/ Does Infinity Exist?
I am pretty happy with the existence of infinity in maths, and being able to handle it is a very useful tool.
But does it exist in nature?
The most definitive I can get is 'maybe'.
The universe may be infinite in extent, but we don't know.
The matter inside a black hole may get infinitely dense, but we don't know.
Can the learned of UKC offer any examples of infinity in nature?
god must be infinite
I have met some people who are infinitely dense.
Fergie time was often infinite, at least until Utd scored.
As you said, I'd say the size and density of a black hole is probably the closest to it in nature (if you define infinitely small as infinite, rather than zero).
The size of the universe really depends on your definition. The universe has a finite 'size' as it is growing from a singular place/moment in space time (though as with the balloon definition, the original location no longer exists), but because nothing 'exists' outside of the fabric of space time, then it could be considered infinite despite having a finite size. Though I'd personally say that I consider it a finite size within a higher set of dimensions.
If the Universe is not infinite, what's 'outside' it?
Nothing. If there is no space and no time outside of the fabric of spacetime, then literally nothing exists.
Only 7 posts in and I can feel myself getting stupider!
If it was possible to travel faster than c to reach boundary, do you think you could go past the boundary in your spaceship. If not, why not? If so, can you be in nothing?
Nope... there is no space so nothing for you to go into! It is a pretty incomprehensible concept and I not something that you can easily get your head around. I know a few physicists who still don't quite comprehend it :)
EDIT: unless m theory proved true and you were able to enter into some higher dimensions after 'uncurling' them (all not really possible) then you might be able to extend past the 4 dimensions of spacetime in one of the higher ones. Though I don't know the details of m theory and maybe the higher dimensions are tied to the lower ones. so perhaps not possible. But that is a rather abstract musing.
Who gives a Shit?
Quantum vacuum zero point energy?
I don't think anything is infinite about that.
> If the Universe is not infinite, what's 'outside' it?
It's possible that it's finite and there is nothing outside it. Just because it is expanding doesn't mean it is expanding into something.
> Can the learned of UKC offer any examples of infinity in nature?
Infinity goes down as well as up and any line you care to draw can be divided into an infinite number of points.
i dont understand that
> i dont understand that
> i dont understand that
A less facetious answer would be that our concept of expanding means expanding in space. When we talk about the universe expanding what we actually mean is space-time itself expanding. By that definition 'outside the universe' is pretty much a meaningless concept.
i just cant get my head around anything, be it spacetime or matter, expanding unless theres something for it to expand to
perhaps the size of the universe is finite but everything within it is shrinking so distances relative to one another are increasing?
just a muse
> > perhaps the size of the universe is finite but everything within it is shrinking so distances relative to one another are increasing?
> just a muse
If the observed rate of expansion was caused by shrinkage all large structures would have disappeared by now!
I reckon not, and that it is purely theoretical. Are there an infinite number of grains of sand on a beach, of stars in the universe, of molecules in the sea ?
Or is that number quantifiable, but that we do not posses the ability to quantify it as yet?
There still remains the question of how many Sunderland fans it would take to write the complete works of Shakespeare, or even read it
> Are there an infinite number of grains of sand on a beach, of stars in the universe, of molecules in the sea ?
All those numbers are just big. Some are even known to some degree of accuracy.
You don't need infinite Sunderland fans. Just one and an infinite amount of time.
If you flew a spaceship at the boundary something would stop it crossing the boundary, but we're not sure what this thing stopping it would be (or do we know?)
are we talking about a finite nothing or an infinite nothing?
Neither applies. When there is literally nothing, you cannot attribute it a size or limit/delimit!
Nothing stopping it at all. You simply couldn't reach it as new space is being created faster than you reach it (speed of light being the maximum).
It might be more correct to say that we are at the boundary. If the universe is expanding like a balloon then we are on the surface. So the boundary is not a physical boundary like a wall but a dimensional one. So just as a 2 dimensional being would not be able to escape a 2 dimensional balloon surface so we would be unable to escape a 3 dimensional expanding universe without being able to cross into a higher dimension.
