I don’t think that people have any idea quite how large infinity is. Good reason for this. Infinity is big. It’s bigger than big. It’s hugeous (yes, hugeous: right out of O’Brian, that word). No: correction: it huger than hugeous. It’s beyond anything you can possibly think of. Here’s some notation invented by Knuth to help us understand what we cannot understand:
10 ↑ 10 = 1010 = ten billion.
10 ↑↑ 10 = 10 to the 10 to the 10 to the 10…ten times.
In other words, a pretty big number. There are 10 exponents here; the last two we know are equal to ten billion, and using the seventh is 1010 billion, and the sixth 10 to the 1010 billionth power, and so on. Hugeous. But that hugeosity pales, absolutely shrivels, next to this:
10 ↑↑↑ 10,
which takes as exponent the hugeous number we just worked out. And you can keep doing this, producing unimaginably big numbers along the way, such as
10 ↑↑↑↑ 10,
10 ↑↑↑↑↑ 10, and so on.
How long can you keep that up? Well, forever. And it turns out that the wee number 10 ↑↑↑↑↑ 10 still hasn’t even come close to infinity. It’s just as far, relatively speaking, from its goal as 10. (Big number fans will enjoy reading about Graham’s number, which uses the Knuth notation.) Strain your brain with this, gentleman and ladies (and you, too, JH). Ponder just how far away infinity is and how, because of its distance, we can never really know what it’s like out there.
So much is preliminary. Now the big question: has the universe eternally existed? By universe I mean all material reality that exists. Eternally means forever, which in turn means an infinitely long time. We have seen that infinitely long is a mighty concept, implying a slice of time so large that we can’t picture or comprehend it. Still, we might be able to deduce consequences of this strange supposition.
If the universe is eternal, then anything that was possible has already happened. How many times? An infinite number of times. Now if—a big if—we are entirely material creatures, that means there already were infinite duplicates of you reading these same words in the same place wearing the same clothes, even. Why? Because we are made only of material things that came together in a certain way, proceeded by other events that also came together in certain ways. This means there were even times when everything was the same except that you forgot to put on your underwear. How embarrassing. Doubly, so, really, because it’s going to happen again. And again. And again…Say, maybe those reincarnation fellows are on to something. As Woody Allen said, we’re going to have to relive the Ice Capades. Of course, it isn’t you each time, but material copies.
Well, that repetition trick only works if we are entirely material creatures, which we are not. For one, our intellects cannot be material, which is why there is only one you, so something more is at work.
But forget about us poor, bare, forked animals for a moment, and figure this. Conditional on some evidence, some physicists say the universe will end in an inescapable heat death, a place where entropy has maxed out and where nothing really happens and from which there is no escape. If that’s true, and the universe is eternal, this heat death should have already happened, and thus we shouldn’t be here. And not only that, suppose there are other possible ways for material existence to stop, cease, or otherwise obliterate itself. Suppose, early on during the initial moments of the big bang, the forces acting on the nascent universe were such that inflation, or whatever, reversed itself and the universe fell back into its singularity. Let your imagination fly, here, but restrain it by what we know of physics, or in those cases where we don’t know, which are many, of what is plausible given what you already know.
(Confused? Don’t be: the only point is to generate a scenario wherein material reality destroys itself. Incidentally, multiverses, or whatever, are no way out of this, because when I use the word universe, I mean all of reality, including those bits, multiverse fans say, that are closed off to our knowledge. The problems and screwiness of infinity and multiverses are also fun to think about—another time.)
If the universe is eternal, then these cataclysms should have already happened, too. They didn’t, so the obvious conclusion is the universe is finite. Meaning it started at a definite point in time, much like many think the Big Bang did. And the only way this could have happened is if material reality popped into existence out of nothing. But nothing, which is the complete utter absence of anything and everything, including whatever you can think of or name, has no causative powers.
One of our foremost metaphysicians, Billy Preston, said it best: You gotta have something if you want to be… “To be” means to exist, and the only way there is to make some thing be is through something actual. If you want a pancake to be, you have to start with actual flour, and the same is true if you want to create a universe from which flour can arise. Since material reality could not have created itself, and nothing sure couldn’t have—nothing is the ultimate slacker—you gotta have something else.
Best answer for this something else is: pure unchanging omnipotent actuality itself. Which is to say, God.
To be continued…
RE: “Best answer for this something else is: pure unchanging omnipotent actuality itself. Which is to say, God.”
Ah, but which God?
As even a superficially cursory review of the theological doctrines of the Catholics, Episcopalians, Baptists, Evangelicals, and so on…will reveal any number of contradictory & irreconcilable doctrines, values & beliefs.
They claim they all know the correct theology, all reference mostly the same documents (the King James version dominates, the Catholic version varies a bit) and ALL point to the very same founder figure…and they have so many inconsistencies that to say they all understand “The” God is debatable vs. they have each defined/created their own particular God.
If there is just a single God, the obvious conclusion is that the nation is comprised of, mostly, heretics — with all those competing & irreconcilably contradictory doctrines, most must be wrong.
But its nice to talk in generalities–it avoids addressing the real, substantive, issue.
I will admit that I don’t understand this. Why, if the universe is actually infinite, would it follow that all material things necessarily have already existed an infinte number of times? I don’t have enough knowledge of math to understand this. I understand the concept but not why it is necessary given the assumption that the universe is infinite.
Ken: Your question/observation seems to have a “gotcha” quality about it, but I’m not certain why you think so. Yes, that IS the implication, which any serious religious believer knows. But you are putting all inconsistencies and disagreements on an equal level (with no justification — people can disagree about all kinds of points with only some, and not necessarily any, meaning that their conception of the overall idea they are talking about is incompatible ). If you and I disagree about the details when we recount the story of something that happened to us both yesterday, that doesn’t mean that the thing didn’t happen or that one or both of us is wrong about everything that happened yesterday.
Wow, a pop music reference. But a nice one.
This is to confuse what can be known by reason unaided by grace, and what can be known by reason only aided by grace (revelation). I can know that Jesus existed as an historical person by reason alone, but I cannot know that he is the second person of the blessed Trinity by reason alone.
Infinity is not quantifiable. It’s outside the methods we use to explain “largeness”.
If the universe is infinite and time is infinite, I don’t see that it follows that everything has already happened (agree with Gail here). Things within time and the universe should be able to change without requiring everything having already happened.
Consider that the universe cannot both be expanding and infinite.
As Captain Janeway noted on Voyager “Time paradoxes give me a headache.” “Nuff said.
Yes, a variety of Christian churches were constructed, and they were all based on the same Scripture (save the Apocrypha/Deuterocanonical books), but one should not lose sight of context when discussing their divergence. Catholic doctrine was influenced by ancient Greek thought applied to Scripture; concepts like transubstantiation are not rejected by the Protestant churches for Scriptural reasons, but because the framework of Aristotelian physics that transubstantiation has been built upon has been largely discredited, or (worse) because they have their *own* ideas to impose upon physics which stand in the way of simply accepting Christ when he says “this is my body” and “this is my blood”.
“If the universe is infinite and time is infinite, I don’t see that it follows that everything has already happened (agree with Gail here).”
If the universe is infinite in time, then everything possible, no matter how unlikely will happen at least once.
Now the universe could have a beginning and extend infinitely into the future. However to say the universe is eternal implies that the universe is symmetrically infinite in time. That is the universe has no beginning, it extends infinitely into the past just as it does into the future.
In an infinite amount of time, everything possible no matter how unlikely will happen at least once.
It has taken an infinite amount of time to get to now.
Therefore everything possible has already happened.
Space is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.
Nice post; I eagerly await the second (or third? or … up to googlplex^googlplex part). I guess you know what St. Augustine and Cantor had to say about infinity and God. If not, there’s a fine article by Adam Drozdek (a computer academic at Duquesne University), “Beyond Infinity”. I thought there was a web link to the original article, but all I can find now is an abstract in French.
So…here’s some more shameless self-promotion– a discussion of this article on my blog:
The best way I know of to bring students to the point that you can get some to start thinking about what “infinity” might mean is to have them figure out what the practical effect of dividing a very small large number (e.g. 10 to the 90) in half is.
For science kids, tell them to produce a list of all primes less than your target number; wait a day or two; and then tell them they only have to do half of them.
For others – ah, don’t bother.
Space is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.
Douglas Adams, The Hitchhiker’s Guide to the Galaxy
English humorist & science fiction novelist (1952 – 2001)
You beat me to it –
However, not everyone knows where that’s from
Thanks for the tip about Rucker’s book. Didn’t know that one (though, of course, I’m no Platonist).
Infinity is not a number!
MattS: I feel that headache coming on…….
JH is not a number, either!
For further evidence of why this gives one a headache, infinity is both a number and not a number. (Banging head on key board now….)
Okay so the idea is that if the universe is infinite, and everything is material, and movement in all material things has a cause, then all things that are possible to happen have happened because causes are infinite (there was never a first cause) and there is no “in the middle” or “toward the beginning or end” of what can happen. There is just endless cause and effect. Is that the idea? If so then I sort of get it…
No, I am not a number. Neither are you. I am not an infinity, neither are you. However, it’s no joke… infinity is a concept.
One funny thing is that, in certain set-up, it’s possible and the goal to seek certainty in infinity (the limit as something goes to infinity) in the study of mathematics and statistics.
Infinity is a concept.
