Excuse the musings of an ignorant for a minute. I am wondering about the possibility of unique events in a massive universe such as ours.
There is the Drake equation (the one about the number of civilizations in our galaxy). In short, it pits very low probabilities against very large numbers and the latter easily overcome the former and comes up with an estimate of 10,000 cases of something as unlikely (when starting from just star dust) as civilizations in our galaxy alone.
Enter Jodi Foster in Contact saying: “if we were alone in the universe, it would be a terrible waste of space”
But then I consider the Mandelbrot Set. An absurdly complex “thing” using just a tiny little bit of the space available to it. Of the infinite plane of complex numbers, it fits in a 2-unit radius circle and inside that little circle it exhibits infinite richness and complexity.
So I have Drake saying, the universe is a big place. Whatever can happen, must happen a lot of times. And then I have Mandelbrot saying that even in an infinite place, a unique (complex) thing can happen.
So, are unique events possible or even probable outside the real of mathematical curio? Can there be unique events in our universe?
(no offense will be taken if the mods decide that this is too far off for a GD and move this elsewhere)
Actually, not only do I think the two ideas (or paradigms) are not contradictory, but, I believe they’re complementary.
On the one hand, “Drake” is saying that even when probabilities are low, wonderful things will still transpire given enough time.
On the other hand, “Mandelbroit” demonstrated that amazingly (infinitely) complex things can arise from very simple programs and certain “initial conditions”.
This is quite reassuring. “Drake” reassures us that we have the space and now, since Mandelbroit, we know that it doesn’t take much at all to program that space such that complex forms emerge from it. (In fact, depending on your philosophy, you could say that the complex forms are already there - waiting for us to reveal them).
Here’s where the argument fails. Regardless of the size and complexity of the universe, there’s no reason to conclude that everything that can happen must happen at all, even once.
True, but the difference between “at all” and “once” is, well, infinite.
Consider your odds of winning the lottery if you buy one ticket. Pretty damn small. But infinitely greater than if you didn’t buy any tickets at all.
The starting point for the Drake equation is an instance of one, with no expectation of direct evidence for or against more than one. That’s a whole different ball game than a “dreamed up” something that “can happen.”
Here are a few unique things in the universe:
The set of all atoms except for the atom to my left
The set of all planets except for earth
The set of all galaxies except for the milky way
etc.
**sunacres ** is, I think, getting the gist of my post. Once you break from zero probability, it would seem that the vastness of the universe multiplies that probability to the point where it can’t happen just once but all over.
**RaftPeople ** is pointing at a set of arbitrarily unique sets, but those are unique by definition, they are not cohesive in the way the Mandelbrot set is and certainly not as complex. A circle of radius 1 and center (0,0) is equally unique but it lacks the depth of complexity the MS has.
KarlGauss’s point is certainly valid. It Doesn’t take much to generate complexity (take a look at all the possibilities of ATCG). But that is part of my point. It takes so little to make complexity that unique complexities are all the more special (if they exist).
dropzone, first week of january. I am not made of money
Here is another way to think of this. If we found a unique event. Would it be truly unique (i.e. unrepeatable) or just with a probability so small that even the vastness of the universe can only bring it to happen once. Is there a real difference between these two?
Assuming the mass and the space in the universe is not infinite, then consider the following:
Cut the universe into 8 sections
Obviously the probability that any of those sections are the same is pretty small, so it seems it’s purely a matter of probability of uniqueness.
Just to expand a little bit on my short post, the size of the “thing” in question clearly influences the probability regarding uniqueness, if you’re stating that for any size “thing” the vastness of the universe overtakes the probabilities of uniqueness I would say that doesn’t seem correct.
Thanks for clarifying RaftPeople, I for one needed it.
The question of uniqueness gets undermined by the fuzziness of “thing”-ness. Every instance of every “thing” is unique, and it is only human pattern finding that groups instances into types and species of things.
So, when you’re looking for two “identical” snowflakes you’re either going to find them or not depending on how you define unique. Same number of molecules in exactly the same relationship to each other? At what moment in time? Same pattern but maybe slightly different scale?
I understand the issue with everything being unique. Every star in the universe is somehow different from all other stars and no two are exactly the same. Yet they are all stars. The concept may be a human one, but the fact remains that stuff in space balls up and that if enough stuff gets into a sufficiently big ball, it goes “wooosh” and shines pretty. It happens slightly differently every time it does but it happens a lot and it is always the same phenomenom. Like this, other phenomena also repeat themselves more or less often.
The definitions of “things” may be human construsts but the things themselves exist (I think). My question is about those things and events that are distinct and complex enough to call our attention (stars, appearances of life, atoms). Fractions of the universe are well, that, fractions of something just as the set of all positive real numbers is just a fraction of a set. Prime numbers on the other hand, are a distinct group.
I may be wrong in believing things exist and then my question would be a pointless one but I am getting a good feeling about this.