Do we actually know that it is nothing though? and not a sea of universe branes?
So if we travelled in one direction we are going round an expanding circle.
What stops us going in to a higher dimension?
people on benefits
> What stops us going in to a higher dimension?
Is it not the case that we could be in them but not perceive them?
Well it's impossible for anyone to know because it's beyond the reach of investigation... ie. can't reach it to find out. But could be Lister's dirty laundry basket for all we know!
Infinity goes down as well as up and any line you care to draw can be divided into an infinite number of points.
No it cannnot, try it!
The infinite divisibility of a line is a mathematical idea, useful as a tool, but not how nature is.
The Planck length is ~10^-35. This is around where we expect quantum effects to be evident. Whilst these lengths are are very small and way below the experimental range, they are a long (infinite) way above an infinite number of points.
A fair point regarding planck length but why are you trying to separate maths from reality in this way? Does maths not describe fundamental properties of reality?
I would say no.
Any so-called infinities in nature are probably more likely to be singularities which an artefact of in incomplete understanding of the real world. For example the singularity at the heart of a black hole: it's only a point of infinite density according to current theories, i reckon a decent version of quantum gravity will 'smooth' out the infinity, making it finite.
Probably you can apply the same logic to the singularity that the big bang came from, smoothing out the start of the universe and making it inherently finite. I admit the concept of 'outside the universe' is still a bit hard...
It could be that a black hole compresses mass to a point smaller than any volume that mass could occupy so that all the mass is converted to energy and hence the singularity is just an almost infinitively small area of pure energy. The energy would have to exist as something small and massless (ie. super high energy photons) which can only occupy a space as small as the plank length. Thus maybe a black hole has a finite size? We'd never know any theory for sure though.
It is easy to do maths in any number of dimensions you want. They can't all reflect reality.
Unless the universe was infinite at the big bang (which happened everywhere) and is now just a whole lot bigger infinity. I think I remember reading that thisis quite a favoured view.
but how many photons can fit into this single energy state and do they become degenerate? And does that have a bearing on their 'final' size, and are they really even there if cant observe them because they're shielded by the horizon?
dunno, but i suspect the answer will be found in cheesy peas....
Yes that is a better explanation than mine... was trying to use the balloon analogy. But the same if you consider it from the perspective of a higher dimension.
EDIT: blond moment... forgot minimum wave length would be plank length so photon energy fixed.
I think at a fundamental level all reality is described by maths, however that does not mean that all of maths is in that reality.
Hence the musing about infinity in nature.
What about circlular infinity? Like for example the endless edge of a Moebius strip. Or even your common or garden circle?
Are there processes or events in nature that are or could be repeated infinitely?
If you're musing on infinity and of a literary bent then I recommend 'The Library of Babel' by Borges. The story itself is quite short.
" Every event must have a cause .Because an infinite number of past events cannot exist , an infinite number of causes cannot exist . Everything has a cause , save the first cause which its self is uncaused .
Philsophical questions like this begin to boggle my mind in about 7 seconds.....
That would be repetition, not infinity. And for something to be repeated infinitely (such as going around your mobius strip) you'd have to continue for an infinite amount of time which can't happen with a finite universe age.
Henman semi-final losses on the grass and cutaways to Lucy
Is a black hole a point of infinite density by current theories? If it is infinitely dense then it has infinite mass so everything in the universe would be gravitationally pulled towards it at infinite speed, no?
I'm no physicist though so I could be very wrong but that seems logical.
Density is mass/volume so only either the mass needs to be infinitely high or the volume infinitely small, not both. It is assumed in many theories to be infinitely small but black holes all have very well defined masses.
That makes sense, forgot only part of the equation needs to be infinite to give an infinite answer.
Slightly less serious question now, what number comes before infinity? Is there an agreed point where we stop counting?
