presumptuousness in answering for someone else. But, it seems that by talking about apatheism, it was suggested that one should not waste time with such inquiries, as they give little to no value.
However, a different view on this, is that it's an exercise in logic. That whether god exists or not, is irrelevant to the actual discussion.
tl;dr: It could seem as the top post was "I don't care about this, because I don't care about religion. I'm an apathist.", which was met with "this can still be valuable, and interesting, from an intellectual point of view".
> As for questions of religious proof, existence, and so on... I'll get to them when I'm done with more important issues, which is basically everything else.
But some of the prevailing religions of our time tell us that one's eternal fate depends on the answers to such questions. To say that such questions aren't important implies you've already decided such religions aren't true; because if any of them were true, that would be important.
Perhaps one should practice all religions in hopes that an an eternal life is not nullified by the practice of another "wrong" religion.
I could also create a religion on the very basis of not contradicting any other religion, and would therefore, by your argument, the be only logical choice?
"one should practice all religions" Apostasy is usually considered worse than out-and-out atheism.
"create a religion" You could, but on those terms I bet it wouldn't get enough interest to become a "religion". Anyway, it's been done before: some 4000 years ago, the Abrahamic god said "Thou shalt love thy neighbour as thyself". Since then, Jews, Christians and Muslims have been happily murdering one another on a regular basis.
They don't just contradict eachother : all but one contradict themselves in trivial matters, and do not see that as a problem. The last one is also not perfectly free from contradiction, but at least it makes an attempt to not be contradictory. That's because Christianity's bible is written by the same people that invented logic. It is generally believed that science developed under Christianity where it would not develop under other religions because Christianity saw nothing wrong : that logical contradictions cannot be is part of the religion after all.
Islam puts it quite directly in the quran : "None of our revelations do We abrogate or cause to be forgotten, but We substitute something better or similar--Knowest thou not that Allah hath power over all things?....Would you question your Apostle as Moses was questioned of old?"
(a lot of verses in the quran have this pseudo question-answer format)
This pretty directly says that
1) there are contradictions in the quran
2) in those cases the later verse is better
(it is known as the principle of "abrogation" and it has recently been used, for example, to scratch allah's three daughters out of the religion in what has become known as the satanic verses incident)
Note that 2) is an islamic legal principle that has been copied into pretty much every legal system on the planet. Earlier law systems, like the Roman one do what Canon law, or Jewish law do : they claim there are no contradictions, and they do work to eliminate them when they find them. America's legal system calls this principle : "lex posterior derogat legi priori".
Also, this is true in at least part of your life : modern law does not try to be contradiction-free. Neither does science. There's a complex history behind that, but it boils down to that the attempts to make science contradiction free were all thoroughly disappointing (e.g. http://en.wikipedia.org/wiki/Constructivism_(mathematics) ) . So now if 2 theories explain 2 different things very well, then we accept them as true even if they obviously contradict one another. And that's in exact sciences.
That is something I always wonder about "religion is contradictory so I don't believe" statements : they preclude belief in pretty much anything. Do you really don't believe math works because of the contradictions that ZFC ? Physics ? Did you ever even bother to check ? The main example I give that people tend to know about is that relativity flat-out says space is continuous, and it's silent on anything other than gravity, and an essential property of the standard model (which also states space is quantized, not continuous) is that there can be no gravity force (the graviton is a contradiction in that theory, so there can be no gravity fields).
Science is thoroughly utilitarian, and when it comes to a utilitarian view, obviously religion has it's worth (one might even say you'd prefer Christianity due to the accomplishments of it's followers).
So the fun part is that for the vast majority of people making statements like "religions contradict" is itself a contradiction in their thinking, they're simply not aware of it. I think this makes it pretty sad, actually.
People don't follow religion because it's (currently) easier. That's reality, and contradictions have nothing to do with it, aside from being an excuse.
any other how-many-angels-can-dance-on-the-head-of-a-pin question
As a software engineer, I've concluded my job is to answer those questions, and then proceed to choreograph said angels ... insofar as CPUs fit on pinheads now, and software constitutes non-corporeal entities which nonetheless have intelligent organized effect on physical objects.
Which is the view that when discuss God they (very largely) do so without any definition of what they mean by "God", and thus their conversations are meaningless.
But it's simple. God does exist. She is a cultural phenomena and his effects can be felt in all our lives very obviously. If nothing else, then because our ancestors used it as an excuse to "civilise" the rest of the world so we can all live now as one big dysfunctional civilisation.
But if you can experience the effects of something it is very likely that it does in fact exist.
Your statement is completely illogical... I get the point that you're trying to make but I think you phrased it quite poorly. Unless of course you literally mean that god(s) physically exist simply because the concept of "gods" helped shape our society... In which case, elves and dwarves must also exist, fairies and dragons as well; after all, the effects of those can be felt in the lives of individuals, although to a much lesser degree.
>But if you can experience the effects of something it is very likely that it does in fact exist.
Once of the most ridiculous things I have read in recent history.
His point is that something doesn't need to literally exist in physical world for it have a real impact on reality.
