A mathematical proof that God exists
September 6th, 2006I apologize to those who fell for the controversial title: I only have a proof that God partially exists, although if you can hang on through the (mildly-)mathematical developments, you will see how it is sufficient.
Lemma
Mathematics does not discover truth - it packages bundles of logically consistent constructs.
Every mathematical theory is based upon a set of axioms, which are arbitrary propositions that are taken for granted as true (at least in this theory). The rest of the theory has to be logically deduced from these axioms by rigorous demonstrations. Any proposition is either true (it is then called a theorem) or false (and its negation is a theorem) — at least this is what mathematicians believed until the twentieth century. Then it was proved that even in well-built theories, there were propositions that could neither be demonstrated as true, nor false. They were called undecidable (or independent) propositions.
If I find an undecidable proposition, I can add it to the axioms, and build a new theory. The new theory extends the initial one. If the initial theory is consistent, then the new one does not have internal contradictions either. In this new theory, the undecidable proposition is now true, as it is now an axiom.
Conversely, I can add the negation of the undecidable proposition (which is also undecidable) to the axioms and make another new theory. This second new theory also extends the initial one. In this theory, the undecidable proposition is now false, as I have added its negation to the axioms. Clearly, both new theories are mutually incompatible, but they are both equally valid.
Theorem
God exists.
Proof
Let us exit the strict mathematical background and consider the proposition ‘God exists’ in the context of science and philosophy. I say that it is undecidable. If it was possible to prove that God existed or did not exist with today’s scientific knowledge, it would have been done already, as there have been so many bright minds intent on doing so.
I can go further, saying that if God exists and wants us to be free to believe, She would make sure the proof of Her existence is not obvious. Indeed, if it was possible to prove that God existed like 1+1 is 2 or like pi is irrational, then we would not be free to believe otherwise.
Conversely, if God does not exist, how can I prove it ? Unlike in mathematics, life’s conceptual universe is not bounded, and there can be no such thing as a proof of non-existence. I cannot prove there is no Flying Spaghetti Monster — I cannot prove there is no God.
Therefore, if the existence of God is an undecidable proposition, I can choose to consider it as true or false with equal validity. QED.
My belief
I personally choose to consider it true. I can make a new ‘life theory’ where I have no doubt, where I am sure that God truly exists, because it is now one of my axioms. It is a personal choice: nobody can take it away from me; nobody can convince me otherwise; but I cannot convince anybody either. If I come across people who made the opposite choice, I know that their theory is as valid as mine, and I certainly cannot convince them that God exists, because this is false in their theory. I myself can even do like mathematicians and not stick to a single theory, switching to the other one when I need: it remains conceptually valid.
I know a lot of rational people who are atheists because they think religious belief is irrational. I have just proved in a very rational way that it can be justified as a logical construct. I have also rationally proved that atheism and religion can coexist in life, like Euclidean and non-Euclidean geometries can coexist in Mathematics. So maybe they would be more inclined to read what I have to say next.
I prefer to believe in God. I admit I could consider life on Earth as a mere accident resulting from the indifferent and universal laws of physics; I could consider that the purpose of our passage in this world is no different from that which dictates the relentless revolution of Titan around Saturn; I could consider that free will is just an impression, allowed by the unpredictability of any deterministic phenomenon too complex. Indeed I could; but I would hate having to.
I had rather see the laws of physics as the precious womb in which life can grow; I had rather see intelligent life on Earth as a superior intention, fortunately mysterious; I had rather see free will as a token of a renewed trust. It gives an overall consistency to my life, which I find more satisfactory, even though I cannot prove it is not culturally biased. However, owing to my initial premise, nobody can prevent me from believing thus, therefore thus I choose to believe.
Read on
An apologia of the scientific mind
nice one….
I can tell that you had loads of fun while writing this. A minor quibble:
I’m not so sure that *every* theory is based on axioms that are *arbitrary* propositions taken as true. I am sure that a consistent geometry could be built using axioms that were the exact opposites of the ones stated by Euclid. But does this make the axioms of Euclid arbitrary? I am fairly confident that a savage who has no idea of geometry would be able to comprehend a theory based on “The whole is greater than the part” more easily than a theory based on “The whole is less than the part”, even though both theories would be logically bulletproof. Considering the way in which the human mind associates, dissociates, categorizes, etc., at least that one axiom of Euclid does not seem arbitrary to me.