Nope... can add as many zero's as you like! The only issue being that you'll start running out of names for the numbers (I know the prefixes up to 10^21 but there are more after that), but still easy to quote the numbers in scientific format.
I think the days of people saying 'a near infinite amount/quantity' are numbered (excuse the pun) in science. Even the number of atoms in the visible universe can be estimated with a certain accuracy; by estimating the mass in the visible universe and assuming a 75-25 split between hydrogen and helium (probably ignoring heavier elements due to limited quantities, but could still be accounted for) you could work out this number. Though I imagine it to be between 10^50 and 10^200 at a guess.
That doesn't make sense to me at all (not your explanation just the fact), surely there must be a defined point at which the number becomes infinite? Otherwise it has no definition of any real use (in my mind anyway)
Numbers do get so big that it's a pain in the arse to express them in a format people can understand. For example, Graham's number is too big to be expressed as 10^10^10^10^etc without needing a lot of paper.
Depends what you mean by real use. There are numbers that exist as a solution to a mathematical problem which are so vast as to make, say, "the number of atoms in the known universe" look tiny, but they aren't infinite. These numbers have real use to some people.
But that is kinda the point... there are no real examples of 'infinity' in nature other than places where the known laws of physics are expected to break down. But in practice infinity can sometimes be used as a term where a number is so large but there is absolutely no indicator of the approximate size so you can't even give an estimation. Then you might say 'almost infinity' etc.
And thanks to kantrak for bringing up Graham's number. I didn't know about that and it was interesting to look up briefly. Yes, such a number would make even the most enormous of quantities or sizes in the universe seem incredibly small. I am guessing the largest 'natural' number conceivable would be the expression of the volume of the observable universe using the planck length as your base unit. Anyone care to do the math? :)
Edit: around 10^186
Thanks for trying to help me get my head round this. In my mind as long as a number has a value it doesn't matter how long it is it still has value but without a real quantifiable value to infinite it is pointless as a construct as it can't actually tell us anything, basically to me it seems to just be a point of giving up trying to comprehend or measure something.
Unless of course the definition of infinite is just a value beyond our current understanding or knowledge, in which case there is nothing which is universally infinite?
Even by that definition a singularity would be considered infinite I'd say.
Yeah I agree (just realised my poor wording at the end) but it's a definition that basically means nothing other than more research needed. Thinking about this now I'm quite happy for infinite to be the value 10^186 (as you worked out) as it is all we can know about from our observation point in the universe as long as that number expands if we do find out more.
That number being the largest in nature was just a guess on my part; I am not a physicist so I could well be wrong. It *could* be that the singularity of a blackhole has an infinite density (particularly something as old and large as a quasar) but as others have suggested, it will probably turn out to not be the case as a result of quantum mechanics. Personally I think it can't be infinite in reality as once the density runs away from a finite point then it can never reach infinity, though the number could well be higher than one could ever possibly consider (maybe more than Graham's number). Who knows.
You shouldn't be at all happy! Infinity is definitely not a number. Infinite is perfectly well defined mathematically; if you give me any number, something is infinite if it is bigger than whatever number you give me. In the case of the universe, it is infinite in size if, whatever distance you give me, I can confidently say it is bigger.
Not sure that is the mathematical definition. If I say 10^186, then you could say 10^200... just because it is bigger it doesn't mean it is infinite.
And of course a mathematical infinity can be considered slightly different to a 'real' infinity. I think think what ajsteele said about anything being bigger than say 10^186 could be considered infinite, while *technically* not correct, for the purposes of discussion and comprehension is not a bad idea I think. What that is saying is that while not infinite, for all intents and purposes it is infinite; which is more appropriate for the real world. but obviously in the mathematical world that does not stand true at all.
Can you explain how infinity as a "number" can be useful though? In your example you can claim the universe is infinite if I say it's 10 miles wide but you can't say it's infinite if I say it's Grahams Number miles wide' it seems like infinite doesn't exist but makes up for not knowing?