(oh and I am not secretly convinced of either answer and waiting to pounce on you when you start getting closer to the other. It is an honest question that I stumbled on in my thoughts)
ultrafilter, it doesn’t make much sense to me either. Hence my question. I can see primes being a bit more glamourous than positive numbers but glamour does not science make. I guess you could make a stronger case for saying number greater than 3 which makes it sound more arbitrary than positives (which is nothing more than greater than 0, really but has a name of its own).
Still, primes do have a bit more apparent complexity (to an ignorant of mathematics such as me) than positives. Hardly the stuff of the Mandelbrot set but a notch higher. Don’t ask me at which point of complexity it is safe to say they make an interesting set. I will be glad to take a look at the super complex ones right now and turn my blind eye to the grey area for a while.
Sapo, I think you’re missing the point about the Mandelbrot set. It’s very pretty to look at, but it’s really just a map to the infinitely larger collection of Julia sets. Benoit Mandelbrot played with it for quite a while, using it to find interesting Julia sets, before he actually saw the familiar shape that now bears his name. Its uniqueness is an artifact of its definition.
Many Julia sets (those whose seed points are inside basin of the Mandelbrot set, roughly 12% of the points within the 2-unit circle) are single-unit islands much like the Mandelbrot set. However, the vast majority of Julia sets are comprised of clumps of related but disconnected basins, resembling Cantor dust. Or, more abstractly, resembling the structure of the universe.
I think that when you compare the Drake equation against the Julia sets, you may find that they are more complimentary than at first glance.
Prophet, you are very right in your first sentence. To me, the Mandelbrot set is mostly pretty colours in a piece of paper. I have no idea of what is going on behind it. What it DID to me, though was to prod me into considering the idea of a complex event as an island in an ocean of non-complexity. A peerless instance.
Don’t think my understanding of Drake is much better, either. I find that the selection of variables and the values assigned to them are very arbitrary (but that could just be because I am not current with the topic). What it DID to me was highlight the fact that even the smallest probability comes through repeatedly if given enough chances.
Is ALL uniqueness an artifact of definition?
or an opposite question
Are all things unique?
Snowflakes may be unique but are Hydrogen atoms unique at some level? or are they identical to each other?
Keeping in mind that a lone island in the middle of the ocean can also be viewed as a collection of materials and processes. The materials aren’t homogenous either - it’s part silica, part coral, part basalt, and so on. What you see above the surface is merely a continuation of the landmass beneath it, which is the result of geological and biological processes, etc. When observed across geologic time, it is seen to grow and change and eventually sink, and probably has “siblings” that go through similar processes at different times.
I’m sure someone more familiar with quantum physics will come by and correct me, but I understand that yes, individual hydrogen atoms are unique in measurable ways. Ignoring such details as whether they compose a larger mocule, and further ignoring head-numbing topics such as quantum electrodynamics, individual hydrogen atoms can differ at least in terms of electrical charge (ions) and atomic mass (isotopes).
As to whether all uniqueness is an artifact of definition - how do you (you, Sapo) define an electron? How do you define a mountain? You can’t, not in the sense that you can define a mathematical set such as Mandelbrot’s, because these things were in place before you. You can, however, observe these things. Given sufficiently high resolution in your tools, I’d wager that you can find uniqueness in observing any natural phenomenon.
As to how this all relates to Drake’s equation, I really don’t know. Drake was adding together odds, and I’ve never really been comfortable with statistics as a model for the universe. The problem with using probability to predict events is: it’s just a probability until observation trumps it. It all seems like (educated) guesswork to me. Hopefully someone can help fix my ignorance here.
Sidenote: Douglas Adams’ Hitchhiker’s Guide stories humorously highlighted what I see as a serious flaw in Drake’s equation. When the aliens showed up, mankind lacked the means to detect the aliens. So while Drake’s hopes were vindicated, his equation was moot because it didn’t take into account our capability to observe.
I don’t think there are enough variations on ions and isotopes (6?) to make ALL hydrogen atoms in the universe unique. Ditto for whatever other variables might be there. Anyone care to take a stab at how many atoms of Hydrogen are out there? This could be a nice example of the vastness of the universe multiplying even the smallest probability.
Probably, the trend is the contrary. We make definitions to bundle stuff together more than to take them apart. We have a rough time remembering Blackie, Zeus, Ginger and Lassie and we make up the concept of dog to have just one name for them all. But most certainly, yes, you are right, zoom in enough and most everything is unique (maybe)
There must be a pretty good definition of electron out there, though. One that includes ALL electrons and leaves out all non-electrons (my bad definition of a good definition). Ditto for stars, black holes, water molecules. Is there?
I think some things are really universal enough that if Drake’s 10K civilizations encountered them, they would all have a concept for them. Something so self-evident that its definition is more a description than a creation. A proton or a star are my top choices for obvious examples. Are they?
