Some 'proofs' I've already addressed elsewhere, such as Pascal's Wager or the Argument From Design. But there are a few others that are notable which people occasionally send my way.
It's hard to overstate the impact Greek thought - particularly that of Socrates, Plato, and Aristotle - has had on the development of Western philosophy. There's a reason for that - a lot of really intelligent and thoughtful people have built on the foundation laid early on in Greece.
However, some foundational aspects of Greek philosophy have major issues. One of the key concepts explicated by Aristotle is the notion of the 'four causes'; roughly speaking, the four kinds of things someone might want to know about a thing or a phenomena.
The classical example is a statue. Its 'material cause', what it's made of, might be brass. Its 'formal cause' is the form it takes - say, the shape of a fallen soldier. Its 'efficient cause' is who or what made it, the process that produced it - in this case, the artisan melting and casting some brass. Its 'final cause', 'telos', is the purpose for which it's being made - to be a decorative memorial to that fallen soldier, perchance.
There's a central problem with the classical Aristotelian understanding, however. The first three 'causes' only need one answer. Something can only be made of what it's made of, or have one process that actually made it. Depending on how you understand 'form', you can say something only has one form, maybe.
But why should something have only one 'telos'? Take a simple example. The White House is where the President of the United States lives when in office. Is that its 'telos'? But wait - it's also an office building; the center of the executive branch. Is that its telos? And, hey, wait a minute - I've taken a tour of the public areas. Is the telos of the White House to be a museum? And then there's the fact that it was designed and architected specifically to impress visiting foreign dignitaries - is that its telos?
The White House fulfills all the above purposes at once, and pretty well, too. Indeed, practically everything humans have designed has been fitted to more than one purpose. Even the most severely functional weapon is simultaneously a threat and a status display, even if it's never used to inflict harm on an enemy. In theory, our arsenal of nuclear weapons were built so that they would never have to be used. What kind of 'telos' is that?
Speaking of weapons, check out any random Jackie Chan movie. He makes anything - frying pans, bikes, ladders, ladles, garden hoses, anything - into a weapon. Exactly why is the 'telos' he gives those objects any less valid than the purposes they are marketed under? Hey, speaking of marketing - is the telos of, say, a frying pan 'to cook food'? Not from the perspective of the people making it - they are making the frying pans for the 'telos' of selling them to other people!
What about natural resources? What is the 'telos' of sand? To be made into static art, like a sand painting? To be made into a dynamic tool, like an hourglass? To be made into sandbags to hold back a flood? To be squirted onto rails to improve a railroad car's traction? To be used as ballast for a balloon? To be purified to make a silicon chip? If for a silicon chip, what kind - in a Wii or an Xbox or a Playstation?
Even the case of the statue isn't quite so clear-cut. A statue in a city square fulfills a purpose - decoration, memorial, whatever - but not only that purpose. It can serve as a landmark for navigating about the city. It can serve as a jungle gym for kids to climb on. It can serve as a perch for birds. It can serve as a blind from behind which to spy on someone. It can serve a fleeing pickpocket as an obstacle to slow pursuit. It can serve as a source of metal to melt down into cannons to defend the city. The number of different 'causes' served by anything in the real world is probably at least equal to the number of agents that interact with it.
The key issue is a very fundamental misconception. Objects - even designed and constructed objects - do not have inherent purposes. Agents have purposes to which objects are put. Any 'telos', any 'final cause', any 'purpose' is always relative to the agent that's doing the intending.
Let me quote this insightful passage from The Bad Idea Blog:
To say that some event means something without at least some implicit understanding of who it means something to is to express an incomplete idea, no different than sentence fragments declaring that "Went to the bank" or "Exploded." Without first specifying a particular subject and/or object, the very idea of meaning is incoherent.
It's worth reading that whole essay. It establishes very clearly that things don't - can't - just "mean" something, hovering there unsupported. Something has to mean something to someone. And the same thing can mean different things to different people. The purpose of an object is the same way - an object has a purpose to someone, in support of some goal. Value, too, is 'relative' in this manner - it's inherent in the very concept of value. Something is of value to someone, and for some purpose.
