Metaphor Shmetaphor
Is God a metaphor for an ineffable mystery, or is He really a bearded, imposing sky-stud with a wicked temper and a mean right hook? In past discussions of religion on this blog I've been accused by my religious co-debaters of positing a cartoon cutout strawman God who, I'm told, is not representative of the modern, sophisticated, rational Judeo-Christian believer, but is rather a throwback to some bygone day when wily churchmen indulged the wooly idolatries of their inbred pagan converts to speed them, and their descendants, into the saving graces of the Church. God's personal nature is merely a metaphor, I am told, a way for the human mind to approach the ineffable nature of eternal truth.
Well I don't know if the wine-sipping New York sophisticates who write for Father John Neuhaus' magazine First Things are still in outreach mode for wooly, inbred pagans, but you would have to conclude so after reading this post from Neuhaus:
So which is it? Hairy Thunderer or Cosmic Muffin?
Well I don't know if the wine-sipping New York sophisticates who write for Father John Neuhaus' magazine First Things are still in outreach mode for wooly, inbred pagans, but you would have to conclude so after reading this post from Neuhaus:
In the pages of First Things, I’ll be coming back to Gerald McDermott new book, God’s Rivals: Why Has God Allowed Different Religions? (InterVarsity). Meanwhile, a comment on an appendix to the book in which he explains why he uses the masculine pronoun in referring to God. This, as you know, is a very big issue with some feminist or “womanist” theologians. Some of them simply switch genders, using “she” and “her, or using them alternately with “he” and “him,” but that results only in highlighting the gender-specificity that they want to overcome.
As McDermott notes, others prefer expressions such as “Godself,” but this undermines the understanding that God is a person. “It is particularly important,” he writes, “to highlight God’s personhood when discussing religions that deny it. Philosophical Hindus and Buddhists, for example, insist there is no personal God because there is finally no distinction between God and the cosmos.”
The Christian God is not “an amorphous essence” but the Father whose Son died on the cross. God is Father, Son, and Holy Spirit. “[These] are not human constructions in response to ineffable religious experiences, but names for God given to humans by God himself. The very names encapsulate the entire story of the triune God.” The names are not our “metaphors” but God’s self-revelation as Father, Son, and Spirit.
So which is it? Hairy Thunderer or Cosmic Muffin?
posted by Anonymous at 5:14 PM
67 Comments:
Boy, do you guys hate.
Peter:
I think Dalrymple is a brilliant writer.
However, having read a couple of the books he talks about (Harris and Hitchens), near as I can tell the closes he came to reading them was glancing at a book store as he drove by.
As for Neuhaus, he exemplifies the peril of devotional thinking -- the stringing together of words in ways that are grammatically secure, thereby giving the appearance of conveying meaning, while actually containing none at all.
It is the consequence of habitually writing authoritatively on a subject that allows no authority whatsoever.
An Imam, using the same devotional thinking, would just as authoritatively come to a very different conclusion.
Yes, well there are three things every religious person has to confront: a) Bad things happen; b) People believe different things; and C)Divine nature and purpose are largely incscrutable. For some, these are killers and so they become non-believers. For others, they are not and so they hold to their faith. Crazy world, eh?
But why do non-believers persist in thinking they have discovered something new when they confront these things, why do they assume they come as a surprise to religious people and why do they make you so angry?
Peter, stop changing the subject of my posts. I'm asking a simple question, Hairy Thunderer or Cosmic Muffin? There is no hate implied in the question. This is one of those questions that has been around since classical times, and still it is unresolved. We don't need to drag Dawkins into every debate, it's not about him. If you want to play the Hate Sweepstakes, then we can round up the usual suspects as well, but its a tiresome game.
So you must admit that the comic book Hairy Thunderer god is alive and well among sophisticated believers, no? This post is directed more at David, though.
This post is directed more at David, though.
Sorry, I should have known. David's my expert on the nature of the Christian God too. But the answer to your question is Hairy Thunderer with a soft spot for a good sob story. There, The Daily Duck resolves another one.
Judeo-Christian God. Or G-d. Don't forget your roots!
How is it directed at me? I didn't comment because it seemed aimed directly away from me.
First, Neuhaus is writing from an inherently Christian position, as he acknowledges. Since I don't believe in G-d the Father and G-d the Son, I'm not on the horns of Neuhaus' particular dilemma.
Second, I've never denied that people -- particularly Christians -- believe in what you call the Hairy Thunderer. What I have said is that I don't; that Judaism, at least as I understand it, doesn't; and that I'm not even sure what people mean when they say that theirs' is a personal G-d.
Third, even Neuhaus is speaking about G-d's essential nature, not what He looks like. He is saying that G-d embodies existential Fatherhood. He's not saying that G-d has a penis, circumcised or not.
Fourth, no one on this side of the aisle, and certainly not me and Peter, is suggesting that G-d is a metaphor. That's your position. We're saying that G-d is real, that G-d actually exists, and that the bearded guy/Lord/King/father/etc. is a metaphor.
Fifth, nor am I saying that G-d is a cosmic muffin, although I'm not sure what a cosmic muffin is. I reject both the god of the gaps and the wholly benevolent god as being false gods. I suppose that my current metaphor (which is simply a metaphor and thus is useful only to the extent that it emphasizes the points that currently interest me) is that G-d is the dreamer and we are the dream or, if you prefer, G-d is the hardware and all that we perceive as real is the software.
That's pretty good, David. I'd jump in and help, but I'm saving my fire for when Duck has a question about The Talmud.
But now that you have cleared up that we don't think God is a metaphor, let's make a bet on how long it takes Skipper to roar in and demand to know how tall we think He is.
David,
I think I understand where you fall on the spectrum, and it's Cosmic Muffin. But I think you have chided me in the past for arguing against a strawman personal God, saying that the personal aspect of God is a metaphor. But as Neuhaus points out, the personal is not a metaphor for most Christians, and so I haven't been shooting flaming arrows at a strawman, but the real belief.
That's your position. We're saying that G-d is real, that G-d actually exists, and that the bearded guy/Lord/King/father/etc. is a metaphor.
This is where it gets tricky. What does it mean to say that God is real but that every way of expressing that reality is a metaphor? What you're really saying with a statement like that is that the ultimate nature of existence is unknowable, which would make you a dunnoist, or an ineffablist. I'd describe my own metaphysical view the same way, which is why for me the word God has no content. To say that you believe that God exists is to say that you affirm some knowable attributes to the source of existence. You can't have it both ways. You can't say "I believe" and then when pressed for details of what you believe in say "there's no way to say".
Third, even Neuhaus is speaking about G-d's essential nature, not what He looks like. He is saying that G-d embodies existential Fatherhood. He's not saying that G-d has a penis, circumcised or not.
You're getting your essences and existences mixed up. Essential fatherhood is the abstracted, Platonic notion or form of fatherhood, the thing that all fathers have in common, which from a knowledge of human fathers boils down to one and only one thing: the donation of semen to a female that conceived and bore a child from that donation. Existential fatherhood is fatherhood in the flesh. It's the fatherhood that exists, hence existential.
Arguing as a good nominalist, I have to say that essential fatherhood only belongs to existential fathers. You can't separate essence and existence. Essence is just an idea, a notion. So if God has the essence of fatherhood, then he must have the existential hardware to back it up.
Peter, did you catch the fact that Dalrymple is himself a non-believer? I also think it is interesting that Nat Hentoff, an atheist, is teaming up with the First Things social conservatives on right to life issues. How can we run an orderly culture war when the players just won't line up on the right sides?
What does it mean to say that God is real but that every way of expressing that reality is a metaphor?
Now that is a profound question. It may be the profound question. I agree that it's an issue, but it is an issue that tells us more about ourselves than about G-d. But you assume a symmetry that doesn't exist. We can't know G-d directly, He knows us perfectly. As a result, He has given us revelations that are suited to our understanding -- revelations that are, necessarily, metaphors because we can't understand reality. That doesn't mean that they are contentless.
As for the essence of fatherhood being sperm donation, I know that you don't believe that.