This is all in good fun, so I hope to indulge a little bit in the proof of God’s existence. I don’t find it useful to say “God exists” or “God does not exist.” I prefer to say that “The probability that God exists is negligibly small and for all practical purposes, it is zero.” It still amounts to the same thing, but I am somehow more comfortable with that view. I am uncomfortable with the whole atheist/agnostic terminology. Introducing probability probably makes me agnostic, but since I say that the probability is negligibly small, it makes me an atheist for all practical purposes ;).
Incidentally, your viewpoint (the reason you want to believe in God) is very similar to Margaret Atwood’s, if we are to believe a recent interview she gave to Bill Moyers. I don’t agree though. I don’t find a superior intelligence necessary in order to appreciate the wonders of the universe and of life in it. However, I cannot truthfully say that my perspective has made my life any easier.
C.S. Lewis agrees with you (as do I) that God left us the choice to (dis)believe.
What I want to get at is the motive of the nonbelievers. Is it always selfish? Arrogant? Dr. Paul Vitz has done some research in psychology indicating that a poor/absent relationship with one’s earthly father can lead to disbelief in our Heavenly one. He cites some prominent atheist figures and points out the father connection.
Of course I noticed you referred to God as “she”. Reason?
Please send any reply to my email, as I may not get back to this site.
Lord be with you,
Bit
If all children who had an absent father had turned atheists, I doubt religion could have happened in the first place…
I referred to God as a she for no other reason than fun.
I will not reply by e-mail — I do hope you can come back to this site, although you might find me a strange sort of believer.
Here is my “proof” that Judeo-Christian God can’t exist. Note by redefining what you consider the Judeo-Christian god to be, you can say that this proof is incorrect. i.e. If you just make it up in other words.
Okay, so imagine you’re omnipotent, ubiquitous and omniscient - as anyone who is all of these things will tell you, that means you’re eall-powerful, everpresent and you know everything. The first two aren’t a problem, but this last is a logical contradiction. Knowing something requires you to (a) know it, and (b) know that you actually know it as opposed to believe it.
The only thing I know is that my mind exists. I know this because my knowlegde of it fulfils both criteria. Someone told me the cogito (part a) and I thought it was nonesense so I went throught the process of trying to work out a way in which it would be possible for me to be only imagining that I was thinking… and failed (part b).
Returning to our supreme being: this character automatically knows everything. In which case there is no reason to think and no possibility to think. Every thought whether correct or erroneous is part of the body of knowledge and having any one of them would be recall rather than thought. So this character can not know that he knows that he exists. Or rather there is no logical proof that he can give that he exists. Therefore this God who knows everything can’t actually know it. It is a logical contradiction in the biblical formulation of God.
You can of course say “Aha… but he/she/it knows this because he/she/it has special God proof stuff that they can spray onto tenets to show if they are true or false. “. However, you’d be making stuff up that doesn’t appear in the bible. Still, the bible appears to have been made up over many many years so there’s no reason why people should stop changing the story every time someone spots that’s it’s nonsense.
Incidentally, in response to the guy who posted about motives of non-believers, the clue is in the question. I’ve just never believed that Gods exist - simple as that. I would hate to be blanketed in with every other non-believer as being part of a group that have a shared set of beliefs. We’re people, we all have our reasons.
Derm, you may be onto something with your proof, and I agree there must be some internal contradiction with the concept of an omniscient God, although I do not really follow your path and the thought part.
I’d like to read more of it when you’ve sharpened it some.
In my post, I was careful enough to avoid defining God. Obviously, as soon as I give a definition of God, there will be a way to find an internal contradiction. My post is more about the right to believe than it is about the existence of God, but I assume you’ll have guessed that much.
Thanks for commenting. See ya around.
Thanks Mand,
My “proof” is just a bit of fun also. You ought to hear my take on Buddhism: The easiest and best way to ensure that you come back as the enlightened one is to make every other living creature (including humans) extinct. It’s a no-brainer - process of elimination.
And don’t get me started on the Humanists.
I won’t get you started on the Humanists
[…] I am a believer deep down, I am still an engineer with principles, and I never buy into a ‘God made it so’ or […]
Derm said:
“Every thought whether correct or erroneous is part of the body of knowledge and having any one of them would be recall rather than thought. So this character can not know that he knows that he exists.”
If God knows everything, He must have some manner of containing knowledge. Let’s suppose it’s a notebook called “Knowledge I Possess”. To get started, God writes down
1. I exist.
From 1, he derives
2. I know that I exist.
He reads His notebook and finds a third new piece of knowledge.