Consider a wooden chair. What value does it have? It depends on the purpose you have for it. It might be something to sit on; it might be an heirloom; you might be using it to ward off a lion; you might be using it for kindling during a blizzard. It might be of only middling worth in the first case and literally worth your life in the last. Which purpose is the "real purpose" - and why?
This confusion has saturated a lot of philosophy, particularly theistic philosopy, and leads to a lot of problems with the traditional arguments for God, as we'll see.
Anselm's Ontological Argument historically enjoyed wide currency, but is now regarded as rather weak. A summary would go something like: "We can imagine the greatest possible being, so it exists in 'our understanding'. But if it only existed in our understanding, then we could imagine something even greater, that existed in reality, too. Since we were trying to imagine the greatest possible thing, then it must exist in reality, or else we'd have a contradiction."
As I said, this isn't widely accepted today among philosophers. Even back in the day, there were objections to this argument - see Guanilo's Island ("A perfect island wouldn't be perfect if it didn't exist, so a perfect island must exist in reality"). Nowadays it's generally agreed that "existence" isn't really a "property" something can have. (See here.)
But there's another problem with this argument that isn't generally pointed out. Can anyone really imagine a 'greatest possible being'?
Back in Medieval times it was generally accepted that there was a definite hierarchy of 'greatness' - the Great Chain Of Being. God was at the top, followed by various levels of angels, followed by kings, lords, peasants, animals, plants, rocks...
This isn't generally accepted now, and for good reason. I don't think that there is, or can be, some absolute, objective standard of 'greatness'. We just discussed above the idea that things don't just have objective, unsupported 'value' - things have value to someone, for some purpose.
The idea of 'objective value' runs into all kinds of problems. Consider - which is objectively better, the color blue or mozzarella cheese? And if God is greater than both, are we saying that the sky would be better if it were God-colored? That a pizza topped with pepperoni and God would taste better than a pizza which settled for pepperoni and cheese?
Recall the wooden chair example above. If I trade some gold away to keep a simple wooden chair, break the chair up and burn it to keep my child warm... have I erred in assessing the value of the gold, or the chair? (Or the child?) The guy who made the chair intended it for sitting on (well, actually, as I pointed out, he made it to sell to people, probably expecting them to sit on it) but was I wrong that it would make a warm fire?
Even worse for the concept of 'objective value', different people will assign different values to the same things. A woodworker might trade you a chair for some of the corn you grew. Who came out better on the deal? You both did - you both have more value (by your personal estimates) than before. (Or else why did you trade at all?) Differential valuing is what makes economics possible. But think - if there's some kind of 'objective value', then at least one of you is wrong. Either the chair was worth 'objectively' more than the corn, in which case you cheated the carpenter - or else the corn was 'objectively' worth more than the chair, in which case the carpenter cheated you. (Or else they are 'objectively' equal, in which case you're both wrong about having more value than you did before.)
Value and greatness are relative, personal measures. My wife is probably worth a lot more to me than she is to you, and me to her, and our kids are more important yet. I wouldn't be surprised if you didn't find them quite so amazing as your own spouse and children, though.
Back in the Middle Ages, St. Thomas Aquinas proposed his "Quinquae Viae" ("Five Ways"), or five 'proofs' of God. None of them hold up terribly well to examination. The fifth, the argument from design, I've already addressed. Of the others...
Three of the "Five Ways" depend critically on his claim that an infinite regress (of movers, or causes, or 'contingencies') is impossible. It's not at all clear that this is the case.
We accept infinite "progress", for example. We have no problem imagining time going on infinitely into the future. (Especially theists, who usually say they'll be there.) We have a hard time imagining space being finite - "What happens when you get to the edge?" children ask. "What's after that?" We don't reject infinite sequences in that case. Why suddenly insist they are impossible when it comes to the past?
Part of the problem comes from our normal understanding of time, that it proceeds by 'addition of successive moments'. That's intuitive... but it appears to be wrong. In Relativity, time really is an actual, physical dimension. The past exists, a literal distance away - one second ago is one light-second away. And experiments bear this out - it's hard to explain the relativity of simultaneity any other way. As Einstein put it, "The distinction between past, present, and future is only a stubbornly persistent illusion". If time is like space, then the problem with 'infinite regress' vanishes.
