• Our understanding of a raspberry-flavoured marshmallow asteroid is a marshmallow than which no greater can be conceived.
• The idea of a raspberry-flavoured marshmallow asteroid exists in the mind.
• A marshmallow which exists both in the mind and in reality is greater than a marshmallow which exists only in the mind.
• If a raspberry-flavoured marshmallow asteroid only exists in the mind, then we can conceive of a greater raspberry-flavoured marshmallow asteroid—that which exists in reality.
• We cannot be imagining a marshmallow which is greater than a raspberry-flavoured marshmallow asteroid.
• Therefore, a raspberry-flavoured marshmallow asteroid exists.
BTW is there a way for us to pay the vast sum of $30 for adlessness without PayPaling?
sawellssays
Alternative angle of attack: “a being than which none greater can be conceived” makes exactly as much sense as “a rational number which is the square root of 2.” Just because the words are lined up in a row doesn’t mean they define a thing that exists.
Also, if I apply the argument to my bank account, does that make me a multi-billionaire? I shall try it on my bank manager, and report the result—hopefully from a tropical beach somewhere, using my brand new top of the range mobile device.
latveriandiplomatsays
The only problem that I have with this is that it seems to accept that the ontological argument is hard to find a flaw in. I think the difficult thing is more like this:
“But wouldn’t a God who could find a flaw in the ontological argument so convincing that even Alvin Plantinga would shut up about it be even greater?”
Though in fairness:
Critiques of ontological arguments begin with Gaunilo, a contemporary of St. Anselm. Perhaps the best known criticisms of ontological arguments are due to Immanuel Kant, in his Critique of Pure Reason. Most famously, Kant claims that ontological arguments are vitiated by their reliance upon the implicit assumption that “existence” is a predicate. However, as Bertrand Russell observed, it is much easier to be persuaded that ontological arguments are no good than it is to say exactly what is wrong with them. This helps to explain why ontological arguments have fascinated philosophers for almost a thousand years.
A mathematician colleague points out another flaw in the argument: namely, and I’m quoting here, it assumes that any set which is bounded above has a least upper bound. Apparently that turns out not to be the case.
antepreprosays
The flaw with the ontological argument is basically more common sense than logic: If you find someone arguing for the existence of something, and the best argument they can muster is that it simply must exist by definition, then you should immediately start becoming suspicious.
Saadsays
richardh, #7
I’m curious. Did they give a counterexample to that (the least upper bound property)?
woozysays
Daz @1:
As humorous as your argument is, it wouldn’t follow as our concept of a raspberry-flavored marshmallow asteroid is *not* such that it can’t be greater; it’s merely that it be marshmallow, raspberry flavor and an asteroid. It need not be in any sense good or great.
However any argument for anything perfect; a perfect sandwich, or (as Saturday Morning Breakfast Theorem noted) the perfect girlfriend would follow must exist.
I always countered “If a god who exists in conception is perfect , then does an elephant which exists in conception weigh several tons. How do we survive the encounter when we read Babar? We let our children hold the book.”
@4. But a rational square root of two can be conceived so it “exists in conception”. There is nothing in it’s definition that implies it ought to exist in actuality. Of course, just as I can conceive of “the deadliest joke in the world” I can also conceive of “a thought that is so horrible that if a single person conceived of it the world would end” and … oops… sorry everybody.
Obviously things “in concept” do not have the “principles in actualization” and it’d be absurd to think they would. Otherwise a thought of an elephant would weigh several tons. Seriously, that’s *really* obvious isn’t it? I can only assume wishful thinking and fear of being branded a heretic kept people from pointing it out from the beginning.
@7. “least upper bound property”. Heh, heh, that’s cute. But not really. But it is cute.
latveriandiplomatsays
@9. The real numbers less than 1 is an example. There is no “greatest real number that is less than 1”.
Also works for the rationals.
It’s not even clear to me that a well defined upper bound exists in this case. I’m not sure “maximally great” is well defined.
Saadsays
latveriandiplomat, #11
The real numbers less than 1 is an example
1 is the least upper bound of that set.
caselowerazsays
Woozy: I always countered “If a god who exists in conception is perfect , then does an elephant which exists in conception weigh several tons. How do we survive the encounter when we read Babar? We let our children hold the book.”
Exactly. Yesterday I was reading a book about a steam train heading right down the tracks toward someone, who I conceived of as myself. It was close, but i managed to call out “Break ideational bridge!” in the nick of time.
(Okay, that’s a reference to an obscure SF story in which technology that turns thoughts into reality exists. Probably amounts to a “me too” comment.)
If an ordered set S has the property that every nonempty subset of S having an upper bound also has a least upper bound, then S is said to have the least-upper-bound property. As noted above, the set ℝ of all real numbers has the least-upper-bound property. Similarly, the set ℤ of integers has the least-upper-bound property; if S is a nonempty subset of ℤ and there is some number n such that every element s of S is less than or equal to n, then there is a least upper bound u for S, an integer that is an upper bound for S and is less than or equal to every other upper bound for S. A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.
An example of a set that lacks the least-upper-bound property is ℚ, the set of rational numbers. Let S be the set of all rational numbers q such that q2 √2) or a member of S greater than p (if p < √2). Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.
As humorous as your argument is, it wouldn’t follow
Well that’s kinda the point. Nothing in the theistic ontological argument logically follows from the premises, either.
as our concept of a raspberry-flavored marshmallow asteroid is *not* such that it can’t be greater; it’s merely that it be marshmallow, raspberry flavor and an asteroid. It need not be in any sense good or great.
My argument is a parallel to that made by the theistic version. I set out my personal definition of what the greatest possible marshmallow would be (it would be an asteroid, and raspberry-flavoured), and then ‘prove’ that it exists, by the exact same ‘reasoning’ that the original uses.
Stripped of the silliness of marshmallows, I’m basically saying that if we allow the flawed logic, the argument says that any definition of ‘maximally greatest’ anything can be ‘proved’ to exist—even to obviously silly or mutually contradictory definitions of ‘maximally great.’
Akira MacKenziesays
Ugh, the ontological argument. I could never understand it, much less take it seriously. So what if I can imagine something greater than something else? It’s the sort of argument I’d expect to hear from a college stoner after their fifth bong.
I fear it’s navel gazing crap like this is what puts a lot of otherwise rational people off of philosophy.
woozysays
@9
Putting the numbers in alphabetical order. Or ordering the complex numbers by comparing first their real components and then, if equal, their complex components.
Actually far simpler is if “the universe” is the rationals, than all rationals whose squares are less than 2 is bounded above by any rational whose square is greater than or equal two but there is no least such rational. However the rationals can be extended into the reals which, almost by fiat, must certainly does have the least upper bound property.
I don’t think this argument actually *does* assume the least upper bound property. It assumes that by def. God has no greater, and that actual existence is greater mere conception. Yes, god is an upper bound and, one presumes, a least upper bound but the argument refuses to accept a universal set in which God doesn’t exist, rather than assumes a property that allows his existence.
Damnit, Bork Quotia shall receive the appropriate sacrifice of a suitably charred HTML For Dummies book, immediately.
latveriandiplomatsays
@12: You’re right, sorry. A correct example would be rational numbers whose square is less than two. There is no upper bound to that set of rationals, in the rationals.
twas brillig (stevem)says
There is no “greatest real number that is less than 1″
.99 > .9
.999 > .99
.9999 > .999
…
.9999999999999… = 1 ah, so. You got me. There are an infinite number of ‘reals less than one’, so there is no “greatest real less than one” My proposed counterexample just shot me in my foot. [limping away]
latveriandiplomatsays
@19: no least upper bound. Argh.
woozysays
Daz @15.
I’m too lazy to search but Saturday Morning Breakfast Cereal has a Bishop Anselm’s Girlfriend cartoon. “My perfect girlfriend is perfect so she totally exists (or else she wouldn’t be perfect) and we totally do it all the time”.
“Maximally great” is silly but I think Anselm would argue that “greatness” is a requirement for God, but not for marshmallow asteroids or even for girlfriends. It’s a dumb argument because bootstrapping existence by fiat is well… dumb.
Nick Gottssays
The real numbers less than 1 is an example. There is no “greatest real number that is less than 1″. – latveriandiplomat
No, but that’s not the definition of a “least upper bound”, which depends on the superset you’re considering – see #14. In your example, the set has no upper bound within the set at all (no member that is as great or greater than any member), but it does have a least upper bound within the reals: 1. I think richardh’s mathematician friend hasn’t got the terminology quite right. Maybe the simple point that sets with an ordering defined on them don’t always have a greatest member – see sawells@4 – is nearer what’s wanted; or the fact that mathematicians can always come up with an infinite cardinal larger than those produced by any proposed schema.
scienceavengersays
@16 Seconded. “Ontological” has been so often misused by speakers in my presence that just the mention of the word gets my eyes rolling.
consciousness razorsays
woozy, #10:
Obviously things “in concept” do not have the “principles in actualization” and it’d be absurd to think they would. Otherwise a thought of an elephant would weigh several tons. Seriously, that’s *really* obvious isn’t it? I can only assume wishful thinking and fear of being branded a heretic kept people from pointing it out from the beginning.
They’re both silly and ill-motivated, but this has almost no relationship to ontological arguments, whether we’re talking about those given by Anselm, Descartes, Plantinga or anyone else. They did think there is a being which must exist, but they didn’t argue their concepts of something have the properties of that thing, like weight or whatever. They, for example, didn’t argue their concept of this being was the greatest concept ever, but instead that what they were conceiving was the greatest being.
Leaving aside the superfluous crap about greatness and so forth, they basically thought there was a contradiction involved claiming that “the thing which must exist doesn’t exist.” That’s what a fool would say. However, you can say that there is no “thing which must exist” without contradiction, so the argument isn’t convincing to non-foolish people. They were wrong about that.