If the universe is eternal, then these cataclysms should have already happened, too. They didn’t, so the obvious conclusion is the universe is finite.
Should or must have happened?
How do we know they didn’t happen since parts of the “universe” (including the possibilities of multiverses and perpetually budding subsequent universes) are not knowable?
If we can’t know, then how can the conclusion be obvious?
Briggs, would you clear this up? It seems a weak point in the argument.
JH: I agree that infinity is a concept, but it’s also a number as far as many mathematicians are concerned. It all depends on semantics. Wiki calls it concept. There is a discussion here on the idea: https://www.quora.com/Is-infinity-a-number
One of the commenters at quora also points out imaginary numbers are not really numbers. I think we could argue to infinity on this (pun intended).
I’m separating this comment:
I will try to explain why I think there is not a problem in time being infinite and everything not having already occurred. First, if you accept the time-space fabric explanation of the universe, this is probably not going to work for you. Time-space fabric does seem to say everything has already happened somewhere, though maybe not everything that “can”. Things can change within the fabric, I think, without having to already occurred. I’m not sure on that one. However, the time-space fabric has a multitude of problems with the infinite outcomes, time travel, etc. (Thus the headaches….)
If you consider time to be never ending and without a beginning, yet linear, that does allow for things to have not happened yet. Yes, I realize how far out of the mainstream physics this is. I didn’t come up with the idea after drinking and drugs nor after watching too many sci fi shows. It is a carefully considered position. It explains a lot of things space-time fabric does not.
I’m not trying to talk anyone into believing this and i cannot fully explain it—I have tried and continue to work on the explanation. (So Will, leave it be.)
Infinity not a number.
Go knock an eight over* and tell me again.
* we used to have great fun on the farm in our high school years getting drunk and tipping over the neighbors’ eights.
… and I just remembered I have to go to the fabric store.
Ah, but which God?
Obviously, if nine blind men disagree on the nature of an elephant, that proves there is no elephant.
And if Copenhagen, many-worlds, standing-wave, transactionalists, and others disagree on the nature of quantum mechanics, that means there is no quantum mechanics.
the framework of Aristotelian physics that transubstantiation has been built upon has been largely discredited
a) transubstantiation is not built upon Aristotelian physics.
b) Aristotelian physics is equivalent to Newtonian physics for motion of location within a plenum.
c) Aristotelian metaphysics has not been discredited, only disparaged. It is however making a clandestine comeback.
d) Transubstantiation is just like transformation, except it involves substance rather than form.
“Now my own suspicion is that the Universe is not only stranger than we suppose, but stranger than we can suppose.” J.B.S Haldane
It’s my suspicion, too, hence IMO it’s ultimately futile to contemplate “infinity” (and, while we’re at, “divine” beings…).
Note that I had to substitute “stranger”for the word in the original quote to get past moderation…
No, it’s because you’re new here. All new readers automatically go to moderation. You’ll be set from now on.
I’ll have to try to look it up.
I believe Peter Woit brought it up in his book titled “Not even wrong”.
There is a quantum theory out there that the reason the universe exists is because it turns out that Nothing is unstable.
Seems to me that Brigg’s has to come up with a theory as to why God picked 15 billion (or so) years ago to create the universe and my unnamed quantum physicist has to explain why Nothing got unstable enough to enable the big bang only 14 billion (or so) years ago.
As for me, I going to go have that headache now.
But nothing is the epitome of stability. Nothing can be stabler than nothing. Of course, I don’t know the details, but any attempt to ascribe properties to nothing must necessarily fail.
@Mr. Briggs: “But nothing […] has no causative powers”
Why would ‘nothing’ need to have causative powers? What’s wrong with the universe being self-contained? If the universe had an edge, you would presumably accept that there was ‘nothing’ outside – just think of the big bang as an edge in time.
“If you consider time to be never ending and without a beginning, yet linear, that does allow for things to have not happened yet.”
Sorry, but no that doesn’t work. Even assuming time is strictly linear, in an infinite amount of time everything possible must happen at least once.
Therefore if time is symmetrically infinite they everything possible has already happened at least once.
The only way to allow for infinite time and having things that have never happened before is for time to be asymmetrically infinite. That is time is without end, but it does have a definite beginning.
The best way to think about symmetrically infinite time is time as a closed loop, where everything happens over and over again in cycles. You can have things that haven’t yet happened this cycle, but you can not have things that have never happened before.
Bill S: Why God picked 15 billion years ago to create the universe (assuming that is correct), it’s probably because he was busy with something else.
Matt S: Why must everything that can happen, happen? If infinity is a valid concept, that means that nothing “new” can possibly happen? Is that what you’re saying. Is it the hundreds of possible outcomes actually exist somewhere? Not exactly multiverses, but every outcome—and what about “impossible” outcomes? Could they have been possible at some points in infinity? So many questions, I know. I’m going to have to ponder more, after the aspirin kicks in. I kind of get the asymmetrically infinite, but there are problems with infinity in one direction (as in how can it exist?). Are we sure heads can’t explode exploring this stuff?
As a possible answer to my last comment, has anyone seen the movie “Pi”. That did not have a good outcome…….
How to tell the difference between an infinite universe with lots of identical copies of places, and a universe that is closed, allowing you to return to the same spot?
If the universe is infinite going back in time, that doesn’t prove that everything must have happened. E.g., think of a rock sitting in space doing nothing forever.
In an eternal infinite universe an infinite number of situations (or combinations) will occur. Repeats are not essential in an infinite universe. In the context of the infinite universe everything that occurs is a fresh instance.
Time is an attribute of the universe. The universe is occurring in a realm outside of time. Therefore, what does *eternal* mean?
If the universe collapses in on itself, then it expands and contracts for an infinite period of time, which easily confounds the argument put forward in this article. I’m not inclined to think the universe is infinite either, but that doesn’t mean I’ll agree with a bad argument for why it isn’t.
And of course, as any atheist will tell you, pushing the problem back onto an omnipotent being solves no mysteries, and in fact creates a greater one, which is where did the omnipotent being come from? The stock answer will be, he always existed, and the atheist will then quite reasonably apply Occam’s Razor.
If the universe collapses in on itself, then it expands and contracts for an infinite period of time…
Ah, the good old Great Cycle that recently gave us the Mayan Calendar foolishness, gave rise to astrology, and stifled the emergence of natural science in countless cultures from China to Mexico. Maybe this time will be different. Has anyone explained how it avoids entropy?
pushing the problem back onto an omnipotent being solves no mysteries
What mystery is that? It’s not a hypothesis put forward to explain some puzzle in physics, after all. A purely actual being is simply the logical consequence of kinesis in the world, and the rest of it follows deductively from that.
where did the omnipotent being come from?
Some folks here have argued that not everything needs to have a cause. Perhaps you can tell us why this one does.
The stock answer will be, he always existed, and the atheist will then quite reasonably apply Occam’s Razor.
Actually, he will misuse the Razor. Brother William intended an epistemological principle, not an ontological one.
Although it’s not clear how you imagine this idealized atheist would apply it.
Poincaré’s recurrence theorem guarantees that for classical systems, under certain technical hypothesis that I will not specify, that almost every point in the phase space (in the sense of measure theory) returns to a point arbitrarily close to it. Assuming that time is infinite in the past direction, it will do it infinitely often.
Slightly less facetious, if someone disagrees with (in MattS formulation) “if time is symmetrically infinite they [sic.] everything possible has already happened at least once” then there is a possible state of affairs that has not been actualized in such an infinite amount of time. Then said person has to enlighten us what “possible” means, since it seems not even an infinite amount of time has managed to bring it about. How and why is such a state of affairs a real possibility?
LOL! Because things exist therefore there is God. Now there’s some logic for ya’! Here’s some better logic for ya’ – because people have a hard time coming to grasp with the scope and breadth of reality, they invent God to make them feel comfortable.
Dr Brigg’s is indulging in these various scientific/metaphysical speculations. I merely pointed out the expansion/contraction speculation, that he conveniently avoided, which tears down his whole argument. If it was up to me, I wouldn’t bother with any of these speculations, much less be invested in any of them.
“A purely actual being is simply the logical consequence of kinesis in the world”
Nope, that’s just random gibberish words you’ve strung together in the hope that you’ll sound profound and nobody will point out that this little emperor has no clothes. Once again you confuse your feelings with rationality.
To all: I haven’t seen anyone mention Roger Penrose’s “Cyclic Conformal Cosmology”, set out in his “Cycles of Time”. There was presumed experimental evidence in “circles” in the microwave background radiation, but whether these are “real” is disputed. (By the way the purported evidence invokes wee p-values–as do the disputants–an interesting statistical subject.)
for more details. Intelligent, informed comments given in the spirit of gracious discourse from anyone?
In the true spirit of the trilogy, lettuce remember to refocus on the fact that Dr. Brigg’s has written an extremely good article on the ramifications of the word infinity.
I do need to ask if vacuum is the same as nothing inasmuchas it is the infinite potential of vacuum energy that allows some to believe they understand the big bang.
Me, myself, and I have a great deal of difficulty with Feynmen’s renormilization (repugnant comes to mind) but as a fizz cyst I have to go with what works.
And I have not even started looking for Asimov’s book that talks about aleph naught.
But I don’t seem to care if time goes back infinitely or if time started with the big bang. Either way I come to like Brigg’s assertion that there must be God.