Is there a mathematical use for infinite that actually leaves you with some rock soild knowledge or proof, I can't think of one and therefore I can't think of a reason for it to be practically useful?
There are examples in physics. A superconductor has infinite conductivity below a critical temperature and helium becomes a superfluid at low temperatures so it has infinite 'flow' ie no friction. Both are arguably the inverse of properties shown to be zero.
Many examples... the most common probably being the asymptote. If you graphically display 1/x on a graph then each 'end' of the line tends towards zero on the corresponding axis but never reaches it. Therefore it continues to infinity before reaching the axis. I probably didn't explain that well but suffering from too much coffee in a bid to work.
A classic one is infinite series representations of numbers, eg
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
or various ways of representing functions as infinite sums of other, better understood functions. These have applications all over the place. I suppose you could in theory get the applications out without using the concept of "infinity", but you'd basically just be using lengthy circumlocutions around essentially the same concept.
Another is that using "infinite" when you actually mean "so big it makes no difference" can make more or less impossible problems into more tractable ones - eg in calculating how a fish moves in water, it's useful to treat the ocean around it as being infinite rather than having to worry about exactly what it's boundary is.
In general, it depends a bit on what sort of maths you're talking about - a set theorist and an algebraic geometer will have very different concepts of "infinity", relevant to the system they work with.
I think I am coming at this wrong, it started as a facetious question but I'm now really intrigued by the concept of infinity. I think I was imagining a massive number earlier, 1 so long it is currently incomprehensible but after thinking for a bit I'm now thinking it's more likely to be a language construct (thats maybe not the right terminology but I'm typing this as I try to think through it). What I mean by that is, infinite can mean different things at different times depending on the context, I'm thinking a low type of "infinity" could be -275 celcius as that is below absolute zero so is infinitely colder than anything we know possible. Am I on the right lines here?
Does anyone know any good books I could read which deal with the idea of infinity or the maths associated with it?
Before Saturday I would have said the infinity was the amount of time it would take Stoke to beat ManU at OT. But that was then, this is now
You're on the right lines but the temperature thing was not the best example to use... you physically can't have anything below absolute zero as temperature is a scalar measure of kinetic energy of atoms effectively. Think of it as kinetic energy in everyday life... you can't have something that less speed than 0 and the same applies to temperature.
I have a book at home called infinity, or something similar, by one of the usual science writers; I can't remember the exact name or who it was by, but I'll look it up when I am home if you're interested. It was a good read if I recall.
Infinity is concept I love. When you get deeper into the mathematics is gets really weird. Concepts like something of finite size can have an infinite boundary, or that you can demonstrate that there are more irrational numbers than there are rational ones, even though there are an infinity of both.
My own view is that infinities exist in neither conceptual or actual physical form. By conceptual I mean that I don't believe that mathematics is a real inherent property of the universe, so while the ideas may exist and be coherent and consistent, there is no property of the universe which makes that so.
Probably John Barrow? Interesting book, a bit waffley and some weirdly chosen quotes I thought.
Possibly, but don't think so... I think the book was called simply 'infinity' which doesn't come up under his name. But searching for that with no author does not bring up results.
Yes, sort of. Insofar as a lot of maths doesn't talk about "infinity" as a number, but as part of a concept, like "f(x) does this as x tends to infinity" meaning roughly "you can find some X such that f(x) does this for all x > X".
Other times, in a computation, you might use "infinity" as a shorthand for "something that's automatically bigger than anything except another infinity".
Meanwhile, set theory, and other areas of maths that are closely linked to it, start from the definition of an infinite collection of objects. Specifically, an infinite collection of objects is the one that can be put into one-to-one correspondence with "itself minus some stuff" - eg the set of natural numbers (1, 2, 3, 4...) is in one to one correspondence with the set of multiples of ten (10, 20, 30, 40...), even though intuitively the set of natural numbers includes "more stuff". The fact that you can keep mapping 70 -> 700, 1,000,000 -> 10,000,000, Graham's number -> Graham's number * 10 etc without running out of stuff is basically what we mean by it being infinite. "Infinite numbers" are then defined to be (roughly) the sizes of infinite sets.