If you would admit that is contradictory, or you’re simply failing to deal with that by arguing against other stuff which isn’t even on the table, then you don’t have an effective response to the argument. Then, they will keep blathering about it and distracting us from real issues, and we don’t want that.
Saadsays
woozy, #17; latveriandiplomat, #19
Ah, didn’t think along those lines. Thanks.
woozysays
Real numbers have the least upper bound property.
I is not less than 1 but it *is* the least number that is larger than all possible numbers that are less than 1. So 1 is the least upper bound of the set of all numbers less than 1.
But *every* set of real numbers that is bounded above *does* have a least upper bound. This is how the real numbers were constructed.
The Rationals do *not* have the least upper bound property. The set of rationals less than 1 has a least upper bound but the set of rationals whose square is less than two does not have a least upper bound within the rationals. Within the reals the set of rationals whose square is less than two *does* have a least upper bound. It is the square root of 2.
brucegee1962says
Here’s my case against the ontological argument:
Imagine a perfectly round circle, or a perfectly straight line, or an angle with precisely 90 degrees, or a point with 0 dimensions.
Now imagine any of those things existing in the real world.
It’s quite easy to do the former — in fact, the tools of math and geometry exist to allow us to reach all kinds of conclusions about these things.
But we cannot imagine any of them existing in reality. Every circle drawn will have at least some deviation, at least at the quantum level.
Therefore we can conclude that one of the qualities of any perfect being is non-existence anywhere outside the world of ideas.
latveriandiplomatsays
@20. Actually the least upper bound property is more complex than what I said, I blew it. My example says a set bounded above doesn’t have to contain its own upper bound (true for dense sets like the rationals and reals, as you demonstrated).
The original statement was that a bounded above subset of a set A doesn’t necessarily have a least upper bound in A.
The bounded above subsets of the rational numbers don’t necessarily have least upper bound in the rational numbers.
The bounded above subsets of the real numbers do have least upper bounds in the reals, this is a critical property of the reals.
All of the above, assumes the usual ordering (<).
woozysays
consciousness razor:
The common counter-argument is that these ontological argument pre-suppose existence. Anselm directly avoids this by pointing out as his second proposition “God exists as an idea in the mind”. He then presumes existence in the mind requires the same definitions and properties as existing in actuality, namely that the God that exists in the mind is “that which no greater can be imagined”.
A Masked Avengersays
Disclaimer: I’m a mathematician, which means I live in the world of “naïve set theory,” which is another way of saying that philosophers routinely dismiss my kind as misguided souls. That said…
The flaw in the ontological argument is quite clear, and it doesn’t involve “existence not being a predicate.” The flaw is that the definition merely tells you that if there’s a god, then in particular it must exist, else it would fail the “greatness” part of the definition. The argument doesn’t tell you whether said god exists; merely that a nonexistent one fails the test of godhood, for the reasons outlined in the argument.
A mathematician will probably point out that “greatness” is not well enough defined to use in definitions or proofs, though. Why is existence required for “greatness”? Why isn’t “stays crispy in milk, and is part of this complete breakfast” also part of the definition? I.e., why does our argument fail to prove that God is a tasty breakfast food? After all, a god who meets all the other aspects of the definition and is a tasty breakfast food, clearly goes the alternative one better…
I second the mathematician quoted above, though: it is also unclear whether beings, ordered by greatness, actually have the least upper bound property. I’d respond that that’s a nit, though, since we have reason to believe that the collection of all beings is finite, and so necessarily has the least upper bound property. For the same reason, there is theoretically a fastest human and a smartest human–which brings us back to the problem of defining “greatness.” Lack of precise measurement might prevent us from defining an ordering at all. We routinely pair the same two humans in a race and have different outcomes due to a variety of factors, so it’s difficult to define “fastest” in such a way as to get unambiguous results.
Bringing that back to the ontological argument, we might discover that not only does Yahweh exist, but all the gods of Olympus, and we might find that it’s difficult to sort them by greatness. Does “able to kick the other one’s ass” suffice? And if so, does one contest suffice, or should it be a best of 7 kind of deal? Do they need to fight to the death? What if both are unkillable?
woozysays
Upper bounds:
Part of the problem is we modern kids in grade school learn about the Reals as an absolute. It’s true that the rationals don’t have the least upper bound property and the square root of two is not in the rationals, but if I give this as a counter-example it’s reasonable that the intuitive response is “yeah, but the rationals aren’t *everything*; just toss in the square root of two”. So it’s a poor teaching example, in that it makes it seem mathematicians are just playing with words to make us look dumb.
There are no real number examples of (above) bounded sets without a least upper bounds because real numbers have the “least upper bound property” (i.e. by definition all bounded sets do have a least upper bound). It’s a fundamental theorem of analysis that such a larger set containing the rationals would exist and the reals were constructed precisely for this property. This is why in grade school we are taught that the reals are “like the rationals but with all the holes filled in”.
Thus we have to look elsewhere for Saad’s counter example. A common thing is to define a different type of order. My personal favorite is the p-adic system where the “size” of a number is inversely proportional to its divisibility by a specific prime number p. But this is a wee bit abstract. (The numbers 75 and 25 would both be of size 1/25 because 5^2 divides them whereas 15 would be a larger size 1/5 as 5^1 divides it. Meanwhile .4 would be of even larger size 5 because 5^-1 divides it. So .4 > 1 > 15 > 75. Confused?)
The complex numbers meanwhile do not have a good order. (I is i greater or less than 1, -1, or -i? What does that question even mean?) We can define a pretty intuitive “dictionary order” by comparing the real components, and if equal, than comparing the imaginary components. In is way 5 + 2i > 3 + 4i but 5 + 3i > 5 + 2i and by associativity 5 + 3i > 3 + 4i. It works and satisfies all the requirements of order.
But it doesn’t have the least upper bound property. For example:
The set of all 4 + xi; The set of all numbers where the real component is 4 and the imaginary component could be anything. 4.1 > 4 + xi for all x. 4.000001 > 4 + xi for all x. So the set has many, many upper bounds. But it doesn’t have a single *least* upper bound. (Any real number greater than 4 would be an upper bound. But 4, itself is not an upper bound as 4 + i > 4, but 4.00000000001 > 4 + i > 4.)
consciousness razorsays
woozy:
I don’t understand. Is it contradictory or not to claim “there is no being-which-must-exist”? If you allow that it’s contradictory, they can very easily fix anything else in their methods of communicating the issue, which seem only like linguistic ambiguities to me. It does not hinge on definitions of greatness or other adjectives, concepts necessarily having all the properties of their referents, or things like that. Correcting trivial things like that would not force us on pain of irrationality to accept the existence of a god. What is really central to it is the charge that atheism is internally inconsistent. The rest is window dressing, which you may separately criticize if you like, but it won’t get you anywhere with the best and clearest form of the argument. If you agree that it’s false that atheism is inconsistent or irrational on those grounds, then I just do not see how that is being demonstrated. You should confront the logical point that they’re attempting (however badly) to make, which isn’t hard, and it doesn’t need to be any more complicated than that.
woozysays
conciousness razor:
I’m not sure I understand. No, it is not contradictory to claim “there is no being-which-must-exist” (Or more accurately to claim “there is no need to assume there must be a being-which-must-exist”) Do you have a specific ontological argument in mind? I have Anselm’s argument in mind. His fault is *not* that it presupposed god’s existence. He specifically gets around that by saying God exists as a thought, which, well, duh. *BUT* either it does exists as a thought that is consistent with its definition or it exists as a thought that is inconsistent with its definition. If it’s inconsistent with its definition the argument goes nowhere. Elephants exist as thoughts inconsistent with their definitions. (The thoughts of elephants don’t have weight.) He has failed to demonstrate that God exists as a thought consistent with definition. My thoughts of God could be bigger; his thoughts of God could be bigger; Perhaps the pope’s thoughts of God can not be bigger because the pope is divinely inspired but he has not adequately addressed that. Thus no, he has failed to show God, consistent with the definition, exists in the mind.
Other ontological arguments may have other faults but this is the one I see with Anselm’s.
llamaherdersays
God is the awesomest thing imaginable, and God would be even awesomer if he existed in reality.
Therefore, if you think about God, he pops into existence.
This is some next-level Secreteering Law of Attraction bullshit.
brettsays
I think of it as the Most Perfect Apple argument. If I imagine the most perfect apple, then it must exist because if it didn’t exist, it would be possible to think of a more perfect apple.
screechymonkeysays
I was always partial to the Ontological Argument for Atheism:
1. Imagine the most perfect argument for atheism.
2. The most perfect argument for atheism must be valid, sound, persuasive, and utterly irrefutable, or else it would not be perfect.
3. Therefore, there exists a valid, sound, persuasive, and utterly irrefutable argument for atheism.
woozysays
@37
I’m fond of the Raymond Smullyen “A living unicorn” exists argument. It consists of a single line: “A living unicorn exists”. …. Think about it.
“Maximally great” is silly but I think Anselm would argue that “greatness” is a requirement for God, but not for marshmallow asteroids or even for girlfriends.
That’s Anselm’s problem, not mine. I see nothing in the ontological argument which demands that it should only be applied to the class of beings known as gods, or which precludes its application to other things. That’s the main point of the marshmallow asteroid, or indeed the ‘greatest possible bank account’ argument. They point out how absurd the reasoning is by showing, without need to address the flawed internal logic, that it leads to absurd conclusions unless arbitrary constrictions are placed on the initial premise.
It’s a dumb argument because bootstrapping existence by fiat is well… dumb.
On this, we agree. ‘Anything you can imagine, I can imagine better,’ is a silly basis for an argument.
woozysays
… He also had the “If I not mistaken, then…” argument. Well, if you aren’t mistaken then *of course*.
woozysays
@39
Oh, we agree on *everything*.