Of course, the bigger point is that everything boils down to what you mean by "exist." The predominant opinion of today is strongly biased toward physical sciences, so if it doesn't exist in flesh and bone it's meaningless. Which is strange coming from a site catering to software engineers.
Yep, the disagreement stemmed from our personal definitions of exist in the context of the subject at hand.
When you're talking about god I would argue that the typical(most commonly used) definition of "exist" would be "physically", or even "spiritually", but not just mentally or based on a concepts impact on the world.
I would never argue that religion hasn't had a huge impact on society, but I think his original statement could definitely have used a qualifying statement as to his actual intended meaning.
I think a more accurate representation of his point would be "the concept of god exists" because the _actual_ diety itself doesn't exist in the traditional physical sense.
I didn't say physically exist, I said exist. Two different concepts.
Yes I do mean to say that gods exist simply because the concept of gods exists. Because that's all that they are - concepts/ideas.
You see the problem is, unless you accept that is enough for gods to exist, then you must also deny the existence of things like justice, law, copyright, mathematics, software, et cetera. Otherwise your belief system isn't even consistent with itself.
I maintain my stance that your original statement was phrased poorly.
It's a semantic argument at this point. We're both using the term "exists", but our personal meanings are different. I personally believe that stating something "exists" in the context of religion/mythological-creatures/magic/otherfakethings usually means that you believe it exists physically.
Obviously I don't deny god exists in the sense that he/she/it is a delusion held by a large portion of the population which effects the world around us; but I do deny that god "exists" in the physical sense...
I don't adhere to your train of logic... as I stated before, for your belief system to be consistent with itself elves/dragons/magic/fairies/etc.. "exist"; Of course the concept of them exists and the world is altered as a result, but your original statement seemed to imply a more literal meaning to the term "exists".
I concede that my original statement was written in haste and thus not phrased the best.
But here's the thing: As soon as you start talking about existence of things, it's nothing but a semantic argument.
I will finish with a youtube clip from a Prattchet-based movie: http://www.youtube.com/watch?v=cIfCGf_suxs The gist is that a lot of things exist solely because we believe in them.
Of course most things do not "exist" if you demand physical material existence as semantics.
soul -> doesn't exist.
law -> doesn't exist.
countries -> don't exist
justice, peace, war, good, evil -> don't exist
computer programs -> don't exist
hacker news -> doesn't exist
and one might even say
You don't exist. You're merely a set of partially filled ion channels in a bag of liquid some existing system likes to simulate for some reason. Just like minds seem to like to simulate God (which is independant of whether God has a physical manifestation or not). Your mind only exists in the same way God exists, no more, no less. God is simply found in more than a single bag of liquid. (I've studied AI, so I would argue the same -partially- applies to your mind, in that people with memories of you effectively simulate your mind to e.g. know if you'd agree or disagree with a proposal of theirs)
And frankly, if I were free to choose, I wouldn't have a physical manifestation at all, if I didn't need it to affect the real world. In fact, I am a computer programmer, one way of describing that job would be that I spend my days making other physical systems do what I'd have to do myself otherwise. I'd love to cut out the middleman that is my body. It's fragile, it's distracting, it fails from time to time, it hurts sometimes, it communicates through talking and typing and that makes it fucking slow to communicate with, it's ... Why would you want that if you can avoid it ?
> He does not extensively discuss what positive properties are, but instead he states a few reasonable (but debatable) axioms that they should satisfy. Various slightly different versions of axioms and definitions have been considered by Godel and by several philosophers who commented on his proof.
Such is the world of philosophy. Remember that some of the greatest scientists and mathematicians lived when all of it was just philosophy. Philosophy is a training exercise to think imaginatively and some of our best science stems from an imaginative spark. It's like saying that people professional athletes running around a field (philosophy) is pointless because they are not running in a race for gold (science) at that specific point in time.
This is by all means not scientific, precisely because you can't address that specific subject matter scientifically:
1. God exists if he makes himself/herself visibly and measurably known and is observed and measured by a human being and the observations are confirmed by peers within the scientific community.
2. God does not exist if a human is present to see the end of the universe (or variant there-of) and he has not been yet observed (assuming he/she is tied to the existence of the universe) and the lack of observation is confirmed by peers within the scientific community.
We simply don't know until either (1) or (2) is satisfied, therefore this is exactly the type of stuff philosophy is good at. It's not saying "can we prove it" it's saying "let's think about creative ways to prove it if we can get a little lax about the scientific method." Were "getting lax about the scientific method" means that "proof" can be derived from either common sense or logical arguments.
> Remember that some of the greatest scientists and mathematicians lived when all of it was just philosophy.
Not really. Galileo and Newton turned philosophy into science by doing experiments, and scientists never looked back. Science differs from philosophy by being steered -- defined -- by empirical evidence, and by falsifiability, the idea that any idea must be testable against reality and must be discarded if the test fails.
That's certainly not philosophy.
> It's not saying "can we prove it" it's saying "let's think about creative ways to prove it if we can get a little lax about the scientific method."
There's no middle ground such as you're suggesting here. There's no overlap between science and philosophy. They are distinct ways of processing ideas. One of them must be compared to reality at every turn.