Algebraic geometry uses infinity to mean something like "the point that you add beyond both ends of a number line so you can think of it as looping around into a circle." Although done properly it uses a slightly more rigorous definition than that!
Yes, sort of.
Not in a mathematical sense for that one - "infinitely cold" doesn't really mean anything, and I can't really see why it would mean "colder than is physically possible".
One normal situation where "infinite" comes into play in talking about reality is probably when you pretend something's infinitely big or far away because it's big enough or far enough away to make no difference - so if you're doing calculations on the trajectory of a cricket ball, you can treat alpha-centauri as being infinitely far away, because it's so far away that its gravitational pull is irrelevant.
Similarly when calculating how a sardine propels itself, you can treat the ocean as infinite (because the effects on the sardine caused by the precise size and shape of the ocean are negligible) whereas when calculating the likely fluctuations in the gulf stream this summer you can't because they aren't.
I'm not certain, as going through all the posts above caused my brain-cell to self combust, but I don't think anyone has mentioned the most clearly documented example of infinity in nature, which is also the most relevant to us - the Inaccessible Pinnacle!
One edition of the SMC guide (and they wouldn't lie would they? - that would be like God lying!) stated it to have "an overhanging and infinite drop on one side, and on the other, a drop both steeper and longer"! (may not be an exact quote, but close).
Not sure just how they measured it though, there may have been a smidgeon of inaccuracy, just like in some of the Munros at the time.
You are missing the point. If it is bigger than ANY number you care to give me, then it is infinite.
I don't think so. If however far you travel from earth you can always travel further, then the universe could be said to be infinite. Same idea.
> pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
I think that using the word "infinite" here is an understandably common but misleading and sloppy way to describe something that is perfectly well defined without mentioning infinity.
By that logic then any irrational number has an infinite number of decimal places.
The way I explain it to my students is that infinity isn't a number, it's a bound bigger than all numbers.
Most kids seem to be able to get their heads round that.
The joys of cardinal sets! The best approach to explaining how some infinities are bigger than others, in my experience.
But now you're saying that it's infinite if it's bigger than an infinite amount of numbers I give you. That's not a good way to define something. I much prefer ablacketts definition.
Nope that is wrong. The universe has a finite size, you just can't reach the edge travelling less than the speed of light because spacetime is expanding. Totally different thing there dude.
The universe *might* have a finite size.
It is fairly well established I'd say. Even if due to the 'shape' it would be meaningless to calculate, it would have a finite size when viewed from the point of a higher dimension.
Either way, even if it didn't have a finite size, the definition provided I was referring to still does not work. It is a consequence of expansion of spacetime and nothing to do with being infinite.
I believe the jury is very much out on whether the universe is infinite or not. I've just read Brian Greene's multiverse book and he seemd to be saying that, on balance, it's probably infinite and indeed therefore always has been.
I also liiked the idea that, if our univerese is a bubble universe, then it can be infinite from our own perspective but finite from the perspective of an observer outside the bubble. A bit mental really....
No. Bigger than ANY number you care to give me right now. It is important to define "infinite" without reference to infinity (obviously!).
Infinity is how long/far Pacman can travel east (if there were no walls) in 'forever'.
Note that 'forever' is 'infinite time'.
Infinity is a well defined mathematical concept, but (unless the Universe really is infinite) you can't have an infinite amount of something - where would you put it?
Once you have the hang of infinities, time to move onto their counterparts - infinitesimals :)
But that's my point. By saying bigget than 'any' number you have to assume I don't just give one number (as then it can just be a finite number that is bigger)... bigger than any and all numbers I give you would be true but then to do that I'd have to give you an infinite number of numbers and thus you are defining infinity using the term in the definition.