I was just nitpicking. The concept of “greatness” and “things are greater if they exist” and “things exist as ideas” are key to Anselm’s argument. To make a silly counter argument you need to include those which my nitpicky critique of your original silliness I claim your marshmallow asteroid didn’t address. Although it *could*… but then it loses its punch.
I like the “perfect apple” or “perfect girlfriend” arguments slightly better (the “perfect girlfriend” adds an air of geekish loser desperation as a bonus).
consciousness razorsays
I’m not sure I understand. No, it is not contradictory to claim “there is no being-which-must-exist” (Or more accurately to claim “there is no need to assume there must be a being-which-must-exist”)
That isn’t more accurate. It’s confusing. I don’t care at the moment what we do or don’t need to assume. There’s no reason to go down that rabbit hole. What are the facts about reality, and how can someone coherently describe those, as distinct from their methodology? I think it is true that there is no being-which-must exist. That is very plainly what I think. If it’s true, that’s simply a fact about the category of real things: the category doesn’t contain any of the beings-which-must-exist while it does contain other sorts of existing stuff.
That fact doesn’t concern what we need to do (or should do) about anything. It is simply a way to describe what there really is. Can that be consistently done without including beings-which-must-exist or necessary beings or anything of the sort? Yes, as I’ve said, it certainly can, and that is how the argument fails, since it was intended to make precisely that point about what there is.
Suppose somebody does think presupposing a god (or whatever) is acceptable. The best you can say is that we don’t need their methodology? That’s pretty silly. You could very well say that they have a false belief: that a belief in a being-which-must-exist is a belief about something which doesn’t exist. Whatever methodology they used or however they may have come to that, it is a further issue that it’s a false belief (or that it’s a true belief, if that were the case). That is certainly relevant, this being an argument about ontology and all, not simply an exercise in epistemology.
tulsesays
If a being must exist, then doesn’t that mean the being is not omnipotent? Heck, it makes the being less powerful than me, since I could make myself not exist.
woozysays
That (““there is no need to assume there must be a being-which-must-exist”) isn’t more accurate. It’s confusing. I don’t care at the moment what we do or don’t need to assume.
I was saying that the refuter may not know whether or not that-which-must-exists exists or not. The burden is on the arguer to argue that that-which-must-exist actually must exist. An agnostic should be able to refute a bad argument just as well as an atheist.
Suppose somebody does think presupposing a god (or whatever) is acceptable. The best you can say is that we don’t need their methodology? That’s pretty silly.
No. I’m saying their methodology is wrong.
You could very well say that they have a false belief: that a belief in a being-which-must-exist is a belief about something which doesn’t exist.
But that’s *not* what a refuter is saying. By pointing out the logical fallacy (and it’s *huge*) of Anselm’s argument the refuter hasn’t demonstrated anything whatsoever about the actual existence of gods or whatever; s/he’s merely demonstrated that Anselm hasn’t either.
Anselm’s proof is invalid. Anselm’s belief … has not been addressed in any way shape or form.
Kant (if @6 source is accurate which I assume it is; I’ve never read Kant or studied philosophy) argues that most ontological argument pre-suppose. That’s the case with Anselm (although he, invalidly, talks his way out of this). But it need not always be the case. Ontological arguments try to argue an existence is a tautology but although many fail because they pre-suppose that doesn’t mean all will. (“I think, therefore I am” is actually a valid ontological argument; so is Godel’s incompleteness theorem.) I won’t say *all* invalid ontological arguments abuse a linguistic descriptive/prescriptive ambiguity (which can always be created a la Godel [that-which-must-exist by definition must exist; ta-da!]) but most do. Anselm definitely does. Descartes’ proof of God is a bit more subtle and I’m not entirely familiar with it. I suspect it does.
David Marjanovićsays
O hai, tulse! I maded you an internet out of lavender cookies. And I did not eated it.
consciousness razorsays
An agnostic should be able to refute a bad argument just as well as an atheist.
While you don’t support this, even if that’s so, I don’t care what agnostics should be able to do (or how you think arguments should proceed, in a way that makes you happy). Again, the argument is about claiming in fact there really is a god. If you’re not willing to talk about what there in fact really is, then that’s your problem. I don’t think there is any controversy in saying your willingness isn’t relevant to the facts about reality.
No. I’m saying their methodology is wrong.
Yet the conclusion they fumbled toward happens to be right? Or is it not right?
But that’s *not* what a refuter is saying.
But I am saying that: there isn’t a god. So, are you claiming here that I’m wrong to do that? I think that’s the truth, and the alternative is that it is false. If I’m wrong, then I’m wrong, but as a fact about reality it’s either one or the other, not both or neither or something else. Indeed, admitting that I may be wrong is something that I’m willing to do and be very explicit about. It seems like you’re avoiding that with these fence-sitting shenanigans about agnosticism, while at the same time ironically claiming to have some kind of methodological high ground where presuppositions are ostensibly a bad thing (but only when others do it). But it isn’t even clear what that high ground is supposed to be or why we should be going there in the argument in the first place.
I don’t know what else to say. If you really want to go into all of the hairy details, you can check out J.L. Mackie’s book on the broader subject of arguments for gods, Miracle of Theism. It’s all worth reading, but OAs specifically are discussed in pages 41-63, where he talks about Descartes, Anselm and Plantinga. As I tried to explain, there is a fairly simple reason why they all fail. You’re apparently not accepting that at the moment, and it’s not at all clear what you think you have as a replacement. Even if it works somehow, it certainly doesn’t look like an improvement.
I actually have an ongoing series on ontological arguments, which may interest anyone who wants to dig into the logic.
IMHO, the whole “maximally great” thing is stupid. Why bother defining God as a being which is maximally great? Maximal greatness specifies a lot of different properties for God, and the ontological argument only uses one property: existence. You might as well define God as an entity which exists and be done with it.
The reason ontological arguments don’t do this is because then it would be too obvious that you can replace God with any old object. And then people would realize the argument is stupid.
zetopansays
I can conceive of a greatest universe which is greater than the greatest god, and unlike the latter it exists.
Partially addressing arguments between @Consciousness Razor #46 and Woozy #44
Most ontological arguments have the interesting property that if God happened to exist, then the argument is sound. “Sound” in the sense that the premises are true and the reasoning is valid. God exists. Therefore God exists. Sound!
It logically follows that if the ontological arguments are unsound, then God does not exist. So if I haven’t absolutely proven that God doesn’t exist, then does that mean I haven’t entirely refuted the ontological argument yet? This seems like too high a burden of proof.
While the probability of God is exceedingly small, the probability that the ontological argument is any good is exactly nil. I can say this because even if the argument were sound, it would still be terrible. As Woozy said, the methodology is wrong.
twas brillig (stevem)says
sidetrack:
summary of ontological argument:
1) God is, by definition, a perfect being. (our definition of God is, “perfect being”)
2) non-existence is a flaw
3) perfect = flawless (i.e. without a single flaw)
4) therefore God exists
if that summarizes ontological argument, that seems to be circular logic. Line 1 defines God as already being existant, so the further lines are superfluous, “begging the question”, etc.
That looks too easy, so my summary has to be wrong. Please correct my ontological wrongness.
woozysays
@46
Um, what are you talking about? I thought we were specifically discussing the ontological argument that god exists (because by definition god is the greatest and existing is greater than non-existing). Kant points out one fault (and there are others) is that this presupposes its conclusion. You claimed this isn’t relevant. Of course, it’s relevant. It’s the entire gyst of the argument; a greatest of all beings exists a priori. And that’s the fallacy of the argument.
I’m not trying to argue completely from ground up that god does not exist. I am simply analyzing this one very very specific argument. And my analysis stands that it is fallacious in that in confuses descriptive properties of God as a concept with prescriptive (god would be the greatest thing ever) requires with god as existent (the greatest thing ever must exist). I do not care about any other argument and I am not making any argument of my own. I’m merely examing this one argument before us. What the hell are you doing?
woozysays
@50. The trick/delusion/fallacy is to bulster the concept of a God to nescessity. Consciousness Razor was correct in that the goal is to present God existence as a logical tautology; where god not existing is a logical contradiction. Anselm does it by claiming God exists already as a concept and now that he exists as a concept … Descartes punts and argues that there is a hierarchy of non-perfect to more perfect hierarchies. The hierarchy must have an upper limit and that upper limit must be perfect else it wouldn’t be an upper limit. Yes, they are all circular but the hide/waffle/get lost in other misleading arguments of whether truth exists and can be implied simply by a sentence existing. It sounds like “tricks” but ontology we can’t dismiss everything as “mere semantics” but the actual lines of were semantics end and true logic begin are vague and subtle and weren’t really resolved until the 20th century.
But no. God exists because his definition requires it really is circular as is the liars paradox and in analysis both are very similar in that the issue is where the meaning of the concepts actually lie (and that the arguments usually punt the actual concept out of the park).
consciousness razorsays
Kant points out one fault (and there are others) is that this presupposes its conclusion. You claimed this isn’t relevant.
What Kant actually said boils down to this: “existence isn’t a predicate.” It’s not some generic hand-waving crap about presupposing or begging the question, and how that’s a bad thing to do. So what does that mean?
Someone makes an ontological argument: we’re supposedly not logically allowed to say “X doesn’t exist,” because X itself involves existence. That is what can’t be consistently done, according to the argument.
That’s a stupid argument, but you have to say more than “it’s stupid.” You have to explain what’s stupid about it. So what do you do?
Kant said existence is not a proper predicate. It’s just putting some X on the table for discussion as it were, but it doesn’t describe X in any substantial way like a normal predicate should. That’s one thing to say, but it obviously bears no resemblance to anything you’ve been saying. This is apparently news to you, since you’ve said you haven’t read any Kant, but you can verify that yourself if you like.