1) Stand at the southernmost point of Jenny lake and look north. Then ponder infinity.
2) Sorry, but if time extends back infinitely, the mathematicians will tell you that the probability of the big bang happening 13.765 million years ago = 0. ergo God was not busy. It would be best if God told us why something out of time created time.
Will: “It just popped into existence or always existed or expands and contracts” for the universe is a simpler explanation than an omnipotent being creating it? I’m not seeing it.
JMJ: How about people are uncomfortable with rules and the idea that they are not God, and thus rejected the idea?
Bill S: Which Jenny Lake?
Okay, mathematicians will tell me the chances that the Big Bang happened 13.76 million years ago if time extends back to infinity, why?
Why was God not busy? (You do realize I was being sarcastic, however, now I’m not.)
I seriously doubt God can tell us why something out of time created time—we’re not that smart. That is assuming that God is outside of time, of course.
By the way, it’s worth mentioning that there are different kinds of infinity. That some infinities are bigger than others in a rigorously definable way. This was discovered by Georg Kantor when he proved, using a very ingenious “diagonal argument” that the totality of real numbers (even in a small interval like (0,1)) cannot be listed in one-to-one correspondence with the integers. The cardinality of the reals is strictly greater than the cardinality of the integers. The cardinality of functions with domain and codomain the reals is strictly larger than the cardinality of the reals, and so on. There are infinitely many kinds of infinity.
Mathematicians just learn to live with this and accept it. As the theory of finitely generated abelian groups is all worked out, those who work in abelian groups turn their attention to these mega, hyper infinities.
Still and all, there is something profoundly mysterious and inscrutable in this to me. Thank God. (Ecclesiastes 3:11).
In philosophy it’s called the homunculus problem. You’re positing something more complex to explain something less complex. (Or just as complex.) The reason why you can’t “see it”, is a failing of your critical thinking skills.
Dr Brigg’s argument is poor because he decides he doesn’t like infinity, so tries to muster arguments against it. Look at the arguments first, then decide which are the better ones. The problem is, he mentions the claims that supports his argument and conveniently doesn’t mention the claims that don’t. Hence he gets nowhere as soon as you point out the cosmological schemes he avoided mentioning.
At the end he concludes God is a better explanation than there not being a God. Which is merely a statement and not an argument.
Of course Dr Brigg’s sets himself an impossible task. He can’t possibly decide such a matter by comparing cosmological schemes.
Good article. For a couple of minutes, there, I thought Briggs was going into the jock-speak mode where athletes can make a 1,000 percent effort, and something like true freshmen exist. On some dusty bench, somewhere, a whole bunch of false freshmen can’t make the scene. Our inability to grasp a concept like infinity is exacerbated by our inability to describe such a beast.
True freshmen 🙂
I think somebody said we need to deal with things, and infinity is only a concept. Mathematicians have learned to not only describe infinity (a lazy figure 8), but they deal with it, too. Everybody knows that math is not real. It becomes a thing when you get your transcript, or applying for a decent job.
How can we even claim to know about infinity if we cannot imagine, or describe it. Is infinity an it? Maybe it’s not important to life, living, and the infinite universe. OMG. How can infinity not be important to an infinite universe? What is a universe? Have we seen this movie?
Our government pays people to sit around all day and think of nothing. Some are called scientists. Or, maybe these are Briggs’s philosophers. Dark matter and dark energy has to come from somewhere. Since we cannot imagine or describe infinity, some people have come up with the ultimate fudge factors. Dark matter. Ha! Maybe there’s a dark infinity, too. We can call it HELL.
Sorry, Briggs, but your article today just demands that I pour a Bourbon and consider this infinity thing.
“which is where did the omnipotent being come from? The stock answer will be, he always existed, and the atheist will then quite reasonably apply Occam’s Razor.”
How is this for an answer. The omnipotent being created the entire universe in which we exist. That includes time as we understand an perceive it.
Before is a time concept, so “what came before the omnipotent being?” can not be rationally answered from our perspective because from our perspective there is simply no before before the omnipotent being created our universe.
By the way, Occam’s Razor, can not help in answering the issue of God and the origin of the universe, because none of the other possible answers are any simpler.
You’re borrowing the arguments of the atheists. If I ask an atheist, where did the universe come from after the Big Bang…? They will sniff their noses and declare that this was before time existed, hence I’m not allowed to ask such an impertinent question. It’s not a convincing trick. Ask a politician how his bank balance grew so quickly, and he may well tell you that your question has no answer, but that doesn’t convince, does it? It’s not a convincing reply in philosophy either.
Both the theist and the atheist believe in a Creator. For the atheist the Creator lacks certain qualities; such as self awareness, and personal interest in your prayers at night. That doesn’t, of course, mean there is a huge difference between either conception, but certainly the theist posits more than the atheist.
(Although I think it would be more productive to spend time on the infinity question rather than the ontological argument.)
Occam’s Razor, can not help in answering
The principle of parsimony as Brother William used it was an epistemological principle, not an ontological one. He did not say that the simplest hypothesis was more likely true. He said the real world could be as complex as it pleased God to make it. But in our theories about the world, we should not make them so complex that we fail to understand them. In modern terms: don’t have so many variables in your models that you can’t grasp the models.
Isn’t Eternity outside of time.?
My attitude is always: “infinity makes perfect sense, as long as you don’t take it too serious. It is the journey toward it that matters.”
Works great, both in mathematics and as a philosophy.
Nothing like thinking about infinity and “reality” late at night. It is hard to get into infinity without becoming mired in the discovered versus invented thinking about math in general, but so be it. I just finished reading Mario Livio’s little book “Is God a Mathematician?” which was a pleasant historical pass but not very satisfying from a mathematical perspective. Hawking’s “God Created the Integers” is much more satisfying in that regard.
In any event, thinking about the role of infinity with respect to material reality needs some care. First, from a physical science perspective, a concept is defined by its measurement. More strongly, to qualify as a physical entity, it must be measurable. By this definition, anything we mortals construct to measure any quantity must necessarily be finite, subject to finite limits on tensile strength, volume, etc. thus by definition we cannot measure infinity and hence a physical entity cannot be infinite. This would include measurements of distance and time. This problem recurs in cosmology and field theory a good bit. A lot of hair pulling has occurred over whether singularities (meaning places where infinite mass-energy density occurs) that should occur in black holes can be “naked,” meaning observable. A standard hypothesis argued is the so-called Cosmic Censorship hypothesis, that there are no naked singularities. It is not hard to envision that infinite terms in equations would complicate the predictive ability of the theory. (This lies at the root of various problems in field theory also). A popular consensus is that such infinities are a symptom of the mismatch between the mathematical theory and physical reality. The only mathematical solutions obtainable for black holes are for an isolated black hole in a vacuum, i.e. alone in the universe, whereas real objects are not alone and always seem to have matter and energy flying about, complicating and invalidating the vacuum equations. The hope seems to be that quantum gravity or entropic principles will somehow keep singularities from happening in real, physical black holes.
But, back to infinite time and whether that means “everything” has happened or will happen in the future. I think the problem is not with the concept of infinity so much as the concept of “everything.” We have to be clear what we mean by “everything.” Dealing with physical reality, by “everything”, we usually mean “all available states of the system.” Now if we first forget what we said earlier about a physical concept requiring measurement to be part of reality and let our physical universe be infinite in space-time (and mass-energy as well). Then by our limitation as mortals we cannot enumerate by measurement all of the (infinite and distinct) available states. Hence going back in time, if we allow time backwards to be infinite, we cannot be sure which infinity is “bigger,” i.e. whether there will always be more available states than we indexed that can have been reached by going backward in time. For example, if because of our limitation of being mortal, we enumerate available states by prime numbers but for non-mortals the index would be by all integers, no matter how far back in time the universe was taken, there would always be states available that we had not indexed, even though we indexed an infinite number of states. As mere, physical, mortals we could never guarantee it.
It is a little worse if we get even more physical. If the universe is finite we can consider the whole universe. If the universe is infinite, we can consider the region of the universe with which we are causally connected by light travel time. In either case we can define the number of available states in the universe so restricted and the universe so defined is a closed system. Associated with that number of available states is an entropy, defined as the log of the number of states. In the physical reality, the second law of thermodynamics gives an arrow of time, in particular, the entropy of a closed system necessarily cannot decrease in time, only increase. It can only stay the same if the system is already in its highest state of disorder, which cannot be the case for the universe since we exist. This means that as time evolves into the future, there are incontrovertibly more states available than at earlier times. Pick some time t2 > t1 in the evolution of the universe. This means that at time t2, there are more states available to the universe than at t1. Pick one of those additional states, say S1. Since that state was not available before, it could not have happened already by going back in time. Thus going backward in time will not result in “everything” happening infinitely many times. I think it gets a little tricky to define this process if the universe did not have a time zero, but it should be possible as long as we are allowed to let the universe get infinitely small (but a “lesser” infinity than the time), i.e. to allow space-time increments to become infinitesimal.