Sperm donation is the least common denominator of fatherhood. If you read my statement carefully, you'll see that I was defining it in those terms. There are many positive aspects of fatherhood that are applied to the metaphor of God as father, but those positive aspects are not universal. With the growth of absentee fatherhood and single parent (mother) families today, the least common denominator definition is becoming more operative.
And as a metaphor for God, the absentee father model is also worth considering. Jesus even gave voice to this image in his dying words on the cross. Where is God the Father when bad things are happening to good people? No matter how many books are written to explain that God really cares for his children and is always looking out for their best interests, this has always been (Job)and will always remain a challenge and a stumbling block to many who are brought up to believe in God the Father.
The cool thing about being a materialist is that you leap past all these struggles with a single bound.
I read Dalrymple as far as where he said atheists imagine a coolly rational place. Not me. I am impressed by the messiness of belief, whatever is being believed in.
Anyhow, I had a thought the other day while reading Weitz' 'Weimar Germany.'
Weitz is a great admirer of Heidegger. Now I've never read Heidegger. I'm choosy about who I take moral advice from.
But Weitz translated whole pages of H. And although this trick would not work in the original Nazi, it occurred to me that it had been composed by a RANG (TM)subassembler: a Random Abstract Noun Generator.
Replace the abstract nouns in any sentence or in any paragraph in any order, and the meaning is not changed.
That's pretty much how I read theology. What could it possibly mean to have a son and a father if both exist from all eternity?
As metaphors, these remind me of a simile the poet/naturalist/nincompoop Diane Ackerman wrote. She was caught in a flimsy shack during a storm, and she described how the walls shook like (quoting from memory) the hull of a rocketship during liftoff.
There are materialist conundrums too. Why are there different sizes of infinities? Loosely speaking (and this may be the point) there are more irrational numbers than whole integers. Or how are their continuous functions ("smooth") that are nowhere differentiable ("the exact opposite of smooth"). Or take your pick of other measure theory paradoxes, or quantum, relativity, E&M, etc.
And if that's okay for mathematics, why not theology? To paraphrase von Neumann, you don't understand things, you just get used to them.
Dang you, Mike! That was the point I was about to make, because (once again) it was a subject in my doctoral thesis, with regard to quantum mechanics. One need only look around the physics community to see that QM is no better known than God. It is really well named as "mechanics", because what it really is a collection of mathematical tools that are used mechanically to generate predictions. What the underlying reality is that they describe is far more debatable. That reality may, in fact, be fundamentally unknowable to use. Yet that doesn't preclude us knowing something about some aspects of it.
Peter, did you catch the fact that Dalrymple is himself a non-believer?
Yes, as was my hero, Stove and that scientific fellow who got off that wonderful line about how reading Dawkins et al. on theology is like reading a treatise on biology by someone whose only connection to the natural world was a mastery of The Book of British Birds. Several others, too, who do science and philosophical scepticism great credit, in that, unlike certain folks around here, they are acutely aware of the sense of loss and what it means. Hats off to Mike and SH as well. Whether they will be able to hold off the scientifc (as in scientism) tide in the face of all the populist second-raters who write the just-so stories, we'll see.
In the end, it is about humility and understanding how much we don't or cannot know. But, getting back to the nature of God, please consider Bertrand Russell's comment that, if he died and found out he was wrong, he would immediately demand to know why He didn't reveal himself better. Never mind theology, isn't our debate between those who think he was a brilliant iconoclastic logician who saw through the scam and stuck it to the Big Guy and those who see him as the eternal, shallow and rebellious adolescent? As with misogynist old Sarte, his life bespeaks the latter, as his estranged, believing daughter would attest.
Neither QM nor mathematics presents a problem to a simple materialist like me.
As my physics adviser puts it, someday somebody will reconcile the contradiction between QM and relativity, and when he does, it will be really, really simple.
Until then, they are both provisional hypotheses and pose no difficulties for materialism.
Although they appear to contradict each other, they do not each contradict themselves, which separates them from theological arguments like the triune monotheistic god.
Perhaps QM and relativity will never be reconciled. With Christian theology, there is no perhaps about it. The son who is as old as his father is not going to be reconcilable, to take an example.
Several others, too, who do science and philosophical scepticism great credit, in that, unlike certain folks around here, they are acutely aware of the sense of loss and what it means.
Sorry, I just don't get the sense of loss. What did I lose?
Duck:
What did I lose?
I take it you mean aside from everlasting life?
Ok, ok, let's stick with the here and now. Whay do you think folks like Dalrymple mean when they talk about Western civilization being grounded in religion and the atheist project being a rejection of it? I could try and pinpoint stuff like art and music, philosophy, justice, a common literary language, and also point to how every notion of decency and morality we have, many of them rapidly disappearing, is grounded in faith. But, Duck, surely you are being disingenuous here. You are too smart for that. It is one thing to say that, on balance, you prefer the modern, quite another to stand before the catherdral at Chartres and say: "I don't get it, what have I lost?"
If I had to sum it up, I would say the loss is the rapidly fading sense of an outward-looking life--a life whose parameters are defined from without as well as from within. Surely it is obvious that we are all becoming what Taylor calls "buffered individuals". Not only do we demand our objective, material freedom, more and more we define things like morality and reality itself as having no meaning beyond what is inside wonderful, deserving, endlessly complicated me. Personally, I think we are all slowly going insane for it, but that's just me. Look at how we reduce the world around us, including others, to rote and simplistic biological and psychological laws, all the while seeing the subjective me as so dense and rich only a five volume Russian novel could unlock its mysteries.
But you get it or you don't. Modern Western minds are very closed and unmoored, and this is a little like trying to convey the sense of loss from not learning Latin anymore to someone who never studied Latin. But I do believe the loss is felt more acutely and widely than we sunny-faced "have-a-good-day" North American moderns or the "nothing matters, all is a dark joke" gloomy Europeans admit to ourselves, which is why the defence of the modern world must rest on so many simplistic distortions and even egregious misreadings of history. We must tell ourselves everyone in the past lived in bloody terror of superstition and the dictates of murderous religion, even though a reading on one Shakespeare play reveals that as absurd. By making the past horrific, we can overrule all doubts about the present and "the modern" doesn't do doubt. Look at you, Duck, with your wholesale dismissal of idealism. Again, you may think today is better than yesterday, but do you honestly think the disappearance of romantic odes and poems isn't a loss? The modern cynicism about justice and patriotism? The disdain for honour and duty, yada, yada?
Peter, you say "gloomy Europeans admit to ourselves" -- do you consider yourself European?
Funny, you can read all of Shakespeare's plays, not just one of 'em, and find not a single mention of religious doctrine.
And from that you conclude he did not live in terror?
The fires of Smithfield were never far from his mind.
Besides, all societies have their cathedrals, their art, music and laws, not just the west. If the west is better, it could be because it has better religion, or it could be because it has embraced secular values that trump religion, at least as far as public policy goes.
If it isn't secularism, then there's no reason for westerners not to adopt Hinduism or Islam. And, of course, many have. Or pretend they have.
Like Christians, they mostly welcome the benefits of secularism for themselves while damning its bad effects on the weaker individuals.
Yes, erp, I could have done with a good editor with that sentence. I was going to call you, but it was only 6:00 am.
The "ourselves" was intended to relate back to "Western minds" in the previous sentence. Didn't really work, did it?
Harry: That's why I asked about mathematics and not just QM vs relativity. And actually, there are fundamental logically inconsistencies in mathematics: Either horn of the Banach-Tarski paradox is deeply disturbing.
But that aside, people cannot comprehend infinity, and the statements that we can make about it strongly violate our common sense. Again, what does it mean that a countable infinity is smaller than a non-countable set? How is that different than saying one eternal being preceeded another? And if I were going to take a swing at your example, I'd just suggest that time doesn't apply to God, and therefore precedence and causality are undefined.
Funny, you can read all of Shakespeare's plays, not just one of 'em, and find not a single mention of religious doctrine.