(There are those who claim that the 'Big Bang' marks a definite starting point to our universe, but things are more complicated than that. See, for example, Hawkings' "Brief History Of Time". As I've said, I don't think we've had the right insights yet to figure this out.)
Aquinas thought he'd proven that an actual infinite regress like that was impossible. However, he was even willing to grant one, if only for the sake of the argument. But he moved on to another level...
Aquinas was following Aristotle, and wasn't using terms like 'motion' or even 'cause' in quite the manner we do today. They talked of 'motion' as things moving from 'potential' to 'actuality' - the example of a lump of bronze having the potential to be a statue, and then being moved (worked) by an artisan into actually being a statue. If motion is understood this way, then nothing can 'move' without something to move it. Aquinas (after Aristotle) argued that you couldn't have an infinite regress of such causes.
Even in Aristotle's time, though, there were problems with this understanding. An acorn 'moves' to become a tree, but does something - in particular, an agent with a cause or 'telos' in mind - 'move' the acorn? Evolution undercuts this idea rather alarmingly. We can now see how evolution can produce things like multiple-drug-resistant superbugs in spite of - even directly contrary to - the wishes of doctors and biochemists.
I pointed out, above, the multifarious purposes to which a statue may be put. But even the artisan who made the statue may have multiple purposes in constructing it. He may wish to commemorate a fallen soldier... while at the same time subtly castigating the generals who ordered the march the soldier died in. The artisan may also choose a particular artistic style in order to make a statement to some of his fellow artisans, and have chosen brass as the medium to help out his brother-in-law the metal merchant. Plus, the artisan no doubt intends to be be paid for the statue... to help keep his children fed.
Nothing in the real world ever serves only one purpose, ever has only one meaning, ever has just one 'objective' it can or does serve. Any 'telos', any 'final cause', any 'purpose' is always relative to the agent that's doing the intending.
The idea behind the 'ontological argument' is that all 'values' have to converge, have to all wind up in the same place, in the same Being. And I argued above that such a concept rests on a fundamental misunderstanding, an ultimately incoherent concept of an 'objective value'. The idea of 'objective purpose' is likewise incoherent, and for the same reasons. Without that assumption, there's no reason to suppose that all purposes, all telê, must converge to an Ultimate Telos. Indeed, it becomes clear that they cannot converge.
Agents are what give rise to purpose. And evolution sure seems to indicate that agents can arise without an agent purposing them. So it seems there's no need for an infinite regress of agents and causes - nor even a need for an 'Unmoved Mover' or 'Ultimate Agent'. Historically it seems there must have been a 'first agent', but that agent can have been caused by agentless, purposeless processes. An infinite regress of them, in fact...
(For more on what an evolutionary perspective does to traditional notions of teleology, see Dennett's "Darwin's Dangerous Idea".)
This one is quite muddled, but basically claims that since we can, e.g., say that one statue is more beautiful than another, there must be something that is maximally beautiful. Even granting this, he then goes on to claim, with no justification that I can see, that some one thing must have all the perfections at the same time.
There are so many things wrong with this that it's hard to know where to begin. First, of course, this really is just a fuzzy version of the Ontological Argument handled above. (In particular, see here.) Second, what about the "de gustibus" princible - "There's no accounting for taste"? In the case of beauty, people can, and do, disagree vehemently on what's more or less beautiful. Third, not all scales need to have a maximum. Consider temperature. There's definitely a coldest temperature (absolute zero) but no "hottest" temperature. (And certainly no one 'ideal' temperature - 'ideal' for what?)
Fourth - and most problematic - even if each 'scale of perfection' has a maximum, and some example of those maxima must (for some reason never spelled out) actually exist... why must all these maxima be instantiated by the same individual? Is there a logical reason why the "most beautiful" sandwich would also, by some necessity, be the "tastiest" sandwich? (Consider that, in food commercials, what's actually shown is very often not food at all, just something that looks appetizing - like motor oil instead of chocolate syrup, or Elmer's glue instead of milk.)