> you can't address [the existence of God] scientifically
Maybe one day science and philosophy will meld to the point where this might be possible in a peer reviewed in the Nature journal kind-of possible way, God cannot be addressed by the current scientific method.
What's valuable about Paul Graham's line-of-thinking is that he can distinguish between science and religion - something that very few can do (and is likely going to make this HN post explode into a pointless evolution vs. creation argument - which is missing the point of the experiment entirely).
> Maybe one day science and philosophy will meld to the point where this might be possible in a peer reviewed in the Nature journal kind-of possible way ...
No, that's not possible, because of two essential properties of science that distinguish it from philosophy -- empirical evidence, and falsifability. The first (empirical evidence) means science is steered by evidence drawn from everyday reality in practical experiments or observations. The second (falsifiability) means it must be possible to test scientific ideas against reality, and those ideas that fail the test must be discarded.
These properties of science mean that science and philosophy will never merge.
> God cannot be addressed by the current scientific method.
And the "current scientific method" is how science is defined. The only way to get around that definition is to abandon science.
That's not quite the same thing. The 4 colour theorem was proved by boiling it down to several thousand test cases and using a computer program to test those by colouring them. This paper is about using theorem provers to check axioms for consistency and proofs derived from those.
Important to note Gödel's completeness theorem here!
You might be even more interested in Prolog, a language which has a close relationship with this stuff: "Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of relations, represented as facts and rules. A computation is initiated by running a query over these relations." (http://en.wikipedia.org/wiki/Prolog)
This would appear to be a more formal version of Thomas Aquinas' Argument from Gradation of Being, borrowing some elements of the Argument from Possibility and Necessity.
Logically proving the existence of God is a very old game. Modern atheists tend to ignore these proofs in favor of attacking superstition and the more obvious theological nonsense.
Which is a shame, because they can be refuted, with interesting consequences for one's worldview. The stance that most powerfully refutes any Ultimate argument for God (of which there are a few good ones) is, imho, Madhyamaka Buddhism, which holds that everything is empty because all phenomena dependently co-arise.
If there were a God and he was all powerful, and he wanted you to believe before you knew He existed, it makes sense that you would be unable to prove or disprove His existence.
This was paid for by tax-payer money? Wow Germany. Awesome.
I've only just begun studying Gödel's proof of incompleteness and so I'm not qualified in any way to verify this paper. However it seems to me, as a layman, to be a very large stretch and mostly a tongue-in-cheek exercise of various logic proof verification software.
From the paper: "Gödel defines God as a being who possesses all positive properties. He does not extensively discuss what positive properties are, but instead he states a few reasonable (but debatable) axioms that they should satisfy"
I'm not an atheist, but for truth's sake you would notice that Gödel definition of God is very wide and not necessarily aligned with any mainstream religion.
I am a Christian, and I think this argument (and more simple ones of the same general form -- notably St. Anselm's) are good examples of the limitations and abuses of logic; they are superficially attractive in isolation, but if you accept the basic form (and, particularly -- as pointed out Kant's critique of Anselm's argument, which is equally applicable to Godel's -- accept the erroneous idea that existence can be treated as a predicate) you can "prove" the existence of pretty much anything.
Reading the Wikipedia version of that, it is full of HUGE assumptions:
> Axiom 5: Necessary existence is a positive property
That's in total disagreement with evolution. No being is really necessary.
> Corollary 1: The property of being God-like is consistent.
Now that's in huge disagreement with any religious book, as all of them are full of inconsistencies.
Any proof lacks any value if the starting axioms lack value by themselves.
Newton wasted years of his life studying the bible. We shall simply view all this as the waste of time it was and simply expect future genius to not do the same.
I wouldn't worry, Gödel himself was only trying to prove that mathematics could be used to determine if God existed and he himself did not publish it until he was near death because people would think he believed in God. If anything it was a more formal approach to the Ontological Argument put forward by Anselm of Canterbury, and he was trying to show how modal logic can be used as language in proof theory, and "the study of what can and cannot be proved in mathematical systems of deduction. Issues of completeness of mathematical systems, the independence of axioms from other axioms, and issue of the consistency of formal mathematical systems are all part of proof theory."
Now what would be really interesting is some form of comparison between this proof and the conjecture by Brian Greene (and others) that it is highly likely that we are all bit players in a massive computer simulation of a Universe...
It's an argument that, if certain logical conditions are met, then the principle of the existence of God is inescapable. Note that I said the principle, not the fact -- the fact was not the matter under discussion.
It's like discussing the concept of an honest politician -- such a thing can obviously be proven to exist in principle, but that doesn't require that it exist in fact, in everyday reality. A good thing, too.
People who read these tracts need to understand that the goal is not to literally prove that God exists (all superficial appearances to the contrary), only to show that the argument can be made and is logically consistent. The same argument can be made with equal justification about Bigfoot, the Loch Ness monster, or an honest politician.
All other issues aside, the Holocaust proves that either (a) God doesn't exist, or (b) he's Spinoza's God, the one who can't be bothered about humans and their problems.
As someone who's read the first 2/3 of GEB at least twice, I'm planning to reorganise my entire worldview around this.