It sure is funny, Harry, especially when you realize that Shakespeare was very popular entertainment at the time--the Steven Spielberg of his day-- and that the party line for you folks is that pre-Enlightenment man was the total prisoner of Christian religious authority and indoctrination, enforced by the rack and/or threats of damnation, and was constitutionally unable and politically forbidden to access or employ reason as we understand it today. You would think the Church would have drawn and quartered him after the opening night of his first play. They must have been throughly at sea in the balcony of The Globe, no?
However, although there is little or any Christian theology, his plays are awash in religious themes and references to the spiritual world and actors that inhabit them. He wasn't a proto-Duckian asking himself "What works?" in the face of every drama life threw his way.
It might have worked, but my short memory is so shot, I couldn't remember it.
6 AM? Middle of the morning.
I’m so relieved you don’t consider yourself European. No offense to monix I hope. In my day, the English didn’t consider themselves European but I don’t know if that has changed. In any case, no offense intended to those who reside in Europe through no fault of their own. (Just kidding!)
Peter:
But why do non-believers persist in thinking they have discovered something new when they confront these things, why do they assume they come as a surprise to religious people and why do they make you so angry?
Its funny. When the religious endlessly reiterate the contents of books over a thousand years old (and to which nothing has been added by all the writing about those books ever since), that is just fine.
But reiterating arguments cautioning against that state of affairs is not?
I suspect that for a great many committed religionists, there is no possible way one could repeat too often the Provenance Problem. Islam, to take one example, cannot be the Religion of Peace, while simultaneously claiming divine diktat to kill unbelievers wherever they might be found.
Not much in contemporary Christianity makes me particularly angry, or even uncomfortable (the sui generis evil accorded to homosexuality being one exception). However, if there is a reason why I should not be furious with a religion that explicitly wants me dead, then it has surely escaped me.
But that gets right back to the Provenance Problem. Presuming there is, in fact, no reason to avoid fury in the face of calls for my murder, the next problem that beggars the imagination is how to challenge some of Islam's divine diktats without undermining them all. Keep in mind, though, that the Provenance Problem extends to all religions. One cannot attack divine diktat in particular without undermining divine diktats in general.
BTW, of Hitchens, Harris, Dennett and Dawkins, which have you read?
Whay do you think folks like Dalrymple mean when they talk about Western civilization being grounded in religion and the atheist project being a rejection of it?
Well, given the complete hatchet job he did on those books (at least the two I have read) he reviewed, I have no idea.
I suspect you, like Dalrymple, does not understand what the anti-theist project is about. Condensed to a sentence, it is this: religious claims should be open to the same analysis and criticism to which non-religious claims are subject.
David:
Since I don't believe in G-d the Father and G-d the Son, I'm not on the horns of Neuhaus' particular dilemma.
Yes, you are. His ability to assert, without equivocation, some of God's essential characteristics that are in direct conflict with yours means at least one of you is wrong. He has made a statement about objective reality; that you don't agree with it is no relief whatsoever.
You also cannot derive any relief by the more limited insistence that G-d is real, and actually exists.
Fine. I will take that as stipulated. Now, proceed from there to any instance of religion without erecting a very rickety scaffold of human invention and delusion.
What is new about the latest crop of "atheist" books is that they are really anti-theist, a different kettle of fish altogether. Your unwillingness to demonstrate how the metaphors of Hairy Thunderer or Cosmic Muffin are objectively incorrect, while your metaphor for G-d more closely encompasses objective reality.
Never mind that your metaphor makes you as much of an anti-theist as Hitchens or Harris.
Mike:
And if that's okay for mathematics, why not theology?
Because, if I was to fire up the RANG and insist there are fewer irrational numbers than integers, I should be well and truly shouted down.
That stands in stark contrast with theism, which (taken generally, and often even within a specific religion) makes any number of mutually exclusive statements about which it is impossible to assign a truth value to any.
This is also true about QM. Keeping in mind the distinction between descriptive and explanatory, QM very much specifies which descriptive statements are true, and which are false.
There is no statement about G-d for which this can be said.
hey skipper wrote: "Keeping in mind the distinction between descriptive and explanatory, QM very much specifies which descriptive statements are true, and which are false."
Is the multiverse concept descriptive or explanatory? Is it (or one of the other interpretations) true or false?
Skipper: If you think that, you misunderstand the metaphor. In particular, Duck is wrong when he suggests that I believe in something that can rationally be called the Cosmic Muffin.
David:
How do you get from your metaphor to any particular religion which is, by definition, at the exclusion of all others?
Hey Skipper: I wasn't arguing that theology is provable, but rather that uncertainty and vagueness exist outside of theology, even in things that we think are pretty obvious.
The Banach-Tarski paradox shows that there are mentally unpleasant consequences to the concept of infinite sets (you can cut a sphere up into a countable set of pieces, and through Euclidean motions reassemble them into two seamless, whole spheres the same size as the original - so conservation of volume/mass doesn't hold with infinite sets). And like the sizes of infinity (again, try thinking about what that might even mean?), these difficulties arise because the concept of infinity is incomprehensible.
My point is that when we don't understand something, we cannot be precise in our language, whether thats about God or infinitesimals. Being a materialist does not remove the concept of infinity, and so there are still fundamentally fuzzy concepts that defy simple description. And that's true whether or not you attach them to a metaphore for god.
Second point: The people who would shout you down about the irrational number's cardinality would still be arguing axiomatically, and deep down at the core of set theory, there are legitimate questions and paradoxes that leave open avenues of philosophical uncertainty. Bertrand Russel didn't spend 300 pages trying to prove 1+1=2 for nothing.
And many feel that, given likewise obvious axioms, you can prove the existence of God.
His ability to assert, without equivocation, some of God's essential characteristics that are in direct conflict with yours means at least one of you is wrong.
Skipper, just as a matter of rational logic, have you read into your analysis that both Neuhaus and David would agree that the nature and will of the Divine is, in large part, inaccessible and inscrutable, that we can only get imperfect glimpses?
The "many worlds" theory is explanatory, not descriptive. It's an opinion about the reality described by QM, it's not part of QM itself.
Mr. Cohen;
We can say that Islam and Christianity are mutually exclusive, because one of the greatest sins in Islam is believing in anything like the Holy Trinity, e.g. that God has a Son. And if you get rid of the Son, you've pretty much eliminated the entirety of Christianity. There are other points of equally deep incompatibility between Islam and Judeo-Christianity, which I can elaborate on if you want.
And with that, we come down squarely on the side of dogma. So, somebody explain to me what the conflict is between the Ducks and dogmatic religiosity.
Skipper:
it is this: religious claims should be open to the same analysis and criticism to which non-religious claims are subject.
C'mon, Rip Van Winkle, that hasn't been an issue in the West for several hundred years, except maybe on Harry's street. It's hard to argue with you when you make up your own definitions of words or claim authors don't really understand their own arguments. Hitches, Dawkins et. al. aren't really atheists? That will come as news to them. So will the argument that they think it's just dandy if believers do their thing without let or hindrance provided only they don't crash those wild and crazy non-believing parties.
Put your slide rule away and take a look at what is going on in the Western intellectual community. One of the problems with the 'Age of Faith/Age of Reason dichotomy so beloved by liberal 20th century popular historians like the Durants is that it fails to understand that the West was born and flourished through constant tensions between apparent incompatibles. Athens/Jerusalem, lord/bishop, church/state, reason/belief, science/faith, freedom/order, etc. Even the U.S. obviously grew to heady heights on a mixture of public freedom and private piety. The list is endless. The real trouble comes when either side tries to eradicate the other, as in Calvinist Geneva or the French and Soviet Revolutions. This is the project of the merry gang of atheists and they aren't mincing their words. They want the whole pie for themselves. You can't write best sellers that call religion a "genetic misfiring" or "parasitic virus", and religious education child abuse, and then pretend you are just running an undergraduate seminar in philosophy. And if you don't think these arguments are having and will continue to have some influence in family courts and child protection authorities you should get out more.
Geez, Skipper, this reminds me of the debates in the years after the war when everyone asked themselves how they could have been so blind and not believed that Hitler would do what he said--he laid it out clearly enough. I wonder if you had been there whether you wouldn't have said: "Well, hold on now, he never actually named that little town in Poland where he would build a big camp..."