The point is that he picks his "number", then tells you, then you get to pick yours. If he picks ten million, you can pick ten million and one. If he picks 10^257 you can pick 10^257 + 1. And so on. One definition of infinity is that it's a number for which you can't do that.
Saying that no number is bigger than it means that you cannot find any one single number that is bigger than it. There may or may not be an infinite number of numbers that are smaller than it but the definition only claims that you can't find one that's bigger.
> But that's my point. By saying bigget than 'any' number you have to assume I don't just give one number.
No. I don't know in advance what number you are going to give me, but I know that, whatever you do give me, the size of my "infinite" thing is bigger (I can, say, travel further across the universe or, by going far enough, find a bigger number in my sequence of numbers).
But now you're using infinity as a finite thing... which is contradictory.
The singularity at the centre of a black hole being infinitely dense.
The universe being infinite in spacial and/or temporal extent.
Maybe nature abhors infinity and these possibilities will be become more certain finite phenomena.
> But now you're using infinity as a finite thing... which is contradictory.
No I'm not.
Robert is defining infinite as "larger than any finite number". The thing you seem to be having trouble with is that there are infinitely many of those finite numbers. Which is true, there are: it's implicit in the way they're defined and used.
But you don't have to list them all out for Robert's definition to make sense. Its enough to state that "infinite" means larger than any of them.
What was wrong with my suggestion of a superconductor having infinite conductance. A current will flow in a loop forever in such a material.
"....any ONE of them" is perhaps clearer.
Is there any such thing as a perfect superconductor in reality?
I'd be surprised if you could actually have a current flow in a loop forever in anything.
I think there is nothing really physically infinite about it.
It is no more infinite than the the time taken for a stationary car to travel a set distance.
The idea of infinite conductivity is equivalent to zero resistance. In fact we can always define a new quantity to be the reciprocal of an existing one, and if the existing one can take zero value, then the new one can take "infinite" value. But this is really just a mathematical trick and I don't think corresponds to a true physical infinity.
It's all a mathematical trick :)
> It's all a mathematical trick :)
Not if the universe really does go on for ever........
What do you mean, does infinity exist in nature? Do numbers exist in nature? Not really; they're abstract things. So is the concept of infinity, so infinity doesn't exist as a "thing" in nature. But just as numbers are useful in describing nature, so is infinity.
Infinity does not exist. (Yet) it is still being made.
I cant give an estimate when it will be finished
I don't mean do numbers exist in nature. I am not addressing Platonism or even the usefulness of maths in describing nature.
I mean in physical reality, does the infinite exist?
For instance the natural number 7 exists in nature. I can line up seven objects. Done.
I don't think I can do that with a countable infinity and am unsure that I can do it with uncountable numbers at all. That's the sort of idea I am looking at.
Every snowflake is unique, therefore there are infinite snowflake possibilities.
However this is probably only true at a molecular level.
Therefore, this leads to the thought that everything in nature has infinite potential.!
There are not an infinite number of possible snowflakes unless there is an infinite number of water molecules.
Are you talking specifically about cardinalities then, i.e. are you asking if there are an infinite number of things in the universe?
Here's who to ask.
I assume so. In which case is it just a matter of whether the universe is infinite and doesn't have all the particles confined to a finite region of it?
I am pretty confident that there are not an infinite number of things in the observable universe and am not wondering about that.
The universe maybe infinite in extent, time may be infinite, there may be regions where matter is infinitely dense. These are three possibilities for the infinite in nature.
I have a suspicion, that there are no such physical infinities.
I agree, however it is not just because an infinite number of something (in this case water molecules) that are required to produce an infinite variety.
A different way of looking at it would be how small things are put together to make big things. Just because a constraint, assumption or simplification etc of some kind is applied, does not mean that an infinite variety is not possible.
Think of it another way. Everything is unique, nothing that I am aware of has been proved to be identical, nor is any copy 100% exact (if considered down to the sub atomic level). Even the same object at two different points in time is slightly different to before.