There is no such thing as the infinite. The term is undefined and has been shown to either be paradoxical or cause very confused thinking. (One might conclude that a smaller number is further away from infinity than a larger number, which of course, it isn’t.) Wittgenstein made the point, and I agree, that there are only processes that are in an abstract sense, infinite. 1 + 1 , 1 + 1 + 1, is a finite series. 1 + 1, 1 + 1 + 1, and so on, describes a process that in an abstract sense is infinite, because “and so on” was appended at the end, but that does not mean there are actually infinite things. If you can’t even define what you are talking about, but at best describe it as something plus something undefined added to it, then you’re not talking about anything. One can ponder what might happen if an unstoppable force struck an immovable object, and this might give you some sense of satisfaction if you smoke a lot of marijuana, or you’re a modern day Aristotelian, but for the rest of us, we know there are no such things as unstoppable or immovable or infinite, hence little point in contemplating the consequences.
In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena. Here are three examples (there are more).
(1) The first example, and the one given by Fr. John Rickert, is the one associated to sizes of sets. Just as natural numbers are the answer to the question “What is the size of finite set X”, infinite cardinals are the answer to the question “What is the size of set X, finite or not” (or maybe more precisely, they were invented so that such a question has an answer). But cardinal arithmetic is both boring and complicated in the extreme. Boring because, we can indeed define the sum of two cardinals, compare two cardinals, etc. but for example a + b is simply the maximum of a and b. Complicated in the extreme because even the value 2^w_0, for w_0 the first infinite cardinal, is independent of ZFC. Even more, ZFC puts very little constraints on what the value of 2^w_0 can be and almost any value is consistent with it. Cantor hypothesized that 2^w_0 = w_1 where w_1 is the first uncountable cardinal, this is the hypothesis of the Continuum. Results, first of Göedel, then Cohen, showed that the Continuum hypothesis is independent of ZFC.
(2) A second concept is more algebraic. In the field of real numbers there is no number r that is greater than every natural number n; in symbols there is no r such that for every n, n < r. Using the ordered field properties, this is equivalent to the following fact: if t is a positive number t such that for every natural n, t < 1 / n, then t = 0. This is called the Archimedean property. In the 1960's A. Robinson constructed an extension of the real numbers that violated the Archimedean property. In such fields, there are "infinitely big" numbers, that is, numbers t larger than every natural n. Their multiplicative inverses are "infinitesimally small" and give one possible rigorous formulation of infinitesimals a la Newton and Leibniz. In the most plausible construal of unicity, such extension is *not* unique because it depends on the choice of an ultrafilter on w_0.
(3) A third concept is a topological-geometric one. The real line is sometimes in interval notation denoted by ]-\infty, \infty[ with \infty Latex notation for the infinity symbol. But the infinity symbol here is not naming anything. But it is sometimes useful to indeed have objects standing in for -\infty and \infty (believe me, it just is) so that we can speak of the extended real line [-\infty, \infty]. More generally, and suggestively, we are "closing the real line by adding points at infinity", or precisely but rather more obscurely, "constructing a compactification of the real line". There are many possible compactifications of the real line (or even of a general topological space) and they can be ordered, in a sense. The minimal compactification is the one-point compactification (equivalently, force the equality, -\infty = \infty. By stereographic projection, what you obtain is the circle) and the maximal one, the so-called Stone-Cech compactification (don't even ask).
Yes. Eternity is not a really long stretch of time; nor is time a subset of eternity. Time is the measure of change in changeable things; eternity is the measure of permanent being. Time possesses ‘before’ and ‘after,’ while eternity is simultaneous:
a) Recall that “movement” meant “change” in them thar days. I like to use the original term kinesis because moderns often confuse “motion” with change-of-location only. Even though that reminder irritates Will.
b) Boethius also defined aeviternity, which was like time in having ‘before’ and ‘after’ but like eternity in having no end. This is actually what many moderns mean when they say ‘eternity’: time that has a beginning but then runs off to infinity.
Nice to see Will agreeing with Aristotle that there is no physically realized infinity. The N2 content of steel billets can be modeled with a normal distribution. But the distribution runs off to infinity, and no billet will contain infinite ppm N2, and certainly not negative-inf ppm N2. Just as the fish stinks from the head down, mathematical and statistical models stink from the extreme values. That’s why remote tail probabilities should be taken no seriously than a mime in Central Park. It’s one reason why Box said that models can be useful, but they are always wrong.
Of course, the irresistable force/immovable object paradox is ancient Chinese and ancient Greek. Aristotle did not, so far as I know ever contemplate it. In Chinese it was the irresistable spear and the impervious shield. (And what happens if I use this spear to strike that shield?) The Chinese term for contradiction is spear-shield, a reference to this fable. In Greek, it was the rabbit that always escapes and the hound that always catches. Atheists often use it against God’s omnipotence — Can God make a stone so heavy he cannot lift it? — perhaps not aware they are using a logical contradiction.
That is, it’s a logical contradiction long before it is a physical impossibility. (A force is irresistible insofar as nothing can resist it. But the immovable object can resist it. Therefore, the force is not irresistible. Et. contra.) It’s a nice illustration that not everything imaginable is possible.
Somewhat apropos: Why Can’t Mathematics Be A Happy Coincidence? (We differ on probability, naturally.)
Listening to Craig’s exposition, I think 1) his comment is that naturalism is “explanatorily impoverished” is the best refutation of materialism/naturalism/scientism, 2) is comments about probability and multiverses are not all that good (as Briggs pointed out).
YOS presents for us a splendid example of the hazards of applying philosophy — it allows one to delude oneself into thinking oneself is objective when the reality is that one is merely deluding oneself into hanging on to a particular belief in lieu of even acknowledging the existence of, much less actually evaluating, evidence to the contrary.
CONSIDER YOS’ remarks [with clarifications added within the brackets]:
To the remark, “Ah, but which God?” (see 1st post) YOS responds:
“Obviously, if nine blind men disagree on the nature of an elephant, that proves there is no elephant.” [NO–it only shows that at least eight of the nine are wrong]
“And if Copenhagen, many-worlds, standing-wave, transactionalists, and others disagree on the nature of quantum mechanics, that means there is no quantum mechanics.” [NO–if people disagree it only means the facts & analysis are insufficient to settle the issue conclusively; some or all may be wrong, partly wrong/correct to varying degrees on various details]
Such “either or”/”black or white” thinking is referred to as “splitting.” It is a type of psychological defense mechanism.
This is also illustrated in the old story about determining the number of teeth horses have, variously attributed to Aristotle or Francis Bacon; both versions follow:
The example attributed to Aristotle, which he established when he was faced with a raging debate among early Greeks about how many teeth were in the mouth of a horse. Aristotle, whose wisdom exceeded his years, settled the argument once and for all by declaring, â€œLetâ€,s look.â€ Thus was born the empirical method of examining things.
In the year of our Lord 1432, there arose a grievous quarrel among the brethren over the number of teeth in the mouth of a horse. For thirteen days the disputation raged without ceasing. All the ancient books and chronicles were fetched out, and wonderful and ponderous erudition such as was never before heard of in this region was made manifest. At the beginning of the fourteenth day, a youthful friar of goodly bearing asked his learned superiors for permission to add a word, and straightway, to the wonderment of the disputants, whose deep wisdom he sore vexed, he beseeched them to unbend in a manner coarse and unheard-of and to look in the open mouth of a horse and find answer to their questionings. At this, their dignity being grievously hurt, they waxed exceeding wroth; and, joining in a mighty uproar, they flew upon him and smote him, hip and thigh, and cast him out forthwith. For, said they, surely Satan hath tempted this bold neophyte to declare unholy and unheard-of ways of finding truth, contrary to all the teachings of the fathers. After many days more of grievous strife, the dove of peace sat on the assembly, and they as one man declaring the problem to be an everlasting mystery because of a grievous dearth of historical and theological evidence thereof, so ordered the same writ down.
Here we routinely read ad hominem [a type of logical fallacy] attacks against science…for all the verbal gymnastics to create the illusion to the contrary, is reduced to name-calling — “scientism” with similar philosophical appeals, based on selective cherry-picking & fact-twisting, no different in principle from claiming that counting the number of teeth in a horse (actually examining reality) constitutes “unholy and unheard-of ways of finding truth.”
In this commentary, Gail Finke, YOS, and routinely Briggs, apply the typical mental machinations to reach a desired (not necessarily the actual) conclusion. The delusion starts with the person presenting the case deluding themselves.
Robert DeNiro co-starred in a movie, full of symbolism, on this theme — Angel Heart. There we see a human, who had deluded himself into believing he’d hide from Satan (played by DeNiro) to avoid paying a debt (with his soul) embark on a journey to find the truth.
Will: “The reason you can’t see it is failing of your critical thinking skills”. I guess then the reason you can’t understand or see why certain trolls are right is failure of your critical thinking skills? (Is it just me, or has Will deteriorated to troll status as of late? )
Okay, I get this from warmists all the time—I cannot present an argument against their points unless I also include those for their points. I cannot report on an experiment unless I report every possible exception and alternate theory. This is not realistic in any sense of the word. While including alternative theories is good, no one can cover them all and you really hate those “lousy” arguments, so if someone is including all the arguments, you’re not going to like that either.
Bob: “Everybody knows math is not real.” You would think so, but it seems to not be the case quite often.
Will: Answer to “What happened before the Big Bang” or “Where did the omnipotent being come from?” is WE DON’T KNOW. We probably never will know. This is all one of those philosophical mind games we used to play in the dorm room at two in the morning. There’s no real answer and it really doesn’t matter. It’s just a thought exercise. Something science fiction writers love.
FAH: Quite interesting comment.