Duck: I know, I know, I'm at it again, but Skipper started it.
Joe: The Duckians are all Jesuits in mufti.
What did I lose?
I take it you mean aside from everlasting life?
Never had it, so I can't miss it.
Ok, ok, let's stick with the here and now. Whay do you think folks like Dalrymple mean when they talk about Western civilization being grounded in religion and the atheist project being a rejection of it? I could try and pinpoint stuff like art and music, philosophy, justice, a common literary language, and also point to how every notion of decency and morality we have, many of them rapidly disappearing, is grounded in faith.
Being an atheist and taking part in the athiest "project", whatever that is, aren't the same thing. I'm fully part of the Western project, just without the extraneous, unnecessary metaphysical bits added by the theologians for effect. You complain that we gloss over all the intricate tension and drama that took place within Christianity, all the debates and questions, yet by saying that Western civilization is grounded in faith you are glossing over that very dirty, untidy process of tension and acting as if it was immaculately conceived and born within the confines of the Holy See. Western civ predates Christianity. Christianity grafted itself onto a fully functioning and mature civilization, the Greco-Roman world, and the more I read about the early years of Christianity the more I'm tempted to think that it was Rome that conquered the Church instead of the other way around. But anyhow, that history of tension includes a rich skeptical tradition that goes back to the early Greek philosophers, and which resurfaced during the Renaissance and the Enlightenment and is with us today. We skeptics are as much responsible for the shape of Western Civilization as you church builders.
But, Duck, surely you are being disingenuous here. You are too smart for that. It is one thing to say that, on balance, you prefer the modern, quite another to stand before the catherdral at Chartres and say: "I don't get it, what have I lost?"
Chartres is still there, isn't it?
If I had to sum it up, I would say the loss is the rapidly fading sense of an outward-looking life--a life whose parameters are defined from without as well as from within. Surely it is obvious that we are all becoming what Taylor calls "buffered individuals". Not only do we demand our objective, material freedom, more and more we define things like morality and reality itself as having no meaning beyond what is inside wonderful, deserving, endlessly complicated me. Personally, I think we are all slowly going insane for it, but that's just me. Look at how we reduce the world around us, including others, to rote and simplistic biological and psychological laws, all the while seeing the subjective me as so dense and rich only a five volume Russian novel could unlock its mysteries.
Mankind has been slowly going insane since the dawn of civilization. Your lament has been expressed by conservatives since Adam ate from the forbidden tree.
But you get it or you don't. Modern Western minds are very closed and unmoored, and this is a little like trying to convey the sense of loss from not learning Latin anymore to someone who never studied Latin. But I do believe the loss is felt more acutely and widely than we sunny-faced "have-a-good-day" North American moderns or the "nothing matters, all is a dark joke" gloomy Europeans admit to ourselves, which is why the defence of the modern world must rest on so many simplistic distortions and even egregious misreadings of history.
Christianity and Judaism themselves are built upon a misreading of history, beginning with the history of the Jewish people, who were never enslaved in Egypt, and therefore never escaped to conquer Canaan. But all this dwelling on loss is what drove me from the Church to begin with. So much pessimism is not healthy, especially when we are supposed to be thankful for the blessings of life. I'm thankful, I just don't know who to address that thankfulness to. You don't have to misread history to conclude that we have it better now here in the West than in most of the history of this world. Cheer up!
The modern cynicism about justice and patriotism? The disdain for honour and duty, yada, yada?
That's not dead, at least not here in the US. You're too quick to call things dead.
First, you guys are right out if you think that numbers and ideas are not real.
Second, all three of the Abrahamic religions accept that worshipers of the other two are on to a version of the truth. I might not understand how Christians can square the Trinity with monotheism, but I accept that they do in their own minds, that they are monotheists and that they worship the One G-d. I accept that there are righteous gentiles and that, if there is a heaven, it won't just be full of Jews. Christians, although many don't realize it, have analogous beliefs.
Islam, because it is unreformed, is a little more difficult. Muslims have not yet recognized that their beliefs make Allah subject to man's will. But they'll get there eventually.
If they do, they'll no longer be Muslims, just as the Christians of today would not have been accepted a Christians by their great grandparents.
Not all Christians are 10 steps along in a 12-step program toward becoming Unitarian-Universalists. Some are yet unreconstructed
I see reliance on divine scriptures and all that goes with that mindset as a continuous reversion toward antimodernism.
Raise hands all who prefer to live in an unmodern world.
Yeah, I thought so.
++++
I did not say mathematics are unreal, just unmaterial. I don't know what you mean by real. It's a dimensionless relation. So is religion.
Mathematics is not self-contradictory, though.
Would it exist without reflective intelligences to work out its theorems? Well, yes in the sense that the relationships would be there to discover. But no, in the sense that the mindless material world would not be changed at all.
Curiously, the discovery of non-Euclidean geometries did not change anything except how we behave. The sun did not stop shining.
"Would it [mathematics] exist without reflective intelligences to work out its theorems?"
Would anything? Does anything? How do you know?
Mathematics both exists and is real in my worldview.
A "mind" is very possibly just the collection of huge numbers of arithmetic operations.
If you agreed with the previous statement, would you conclude that minds not real?
Raise hands all who prefer to live in an unmodern world.
Yeah, I thought so.
Hold on a second, Harry. You and Duck have spent the bulk of this thread demonstrating that the world can be as modern or as unmodern or postmodern as we like, in piecewise fashion. (You, by pointing out that the various departments of science can look after themselves quite nicely; Duck, by pointing out that there will always be a Chartres.) I see no reason to think that we can't simply do unto modernism as modernism did unto Christendom. So, we keep the labor-saving gadgets; we ditch the salvation-by-works secular theology and the welfare state that rides on it, just as soon as you Great Society types die off; we move to the desert outside Vegas, put up a satellite dish, some claymores and some solar panels, and pop a cold one. Works for me, and that meets the standard around this place.
Mathematics doesn't contradict itself to the extent it's axioms do not contradict each other, which is not always known. When they are found to do so (eg, naive set theory), they have to be replaced with ones that are not.
My point in bringing this up is that there are ill-posed questions about mathematical concepts like infinity, but that this doesn't render the entire concept, as Duck put it, a complete null. You just feel disconcerted if you extrapolate too far from finite concepts. And at that point, you just go on Dogma. So I guess my answer about God's nature is that he's terrifyingly alien ("holy") and personal at the same time. Which I guess is weak, but so be it.
Pardon if this goes back to old turf, but if mathematics isn't material, then is language? How can we discuss a particular measurement without reference to an abstract conception of it?
Duck:
Mankind has been slowly going insane since the dawn of civilization.
Now there is the ultimate "It was just as bad or worse in the past" argument.
The product of two negative numbers is positive.
The product of two positive numbers is positive.
I is the square root of -1.
Besides which, Harry, your whole argument has been that, as a materialist, you can dismiss things that are not material. The only things that exist are those things that are material. We all agree, here, that G-d is not material, but neither is "2" or "freedom."
I don't think I said we can dismiss them. I said we don't have to worry about where they come from.
For personal comfort, we may want to maintain freedom.
'And at that point, you just go on Dogma.'
Does mind exist? Sure, it's particles firing.
That it remains little understood, or that it can imagine things that do not exist is not an argument for the existence of immaterial forces.
It's true that we have no guarantee that the relationships we work out have any more than the apparent structure that they have. Maybe it's all moonshine.
But none of you guys walk through closed doors any more than I do, despite your belief in spooks.
I do not think, though -- going back to the original point -- that most atheists are as firm materialists as I am.
Bret:
Is the multiverse concept descriptive or explanatory?
Both. At one level, it explains how there can be a universe with some astonishingly fine-tuned physical constants.
But peel that onion one more layer, and it is only descriptive.
Is it (or one of the other interpretations) true or false?
By including only two options -- T or F -- you have posed a false dichotomy. It is objectively true that the answer is either T or F. It is also objectively true that there is, for humans, no way of telling which. Therefore, the true set of answers has three elements: {T, F, null}.