You can also say that even if existence counted as some kind of a predicate (maybe a very unique one unlike any other predicate), there’s still no contradiction. We don’t even need to make it as complicated as Kant did — much less whatever the hell you’re doing ignorantly and off the cuff. You can clearly separate X’s internal references to “existence” or “necessary being” (if some theist insists that’s what they mean by a god) from any claims made about whether nor there is an X like that in reality. In other words, there isn’t an X like that. (And since they are claiming things about reality, that is what you should be engaged with here, it hardly needs to be said.) There was this highly misleading idea about how to formally structure arguments and what parts of them entailed about each other, but that can be cleared up very quickly and easily, without introducing anything new.
@twas brillig #50,
No, Anselm’s ontological argument is pretty obtuse, you got that. But it’s sort of like arguing over Zeno’s paradox, or whether 0.9999… = 1, or whether 1 + 2 + 3 + … = -1/12. Ultimately pointless but we do it because it’s fun.
And in case you start thinking it’s too easy, you can move onto other ontological arguments. Lots of people in this thread are talking about the conceptual ontological argument, which I state as charitably as possible:
1) As long as a predicate has no contradictions, you can conceive an object with that predicate.
2) Let G(y) be the predicate “y is a perfect being”. G has no contradictions.
3) Therefore, I can conceive of an object x such that G(x).
4) The definition of G implies existence outside of our minds.
5) Therefore, x exists.
Over the years I’ve encountered only a few people who actually support ontological arguments. These mythical creatures are mostly harmless. Except for Alvin Plantinga, who for some reason is considered a serious philosopher. He uses a different version of the ontological argument based on modal logic. It’s also really obtuse, but you’d need to understand modal logic to pinpoint why.
A Masked Avengersays
“Existence is not a logical predicate” is an asinine statement: it’s possible to conclude that things exist. Mathematicians do it all the time, for some value of “exist.”
The flaw here is that defining something doesn’t make it so. I can articulate the definition of “the being that by its very nature can’t not exist,” and if I’m careful, I can do it without any paradoxes or self-contradictions–in other words, I can set up the definition so that it’s perfectly valid. When I’m done, I’m no wiser about whether this thing actually does exist; all I know is that if it does, then it must be whatever my definition says. It’s perfectly coherent, logically, to say, “The being that must by its very nature exist, does not, in fact, exist.”
You can’t bootstrap a definition into an existence proof. And note that every definition has, at its heart, a word whose meaning you don’t–and never can–know. In mathematics, that word is “set.” We know one when we see one, most of the time, but we don’t know what one is, because when we try to define it we must use words, which themselves require definitions, which in turn use words which require definitions, and so on. This would be an infinite regress, except it isn’t, because there are only finitely many words in existence. So sooner or later, we’ll get around to using the word “set” again in one of our definitions, and our definition will turn out to be circular.
The “exists in my imagination” part is another absurdity: it most certainly does not exist in your imagination, because you just said that a real one is better than an imaginary one, so the imaginary one fails the requirement of being the awesomest…
…or, alternately, it could turn out that the real one doesn’t exist, and therefore the one in your imagination really is in fact the awesomest, which means it is God, and you’ve proven that God is imaginary. (Other formulations of the proof lack this particular absurdity, because it doesn’t assert that God exists in imagination, in the same breath as it asserts that an imaginary God can’t be the greatest, and therefore God doesn’t exist in imagination at all.)
Crip Dyke, Right Reverend Feminist FuckToy of Death & Her Handmaidensays
@Brett:
I think of it as the Most Perfect Apple argument. If I imagine the most perfect apple, then it must exist because if it didn’t exist, it would be possible to think of a more perfect apple.
Actually this is almost my proof that Allah exists as a god separate from Yahweh separate from Jehovah separate from Hanuman, etc.
Imagine the most perfect apple. It must not only exist, but it must be in my mouth right now, coating my tongue with deliciousness and providing me with good nutrition. It must be MY apple. Because it would definitely be better if it were the most perfect apple AND mine! And in my mouth right now!
Likewise, the most perfect being isn’t merely truly awesomely scary and powerful to anyone. The most perfect being also personally takes my side in arguments and delights in feeding me fresh kiwi. Because hell, if the most perfect being wasn’t catering to my wishes when they conflict with the wishes of any other persons in the universe, then I could imagine a being a little more perfect if you know what I mean.
Kalam == the proof that for every imagination, there’s a god who can beat up all the gods of any one else’s imagination.
There’s a very special name for a proof that effective.
Azkyroth, B*Cos[F(u)]==Ysays
I prefer the Argumentum ad Veshkorg:
“In response, I submit the Veshkorg, which I define to be a flying predator which has speed, strength, sensory accuity, and ravenous hunger for the flesh of slow-witted, inattentive, self-absorbed motorists such that nothing more so can be conceived. But all of those properties would be intensified if the creature actually existed, so therefore it must exist! …I will let you know if my commute improves.”
Azkyroth, B*Cos[F(u)]==Ysays
My personal favorite is the p-adic system where the “size” of a number is inversely proportional to its divisibility by a specific prime number p. But this is a wee bit abstract. (The numbers 75 and 25 would both be of size 1/25 because 5^2 divides them whereas 15 would be a larger size 1/5 as 5^1 divides it. Meanwhile .4 would be of even larger size 5 because 5^-1 divides it. So .4 > 1 > 15 > 75. Confused?)
….isn’t the number line a special case of p-adic system with p=1?
Azkyroth, B*Cos[F(u)]==Ysays
Other thoughts:
Just because the words are lined up in a row doesn’t mean they define a thing that exists.
I felt a great disturbance, as if 90% of the formal “philosophy” I’ve ever encountered threw itself on the ground and held its breath while kicking and pounding its fists, and then was suddenly silenced.
If a being must exist, then doesn’t that mean the being is not omnipotent? Heck, it makes the being less powerful than me, since I could make myself not exist.
Stolen.
We don’t even need to make it as complicated as Kant did — much less whatever the hell you’re doing ignorantly and off the cuff.
Frankly, woozy’s making a lot more sense.
Nick Gottssays
“Existence is not a logical predicate” is an asinine statement: it’s possible to conclude that things exist. – A Masked Avenger@55
That doesn’t make existence a predicate. The usual way of expressing “God exists” in predicate calculus would be:
∃x: G(x)
where G symbolises the predicater Godlikeness. “∃” is usually read “There exists”, and is a quantifier, so the whole thing is read as “There exists an x such that x is Godlike.” Most modern defenders of the ontological argument (Norman Malcolm, Charles Hartshorne, Alvin Plantinga) argue that God has a special sort of existence, “necessary existence”, that is a predicate, so there!
Anrisays
It seems to me that the easiest way to defeat this argument is to listen to it, nod, wait a bit for god to manifest, then tilt your head quizzically and say “Hunh. Looks like you got your definition of god wrong. Try again.”
A Masked Avengersays
Nick,
…so the whole thing is read as “There exists an x such that x is Godlike.”
It would help if you defined “predicate” here. As a working mathematician, I never have recourse to Russell and Whitehead, and never attempt to formulate my proofs in rigorous first-order logic–which is itself formally studied more by philosophers than mathematicians. See my disclaimer in my first post.
If I understand your statement, it is that a predicate, strictly speaking, maps entities to truth values, and the domain of a predicate is therefore entities whose existence is already a given. Hence a predicate can only be applied to God if God first exists as one of the entities in the domain.
If I have that right then OK, fair enough. It’s consistent with my experience that “existence” proofs do, in fact, work that way: the elements of the domain already do exist, and the question is whether any of them meet the criteria. Saying “a solution exists” is the same as saying, “one of the points in this space qualifies as a solution.”
If you reformat the ontological proof thusly, you get, “Of all the beings, at least one is godlike.” I’ll cheerfully grant that by definition at most one can be godlike, so uniqueness comes for free; “existence” becomes the problem. But the difficulty of the problem becomes obvious, since there is no way to get any traction–since most of the beings in the universe are unknown to us, and there’s no purely logical method of deducing that the smartest one, say, must necessarily also be the most altruistic one. I like that. It exposes the fundamental difficulty: we know jack shit about what beings do or don’t exist in the universe, and navel-gazing will not provide a method of backing into that knowledge.
Nick Gottssays
A Masked Avenger@62
If I understand your statement, it is that a predicate, strictly speaking, maps entities to truth values, and the domain of a predicate is therefore entities whose existence is already a given. Hence a predicate can only be applied to God if God first exists as one of the entities in the domain.
The first bit certainly: a predicate maps entities onto truth values; and I think that if a predicate is applied to a constant, that implies that what the constant refers to, exists. But you can perfectly well say:
~∃x: G(x)
“There exists no x such that x is Godlike”. Of course at the time the ontological argument was formulated, they were still working with Aristotelian logic, in which there was no clear notion of quantifiers as logical operators. I’m no expert on the history of logic, but my hunch is that it was this lack of clarity which made the ontological argument plausible (to some people – others, like Gaunilo, who essentially came up with Daz’s argument @1, could see it was hogwash). The guys I referred to earlier try to use modal logics (which are notoriously tricky) to resuscitate the argument, but always with some sleight of hand – Plantinga defines “God” as a being who exists in all possible worlds (this being an aspect of his greatness), then works a switch between epistemic possibility (“for all we know, there is such a being”), and logical possibility (“there is some logically possible world in which such a being exists”) in the course of the argument.
Dr Marcus Hill Ph.D. (arguing from his own authority)says
I think all the LUB arguments miss the point a bit. It doesn’t matter a jot if the set of all things with the greatness measure have the LUB property if we don’t first show that the set is actually bounded. Why do we assume there is some theoretical maximum greatness? Anselm considered this sort of “perfection”, but is there any reason to believe that whatever one-dimensional measure of “greatness” we’re using has a theoretical maximum (and remember, according to the argument, there must be such a theoretical maximum as it defines the being in question)? We’re not doing something akin to asking either for the LUB of a set of rationals or reals, we’re potentially asking for the greatest real number.