Therefore, if the smallest possible building block is not identical, this leads to the conclusion that there is an infinite variety of more complex things that can be built. Hence endless snowflakes!
'I have a suspicion' and 'Pretty confident' - Sounds like rather large assumptions, considering infinity is being discussed.
I don't quite follow you or lowersharpnose here.
The potential for an infinite number of different snowflake configurations depends in no way on the actual possibility of them all existing simultaneously, nor necessarily on there being an infinite number of water molecules to make them. Unless that is there are only finitely many arrangements for each finite number of molecules. I.e. If you had 12 water molecules there would be, say 120 ways they could be arranged as a snowflake (I have no idea, just speculating and it seems plausible that there would be construction restraints like this).
So are you saying there are infinitely many ways for a finite number of water molecules to arrange into a snowflake (I don't know anything about the maths of snowflakes, but it seems probable to me that this isn't true), or that given an infinite number of water molecules there are infinitely many ways they can be arranged? The latter seems trivial, as all you are relying on the idea of having an infinity in order to justify the another infinity.
I only made a reference to a molecule to help understand that when considering something very big like infinity, you need to go to the very small to ascertain the differences.
Because infinity is so big, many would immediately consider that a duplicate is inevitable. However, when you examine what a duplicate actually means, you can begin to understand that everything can be considered unique instead. The differences become more apparent when you consider the movement of the smallest component particles and energy transfer.
It follows that every snowflake must be unique at a single point in time. Even more bizarre is that the same snowflake will not the same a moment later either.
If nature cannot create something that is absolutely identical in every respect for even a moment, then there must be infinite possibilities for creating complex structures.
The only constraint I can think of is if time stands still and all particles stop moving.
Ah, now, this is the bit I don't agree with. The number of possibilities can be staggeringly, incomprehensibly large, but still not infinite.
Let us take your snowflake example. Let us assume that the snowflake is 1.6cm x 1.6cm x 1.6cm or smaller. Space within the snowflake is descretised into Planck-length size chunks, the Planck length is 1.6e-35 meters. So our snowflake is 1e33 Planck lengths on a side, and contains 1e99 individual possible points in space. If we ignore the fact that water molecules are much, much, bigger than the Planck length, and allow every point in space to either contain one or not, then there are 2^(10^99) possible snowflakes. That is a ridiculously large number, but it is not infinity.
Thanks. I still don't really see what you are arguing towards though. The possibility of there being infinite variety in something doesn't need to be proved in this way, and you don't need to go to the very small to achieve it. Are you just arguing for there being a potential infinity, or are you arguing that as nothing is the same as itself from one moment to the next that this is an actual, physically represented infinity?
I'd also agree with John B above. You can argue that everything is unique in an instant and never the same thing in the following instant, and I think the idea of infinity would be consistent with this, but not implied by it. The definition of identity you're using seems to rely on the concept of infinity in order to exist.
Infinity died for our sins 2000 years ago
You cannot arrange a finite number of entities in an infinite number of ways.
Your arguments are fallacious.
What are you on about?
You're right, but only if there are finitely many positions that they can be placed in. It seems plausible that a finite number of objects could have an infinite number of positions relative to one another.
You can if you have an infinite space.
How about a Koch snowflake. An example of something that has a finite area, but an infinite perimeter.
Have you got a better way?
You can if you have an infinite space.
Which is one of the possible natural infinities I mentioned above.
As to things like the Koch snowflake, they are mathematical constructs, not natural entities.
Quoted from Wikipedia on Fractals.
'The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth Fractals are not limited to geometric patterns, but can also describe processes in time Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature'
The key phrase there is "various degrees of self-similarity". For something to have weird properties like a Koch snowflake, you'd need that to be infinite degrees of self similarity, which is essentially impossible in nature because at some point you're going to get down to subatomic sizes and not really be able to go any further.
Do you seriously think that snowflakes retain a fractal nature nature at a scale below that of molecules?
Ah, you beat me to it. Cheers.