I'm going with null.
Dunnoists fully understand that a for a great many questions, null is the only intellectually honest answer.
David:
If you think that, you misunderstand the metaphor.
No, I don't misunderstand the metaphor in the least.
What I do understand is that your metaphor cannot be reconciled with, say, Neuhaus's. That makes you, with respect to Neuhaus, an anti-theist.
Mike:
I wasn't arguing that theology is provable, but rather that uncertainty and vagueness exist outside of theology, even in things that we think are pretty obvious.
No doubt. But that isn't what I was talking about.
I'm sure you are aware of the Incompleteness Theorem. Theology perfectly inverts it. Within the realm of theology, there are mutually exclusive statements. Some of them must be objectively false.
However, there is absolutely no means to determine, within theology which those are.
So, for formal systems there is the problem of being able to formulate true statements whose truth is unprovable; in contrast, in theological systems there are false statements whose falsehood is unprovable. Which means there is no way to determine whether all, most, or only some are completely wrong.
Upon that sand, religious castles are built.
Peter:
Of Harris, Hitchens, Dawkins, et al, which have you read?
I completely understand that, in the West, religious claims are (largely, although religious figures get away with saying all manner of nonsense without getting called on it) open to analysis and criticism.
In the West. (Taken as excluding the Islamicized portions of Europe that most assuredly don't see things that way; or any number of Western religious leaders who are backing a UN anti-blasphemy resolution)
Unfortunately, there is that little Provenance Problem. The provenance of all theological claims is the same: how does one challenge Islam on its horrendous divine diktats without simultaneously challenging all revelation?
David:
Second, all three of the Abrahamic religions accept that worshipers of the other two are on to a version of the truth.
One of the words "version" and "truth" do not belong in the sentence.
You may accept certain things about heaven, but that sure doesn't mean either you are right, or your supersessionary co-religionists see it that way at all.
Mike:
So I guess my answer about God's nature is that he's terrifyingly alien ("holy") and personal at the same time. Which I guess is weak, but so be it.
Okay, I'll take that as stipulated.
So how to get from there to, say, Limbo. Or not Llimbo, for that matter.
And all the rest of the byzantine edifices that are religion?
However, although there is little or any Christian theology, his plays are awash in religious themes and references to the spiritual world and actors that inhabit them. He wasn't a proto-Duckian asking himself "What works?" in the face of every drama life threw his way.
Shakespeare was a master of human psychology. He portrayed the human condition, which is also the main concern of materialists, who have no other spiritual realms to distract them from it. Religion, in the end, is also concerned with the human condition, as much as it seeks to fashion an escape hatch from it. We're all humanists, and Shakespeare speaks to all of us.
Hold on a second, Harry. You and Duck have spent the bulk of this thread demonstrating that the world can be as modern or as unmodern or postmodern as we like, in piecewise fashion.
True, to a point. But there are some boundaries past which we leave the real accomplishments of modernism behind. Those accomplishments aren't to be found in the arts or architecture, for sure. But if we give up on science and technology for some medieval, spiritualized notion of physics and biology, then we would have left modernity behind. Likewise with freedom of speech and conscience. There are many groups among us today, some in academia, others in religious communities, who are constructing political ideologies every bit as repressive as that imposed by the medieval church. Freedom of speech and conscience can't be up for grabs, either.
Shorter Duck: absolutism threatens freedom.
So it does.
The inherent problem of all revealed religions -- whether you think the 3 abrahamic religions are cousins or not -- is that to be non-absolutist, at some point you have to say, 'Well, here I have the veritable word from on high, but in this case, I'm gonna ignore it.'
In that respect, to be modern and antiabsolutist is to be no different from me, but with more angst about it.
Duck:
But if we give up on science and technology for some medieval, spiritualized notion of physics and biology, then we would have left modernity behind.
I am speechless, Duck. What in the modern Western intllectual zeistgeist could possibly lead you to believe that would be the remotest possibility, or that anyone is trying to effect that? I know we can all Google the names of hillbilly televanglists trying to raise money for their creationist theme parks in order to forstall a victory by Al Qaeda, but c'mon!
This is similar to trying to have a serious discussion with radical feminists about the fallout of the sexual revolution on modern teenaged girls, only to have them insist that is the cost of doing business and the real battle is to remain ever vigilant lest the Old Testament patriarchs return to bar women from higher education and take away their shoes.
Whoa, calm down Peter. I was merely making a point to Joe that all things modern aren't equally valuable. What defines modernity, in my mind, are certain key political, social and intellectual values - secular, representative government founded on a platform of individual rights, social equality (ie, a classless society) and an unfettered intellectual sphere grounded in science. We can play with the pre or post-modern as much as we want with the peripheral things. It doesn't matter if we wear nose-rings or play the digireedoo or dress like pagans - we're still moderns if we adhere to those values I alluded to above. That's my point.
No greater menace to civilization than an Old Testament patriarch with a shoe fetish, Peter. Those feminists are wise to be so wary.
There's power in shoes. That's why so many of us chicks want more and more of them. Makes us feel secure. Just ask Manolo.
Skipper: I'm sure you are aware of the Incompleteness Theorem. Theology perfectly inverts it. Within the realm of theology, there are mutually exclusive statements. Some of them must be objectively false.
Your right about there existing contradictions in theology, but contradictions exist elsewhere too and they are likewise arbitrary and uncertain. When we don't know how to pick among contradicting mathematical statements we call that a paradox. Zeno's Paradox is a good one still.
Second, I don't think you have the incompleteness theorem's negation correct. The negation would be that every statement that is logically consistent with a set of axioms can always be demonstrated via a chain of 1st order logical arguments to be so. Which is not true about mathematics or theology.
That doesn't really touch on the correctness of the original axioms, though. many of the mathematical hypotheses which people have been trying to prove from our standard set of mathematical axioms, e.g., the Riemann hypothesis, may be true, but since no proofs yet exist, it may be that they are some of the unprovable statements of the standard mathematical axioms. So, if you want just, assume that the Riemann hypothesis is true, and a lot of mathematics does so, but it's not as ascetically pleasing to do so. But it could also be wrong, and we may not be able to ever prove that. And so like religion those axioms are subjective. They seem reasonable, and are chosen to be as few as possible, but they are still unproved.
That said, I think we intuitively feel that they are mostly right, and that not believing them is probably a lot harder than not believing the Bible or in ethics because, if we agree on the axioms, we can discuss them objectively with each other and say we are in agreement or not.
This falls apart when it comes to morality and religion because these truths are subjective. I can shoot someone, and everyone can objectively agree I did so, but whether I'm good or evil is subjective. So while there is some subjectivity in mathematics, I would argue like Hume that every moral system is completely subjective ("no is implies an ought"), including those that do or do not refer to God. You can demonstrate that torturing babies causes them pain, but you cannot prove that torturing babies is wrong. But we intuitively believe it is.
I agree that there are a lot of self-consistent religious/moral choices you can make without being able to differentiate which is correct, or at least not until you're dead and have been judged or evaporated into oblivion. It would be annoying to live as a devout Christian and end up in Shinto hell, and more annoying yet to end up in Reformed Baptist Church of God, Reformation of 1915 hell.
But I think that the uncertainty of the choice of a moral or theological system doesn't get you off the hook for making that choice. Which is irrational, but how else could it be?
Mike:
By the incompleteness theorem, I mean this:
If a proof system for arithmetic is sound (meaning that only true formulas are provable) then there must be a true formula which cannot be proven.
The inverse -- not the negation -- is:
For a system of statements, some of which must be false, but it is not known which ones, then it i impossible to prove which statements within that system are true.
Two consequences follow:
1. All theological metaphors are equally valid. Which is to say that theological metaphors provide no useful explanatory function; in other words, theological metaphors are really instances of wish fulfillment.