It’s like a couple of kids arguing:
“OK, so you’ve imagined a great being – what’s its greatness score?”
“INFINITY!”
“Did you add existence to your being?”
“Uh, no.”
“AHA! Well, if I get your being and add existence, then its greatness score is INFINITY PLUS ONE! Checkmate, atheists!”
Athywren, Social Justice Weretribblesays
So the ontological argument is the one that states that, if god exists, then god exists, therefore god exists, right? I mean, basically, when you get past the densely packed logick.
I kind of love that argument. Not because it’s particularly impressive or convincing, but because it’s reassuring. They’ve had thousands of years to come up with an argument – their god has had thousands of years to give a shred of evidence – and one of the best they can come up with is just a linguistic game of twister. I try to maintain a healthy uncertainty on all topics, just because I know how easily skepticism can turn into a mere title if you don’t, but these arguments… I honestly don’t think I need to worry about gods. At least not gods who care what we believe. Their believers so clearly have nothing of substance.
A Masked Avengersays
So the ontological argument is the one that states that, if god exists, then god exists, therefore god exists, right? I mean, basically, when you get past the densely packed logick.
Yes, that’s how I read it.
sawellssays
Another way of looking at it is this: we can grant that the ontological argument establishes that, if a god exists, then one of its properties is… existence. The sneaky bit is the claim “God exists as an idea”, which is flat wrong. The idea of Anselm’s God might exist in Anselm’s mind – but the idea is not the God, and the argument fails.
woozysays
@58
….isn’t the number line a special case of p-adic system with p=1?
Mmmmm…. I don’t think so. The p-adic norm of a rational q would be 1/p^n where is the highest integer (pos or neg) where p^n divides q. Thus the 1-adic norm would be 1 for all numbers.
Then again it’s been a loooong time since I’ve given this any thought and …. well, although I was well versed in it at one time I’m afraid that is no longer the case.
@64
I think all the LUB arguments miss the point a bit. It doesn’t matter a jot if the set of all things with the greatness measure have the LUB property if we don’t first show that the set is actually bounded.
I think the original claim might have been referring to Descartes proof specifically. Anselm seems to believe all sets are bounded … by INFINITY!
My problem with pointing out the mathematical errors in these proofs, is that the subtleties and rigor involved in understanding the math is exponentially harder and more subtle then the direct logic itself.
“AHA! Well, if I get your being and add existence, then its greatness score is INFINITY PLUS ONE! Checkmate, atheists!”
Anselm defines God as that which can’t be any greater, to which I can’t help thinking Superman holding God in his arms would be greater.
…or less flippantly what I imagine as God can easily be imagined as greater by someone smarter than me.
@54
2) Let G(y) be the predicate “y is a perfect being”. G has no contradictions.
I’ve always felt “a perfect being” was chock full of contradictions, myself, but that’s not the basic flaw I find in the argument.
3) Therefore, I can conceive of an object x such that G(x).
And yet, oddly, my concept of the perfect being is, in itself, not perfect. …. strange….
• Our understanding of a raspberry-flavoured marshmallow asteroid is a marshmallow than which no greater can be conceived.
• The idea of a raspberry-flavoured marshmallow asteroid exists in the mind.
• A marshmallow which exists both in the mind and in reality is greater than a marshmallow which exists only in the mind.
• If a raspberry-flavoured marshmallow asteroid only exists in the mind, then we can conceive of a greater raspberry-flavoured marshmallow asteroid—that which exists in reality.
• We cannot be imagining a marshmallow which is greater than a raspberry-flavoured marshmallow asteroid.
• Therefore, a raspberry-flavoured marshmallow asteroid exists.
Could a god make an ontological argument that even he couldn’t accept?
BTW is there a way for us to pay the vast sum of $30 for adlessness without PayPaling?
Alternative angle of attack: “a being than which none greater can be conceived” makes exactly as much sense as “a rational number which is the square root of 2.” Just because the words are lined up in a row doesn’t mean they define a thing that exists.
Also, if I apply the argument to my bank account, does that make me a multi-billionaire? I shall try it on my bank manager, and report the result—hopefully from a tropical beach somewhere, using my brand new top of the range mobile device.
The only problem that I have with this is that it seems to accept that the ontological argument is hard to find a flaw in. I think the difficult thing is more like this:
“But wouldn’t a God who could find a flaw in the ontological argument so convincing that even Alvin Plantinga would shut up about it be even greater?”
Though in fairness:
from:
http://plato.stanford.edu/entries/ontological-arguments/
A mathematician colleague points out another flaw in the argument: namely, and I’m quoting here, it assumes that any set which is bounded above has a least upper bound. Apparently that turns out not to be the case.
The flaw with the ontological argument is basically more common sense than logic: If you find someone arguing for the existence of something, and the best argument they can muster is that it simply must exist by definition, then you should immediately start becoming suspicious.
richardh, #7
I’m curious. Did they give a counterexample to that (the least upper bound property)?
Daz @1:
As humorous as your argument is, it wouldn’t follow as our concept of a raspberry-flavored marshmallow asteroid is *not* such that it can’t be greater; it’s merely that it be marshmallow, raspberry flavor and an asteroid. It need not be in any sense good or great.
However any argument for anything perfect; a perfect sandwich, or (as Saturday Morning Breakfast Theorem noted) the perfect girlfriend would follow must exist.
I always countered “If a god who exists in conception is perfect , then does an elephant which exists in conception weigh several tons. How do we survive the encounter when we read Babar? We let our children hold the book.”
@4. But a rational square root of two can be conceived so it “exists in conception”. There is nothing in it’s definition that implies it ought to exist in actuality. Of course, just as I can conceive of “the deadliest joke in the world” I can also conceive of “a thought that is so horrible that if a single person conceived of it the world would end” and … oops… sorry everybody.
Obviously things “in concept” do not have the “principles in actualization” and it’d be absurd to think they would. Otherwise a thought of an elephant would weigh several tons. Seriously, that’s *really* obvious isn’t it? I can only assume wishful thinking and fear of being branded a heretic kept people from pointing it out from the beginning.
@7. “least upper bound property”. Heh, heh, that’s cute. But not really. But it is cute.
@9. The real numbers less than 1 is an example. There is no “greatest real number that is less than 1”.
Also works for the rationals.
It’s not even clear to me that a well defined upper bound exists in this case. I’m not sure “maximally great” is well defined.
latveriandiplomat, #11
1 is the least upper bound of that set.
Woozy: I always countered “If a god who exists in conception is perfect , then does an elephant which exists in conception weigh several tons. How do we survive the encounter when we read Babar? We let our children hold the book.”
Exactly. Yesterday I was reading a book about a steam train heading right down the tracks toward someone, who I conceived of as myself. It was close, but i managed to call out “Break ideational bridge!” in the nick of time.
(Okay, that’s a reference to an obscure SF story in which technology that turns thoughts into reality exists. Probably amounts to a “me too” comment.)
richardh@7,
That’s not quite right, or at least, incomplete. According to wikipedia:
woozy #10:
Well that’s kinda the point. Nothing in the theistic ontological argument logically follows from the premises, either.
Ugh, the ontological argument. I could never understand it, much less take it seriously. So what if I can imagine something greater than something else? It’s the sort of argument I’d expect to hear from a college stoner after their fifth bong.
I fear it’s navel gazing crap like this is what puts a lot of otherwise rational people off of philosophy.
@9
Putting the numbers in alphabetical order. Or ordering the complex numbers by comparing first their real components and then, if equal, their complex components.
Actually far simpler is if “the universe” is the rationals, than all rationals whose squares are less than 2 is bounded above by any rational whose square is greater than or equal two but there is no least such rational. However the rationals can be extended into the reals which, almost by fiat, must certainly does have the least upper bound property.
I don’t think this argument actually *does* assume the least upper bound property. It assumes that by def. God has no greater, and that actual existence is greater mere conception. Yes, god is an upper bound and, one presumes, a least upper bound but the argument refuses to accept a universal set in which God doesn’t exist, rather than assumes a property that allows his existence.
Damnit, Bork Quotia shall receive the appropriate sacrifice of a suitably charred HTML For Dummies book, immediately.
@12: You’re right, sorry. A correct example would be rational numbers whose square is less than two. There is no upper bound to that set of rationals, in the rationals.
There is no “greatest real number that is less than 1″
.99 > .9
.999 > .99
.9999 > .999
…
.9999999999999… = 1 ah, so. You got me. There are an infinite number of ‘reals less than one’, so there is no “greatest real less than one” My proposed counterexample just shot me in my foot. [limping away]
@19: no least upper bound. Argh.
Daz @15.
I’m too lazy to search but Saturday Morning Breakfast Cereal has a Bishop Anselm’s Girlfriend cartoon. “My perfect girlfriend is perfect so she totally exists (or else she wouldn’t be perfect) and we totally do it all the time”.
“Maximally great” is silly but I think Anselm would argue that “greatness” is a requirement for God, but not for marshmallow asteroids or even for girlfriends. It’s a dumb argument because bootstrapping existence by fiat is well… dumb.
No, but that’s not the definition of a “least upper bound”, which depends on the superset you’re considering – see #14. In your example, the set has no upper bound within the set at all (no member that is as great or greater than any member), but it does have a least upper bound within the reals: 1. I think richardh’s mathematician friend hasn’t got the terminology quite right. Maybe the simple point that sets with an ordering defined on them don’t always have a greatest member – see sawells@4 – is nearer what’s wanted; or the fact that mathematicians can always come up with an infinite cardinal larger than those produced by any proposed schema.