Ken: Why do people who complain about self-delusion always seem to think everyone else is guilty and they are somehow exempt? It applies to everyone, not just those with whom you disagree, which is why the argument is what is important, not the speaker. Yet, that is a concept completely lost on most of the commenters. Global warming is chock full of this technique. Attack the speaker and declare them guilty of delusion. As I often say, no science here, no psychology here, nada.
I see that someone here invoked the Poincaré recurrence theorem to try to justify Briggs’ assertion that, in an eternity, every possible thing must occur. Some of the assumptions needed to prove this theorem do no apply to an expanding (i.e., our) universe, but nice try anyway. The assertion remains just that, and is stated without proof. There are other weak areas in the argument, but that will do.
Expanding from what to what. Both points are addressed here, too.
At least someone understood the irony! You cannot take the “which God?” business to mean that there is no God any more than you can take the blind men to mean there is no elephant. However, it does not mean that at least eight are wrong. It may mean (as the story does show) that each has hit upon something true about the elephant.
I thought splitting was the making of distinctions, such as splitting “homeless” into “mentally ill,” “drug addicted,” “lost employment,” etc. It is the opposite of “lumping” in which actually diverse things are lumped into a single term.
“Expanding from what to what. Both points are addressed here, too.”
Really? Well, the answer is “from a smaller volume to a larger volume”. The theorem doesn’t apply, and you would have to work much harder to show what you want to show, that every point in the configuration space is visited multiple times.
Enjoyed the writing anyway, and the song.
“I see that someone here invoked the Poincaré recurrence theorem to try to justify Briggs’ assertion that, in an eternity, every possible thing must occur. Some of the assumptions needed to prove this theorem do no apply to an expanding (i.e., our) universe, but nice try anyway. ”
The only one that invoked the Poincaré recurrence theorem was me, so I suppose that this is directed at me. The second paragraph, the one after the theorem, starts with “Slightly less facetious” — is what I mean not obvious? In case it is not, I wasn’t “trying to justify” anything whatsoever.
I’ve been waiting for someone to bring something up, but folks so far have only danced around it, so I guess its up to me to jump in and make a fool of myself.
“If the universe is eternal, then anything that was possible has already happened.”
There is an unstated assumption in this assertion. The assumption is that “possible” is the same as “having a finite probability of occurring”.
I would assert that events that have an infinitely small chance of occurring happen all the time. (In keeping with the theme, I will also irrelevantly assert that there are actually an infinite number of infinitely unlikely events happening infinitely often.)
So, how often can an infinitely unlikely event be expected to occur in an infinite universe? Well, hearkening back to college calculus, the answer could be never, once, some other finite number of times, or infinitely many times, depending on the relative nature of the two infinities.
The only counter-argument I can see to this is quantum mechanics, if everything (as in the Briggsian “every single thing”) is quantized to a finite number of possible discrete states.
For you quantum mechanics types out there, is there any variable in quantum mechanics that isn’t quantized? What about time?
“is there any variable in quantum mechanics that isn’t quantized? What about time?”
Quantization usually arises with bound states, for example the electron bound to the nucleus, whose energy is quantized. Plenty of things are continuously distributed, and time is a continuous variable.
In regard to your other concerns, nothing in Briggs’ article is defined well enough to reason about carefully.
With respect to your QM query: First we need to be clear what we mean by “variable.” If we mean an “observable” capable of being measured, then the answer is generally yes, observables are quantized. Sometimes the argument derives from specific representations of QM, i.e. different math schemes and spaces of states, and sometimes it derives from analysis of the act of observing, or both. If the variable is not an observable, say a phase of a complex wavefunction, or worse, not an observable “yet” meaning we have not yet observed it, then it sometimes happens to represent such variables (not direct observables) as continuous quantities but unless it becomes an observable it is “just” math. Second, it is fair to say the jury is out with respect to space and time being quantized. There is a widely held line of thought that the smallest length scale is the planck distance and the smallest time is the planck time (the time light takes to go the planck length). Google planck length and you will find a bunch of things to read. Answering the question (is space-time or is it not quantized?) is equivalent to deriving the theory of everything, the unified field theory, quantum gravity, etc. all of which have stumped Einstein, Feynman, Hawking, Penrose, to name a few, so don’t feel bad if it is hard to think about. In any event (even a space-time event) if space and time are quantized, then any quantity constructed with space and time would be quantized, which would include virtually every quantity of physical interest. (Some units, such as temperature, are not explicitly constructed from distance and time, but their definitional measurement requires measuring the state of a physical system at various of its available states and deriving them (e.g. temperature) from that.)
@Rodrigues & Others,
You’re confusing a process or procedure that does not halt, with infinity itself. This is a mistake that is always made every time the topic of randomness comes up, and one wastes a lot of time trying to straighten out that confusion. Some are brighter than others, but some always fail to grasp Wittgenstein’s point. Here is an infinite procedure simply defined:
x := x + 1
Because you’re described a process that does not halt, it does not follow that there is a real thing called “infinity” that actually exists in the real world.
I wouldn’t be citing YOS as someone who merely does bad philosophy. He is not even doing philosophy. What he does is more akin to poetic writing, such as the writings of Christian mystics, or the Tao Te Ching, or Sufism. It has its grounding in early Greek philosophy, especially Plato and Aristotle, and then also the neo-Platonists, et al. A new student of philosophy might be given a work such as Plato’s Crito to read. It’s illuminating in the sense that it is easier to see how Plato draws unfounded conclusions merely by using poorly defined words and applying subtle shifts in meaning. Even a non philosopher can often detect how Plato managed to fool himself. Hence there is Perfection, and Manifolds, and Truth, Being, Potentiality… ad nauseam. These word have no meanings and all meanings. It’s an elaborate word game, and no more. The post modernists do similar when they write things such as “This three-part phallogocentric negation and sublation of history can be grasped easily. ” If you point out that this is gibberish you will of course deeply offend the writer, who will declare that you are simply not bright enough to grasp such deep profundity.
“You’re confusing a process or procedure that does not halt, with infinity itself.”
I am not confusing anything. I never spoke anywhere about “processes” or “procedures” or what not. Neither have I said or implied that there is a “real thing called “infinity” that actually exists in the real world”. Neither are you straightening any possible or actual confusion of mine. What I said is strictly and exactly right and anyone can check for himself by opening the relevant books.
Exactly so. Rather, the arguments are that actual infinities cannot exist in the world (universe/multiverse/whatever).
“In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena. Here are three examples (there are more)…”
The above is what you wrote, which is wrong. The concept of infinity WAS NOT defined. You merely described various processes that DO NOT HALT.
Hence what I wrote was correct, and what you wrote just now was nonsense.
I can walk around the block, circling my house. At no point do I run out of block to walk, because my path is roughly circular. It does not, therefore, imply, that I have discovered an infinite path way.
“I can walk around the block, circling my house. At no point do I run out of block to walk, because my path is roughly circular. It does not, therefore, imply, that I have discovered an infinite path way.”
How many points lie along your path?
In spite of some contentious commenters, this discussion has been very interesting. It’s impossible to have a complete discussion on this topic, or anything more than a short introduction really, but people have posted some long, detailed comments that I will be saving so I can further research some of the ideas. I may not end up agreeing with the ideas, but I like that people are giving many ideas to ponder. Infinity is always a fascinating topic.
‘“In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena. Here are three examples (there are more)…”
The above is what you wrote, which is wrong. The concept of infinity WAS NOT defined. You merely described various processes that DO NOT HALT.’
Yes, what you quote is what I wrote. Congratulations, you can quote. No, it is not wrong. Yes, the mathematical concept (or concepts) of infinite is well defined, and there is no (known) paradox in the technical sense of the word. Yes, it is standard mathematics. Want references? No, I did not described, “merely” or otherwise, “processes that DO NOT HALT”.
“Yes, the mathematical concept (or concepts) of infinite is well defined, and there is no (known) paradox in the technical sense of the word.”
Nope, there is no mathematical concept of infinite that is well defined. Even your attempt to weasel out of your mistake is still wrong. All you’ve done is list a bunch of processes that do not halt.
This is not a matter of semantics. The question is, are infinite things possible? Pointing to a circle, or any variation on my example of x = x + 1, loop, do not demonstrate this. Of course, I don’t think you’re confused on this point. Merely that you obstinate in admitting your mistake, hence the bravado.
“How many points lie along your path?”
How many can I observe? Definitely a finite number.
Or by point, perhaps you don’t mean something observable or measurable. Do you mean something abstract like a mathematical process that does not halt?
If you mean the second definition, what relation does this abstract mathematical process have to my afternoon walk?
“Nope, there is no mathematical concept of infinite that is well defined.”
Yes, there are. Several. Once again, this is merely a matter of opening a book; need references?
“The question is, are infinite things possible?”
A question I never addressed.
“How many can I observe? Definitely a finite number.”
That wasn’t the question, and I’m not sure that would be the correct answer.
“Or by point, perhaps you don’t mean something observable or measurable. Do you mean something abstract like a mathematical process that does not halt?”
Attempt #2 at avoiding the question by changing it. I didn’t say anything about a “process”.
“If you mean the second definition, what relation does this abstract mathematical process have to my afternoon walk?”
When I asked “how many points”, I means “how many points.” What is the answer?
I have very little patience for fools. You will go on my ignore list shortly, but before I do that I’ll make one last attempt. A process is merely a procedure or operation. Anything you care to describe is in one way or another a process. So yes, you can’t avoid dealing with processes even if you didn’t use the word process. Childish I need to explain this to you.