2. One of, if not the, primary counters theists make to anti-theists citing instances of religious evil is that those instances are somehow false, or "bad", and distinct from "good" religion. Within the system of theological statements, this response can no more stand than a house of cards on a windy day. This becomes immediately obvious because theists use material considerations to make their case. David, in Objectively Moral completely gave the game away when he provided the criteria for distinguishing between a religion and a cult.
Whether the criteria are useful or not is beside the point. What is very much to the point is that each of the criteria are completely material. One would think that theology's barking inability to provide any means to distinguish good theology from bad would make theologians a great deal more humble.
This has nothing to do with axioms. For example, the traveling salesman problem can be expressed as: For a set of nodes, there exists a linked list where the distance between nodes is shorter than all other linked lists.
Nothing axiomatic about that, or particularly difficult to either formulate or conceive: the statement is prima facie true
It is also true that, except for a trivially small number of nodes, it is (practically speaking) impossible to demonstrate a given list is shorter than all other lists.
Also, note that these are objective, not subjective statements. As in:
-- Jesus, as the Son of God, is divine.
-- Jesus, while a gifted prophet, was neither the Son of God, nor divine.
Those are both objective theological statements (Christian and Muslim, respectively), one of which must be wrong, but it is utterly impossible to know which.
Ya lost me, Mike.
1. Zeno's paradox is only so if you neglect acceleration.
2. But the religious nuts do not accept that morality is subjective. That's what Orrin's all about.
Mike:
Apologies, but my traveling salesman problem is an example of something else entirely, not the incompleteness theorem.
Unfortunately, a good, concrete example doesn't come readily to mind.
Skipper: I agree with you on the uncertainty of theology, but I'm going to insist that it extends to non-theistic thinking as well. All human thought is axiomatic, and all axioms are objectively arbitrary. If someone denies a mathematical axiom, like say the existence of infinite sets, because he does not like some of the logical conclusions that axiom implies, you cannot argue objectively that he's wrong, only that he's subjectively wrong. This is the same crumbling deck of cards and the same shifting sands that theology lies on.
I'm not quite sure what you were trying to get at with that NP-complete problem, but neither the statement of the traveling salesman problem nor any conclusions about it are objectively true because we first have to posit axiomatic set theory, to define what is a node, what is a set of nodes, what does it mean that nodes are connected, what does it mean to order the listing of the connections of these nodes, what does it mean that there is a least element in this ordering? Etc. This may seem too basic or intuitive to even question, but underneath the statement of that problem are these definitions and axioms.
There are two self-consistent mathematical theories in that wikipedia set theory article depending on whether or not you take item #10, the axiom of choice, to be true. This axiom is controversial because of a paradox regarding the conservation of volume. But the axiom itself seems plausible (that there always exists a way to label the elements in a set), and is often used to prove many other theorems.
Philosophers who are interested in having objective discussions about the correct choice of axioms rule out a choice of inconsistent axioms by axiom, but this still leaves a large number to choose between. Incompleteness makes our choice of axioms based on a preference for self-consistent theories somewhat moot because that preference cannot be fully demonstrated. After this, you have to invoke more subjective preferences to choose between them, e.g., Occam's razor. Ultimately, it comes down to "Because I said so." And so it may not be possible to have an objective conversation about axioms.
Regarding Jesus's divinity, by definition, it does not seem objectively decidable, but neither would be the axioms of any moral system, and neither is the axiom of choice.
Harry: Zeno's first paradox (Achilles and the tortoise) is often told now in a way that can be resolved by using calculus to find the limit of an infinite geometric series, but the second (the dichotomy paradox) isn't solvable by calculus because you have to start with the infintesimal instead of finishing with it.
Because I like to think that space and time are continuous, this is odd to me. Anyway, I would say that, whether tacity or explicitly, we all make a choice of axioms. Mathematics seems very plausible to me because of the complexity, richness, symmetry, ie, beauty, that it arrises from it, but there are paradoxes, or at least the appearance of paradoxes.
All uncertainties are not created equal. The uncertainty over whether Hillary Clinton will become the next president is not the same as the uncertainty over whether Stephen Colbert would be.
But I don't know of anyone who is basing their eternal happiness on whether String Theory is true or false.
Lucky for me, materialism doesn't encounter any infinities, not even of dimensionless qualities.
The axioms of set theory are indeed pretty, but there are no sets with infinite members. You can do anything you like with set theory, but it won't affect a thing.
I am not much of a mathematician, but the hidden -- and unacceptable -- statements in Zeno's paradox were obvious on a logical basis. It can be solved by calculus but it can be rejected in logic.
Duck: Yes, sand vs sandstone. Metaphorically speaking.
Harry: I don't think there's a logical solution to that second paradox.
And it's possible to construct infinite sets inductively out of the null set. That's fairly fundamental.
I hate to agree with Mr. Eager in public, but fundamentally all mathematics is just language, without any intrinsic correspondence to reality. It is not the case that "it's possible to construct infinite sets inductively out of the null set". It is the case that you can create a format language that lets you express such a concept. All of the different varieties of set theory aren't more or less true, they are simply dialects of a base language, to be selected on utility for some other, non-mathematical purpose.
Susan's Husband: I agree, actually.
It's just that if you are going to toss out infinity, then you do lose other things too. And those are things that we generally like, which is why not everyone is a hard materialist. I suspect that in order to remove infinity from mathematics, then you have to remove the null set or the principle of induction. Would you?
Mike:
You have pretty much completely lost me.
All human thought is axiomatic, and all axioms are objectively arbitrary.
No, it isn't. Human thought can be inferential, deductive, intuitive, etc. Further, all axioms are not objectively arbitrary. If that were so, no axiom anywhere at any time would be preferable, other than via arbitrary preference, from any other.
Simple example: In plane geometry, a straight line is the shortest distance between two points. That is an axiom. No doubt I could axiomatically assert that a half circle connecting those points is the shortest distance.
Any guess as to which axiomatic assertion is distinctly preferable?
Presuming you prefer the former axiom to the latter, upon what is that preference based?
Upon a whole bunch of good reasons, primary among them being that plane geometry choosing the former as an axiom is internally consistent, where choosing the latter would not be.
Systems to which the incompleteness theorem apply are internally consistent in that the introduction of any contradictory axiom will render the whole system meaningless.
In contrast, in theology, one may cook up any set of "axioms" to derive any conclusion one desires; there is no threat whatsoever of determining whether one axiom is true, or another false.
Other than, that is, through wish fulfillment.
(BTW, you have pretty much elided the difference between an axiom and an observation. Was Jesus' resurrection an observed fact, or axiomatic? If the former, then it had to both actually happen, and also embody some specific meaning. If the latter, than we can certainly unburden ourselves of having to worry about its historical accuracy. That is a position that won't earn much favor in most any Christian church. Are you Unitarian, by any chance?)
Zeno's paradox is inapplicable, in that it is only apparent. In other words, it is fairly safe to say there is abundant evidence of motion. Asserting that it is impossible creates an apparent paradox only with respect to the implicit assumptions. Once upon a time, it was "proven" that bumblebees cannot fly. Yet they do. That was not a paradox, merely the appearance of one, based solely upon the implicit assumptions of the "proof". (Laminar airflow being the erroneous assumption).
Just so with Zeno's paradox. A Scientific American article of some thirteen years ago (back when it was a respectable publication) used the concept of quantum mathematics to put paid to the paradox.
Returning to my original point: Within the scope of theology (that is, all supernatural religious belief systems) it is impossible to ascertain which statements, axiomatic or otherwise, or preferable -- other than through wish fulfillment -- to any other statement. According to Christians, essentially all of Islam is nonsense, root and branch. According to Islam, Christianity is not necessarily total nonsense, but it is woefully incomplete.
Whereupon we are faced two mutually exclusive sets of axioms, observations and metaphors within precisely the same problem space.
One thing is certain: one of them is wrong.
Sorry, two things are certain: there is no way to tell which.
Make that three things: both of them might be wrong.
There may be branches of mathematics that are internally inconsistent -- if there are, I don't know of any -- but I doubt one can get there without creating things that do not exist.
As Harry says, neither the infinite, nor the infinitesimal, exist.
I suspect that in order to remove infinity from mathematics, then you have to remove the null set or the principle of induction.