@16 Seconded. “Ontological” has been so often misused by speakers in my presence that just the mention of the word gets my eyes rolling.
woozy, #10:
They’re both silly and ill-motivated, but this has almost no relationship to ontological arguments, whether we’re talking about those given by Anselm, Descartes, Plantinga or anyone else. They did think there is a being which must exist, but they didn’t argue their concepts of something have the properties of that thing, like weight or whatever. They, for example, didn’t argue their concept of this being was the greatest concept ever, but instead that what they were conceiving was the greatest being.
Leaving aside the superfluous crap about greatness and so forth, they basically thought there was a contradiction involved claiming that “the thing which must exist doesn’t exist.” That’s what a fool would say. However, you can say that there is no “thing which must exist” without contradiction, so the argument isn’t convincing to non-foolish people. They were wrong about that.
If you would admit that is contradictory, or you’re simply failing to deal with that by arguing against other stuff which isn’t even on the table, then you don’t have an effective response to the argument. Then, they will keep blathering about it and distracting us from real issues, and we don’t want that.
woozy, #17; latveriandiplomat, #19
Ah, didn’t think along those lines. Thanks.
Real numbers have the least upper bound property.
I is not less than 1 but it *is* the least number that is larger than all possible numbers that are less than 1. So 1 is the least upper bound of the set of all numbers less than 1.
But *every* set of real numbers that is bounded above *does* have a least upper bound. This is how the real numbers were constructed.
The Rationals do *not* have the least upper bound property. The set of rationals less than 1 has a least upper bound but the set of rationals whose square is less than two does not have a least upper bound within the rationals. Within the reals the set of rationals whose square is less than two *does* have a least upper bound. It is the square root of 2.
Here’s my case against the ontological argument:
Imagine a perfectly round circle, or a perfectly straight line, or an angle with precisely 90 degrees, or a point with 0 dimensions.
Now imagine any of those things existing in the real world.
It’s quite easy to do the former — in fact, the tools of math and geometry exist to allow us to reach all kinds of conclusions about these things.
But we cannot imagine any of them existing in reality. Every circle drawn will have at least some deviation, at least at the quantum level.
Therefore we can conclude that one of the qualities of any perfect being is non-existence anywhere outside the world of ideas.
@20. Actually the least upper bound property is more complex than what I said, I blew it. My example says a set bounded above doesn’t have to contain its own upper bound (true for dense sets like the rationals and reals, as you demonstrated).
The original statement was that a bounded above subset of a set A doesn’t necessarily have a least upper bound in A.
The bounded above subsets of the rational numbers don’t necessarily have least upper bound in the rational numbers.
The bounded above subsets of the real numbers do have least upper bounds in the reals, this is a critical property of the reals.
All of the above, assumes the usual ordering (<).
consciousness razor:
The common counter-argument is that these ontological argument pre-suppose existence. Anselm directly avoids this by pointing out as his second proposition “God exists as an idea in the mind”. He then presumes existence in the mind requires the same definitions and properties as existing in actuality, namely that the God that exists in the mind is “that which no greater can be imagined”.
Disclaimer: I’m a mathematician, which means I live in the world of “naïve set theory,” which is another way of saying that philosophers routinely dismiss my kind as misguided souls. That said…
The flaw in the ontological argument is quite clear, and it doesn’t involve “existence not being a predicate.” The flaw is that the definition merely tells you that if there’s a god, then in particular it must exist, else it would fail the “greatness” part of the definition. The argument doesn’t tell you whether said god exists; merely that a nonexistent one fails the test of godhood, for the reasons outlined in the argument.
A mathematician will probably point out that “greatness” is not well enough defined to use in definitions or proofs, though. Why is existence required for “greatness”? Why isn’t “stays crispy in milk, and is part of this complete breakfast” also part of the definition? I.e., why does our argument fail to prove that God is a tasty breakfast food? After all, a god who meets all the other aspects of the definition and is a tasty breakfast food, clearly goes the alternative one better…
I second the mathematician quoted above, though: it is also unclear whether beings, ordered by greatness, actually have the least upper bound property. I’d respond that that’s a nit, though, since we have reason to believe that the collection of all beings is finite, and so necessarily has the least upper bound property. For the same reason, there is theoretically a fastest human and a smartest human–which brings us back to the problem of defining “greatness.” Lack of precise measurement might prevent us from defining an ordering at all. We routinely pair the same two humans in a race and have different outcomes due to a variety of factors, so it’s difficult to define “fastest” in such a way as to get unambiguous results.
Bringing that back to the ontological argument, we might discover that not only does Yahweh exist, but all the gods of Olympus, and we might find that it’s difficult to sort them by greatness. Does “able to kick the other one’s ass” suffice? And if so, does one contest suffice, or should it be a best of 7 kind of deal? Do they need to fight to the death? What if both are unkillable?
Upper bounds:
Part of the problem is we modern kids in grade school learn about the Reals as an absolute. It’s true that the rationals don’t have the least upper bound property and the square root of two is not in the rationals, but if I give this as a counter-example it’s reasonable that the intuitive response is “yeah, but the rationals aren’t *everything*; just toss in the square root of two”. So it’s a poor teaching example, in that it makes it seem mathematicians are just playing with words to make us look dumb.
There are no real number examples of (above) bounded sets without a least upper bounds because real numbers have the “least upper bound property” (i.e. by definition all bounded sets do have a least upper bound). It’s a fundamental theorem of analysis that such a larger set containing the rationals would exist and the reals were constructed precisely for this property. This is why in grade school we are taught that the reals are “like the rationals but with all the holes filled in”.
Thus we have to look elsewhere for Saad’s counter example. A common thing is to define a different type of order. My personal favorite is the p-adic system where the “size” of a number is inversely proportional to its divisibility by a specific prime number p. But this is a wee bit abstract. (The numbers 75 and 25 would both be of size 1/25 because 5^2 divides them whereas 15 would be a larger size 1/5 as 5^1 divides it. Meanwhile .4 would be of even larger size 5 because 5^-1 divides it. So .4 > 1 > 15 > 75. Confused?)
The complex numbers meanwhile do not have a good order. (I is i greater or less than 1, -1, or -i? What does that question even mean?) We can define a pretty intuitive “dictionary order” by comparing the real components, and if equal, than comparing the imaginary components. In is way 5 + 2i > 3 + 4i but 5 + 3i > 5 + 2i and by associativity 5 + 3i > 3 + 4i. It works and satisfies all the requirements of order.
But it doesn’t have the least upper bound property. For example:
The set of all 4 + xi; The set of all numbers where the real component is 4 and the imaginary component could be anything. 4.1 > 4 + xi for all x. 4.000001 > 4 + xi for all x. So the set has many, many upper bounds. But it doesn’t have a single *least* upper bound. (Any real number greater than 4 would be an upper bound. But 4, itself is not an upper bound as 4 + i > 4, but 4.00000000001 > 4 + i > 4.)
woozy:
I don’t understand. Is it contradictory or not to claim “there is no being-which-must-exist”? If you allow that it’s contradictory, they can very easily fix anything else in their methods of communicating the issue, which seem only like linguistic ambiguities to me. It does not hinge on definitions of greatness or other adjectives, concepts necessarily having all the properties of their referents, or things like that. Correcting trivial things like that would not force us on pain of irrationality to accept the existence of a god. What is really central to it is the charge that atheism is internally inconsistent. The rest is window dressing, which you may separately criticize if you like, but it won’t get you anywhere with the best and clearest form of the argument. If you agree that it’s false that atheism is inconsistent or irrational on those grounds, then I just do not see how that is being demonstrated. You should confront the logical point that they’re attempting (however badly) to make, which isn’t hard, and it doesn’t need to be any more complicated than that.
conciousness razor:
I’m not sure I understand. No, it is not contradictory to claim “there is no being-which-must-exist” (Or more accurately to claim “there is no need to assume there must be a being-which-must-exist”) Do you have a specific ontological argument in mind? I have Anselm’s argument in mind. His fault is *not* that it presupposed god’s existence. He specifically gets around that by saying God exists as a thought, which, well, duh. *BUT* either it does exists as a thought that is consistent with its definition or it exists as a thought that is inconsistent with its definition. If it’s inconsistent with its definition the argument goes nowhere. Elephants exist as thoughts inconsistent with their definitions. (The thoughts of elephants don’t have weight.) He has failed to demonstrate that God exists as a thought consistent with definition. My thoughts of God could be bigger; his thoughts of God could be bigger; Perhaps the pope’s thoughts of God can not be bigger because the pope is divinely inspired but he has not adequately addressed that. Thus no, he has failed to show God, consistent with the definition, exists in the mind.
Other ontological arguments may have other faults but this is the one I see with Anselm’s.
God is the awesomest thing imaginable, and God would be even awesomer if he existed in reality.
Therefore, if you think about God, he pops into existence.
This is some next-level Secreteering Law of Attraction bullshit.
I think of it as the Most Perfect Apple argument. If I imagine the most perfect apple, then it must exist because if it didn’t exist, it would be possible to think of a more perfect apple.
I was always partial to the Ontological Argument for Atheism:
1. Imagine the most perfect argument for atheism.
2. The most perfect argument for atheism must be valid, sound, persuasive, and utterly irrefutable, or else it would not be perfect.
3. Therefore, there exists a valid, sound, persuasive, and utterly irrefutable argument for atheism.
@37
I’m fond of the Raymond Smullyen “A living unicorn” exists argument. It consists of a single line: “A living unicorn exists”. …. Think about it.
woozy #22:
That’s Anselm’s problem, not mine. I see nothing in the ontological argument which demands that it should only be applied to the class of beings known as gods, or which precludes its application to other things. That’s the main point of the marshmallow asteroid, or indeed the ‘greatest possible bank account’ argument. They point out how absurd the reasoning is by showing, without need to address the flawed internal logic, that it leads to absurd conclusions unless arbitrary constrictions are placed on the initial premise.