The answer to your question is none. Points are mathematical abstractions. There is no relationship to my walk around the neighbourhood and your mathematical abstraction. If you think there is a relation, you need to explain what that relationship is and how it’s supposed to connected to my walk. You’re not going to win an argument by declaring that the other side hasn’t given you the answer you wanted, especially when you’re asking the wrong question. If you think I’m wrong, explain the relation between the two. Or do what you’re more likely to do, which is to knock over the chess board.
“Yes, there are. Several. Once again, this is merely a matter of opening a book; need references?”
I don’t want references. I want you give me a definition of the concept of infinity which is what you said you would do. All you gave me was a list of processes that don’t halt. I’d already done that many times, and none of those things explain the concept of infinity. And none of your references will either. Because there is no such thing.
“A question I never addressed.”
You did address that by “correcting” what I originally wrote, by confusing what I wrote with something else. I’ve had such exchanges a hundred times before, in the class room, with other academics, and on the internet. It’s a very common confusion which is why I’m straightening it out now. Even smart people confuse processes that do not halt, with the concept of infinity. That’s why we are even discussing the topic. It boils down to sorting out this confusion, which you are persisting in.
Children in grade school don’t often get hysterical when they are asked how many points are on a line. Why should an adult? I don’t know what terrible things happened to you, but I’m sorry. Bye-bye.
Every day I walk that path and I’ve never seen points. How do I find them? What do they look like?
“I don’t want references.”
Of course you don’t.
“I want you give me a definition of the concept of infinity which is what you said you would do.”
Need references? Because I am not going to fall for this cheap rethorical trick where either I give a “definition of the concept of infinity” in the span of a blog comment, and to your capricious, irrational standards, or else. Go read a book if you want to know how it is done in all the gory details.
“All you gave me was a list of processes that don’t halt.”
I did not gave a list of “processes that don’t halt”, no matter how many times you repeat it. w_0 is the name of the first infinite cardinal, a definite, concrete (ah the pun) mathematical object. The hyperreals are the name of a definite mathematical object constructed as an ultrapower of the real line. Some of its elements are infinitely big in the precise sense that they are larger than any natural number n in the hyperreals order. The Stone-Cech compactification of a topological space is a functor on the category of topological spaces, spitting out a definite topological space some of whose points are the “points at infinity” of the original space. None of these objects are “processes that don’t halt”. All of them flesh out distinct, well-defined, non paradoxical concepts of infinity.
“I’d already done that many times, and none of those things explain the concept of infinity. And none of your references will either. Because there is no such thing.”
Once again, if by “there is no such thing” you mean that there is no actual number of infinite things in the actual world. or some variation thereof, I never pronounced about such a question. But if you want my opinion, no “there is no such thing”. I will spare you the arguments. If you mean something else by it, well, the opinion will be different. Above you asked “The question is, are infinite things possible?”, yet another, different question, that I also did not address, but whose answer will depend on the modality, how one views mathematical objects, etc.
‘“A question I never addressed.”
You did address that by “correcting” what I originally wrote, by confusing what I wrote with something else.’
No, I did not address it anywhere. Period, end of story. That you pile misrepresentations on top of misreadings is your problem, not mine. In fact, it is quite baffling why you insist on putting words in my mouth. Are you asking for a fight? Take some xanax and that too will surely pass. The post you first addressed began with “In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena.” I have just recapitulated this story above. What corrections followed were due mainly to your ignorance and inability to read.
“I’ve had such exchanges a hundred times before, in the class room, with other academics, and on the internet. It’s a very common confusion which is why I’m straightening it out now.”
I am not interested in your experiences. If your discussions are of the caliber of the ones you engage in this blog, then I feel sorry for the class room, the academics and you. The internet is what it is and one more troll will not make it substantially worse. You are not straightening out anything at all, because the only one who is confused here is you.
Now, are we done here, or do I have to correct you yet again in another (pointless, boring) round of misreadings and misrepresentations?
At this point, it may be time for the “ignore the troll” maneuver concerning Will. I don’t like shutting people out, especially not when they have demonstrated in the past that they are intelligent and can engage in reasoned discourse, but looking through this thread, there is insult after insult hurled at virtually everyone. It’s distracting to the actual discussion. Will has said he has a list of those to ignore. He obviously does not want to discuss things that he disagrees with—it upsets him. So let’s not upset him.
You’ve merely described processes that do not halt. The fact that you genuinely don’t understand what you think you’re claiming is a little depressing. Since you are convinced there exist objective mathematical objects that describe infinity (your exact claim) would you care to pick one of your very large but presumably non infinite collection, and write out the algebra here for us?
There are actually only two possibilities –
You introduce some variation on an undefined term.
You describe some type of conventional recursive process
Or is this all your usual empty bluster? Of course it is. All your googling is not going to save you here, because you’re writing utter nonsense and you don’t really understand what you’re doing. Google can provide near infinite references for you but Google can’t think for you.
It just occurred to me that I’ve been making a fatal mistake. I’ve assumed that someone who wants to debate philosophy of mathematics actually possesses at least some sort of rudimentary familiarity with philosophy of mathematics…
Since you’ve made the claim that none of the examples you’ve googled are processes that do not halt, it might perhaps have been constructive for you to first ask me what I meant by that. Since you may not be familiar with the jargon I’m using. (Fair enough.) Let me clarify what I mean by this word –
When I use the word “process” I simply mean any set of operations executed in a symbol manipulation system that obeys axiomatic rules. E.g., when you solve an algebraic equation, you are engaging in a (mathematical) process.
A symbol system may even have a formal, i.e., “objective”, symbol for infinity. This could be as simple as ?
Indirectly, a process such as Pi (C/d, ratio of a circle’s circumference to its diameter) may represent a number of infinite series.
The first example is an undefined term. The second example describes a process that does not halt.
You can, if you wish, describe either as “concrete” or “objective” mathematical objects, or whatever you want. But of course, the first example is merely a placeholder for something undefined, and the second does not refer to anything with an actual defined value either. There is nothing in our universe, for example, that expresses Pi. It’s a mathematical abstraction. Its exact value is unknown. Of course, you might point out that a perfect circle perfectly encapsulates the value of Pi., except there is no such thing as a perfect circle. Although if you want to argue that perfect circles exist because Pi exists, your argument is rather circular. 😉
Anyway, since I mentioned before, that I agreed with Wittgenstein on this point, and you seem convinced that I’m wrong about everything, let’s leave my opinions out of this entirely for the moment. In what respect is Wittgenstein wrong in relation to his critique of infinity, in relation to what I’ve discussed here, which is of course, simply the nth reformulation of what I originally wrote?
(I always like to ask questions that people can’t google the answers to. It separates the children from the thinkers.)
“Will has said he has a list of those to ignore.”
I am supposed to be on that list — just another thing I imagine, where Will is not to be taken seriously.
And here we go again.
“Since you are convinced there exist objective mathematical objects that describe infinity (your exact claim) would you care to pick one of your very large but presumably non infinite collection, and write out the algebra here for us?”
Whether I am convinced “there exist objective mathematical objects that describe infinity”, depends on what you mean by “exist”, “objective” and similar words. What I wrote *exactly*, as in contradistinction to what you imagine “reading” was “In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena. Here are three examples” and then I give the examples. As for the question, go read a book. Need references?
“All your googling is not going to save you here, because you’re writing utter nonsense and you don’t really understand what you’re doing.”
When I offered references, the references are to books. Neither did I use google, nor have a need to, because I do know what I am talking about — it is one of the perks of having a phd in mathematics, done research, attended conferences, talked to mathematicians, having read, you know, books. Need references?
“When I use the word “process” I simply mean any set of operations executed in a symbol manipulation system that obeys axiomatic rules. E.g., when you solve an algebraic equation, you are engaging in a (mathematical) process.”
I am not confused, you are. The existence of the first infinite cardinal w_0 is a consequence of ZFC axioms (much less than ZFC is needed, but nevermind), that is, there is a valid proof (obviously finite), formalizable in ZFc of the proposition stating that there exists a set satisfying such and such properties, some of those being the mathematical formalization of “w_0 is infinite”. A proof in a formal theory is no different in kind — for current purposes, that is — than what you sloppily call “set of operations executed in a symbol manipulation system that obeys axiomatic rules”. The same for all the other examples I gave.
“Indirectly, a process such as Pi (C/d, ratio of a circle’s circumference to its diameter) may represent a number of infinite series.”
The number pi is defined as the ratio between the perimeter of a circle and its diameter. It is a perfectly well-defined mathematical object, not “merely a placeholder for something undefined”. Neither there is anything circular in the definition, because we do *not* have to define pi in order to define circles. Neither it is relevant whether there are perfect circles in physical reality or not (whatever precise meaning one cares to attach to this sentence). The number pi can *also* be defined analytically, say as the sum of specific convergent series. This is also a perfectly valid definition, picking out a unique, determinate object — and *provably* so, with a proof formalizable in ZFC (once again, ZFC is overkill, much less is needed) — and not as you say afterwards that “does not refer to anything with an actual defined value either”.
“There is nothing in our universe, for example, that expresses Pi. It’s a mathematical abstraction. Its exact value is unknown.”