3. I know that I know that I exist. (by statement 2)
God, being a sharp fellow, realizes He’s entering and infinite regress. So, He throws away the notebook and gets another. This time, He writes in bold at the top of the first page
For each piece of knowledge in Knowledge I Possess, define a function f: Knowledge -> Knowledge where f applied to some statement k produces a new sentence “I know that” concatenated with statement k.
This saves God a lot of writing, and also captures the infinite regress.
Incidentally, this is the basis for Kronecker’s observation that the natural numbers are the work of God.
Interesting chain of thoughts. I was almost sure that God was a mathematician
see my web site, ma conception of God :
http://jebara.topcities.com
Adib Ben Jebara.
I find the Kalam Cosmological Argument very convincing proof for the existence of God. Take a read of it here: http://operation513.blogspot.com/2008/07/kalam-cosmological-argument-for-gods.html
I cannot say I am as convinced as you. But, as I said, anything that’s good for you is good for you.
Hey,
I found an old entry of yours as I was searching online, and I found the topic to be interesting as I am intrigued by both (in)completeness (of the Godelian kind) and philosophy of religion, so I hope that this comment will be welcome, though most certainly dated, lol.
If I understand your reasoning correctly, you think that the proposition “God exists” is undecidable and that, therefore, we can reasonably take it to be an axiom of sorts. If we can reasonably take “God exists” to be an axiom, then we can reasonably believe in God (presumably regardless of other considerations).
If I understand you right (and I may not have, lol), then I have some constructive things to submit for consideration.
First, the technical form of (in)completeness that you’re describing, at least in my understanding, requires that one be talking about a specific system of axioms that is sufficiently strong. That is, the proposition “God exists” is undecidable relative to a set of axioms. If this is the case, then we still might be able to choose a different set of axioms such that “God exists” is no longer undecidable. (Consider the Continuum Hypothesis in ZFC or the Axiom of Choice in ZF). For instance, quite vacuously, if we take the set S to be the set of axioms that you normally accept and then form S’ = S U “God exists”, then “God exists” is not undecidable relative to S’. (There are more interesting examples, but this was the first one that came to my head, lol.)
Second, I wonder if we should really consider “God exists” to be undecidable rather than simply undecided. You are totally right when you say that philosophers *still* manage to argue about these things and that they are far from resolved. But, I think the question is whether there is reason to think that questions such as these cannot be resolved *in principle*. Until there is such good reason, why give up so soon? Lol.
Third, I think that perhaps what you’re going for more generally is a limit on what we can know, an epistemic limitation. One might be tempted, given the state of the debate on the existence of God, to put “God exists” in the category of “unknowable” propositions (as opposed to simply unknown), along with “God does not exist”. This is basically the statement of strong agnosticism. If we take this position, then why is it acceptable to hold “God exists” to be true anyway? A lot of things can follow from a belief in God. If someone believes something without good reason to believe it (more specifically, on a desire, whim, or personal feeling), then it could potentially be harmful. Consider the effect of creationism in modern American public schools on the quality of scientific education (although it differs specifically on what kind of God one were to believe in). It seems almost reckless to choose to believe in something that one believes to be unknowable. In general, as competent, rational beings, we ought to maximize the kinds of true things we believe and keep to a minimum a belief in things that are false. The equivalent of a coin flip doesn’t seem to be in accordance with this general rule, as we could do this with more “unknowable” propositions, yet, as we increased the number, the likelihood of us having a set of true beliefs decreases exponentially and the number of false beliefs increases exponentially.
Fourth, I guess, more simply, I don’t agree that “God exists” is an “unknowable” proposition in the epistemic sense. The concern about philosophers debating endlessly about the issue of the existence of God seems to simply be a pragmatic one. If we are to think that “God exists” is “unknowable” in the epistemic sense, then there ought to be good reasons for thinking that “God exists” cannot be known, even in principle (much like what I was talking about earlier). There are some that try to show this, but that goes beyond the scope of this discussion, :). In short, I don’t think that one can simply cop-out of the question because one isn’t in a good epistemic situation.
Anyway, that went much longer than I meant it to, but I hope that I said something interesting… if not, then just ignore my ramblings, lol. Good luck with your blog.
Jake
“I can choose to consider it as true or false with equal validity. QED”
Being uncertain of a state of existence is not the same as proving existence…
This is basic fundamental math : being certain that an assertion cannot be either proved right nor wrong within a given theory allows (urges) a mathematician to decide to choose it either right or wrong and add it to the theory to constitute a bigger theory. Obviously, the other choice is just as valid, but results in a different theory.