On this, we agree. ‘Anything you can imagine, I can imagine better,’ is a silly basis for an argument.
… He also had the “If I not mistaken, then…” argument. Well, if you aren’t mistaken then *of course*.
@39
Oh, we agree on *everything*.
I was just nitpicking. The concept of “greatness” and “things are greater if they exist” and “things exist as ideas” are key to Anselm’s argument. To make a silly counter argument you need to include those which my nitpicky critique of your original silliness I claim your marshmallow asteroid didn’t address. Although it *could*… but then it loses its punch.
I like the “perfect apple” or “perfect girlfriend” arguments slightly better (the “perfect girlfriend” adds an air of geekish loser desperation as a bonus).
That isn’t more accurate. It’s confusing. I don’t care at the moment what we do or don’t need to assume. There’s no reason to go down that rabbit hole. What are the facts about reality, and how can someone coherently describe those, as distinct from their methodology? I think it is true that there is no being-which-must exist. That is very plainly what I think. If it’s true, that’s simply a fact about the category of real things: the category doesn’t contain any of the beings-which-must-exist while it does contain other sorts of existing stuff.
That fact doesn’t concern what we need to do (or should do) about anything. It is simply a way to describe what there really is. Can that be consistently done without including beings-which-must-exist or necessary beings or anything of the sort? Yes, as I’ve said, it certainly can, and that is how the argument fails, since it was intended to make precisely that point about what there is.
Suppose somebody does think presupposing a god (or whatever) is acceptable. The best you can say is that we don’t need their methodology? That’s pretty silly. You could very well say that they have a false belief: that a belief in a being-which-must-exist is a belief about something which doesn’t exist. Whatever methodology they used or however they may have come to that, it is a further issue that it’s a false belief (or that it’s a true belief, if that were the case). That is certainly relevant, this being an argument about ontology and all, not simply an exercise in epistemology.
If a being must exist, then doesn’t that mean the being is not omnipotent? Heck, it makes the being less powerful than me, since I could make myself not exist.
I was saying that the refuter may not know whether or not that-which-must-exists exists or not. The burden is on the arguer to argue that that-which-must-exist actually must exist. An agnostic should be able to refute a bad argument just as well as an atheist.
No. I’m saying their methodology is wrong.
But that’s *not* what a refuter is saying. By pointing out the logical fallacy (and it’s *huge*) of Anselm’s argument the refuter hasn’t demonstrated anything whatsoever about the actual existence of gods or whatever; s/he’s merely demonstrated that Anselm hasn’t either.
Anselm’s proof is invalid. Anselm’s belief … has not been addressed in any way shape or form.
Kant (if @6 source is accurate which I assume it is; I’ve never read Kant or studied philosophy) argues that most ontological argument pre-suppose. That’s the case with Anselm (although he, invalidly, talks his way out of this). But it need not always be the case. Ontological arguments try to argue an existence is a tautology but although many fail because they pre-suppose that doesn’t mean all will. (“I think, therefore I am” is actually a valid ontological argument; so is Godel’s incompleteness theorem.) I won’t say *all* invalid ontological arguments abuse a linguistic descriptive/prescriptive ambiguity (which can always be created a la Godel [that-which-must-exist by definition must exist; ta-da!]) but most do. Anselm definitely does. Descartes’ proof of God is a bit more subtle and I’m not entirely familiar with it. I suspect it does.
O hai, tulse! I maded you an internet out of lavender cookies. And I did not eated it.
While you don’t support this, even if that’s so, I don’t care what agnostics should be able to do (or how you think arguments should proceed, in a way that makes you happy). Again, the argument is about claiming in fact there really is a god. If you’re not willing to talk about what there in fact really is, then that’s your problem. I don’t think there is any controversy in saying your willingness isn’t relevant to the facts about reality.
Yet the conclusion they fumbled toward happens to be right? Or is it not right?
But I am saying that: there isn’t a god. So, are you claiming here that I’m wrong to do that? I think that’s the truth, and the alternative is that it is false. If I’m wrong, then I’m wrong, but as a fact about reality it’s either one or the other, not both or neither or something else. Indeed, admitting that I may be wrong is something that I’m willing to do and be very explicit about. It seems like you’re avoiding that with these fence-sitting shenanigans about agnosticism, while at the same time ironically claiming to have some kind of methodological high ground where presuppositions are ostensibly a bad thing (but only when others do it). But it isn’t even clear what that high ground is supposed to be or why we should be going there in the argument in the first place.
I don’t know what else to say. If you really want to go into all of the hairy details, you can check out J.L. Mackie’s book on the broader subject of arguments for gods, Miracle of Theism. It’s all worth reading, but OAs specifically are discussed in pages 41-63, where he talks about Descartes, Anselm and Plantinga. As I tried to explain, there is a fairly simple reason why they all fail. You’re apparently not accepting that at the moment, and it’s not at all clear what you think you have as a replacement. Even if it works somehow, it certainly doesn’t look like an improvement.
I actually have an ongoing series on ontological arguments, which may interest anyone who wants to dig into the logic.
IMHO, the whole “maximally great” thing is stupid. Why bother defining God as a being which is maximally great? Maximal greatness specifies a lot of different properties for God, and the ontological argument only uses one property: existence. You might as well define God as an entity which exists and be done with it.
The reason ontological arguments don’t do this is because then it would be too obvious that you can replace God with any old object. And then people would realize the argument is stupid.
I can conceive of a greatest universe which is greater than the greatest god, and unlike the latter it exists.
Partially addressing arguments between @Consciousness Razor #46 and Woozy #44
Most ontological arguments have the interesting property that if God happened to exist, then the argument is sound. “Sound” in the sense that the premises are true and the reasoning is valid. God exists. Therefore God exists. Sound!
It logically follows that if the ontological arguments are unsound, then God does not exist. So if I haven’t absolutely proven that God doesn’t exist, then does that mean I haven’t entirely refuted the ontological argument yet? This seems like too high a burden of proof.
While the probability of God is exceedingly small, the probability that the ontological argument is any good is exactly nil. I can say this because even if the argument were sound, it would still be terrible. As Woozy said, the methodology is wrong.
sidetrack:
summary of ontological argument:
1) God is, by definition, a perfect being. (our definition of God is, “perfect being”)
2) non-existence is a flaw
3) perfect = flawless (i.e. without a single flaw)
4) therefore God exists
if that summarizes ontological argument, that seems to be circular logic. Line 1 defines God as already being existant, so the further lines are superfluous, “begging the question”, etc.
That looks too easy, so my summary has to be wrong. Please correct my ontological wrongness.
@46
Um, what are you talking about? I thought we were specifically discussing the ontological argument that god exists (because by definition god is the greatest and existing is greater than non-existing). Kant points out one fault (and there are others) is that this presupposes its conclusion. You claimed this isn’t relevant. Of course, it’s relevant. It’s the entire gyst of the argument; a greatest of all beings exists a priori. And that’s the fallacy of the argument.
I’m not trying to argue completely from ground up that god does not exist. I am simply analyzing this one very very specific argument. And my analysis stands that it is fallacious in that in confuses descriptive properties of God as a concept with prescriptive (god would be the greatest thing ever) requires with god as existent (the greatest thing ever must exist). I do not care about any other argument and I am not making any argument of my own. I’m merely examing this one argument before us. What the hell are you doing?
@50. The trick/delusion/fallacy is to bulster the concept of a God to nescessity. Consciousness Razor was correct in that the goal is to present God existence as a logical tautology; where god not existing is a logical contradiction. Anselm does it by claiming God exists already as a concept and now that he exists as a concept … Descartes punts and argues that there is a hierarchy of non-perfect to more perfect hierarchies. The hierarchy must have an upper limit and that upper limit must be perfect else it wouldn’t be an upper limit. Yes, they are all circular but the hide/waffle/get lost in other misleading arguments of whether truth exists and can be implied simply by a sentence existing. It sounds like “tricks” but ontology we can’t dismiss everything as “mere semantics” but the actual lines of were semantics end and true logic begin are vague and subtle and weren’t really resolved until the 20th century.
But no. God exists because his definition requires it really is circular as is the liars paradox and in analysis both are very similar in that the issue is where the meaning of the concepts actually lie (and that the arguments usually punt the actual concept out of the park).
What Kant actually said boils down to this: “existence isn’t a predicate.” It’s not some generic hand-waving crap about presupposing or begging the question, and how that’s a bad thing to do. So what does that mean?
Someone makes an ontological argument: we’re supposedly not logically allowed to say “X doesn’t exist,” because X itself involves existence. That is what can’t be consistently done, according to the argument.
That’s a stupid argument, but you have to say more than “it’s stupid.” You have to explain what’s stupid about it. So what do you do?
Kant said existence is not a proper predicate. It’s just putting some X on the table for discussion as it were, but it doesn’t describe X in any substantial way like a normal predicate should. That’s one thing to say, but it obviously bears no resemblance to anything you’ve been saying. This is apparently news to you, since you’ve said you haven’t read any Kant, but you can verify that yourself if you like.
You can also say that even if existence counted as some kind of a predicate (maybe a very unique one unlike any other predicate), there’s still no contradiction. We don’t even need to make it as complicated as Kant did — much less whatever the hell you’re doing ignorantly and off the cuff. You can clearly separate X’s internal references to “existence” or “necessary being” (if some theist insists that’s what they mean by a god) from any claims made about whether nor there is an X like that in reality. In other words, there isn’t an X like that. (And since they are claiming things about reality, that is what you should be engaged with here, it hardly needs to be said.) There was this highly misleading idea about how to formally structure arguments and what parts of them entailed about each other, but that can be cleared up very quickly and easily, without introducing anything new.