How does removing infinity affect induction? One could still use it, but one would no longer have to rely upon it, as it would be (at least theoretically) possible to achieve proof by enumeration.
And since I can create the null set out of two mutually exclusive statements (The intersection of the set of all Greeks named Plato are men, and the set of all Greeks named Plato who are not men) within an enumerable universe, I don't know how the lack of the concept of infinity could begin to negate the null set.
I certainly understand why most people shy away from hard materialism. We all know that we can, at any time, create immaterial ideas. Like the Flying Spaghetti Monster.
The FSM we understand is pure invention, a quirk of the way our brains evolved. Presumably octopuses cannot do it.
Where we go astray is to think that FSMs that other people invented are real. Like infinity.
Infinity is not real, though it turns out to be a useful way of thinking about things, which might also be true of FSMs and even, perhaps, of any Big Spook.
Although, most people when they think about the Big Spook come to the wrong deductions.
Anyhow, I don't have to give up infinities. I understand, sort of, the differences among them. I cannot do it myself, but I understand, vaguely, how physicists use infinities to exclude certain otherwise attractive theories about how material universes operate.
My physics adviser tells me that sound does not arise from current fundamental theories. I am not at all clear why not, but so he tells me. Nor, obviously, do imaginary deities.
The fact that we can multiply components unnecessarily -- as William of Ockham so brilliantly recognized -- should give us all to think.
Hey Skipper and Harry: Sorry for the confusion, it's not intentional. Looong reply to follow:
Skipper: Implausible as it may seem, your geometry hypothesis is actually good example of our freedom to choose axioms, because there are in fact Non-Euclidean geometries where the shortest distance between two points isn't a straight line. Space in these geometries is curved and so parallel lines either converge (elliptically curved space) or diverge (hyperbolically curved space). Essentially parallelism is a local property in Non-Euclidean geometry.
That sounds odd in three dimensions, but as that wiki article illustrates, it's easier to understand in two dimensions. A two-dimensional Euclidean space is a flat sheet, and yes, the shortest distance between two points is a straight line. This is by definition, since that's what flat sheet means, and an attempt to explain one in terms of the other is results in circular reasoning. By contrast, the two-dimensional surface of a sphere is a non-Euclidean space with elliptical curvature. The shortest path between two points is given by a "great circle" path. If you draw two points non-overlapping points on the sphere, and then construct two parallel lines through those points, they will intersect elsewhere on the sphere. For antipodal pairs of points there are an infinite number of shortest paths. Elliptical space doesn't quite meet our intuition, but there's nothing wrong with it either.
In three dimensions, the curvature of space causes it to dilate as we move around in it. If lines move together at a constant rate, it's elliptical, but if they grow apart at a constant rate, then it's hyperbolic. If the curvature of space varies from point to point, then the space is called a Riemann manifold.
These four axioms produce self-consistent geometric theories, and furthermore, the Euclidean and non-Euclidean geometries have been shown to be equi-consistent; that is, the axiom of parallelism doesn't affect their relative self-consistency. Beyond this axiom, they (all) may not be self-consistent due to an inherent contradiction in some of the other axioms that go into their construction, and due to the incompleteness theorem, we may never be able to detect this. These theories exist in the same problem space, in that all of the other axioms fixed (well, not for the Riemann geometry, but that is because it is a more general theory in that it doesn't require space to curve isotropically), but they contradict each other and draw radically different conclusions (the interior angles of triangles in elliptic space add to less than 180 degrees, in hyperbolic space, they add to more than 180 degrees, etc) Whether or not you feel that non-Euclidean spaces are unphysical doesn't bear on their existence because the question isn't between Euclidean or non-Euclidean, because they are equivalently physical based on the question of whether parallel lines intersect or not.
Now this might all be academic except that, if you state that some mathematical axioms are preferable to others because they are congruent with physical experiments, then General Relativity states that gravity is a geometrical effect due to the local elliptical curvature in a four-dimensional spacetime manifold generated by objects with mass. All of the measurements of theoretical predictions of this hypothesis that I know of agree with the prediction to a large number of digits. So it would seem that space curves, and by different amounts in different places. Extending this to cosmological measurements, the entire universe seems to be slightly elliptically curved. But Relativity is a model, and it doesn't even tell us why or how mass works, and our measurements have finite precision and accuracy. Relativity is controversial still, and attracts a large number of crackpots (helpful identification guide), because it's so counterintuitve (particularly time-dilation).
So, does that mean that four dimensional Riemann Geometry is fundamentally true because we observe it in attempts to understand physics? Maybe, but it would not surprise me if it was replaced by something more fundamental. In that case, space may not have four dimensions, but who knows, something more. The proponents of superstring theory claim space is ten dimensional because that allows them to unify all of the forces into different manifestations of one force, and that we can't see these extra dimensions because they are very small. It's an open question whether these dimensions are observable at all or if superstring theory can ever be tested (so, currently superstring theory gets 50 points on that crackpot index #37). Consequently, it pains me to say whether a particular mathematical concept is an absolute truth or just an imaginary concept because it's physical or unphysical. I believe that "physical" is not rigorously well defined.
Here's why I think that: let's talk about physics theories that might be true given what we perceive. The question about how fundamentally true a physics theory is hard to decide because there is noise in any measurement. Some measurements are binary and either a phenomenon is there or it isn't. Either the atomic energy levels split in the presence of a magnetic field or they don't, etc. But suppose your theory has enough defined entities and interactions to explain all observed phenomena. Then come questions about the parameters for phenomenon that you observe. How much do they split? Even if you had the really real theory, this noise will causes discrepancies between the theory and the experiment. Consequently, it's hard to know whether the difference arises because of noise or if there is a divergence between the model and reality.
As the precision of the measurement is improved, or more revealing experiments are devised, the discrepancies between the theory and experiment may grow to a point where we can say that, unless there is a flaw in our experiment design or analysis, there is something like a six-sigma confidence interval that something is wrong with the theory. At this point, we have to revise the fundamental axioms of our physical model to curve fit the experiments. So a lot of new hypotheses are generated that attempt to cover all of the known observations, and one is usually chosen because it is able to explain all of the observations and hopefully make additional predictions that are testable and found to be consistent, and then because it's more "beautiful".
Generally speaking, the old theory is generally seen as a limiting case of the new one, but the fundamental elements have changed (eg, space and time vs spacetime; Euclidean additions of velocities vs non-Euclidean additions in Minkowski space, etc), and so maybe spacetime is it, or maybe it's only a part of something else that only manifests itself above a measurement noise floor at higher energies, smaller/larger distances, etc.
Second point: Some theories work well, but have aesthetically unpleasing aspects. Quantum Electrodynamics is one of the most successful theories ever (quantitatively, it has been more precisely and comprehensively tested by measurement than Relativity or anything else). It's predictions for certain parameters can be compared to measurements with many independent determinations (ie, you derive a value from one set of experiments involving electron spins, another. The agreement between theory and measurement goes to 14+ digits, and the linearity of the mode number (the linearity of the infinite collection of harmonic oscillators that comprise the field) has been demonstrated over 22 orders of magnitude. But it has to be renormalized - essentially, getting finite answers for these predictions required adding in offsetting infinite values. And these infinite quantities refer to infinite energies. It's not coincidental that it has been tested so thoroughly.
Now, if you believe a fundamental law of nature exists, our discomfort with an aspect of it is immaterial, but that sort of discomfort has either lead to the prediction of new particles based on the suggestion that they will fulfill symmetry, or new theories. So is QED fundamental or not? Can you physically add two infinite quantities to come up with a finite quantity that you measure? I'd argue for agnosticism here, with the caveat that it seems to be a pretty good working hypothesis. But I certainly am not worried about judging whether mathematical concepts are real or not based on whether they appear in the description of reality given by QED, the opposite method seems better.