For what must be the umpteenth time, I never addressed or commented on the two first sentences — and any comment of mine would depend on what you mean by concrete reality “expressing pi” or “mathematical abstraction”. As far as the third one, it is once again a quite accurate index of your complete and utter ignorance. pi *is* a value, that is, a real number. Your sentence is as dumb as saying that the value of 2 is unknown. Probably what you want to say is that one cannot enumerate all the digits of pi in some numerical base. That is true, but also irrelevant to anything I said. It is also true that pi is computable, in the precise sense that there is an algorithm that will spit out the nth digit of pi in a finite amount of steps. And if you object to this definition, then I should add that there are many (infinite actually) *rational* numbers for which we cannot write down in a finite *and* reasonable amount of time (say before the expected heat death of the universe). And if by enumerate, you mean literally enumerate, than you can even drop “*and* reasonable amount of time”. But as I said, either the computability of pi, or the impossibility of enumerating all its digits, is irrelevant.
As for the rest of your shenanigans, typical of the intellectual scumbag you are, the less said the better.
Perhaps this sentence perfectly articulates your confusion:
“It is a perfectly well-defined mathematical object”
OK, what is the last digit of Pi calculated to one goolgeplex of precision?
Oh, you don’t know?
But you’re claiming that Pi is a perfectly well defined mathematical object? Yet mysteriously, you don’t know its value. Not only don’t you know its value, you have no means of ever calculating its value.
So what do you mean when you toss around words like “objective” and “well defined”? Perhaps you actually mean some sort of documented process or receipt exists? Aren’t we back to my original example:
X := X + 1
Is that the definition of infinity? Or as Wittgenstein phrased it, there is no infinity in the world, hence no definition, only various more elaborate forms of “1+1 and so on.”
I’ll do my best to try to make my examples simpler and simpler until you understand why you are so confused. All the insults in the world don’t save you from making a fool of yourself. The more insults, the more obviously foolish you appear. I presume by your rambling response that you don’t really understand the point Wittgenstein was making, which I iterated here.
(Of course Rodrigues you’re not confused, except when you initially “corrected” my post. I’m good at explaining very complex philosophical ideas in simple terms. So you were able to grasp the point I made, eventually. You’ll never admit your mistake of course, which is why you now go through the rhetorical gymnastics of agreeing with what I wrote while simultaneously trying to weasel your way out of admitting your error. You are a very sad little fellow indeed.)
If you can stand some advice from old Uncle Matt: go read a book. You’re rather embarrassing yourself here. The material G. Rodrigues is discussing is so common in mathematics that it’s not even close to esoteric. But maybe you’re coming at this from a statistics and not a mathematics viewpoint? I recommend Billingsley’s great book (Probability and Measure) which introduces Cantor’s ideas, and how countable infinities are different from uncountable (not that these exhaust the different kinds of infinities, as Rodrigues has been trying in vain to point out).
I’m not going to argue with you, because unless you’re willing to understand the argument, there’s no point.
“But you’re claiming that Pi is a perfectly well defined mathematical object? Yet mysteriously, you don’t know its value. Not only don’t you know its value, you have no means of ever calculating its value.”
Already answered this, in more than way; predictably it all flew past you. At any rate, not going to repeat myself. Go read a book.
“Is that the definition of infinity?”
No. What you wrote is rubbish. The program is ill-defined because you forgot to initialize the variable X (before the loop, to 0 say). The program is ill-defined because it provably does not halt. If we patched these mistakes and added a print X in the appropriate place, you would have an algorithm that enumerates and prints out the natural numbers. But a (constructive) enumeration of w_0 is a distinct thing from w_0 itself. Quite clearly, and demonstrably, you do not have the faintest clue of what I am talking about, or of mathematics in general, and even manage the admirable feat of messing up trivial programs. Maybe it would be more productive to, you know, actually go read a book? Need references?
“I’ll do my best to try to make my examples simpler and simpler until you understand why you are so confused. All the insults in the world don’t save you from making a fool of yourself. The more insults, the more obviously foolish you appear.”
You will do “your best”? I actually smiled at this. Your bravado is duly noted. As for the insults, it is not an insult to declare what everyone else knows it is true. And quite frankly, it is a little too late for you to be whining about it. And are you worried by me “making a fool” of myself? I am not. With you as a term of comparison, I must come off as the Platonic exemplar of the intellectual virtues.
“Of course Rodrigues you’re not confused, except when you initially “corrected” my post. I’m good at explaining very complex philosophical ideas in simple terms.”
Good at “explaining very complex philosophical ideas”? This one made me laugh. We have officially entered the Twilight Zone. At any rate, does this mean that you will *not* do your best to make me understand why I am so confused? Yeah, that is the wiser choice… And “very sad little fellow” — is that an insult? Are you not afraid of “making a fool of yourself”? Silly me, of course you are not.
“So you were able to grasp the point I made, eventually. You’ll never admit your mistake of course, which is why you now go through the rhetorical gymnastics of agreeing with what I wrote while simultaneously trying to weasel your way out of admitting your error.”
Ah I see. Secretly, deep down, I actually agree with you, and just do not publicly recognize your genius tackling “very complex philosophical ideas” out of sheer irrational caprice? Who can argue with this?
Now, are we done here, or is this going to continue until Mr. Briggs mercifully closes the comment thread?
I thought a long quote from Hawking’s book “God Created the Integers” might be useful so I scanned a few pages (starting at about 1131 in the paperback edition) and converted them to text. It is one of the most understandable and enjoyable (and short!) discussions of Cantor’s thoughts I have ever seen. I tried copy/paste into the comment box but the formatting (especially bolding) did not transfer and it is needed for the diagonalization discussion. I can’t see how to copy/paste graphics into comments either, else I would just convert to jpeg and copy. In any event, I recommend pp 1131 to 1135 (2007 paperback edition) of Hawking’s “God Created the Integers” in which Hawking introduces Cantor’s work.
Of course you’re not going to argue because you can’t address the point I am making. Why not state my error? If this consists of the ridiculous gripe that different terms for infinity are found in mathematics, that is self evident, not my point at all, and completely irrelevant to my point. In any symbol manipulation system you can define infinity (an 8 on it’s side is one such example). So what? Would you care then care to explain Wittgenstein’s point about infinities or is your mock cleverness sufficient?
Let’s put aside the 90% of your post was mindless blather. Did you enjoy the rant? Anyway, let’s “correct” the algorithm by adding a little piece to it I didn’t show, since it’s completely unnecessary to show, but this was important to you.
X = 1
X = X + 1
Do you regard this as a sufficient definition of infinity now?
The addition is irrelevant to my point and has nothing to do with infinities. X can be 1, or zero, or one googleplex. What changes? You objection here strikes me yet another confusion. (I’m only pointing this silliness out because you’re so wonderful y pompous.)
I think part of the problem is that you do understand my point but insist on using weasel words to pretend that something obvious isn’t obvious. The symbol for infinity (8 on its side) can be “mathematically objective”. Any symbol can be mathematically objective in that sense. But that means nothing useful in the context of this discussion. Any symbol manipulation system can manipulate symbols, the values of which are undefined. In syllogistic logic for example:
p –> q
We don’t need to know if p or q are real things. They can be undefined things. They need not relate to anything in the real world. You can manipulate symbols with undefined terms to your heart’s content. But that doesn’t mean those symbols correspond to anything in the real world. They may, or may not.
You can yell and scream and kick your feet and turn red and post a million insults, expounding on the fact that p and q have well defined meanings in syllogistic logic. And certainly, the temper tantrum aside, that is true as far as it goes. But is the main point of the explanation still flying over your head? I doubt it.
Now, infinity is worse than that. It’s not something that could exist, that simply may not. It’s a symbol that does not represent anything real nor could it. By real, I mean reducible to a natural number. (All other number series, which you like to quote to demonstrate your cleverness to yourself for some pointless reason, must be derived from natural numbers. Natural numbers are the foundations of everything that follows from them, at least in most mathematical domains.) Infinity is not a natural number, obviously. It’s non computable, if you prefer that term. Now, there exist things that may exist that are non computable, but infinity does not even quality for that.
Sorry, I am starting to run out of ways to dumb this down any further than this. Actually, as I said, by now you perfectly understand the point I’m making, it’s just your inflated ego that prevents acknowledgement.
“Do you regard this as a sufficient definition of infinity now? ”
Already answered the question; predictably it flew past you. Not going to repeat myself. And technically, the program is still incorrect because it provably does not halt — another bit you missed. Maybe you should go read a book in computer science as well?
“They need not relate to anything in the real world. You can manipulate symbols with undefined terms to your heart’s content. But that doesn’t mean those symbols correspond to anything in the real world.”
I never said, implied, or commented about any putative relation “to anything in the real world”. I never said, implied, or commented upon any putative correspondence “to anything in the real world”. It is all irrelevant to what I actually said. But I have already pointed this out, oh how many times? I lost count. And in order for me to say anything definite, you would have to specify what the relation and or correspondence is supposed to be. And for that to happen, that is, for the discussion to have a minimum of intellectual meat in it, you would have to go read a book. Need references?
“All other number series, which you like to quote to demonstrate your cleverness to yourself for some pointless reason”
The reason I quoted the examples I quoted is because the point of my post was, and I will here quote myself: “In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena. Here are three examples”. When one makes a claim, evidence is needed to support the claim, thus the examples. There was nothing “pointless” in it.
On the comments themselves about symbols, syllogisms, number series, etc. I will abstain from qualifying or responding to them. What’s the use? And I have already said more than enough, no need to waste my time and yours wallowing even deeper in your ignorance. Go read a book.