Self-similarity is a property of fractals, which may be manifested as:
1. Exact self-similarity, This I suspect is a pure mathematical construct because it has to be identical at all scales e.g. Koch snowflake
2. Quasi self-similarity. This approximates the same pattern at different scales and may contain small copies of the entire fractal in distorted and degenerate forms.
I have no valid reason to be able to agree with your statement that the process stops at the sub atomic level. If you have knowledge to confirm your assertion, it would make interesting reading.
Perhaps his use of 'sub-atomic' was misleading. Plank scales would be a better term. But in reality it is not that relevant I think as fractals are not hugely relevant in reality due to the fact that they can't really exist due to the limits of the plank length (though they are good for describing structures within a range of sizes, rather than an infinite scale).
That would make more sense. I do find it hard to believe that the Planck scale (which is man made) represents a limit on what nature can do. However I am not arguing a belief.
It might not represent a limit at all. But anything smaller is out of our fundamental measurement capability (which is more the point of it being 'man made') so we'd never know. And christ knows what quantum weirdness is going on at scales much smaller than the Planck length.
How is the Planck length man made?
IMO, it is more a feature of our universe.
I'd say that you're the one making the bold assertion here and as such you're the one that has to back it up with something. But in any case, self-similarity pretty obviously doesn't continue down to the molecular / atomic level, since water molecules aren't snowflake shaped and hydrogen and oxygen atoms aren't water molecule shaped or snowflake shaped, even to the extent that they have a "shape" in a conventional sense. AIUI, beyond about that level, the approximation that we call "shape" breaks down pretty much entirely, so the concept of self-similarity becomes meaningless.
Since you're into quoting wikipedia:
"Infinite iteration is not possible in nature so all 'fractal' patterns are only approximate."
I had given up on the example and the whole aspect of chasing fractals, but as you have put in the effort towards finding this out I thought I would respond. You are right, it does say that.
I wonder how the Wikipedia entry expected the word 'nature' to be interpreted though. Today the concept of nature as a whole has been expanded to include the physical universe. If space itself, is considered fractal, then it can not be stated as a generic term that it is not possible.
Wikipedia: Fractal cosmology:
Basically there is currently a conflict between Relativity and Quantum Mechanics.
"French mathematician Alain Connes has been working for a number of years to reconcile Relativity with Quantum Mechanics, and thereby to unify the laws of Physics, using Noncommutative geometry. Fractality also arises in this approach to Quantum Gravity. An article by Alexander Hellemans in the August 2006 issue of Scientific American quotes Connes as saying that the next important step toward this goal is to "try to understand how space with fractional dimensions couples with gravitation." The work of Connes with physicist Carlo Rovellisuggests that time is an emergent property or arises naturally, in this formulation, whereas in Causal dynamical triangulation,choosing those configurations where adjacent building blocks share the same direction in time is an essential part of the 'recipe.' Both approaches suggest that the fabric of space itself is fractal, however."
Yes and no I think... the planck length is the smallest length we can 'measure' so is mainly relevant from the observer point of view rather than determining behavior at this scale. We literally have no way of knowing anything smaller (or how it behaves).
Crayefish has already given a good explanation.
I consider that anything we have created, and this includes mathematical constructs or measurements that help our understanding of something can be considered man made. To get a better understanding of something so small, our monitoring tools or methods of observing need to become more advanced. We cannot dismiss something, just because of our failure to measure it properly. We have might have unwittingly placed an arbitrary limit on smallness with respect to the Planck length.
To be honest, I am not sure if there is any other fundamental reason for the Planck length being the limit that it is, other than the measurement phenomenon... there could be but I don't know it. If recall, if the photon wavelength was any smaller, it would result in a blackhole but not sure how much significance that has in the real world as these are unrealistic energies as far as I know. We define the Planck length via various constants (c, G and Plank const I think) but I don't think that means it has a physical significance in the real world.
Hazy topic for someone like me who has been out of science for a while :)
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