This is getting long, so I won't go into the problems of the interpretations of the concept of measurements due to quantum mechanics, and pardon the jargon, but in particular, the philosophical issues about identity and causality generated by measurements of the Bell State Inequalities to resolve the Einstein-Posen-Rodensky paradox. I helped a summer student set up the experiment to measure this using parametric down-conversion to produce entangle photon states and measure their superluminal correlations. Needless to say, we didn't overturn the conventional wisdom. The state-of-the-art measurements come down pretty squarely on the side of non-locality.
I don't have a reference, but I went to a talk where Antwain Zellinger reported on experiments where the measured statistical values are 30 standard-deviations away from what a broad class of "realism" theories allows. There's something weird/spooky about the process of making measurements at the quantum level. Now, there's nothing that says are our intuition has to scale across all physical length scales, and so why should microscopic processes conform to our common-sense expectations about identity and causality? However, Schrodinger's cat is a thought-experiment which directly leverages the microscopic into the macroscopic. What does it mean to be in a totally ambiguous macroscopic superposition state? Hence the intense discomfort about the theory.
Anyway, the above is an attempt to explain why I think there are limitations to appeals based on physicality. I have plenty of confidence that most of what we think is true, but if you asked me to shoot someone based on whether infinity physically exists or not, I wouldn't. I do believe that the question of what is physically possible is limited by what is logically and mathematically possible, but those limits are axiomatic.
I think it's philosophically unsound, but if you wanted to, you can doubt these results too. In a pedantic philosophical sense, we are making lots of assumptions, including assumptions that we exist, that things exist, that we can identify and count them, that language means things, that logic exists, that laws of nature exist, and that these laws are isotropic across the universe.
I think it's obvious that we exist; I think it's obvious that the rules of logic are true, because to refute them, you are both rejecting any semblance of rational discourse and invoking them to do so. These things are axioms. I think math exists, and I think there's a fundamental law of nature, even if these things are unknowable.
To follow up on the question of infinity, what if it could be experimentally demonstrated? QED is one place to point to here, but suppose an experiment was devised that tested the many-worlds interpretation of QM. Incidentally, I think this theory is not rigorous because measurement and object are undefined, and it's also really strange because in order to explain an unpleasant aspect of QM, it hypothesizes that there infinitely many invisible worlds. That seems like a Pyrrhic victory at best. But anyway, I don't see why it's not logically possible to have an infinite number of worlds, and suppose it was somehow testable in the way that Bell's Inequalities allows us to test non-locality for QM, and suppose to good experimental evidence, the MW theory is true. Would that change whether you think infinity exists? Did my discussion above change whether or not non-euclidean geometries exist?
Let me get back to you on the other stuff, but I do think that the Zeno's Dichotomy paradox is deeper than people realize, and that the attempts to defuse it don't satisfy me. Here's why:
Cauchy came up with a good way to make infinite sums well defined using an delta proof. For a given sum, it's possible to decide if it converges and to what value it converges to, if for every number delta there is a point in the series for which the finite sums are always within range of the final limit given by delta. . This represents the solution that people give to the paradox. But to do this, they rewrite it, so that the infinite term comes last
1/2 + 1/4 + 1/8 + ... = 1
However, as Zeno gives the paradox, the infinitesimal comes first (you cannot take the first step); this is undefined, and so the series is undefined. They are identical series except for the ordering, and so one is defined and finite, and one does not exist. Hence the paradox.
Re infinity and induction; the existence of infinite sets follows from the existence of finite sets and induction. The null set is a particular example of a finite set. von Neumann's construction follows
0 == {}
1 == {0,1] == {{}}
2 == {0,1,2} == { {},{{}} }
and so forth. The null set exists (by axiom). Induction is true, hence it follows that the infinite set we call the natural numbers exists. If this doesn't follow, then one of the premises must be false.
Mike:
Wow.
Two things:
First, I am nowhere near to hitting in your league.
Second, I have a longer response, but it will have to wait until tomorrow.
The "many worlds" theory doesn't imply a truly infinite number of worlds, only an arbitrarily large set. Each wave collapse generates a finite number of worlds, and there have been a finite number of wave collapses since the start of time, therefore the total, however large, is still finite.
Mike:
Like I said, I am so not hitting in your intellectual league.* I pretty much understood -- save for Bell State inequalities, anyway -- your entire reply; I sure as heck could not have written it.
But I shall press on, regardless.
Implausible as it may seem, your geometry hypothesis is actually good example of our freedom to choose axioms, because there are in fact Non-Euclidean geometries where the shortest distance between two points isn't a straight line.
Which is why I confined my example to plane geometry.
However, so long as I confine the problem space to plane geometry, my choice of axioms has no need of considering other geometries, and if I was to adopt axioms from those other geometries into plane geometry, I would have a heck of a mess on my hands, because I would have mutually exclusive axioms.
Fortunately, anyone with sufficient knowledge of geometry would have good cause to reject the axioms adopted from a different problem space, and retain only those whose use did not lead to absurd results.
This is consistent with my invocation of the incompleteness theorem. Using an example that I hope is closer to the mark than my regrettable reliance upon the traveling salesman problem, within plane geometry it is possible to formally state that only four colors are required to construct any map so that two adjacent regions do not share the same color.
So far as anyone knows, that is true. However, IIRC, no one has been able to prove it: within formal systems it is possible to make statements that are true, but can't be proven (with the corollary being that all false statements are provably so).
Consequently, it pains me to say whether a particular mathematical concept is an absolute truth or just an imaginary concept because it's physical or unphysical. I believe that "physical" is not rigorously well defined.
Here you lost me a bit. IMHO, mathematics cannot possibly be absolutely true, because to assert that is to make a category mistake. Mathematics are an abstraction of reality, not reality itself. The Theory of Relativity is not time, mass, speed and gravity -- it is an abstraction of those things.
As the precision of the measurement is improved ... unless there is a flaw in our experiment design or analysis, there is something like a six-sigma confidence interval that something is wrong with the theory.
As an historical aside, as soon as telescopes sufficiently powerful to distinguish binary star systems were developed, Newtonian mechanics were faced with a fatal contradiction -- the orbiting star should appeared like a ring around the central star.
But anyway, I don't see why it's not logically possible to have an infinite number of worlds.
Or a very large number, or just one. Science, in this regard, is still working in the same realm theology always finds itself: unable to conclusively distinguish between mutually exclusive theories competing within the same problem space.
suppose it was somehow testable in the way that Bell's Inequalities allows us to test non-locality for QM, and suppose to good experimental evidence, the MW theory is true. Would that change whether you think infinity exists?
As AOG said above, that would change my mind from sheer agnosticism on the subject to (depending upon the evidence) agreeing either a very large, or infinite, number of worlds exist.
Did my discussion above change [your mind] whether or not non-euclidean geometries exist?
It would have, had I not already known it. (In fact, as a pilot, I work with non-euclidian geometry all the time.
... but I do think that the Zeno's Dichotomy paradox is deeper than people realize, and that the attempts to defuse it don't satisfy me.
That mathematics is an abstraction of reality is one reason (the other could be blinkered pig ignorance) I find Zeno's paradox relatively uninteresting -- it relies upon the abstraction insist reality is impossible. If Zeno's paradox says motion is impossible, so much the worse for Zeno.
Besides, as Harry noted above, it seems to push F = ma over the side. None of those terms are (SFAIK) the result of infinite sums. Similarly for velocity and time.
Yet somehow the d part of d = vt must be the consequence of an infinite sum with the infinitesimal first.
Like I said, so much the worse for Zeno: as with the bumblebee paradox, it is the abstraction of reality that is wrong.
the existence of infinite sets follows from the existence of finite sets and induction.
True. But what, in a material sense, does that mean? If I was to construct a set consisting of all the fundamental particles in the universe, that would, presumably, be the largest possible material set. An infinitely large set of anything does not exist, but that does no violence to the concept (a useful abstraction of reality) of either induction or infinity.
Your response reminds me very much of the quality of Scientific American before it developed a very bad case of Omni Envy. I subscribed for roughly fifteen years before it dove into a ditch, and for only six months after.
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Full intellectual limits disclosure: my educational background includes several differential equations, several semesters of calculus and physics, and a masters in Computer Science.
All of which is at least 14 years in my deep six.
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