“Sorry, I am starting to run out of ways to dumb this down any further than this.”
Really, there is is no need to dumb down anything for my sake. What flows from your keyboard is already naturally dumb.
“Actually, as I said, by now you perfectly understand the point I’m making, it’s just your inflated ego that prevents acknowledgement.”
So it is my “inflated ego” that prevents me from acknowledging your genius at grasping, extricating and explaining “very complex philosophical ideas”? What can I say? Not all of us had the fortune of being endowed with a naturally humble ego like you have, but must strut and parade around our own, comment on what we know nothing about, not read books, etc. In truth I tell you, this blog does not deserve you.
Ignoring all your insecure ramblings let’s get to the heat of your misunderstanding once again:
“In mathematics, the concept of infinity is neither undefined nor paradoxical, but it is ambiguous because mathematicians use the same word to name very different phenomena.”
1. When Wittgenstein explained why infinity was paradoxical in what way was he wrong?
2. Infinity is undefined. It has no realization in the set of natural numbers. All you’ve done is list a variety of different mathematical recipes that describe processes that do not halt. Dr Brigg’s is completely muddled on this point also, although when he congratulated you on your cleverness, he merely repeated what I had explained earlier. 😉
3. Since you criticized what I wrote, you don’t get to decide what I was talking about. All you’ve done is ramble on about various incarnations of mathematical recipes that involve processes that do not halt.
4. There is a reason why I’ve kept my discussion laser focused on this one issue, because I was only making one very specific point about infinity. But a very important one. That it has no natural number value, it is non computable. It’s value is undefined. It is not a “real” thing. Lots of things in mathematics may be manipulated (it is after all, a symbol manipulation system), which are not “real”.
5. I can manipulate p and q all day in syllogistic logic. I don’t need to define what p and q are. p and q need not exist. In fact, it can very useful to work with abstractions in mathematics, although that should be obvious to everyone here.
6. You’ve confused the symbol for X for the value of X. You can’t mouth off that this is not what you want to talk about. That’s what I was talking about. You criticized my explanation. For all the empty blather, what’s wrong with it? You can’t criticize a position then complain like a petulant child, that this is not what you were talking about. If that’s not what you were talking about, it just means you didn’t understand what you were criticizing. You can’t have it both ways.
I’d also add a final comment, that the reason why I’ve harped on about this, is that the sort of confusion that Rodrigues introduces, is really at the core of the confusion about infinity. When another poster here tried to argue with me that there were an infinite number of points on my walk around the block, he was using a mathematical abstraction to try to prove the material existence of the mathematical abstraction. The argument was rather circular. ;-P
It’s confusing the map for the territory. In philosophy we call this sort of mistake a category error. It’s where you confuse the ontology of one object with the ontology of another. The map isn’t the territory. But as an approximation, simply as a guide to navigation, it can be rather useful.
“Since you criticized what I wrote, you don’t get to decide what I was talking about.”
My original post was not addressed at you, you decided to “respond” to me. It is true that the post starts with “In mathematics, the concept of infinity is neither undefined nor paradoxical”, and the insertion of the words “undefined” and “paradoxical” wass directly motivated by your use of the same words, but I only addressed standard mathematical facts that anyone can check by opening the relevant books. You took umbrage at it and started down the Ignorance road to Nonsense Ville.
“There is a reason why I’ve kept my discussion laser focused on this one issue, because I was only making one very specific point about infinity. But a very important one. That it has no natural number value, it is non computable. It’s value is undefined.”
No. The reason why you focused on this one issue is because you have no idea what you are talking about. But you have put your hoof in your mouth so far down that backtracking would require preternatural moral and intellectual courage. So you shove it even deeper. You do not even have an idea of how utterly ridiculous you sound when you say stuff like “I was only making one very specific point about infinity. But a very important one. That it has no natural number value, it is non computable. It’s value is undefined.” Go read a book.
“You’ve confused the symbol for X for the value of X.”
I have made no such confusion, no matter how many times you repeat this falsehood. In fact, in one of my examples, I explicitly made the distinction. Let this stand also as a comment on other, similar falsehoods.
Pythagorus, and the cult that surrounded him, also believed that there were only whole numbers and their ratios. After Hippasus of Metapontum discovered that this is not so, he mysteriously drowned at sea.
I suppose we should consider ourselves fortunate that Will Nitschke is unlikely to inspire a fanatical cult. Although, one never knows….
I think I understand your question. Once “infinity” is defined, one may ask if it is just a notion or it exists as a mathematical subject. This consequently involves the nature of mathematical objects.
If I remember correctly, you are a mathematical Platonist. I am sure you have a lot to contribute on this topic.
IMO, it’s good to know all thoughts on the philosophical nature of mathematics, to argue about them is fruitless.
“Pythagorus, and the cult that surrounded him, also believed that there were only whole numbers and their ratios.”
It is hard to guess what goes on in his mind (and if I were to guess it would be: “nothing”) but since one of his objections to the number pi was “But you’re claiming that Pi is a perfectly well defined mathematical object? Yet mysteriously, you don’t know its value. Not only don’t you know its value, you have no means of ever calculating its value.” it is not clear if he thinks those rational numbers that have an infinite (but repeating) sequence of digits do not exist, or even whether natural numbers so large that the entire lifetime of the universe would not be enough to write them down (in decimal notation) exist. And of course, part of the problem is what do we mean by “exist” as applied to mathematical objects.
“If I remember correctly, you are a mathematical Platonist.”
Actually no, I am not a Platonist. I am indeed a realist, but of the Aristotelean-Thomist flavor. For comparison James Franklin is an Aristotelean, sort of a midway position, but he has, I think, a poor grasp on abstraction and his take on infinitary objects is quite puzzling, to say the least.
“And of course, part of the problem is what do we mean by “exist” as applied to mathematical objects.”
The definition is, that which is computable. That which resolves to a natural number value. Lots of things in mathematics do none of these things. Different mathematical systems are different flavours of symbol manipulation systems. The question is, does the symbol correspond to anything observable or measurable. It need not be observable or measurable, only that it can be so at least in principle.
I’d also wrap this up by pointing out that your last comment directed at me did not consist of a single logical or rational argument. It was a bunch of statements declaring I was wrong, stupid, ignorant, because you said so. That about sums your intellect up. Of course, I didn’t go to this effort to educate fools on the internet. I find it enjoyable from time to time to re-articulate some of these ideas that I have discussed with other philosophers over the decades. Anyway, you never did manage to explain Wittgenstein’s point so as to further my education. Oh well. ;-P
“The definition is, that which is computable. That which resolves to a natural number value.”
The first time the word “computable” was used in this thread was by me, and it was used in the technical, mathematical sense. You do not know what the word means, neither invoking it like a magic talisman goes anyway towards answering the other questions that supposedly were very important to you (“are infinite things possible?”, “They need not relate to anything in the real world”, “The question is, does the symbol correspond to anything observable or measurable”, etc.). *No* way at all. A mathematical object being computable does not automagically make it “correspond to anything observable or measurable”, not even in principle. For your information, the first infinite cardinal w_0 — and I have already mentioned this — is *trivially* recursively enumerable, *trivially* computable. So by the criteria quoted above w_0 exists. For your information — and I have already said this — the number pi is a computable number. Your choice words on pi? “But you’re claiming that Pi is a perfectly well defined mathematical object? Yet mysteriously, you don’t know its value. Not only don’t you know its value, you have no means of ever calculating its value.” Truly pathetic. Let me predict what will be your next move: you will say you are not using “computable” in the technical sense, but only according to your private little language. Since “That which resolves to a natural number value” is meaningless waffle, you can make it mean anything whatsoever! Now go forth, falsify my prediction and spout more nonsense; just, for the love of God and everything that is holy, go *read a book*.
Reading SOME of the comments it reminds me of ZENO’S Arrow paradox.
I believe however a DIFFERENT perspective on time might be helpful to reflect on this with a greater understanding. Has time not been shown to be generally misunderstood as something entirely separate from space (and HIGHLY warped space at that), when in reality it is an integral of space?
Do Black Holes STOP time?
IF the Universe is expanding faster than the speed of light would that reverse time?
A better understanding of mass requires a better understanding of time!
Is there REALLY such a thing as change?
How does something become something else? Can something become something else? (Becoming/Changing/—Time is implied at the foundation of all change – could it be illusory? ). Is matter (mass and therefore time) continuous? Are the conservational theories of mass and energy as intractable as our senses suggest
“Parmenides had argued that it is impossible for there to be change without something coming from nothing. Since the idea that something could come from nothing was generally agreed to be impossible, Parmenides argued that change is merely illusory. In response, Leucippus and Democritus, along with other Presocratic pluralists such as Empedocles and Anaxagoras, developed systems that made change possible by showing that it does not require that something should come to be from nothing. These responses to Parmenides suppose that there are multiple unchanging material principles, which persist and merely rearrange themselves to form the changing world of appearances. In the atomist version, these unchanging material principles are indivisible particles, the atoms: the atomists are said to have taken the idea that there is a lower limit to divisibility to answer Zeno’s paradoxes about the impossibility of traversing infinitely divisible magnitudes!” http://plato.stanford.edu/index.html ‘Stafford encyclopedia of philosophy’
Could it be that the motion of immutable particles within an immutable void create the illusion of change????????????