@twas brillig #50,
No, Anselm’s ontological argument is pretty obtuse, you got that. But it’s sort of like arguing over Zeno’s paradox, or whether 0.9999… = 1, or whether 1 + 2 + 3 + … = -1/12. Ultimately pointless but we do it because it’s fun.
And in case you start thinking it’s too easy, you can move onto other ontological arguments. Lots of people in this thread are talking about the conceptual ontological argument, which I state as charitably as possible:
1) As long as a predicate has no contradictions, you can conceive an object with that predicate.
2) Let G(y) be the predicate “y is a perfect being”. G has no contradictions.
3) Therefore, I can conceive of an object x such that G(x).
4) The definition of G implies existence outside of our minds.
5) Therefore, x exists.
Over the years I’ve encountered only a few people who actually support ontological arguments. These mythical creatures are mostly harmless. Except for Alvin Plantinga, who for some reason is considered a serious philosopher. He uses a different version of the ontological argument based on modal logic. It’s also really obtuse, but you’d need to understand modal logic to pinpoint why.
“Existence is not a logical predicate” is an asinine statement: it’s possible to conclude that things exist. Mathematicians do it all the time, for some value of “exist.”
The flaw here is that defining something doesn’t make it so. I can articulate the definition of “the being that by its very nature can’t not exist,” and if I’m careful, I can do it without any paradoxes or self-contradictions–in other words, I can set up the definition so that it’s perfectly valid. When I’m done, I’m no wiser about whether this thing actually does exist; all I know is that if it does, then it must be whatever my definition says. It’s perfectly coherent, logically, to say, “The being that must by its very nature exist, does not, in fact, exist.”
You can’t bootstrap a definition into an existence proof. And note that every definition has, at its heart, a word whose meaning you don’t–and never can–know. In mathematics, that word is “set.” We know one when we see one, most of the time, but we don’t know what one is, because when we try to define it we must use words, which themselves require definitions, which in turn use words which require definitions, and so on. This would be an infinite regress, except it isn’t, because there are only finitely many words in existence. So sooner or later, we’ll get around to using the word “set” again in one of our definitions, and our definition will turn out to be circular.
The “exists in my imagination” part is another absurdity: it most certainly does not exist in your imagination, because you just said that a real one is better than an imaginary one, so the imaginary one fails the requirement of being the awesomest…
…or, alternately, it could turn out that the real one doesn’t exist, and therefore the one in your imagination really is in fact the awesomest, which means it is God, and you’ve proven that God is imaginary. (Other formulations of the proof lack this particular absurdity, because it doesn’t assert that God exists in imagination, in the same breath as it asserts that an imaginary God can’t be the greatest, and therefore God doesn’t exist in imagination at all.)
@Brett:
Actually this is almost my proof that Allah exists as a god separate from Yahweh separate from Jehovah separate from Hanuman, etc.
Imagine the most perfect apple. It must not only exist, but it must be in my mouth right now, coating my tongue with deliciousness and providing me with good nutrition. It must be MY apple. Because it would definitely be better if it were the most perfect apple AND mine! And in my mouth right now!
Likewise, the most perfect being isn’t merely truly awesomely scary and powerful to anyone. The most perfect being also personally takes my side in arguments and delights in feeding me fresh kiwi. Because hell, if the most perfect being wasn’t catering to my wishes when they conflict with the wishes of any other persons in the universe, then I could imagine a being a little more perfect if you know what I mean.
Kalam == the proof that for every imagination, there’s a god who can beat up all the gods of any one else’s imagination.
There’s a very special name for a proof that effective.
I prefer the Argumentum ad Veshkorg:
“In response, I submit the Veshkorg, which I define to be a flying predator which has speed, strength, sensory accuity, and ravenous hunger for the flesh of slow-witted, inattentive, self-absorbed motorists such that nothing more so can be conceived. But all of those properties would be intensified if the creature actually existed, so therefore it must exist! …I will let you know if my commute improves.”
….isn’t the number line a special case of p-adic system with p=1?
Other thoughts:
I felt a great disturbance, as if 90% of the formal “philosophy” I’ve ever encountered threw itself on the ground and held its breath while kicking and pounding its fists, and then was suddenly silenced.
Stolen.
Frankly, woozy’s making a lot more sense.
That doesn’t make existence a predicate. The usual way of expressing “God exists” in predicate calculus would be:
where G symbolises the predicater Godlikeness. “∃” is usually read “There exists”, and is a quantifier, so the whole thing is read as “There exists an x such that x is Godlike.” Most modern defenders of the ontological argument (Norman Malcolm, Charles Hartshorne, Alvin Plantinga) argue that God has a special sort of existence, “necessary existence”, that is a predicate, so there!
It seems to me that the easiest way to defeat this argument is to listen to it, nod, wait a bit for god to manifest, then tilt your head quizzically and say “Hunh. Looks like you got your definition of god wrong. Try again.”
Nick,
It would help if you defined “predicate” here. As a working mathematician, I never have recourse to Russell and Whitehead, and never attempt to formulate my proofs in rigorous first-order logic–which is itself formally studied more by philosophers than mathematicians. See my disclaimer in my first post.
If I understand your statement, it is that a predicate, strictly speaking, maps entities to truth values, and the domain of a predicate is therefore entities whose existence is already a given. Hence a predicate can only be applied to God if God first exists as one of the entities in the domain.
If I have that right then OK, fair enough. It’s consistent with my experience that “existence” proofs do, in fact, work that way: the elements of the domain already do exist, and the question is whether any of them meet the criteria. Saying “a solution exists” is the same as saying, “one of the points in this space qualifies as a solution.”
If you reformat the ontological proof thusly, you get, “Of all the beings, at least one is godlike.” I’ll cheerfully grant that by definition at most one can be godlike, so uniqueness comes for free; “existence” becomes the problem. But the difficulty of the problem becomes obvious, since there is no way to get any traction–since most of the beings in the universe are unknown to us, and there’s no purely logical method of deducing that the smartest one, say, must necessarily also be the most altruistic one. I like that. It exposes the fundamental difficulty: we know jack shit about what beings do or don’t exist in the universe, and navel-gazing will not provide a method of backing into that knowledge.
A Masked Avenger@62
The first bit certainly: a predicate maps entities onto truth values; and I think that if a predicate is applied to a constant, that implies that what the constant refers to, exists. But you can perfectly well say:
“There exists no x such that x is Godlike”. Of course at the time the ontological argument was formulated, they were still working with Aristotelian logic, in which there was no clear notion of quantifiers as logical operators. I’m no expert on the history of logic, but my hunch is that it was this lack of clarity which made the ontological argument plausible (to some people – others, like Gaunilo, who essentially came up with Daz’s argument @1, could see it was hogwash). The guys I referred to earlier try to use modal logics (which are notoriously tricky) to resuscitate the argument, but always with some sleight of hand – Plantinga defines “God” as a being who exists in all possible worlds (this being an aspect of his greatness), then works a switch between epistemic possibility (“for all we know, there is such a being”), and logical possibility (“there is some logically possible world in which such a being exists”) in the course of the argument.
I think all the LUB arguments miss the point a bit. It doesn’t matter a jot if the set of all things with the greatness measure have the LUB property if we don’t first show that the set is actually bounded. Why do we assume there is some theoretical maximum greatness? Anselm considered this sort of “perfection”, but is there any reason to believe that whatever one-dimensional measure of “greatness” we’re using has a theoretical maximum (and remember, according to the argument, there must be such a theoretical maximum as it defines the being in question)? We’re not doing something akin to asking either for the LUB of a set of rationals or reals, we’re potentially asking for the greatest real number.
It’s like a couple of kids arguing:
“OK, so you’ve imagined a great being – what’s its greatness score?”
“INFINITY!”
“Did you add existence to your being?”
“Uh, no.”
“AHA! Well, if I get your being and add existence, then its greatness score is INFINITY PLUS ONE! Checkmate, atheists!”
So the ontological argument is the one that states that, if god exists, then god exists, therefore god exists, right? I mean, basically, when you get past the densely packed logick.
I kind of love that argument. Not because it’s particularly impressive or convincing, but because it’s reassuring. They’ve had thousands of years to come up with an argument – their god has had thousands of years to give a shred of evidence – and one of the best they can come up with is just a linguistic game of twister. I try to maintain a healthy uncertainty on all topics, just because I know how easily skepticism can turn into a mere title if you don’t, but these arguments… I honestly don’t think I need to worry about gods. At least not gods who care what we believe. Their believers so clearly have nothing of substance.
Yes, that’s how I read it.
Another way of looking at it is this: we can grant that the ontological argument establishes that, if a god exists, then one of its properties is… existence. The sneaky bit is the claim “God exists as an idea”, which is flat wrong. The idea of Anselm’s God might exist in Anselm’s mind – but the idea is not the God, and the argument fails.
@58
Mmmmm…. I don’t think so. The p-adic norm of a rational q would be 1/p^n where is the highest integer (pos or neg) where p^n divides q. Thus the 1-adic norm would be 1 for all numbers.
Then again it’s been a loooong time since I’ve given this any thought and …. well, although I was well versed in it at one time I’m afraid that is no longer the case.
@64
I think the original claim might have been referring to Descartes proof specifically. Anselm seems to believe all sets are bounded … by INFINITY!
My problem with pointing out the mathematical errors in these proofs, is that the subtleties and rigor involved in understanding the math is exponentially harder and more subtle then the direct logic itself.
Anselm defines God as that which can’t be any greater, to which I can’t help thinking Superman holding God in his arms would be greater.
…or less flippantly what I imagine as God can easily be imagined as greater by someone smarter than me.
@54
I’ve always felt “a perfect being” was chock full of contradictions, myself, but that’s not the basic flaw I find in the argument.
And yet, oddly, my concept of the perfect being is, in itself, not perfect. …. strange….