studiot

Wow I'm completely blown away by this completely incorrect statement from a recognised authority. Wet, Wetting and Wetness and molecules are precisely defined terms in the physical sciences which arise from surface tension effects. Since ST arises from differential interactions between at least five molecules and wetting between at least two of these, ascribing any wetness to a single molecule is nonsense. Incidentally I prefer to use two words rather than strong and weak plus one word (which actually inefficiently makes three words to learn) for different concepts. So I use emergent and arising.

I didn't think of that. +1 There is a more detailed discussion here. I am not an expert on number theory, I find the minutiae rather boring after a while. https://math.stackexchange.com/questions/823970/is-i-irrational

Can't see why. e and pi are irrational (and also trancendental but that is irrelevant here). i and 1 and 0 are all rational. So proceed as follows [math]{e^{\pi i}} + 1 = 0[/math] Euler Rearrange [math]{e^{\pi i}} = - 1[/math] Raise each side to the power -i [math]{\left( {{e^{\pi i}}} \right)^{ - i}} = {\left( { - 1} \right)^{ - i}}[/math] Which is [math]\left( {{e^{ - i\pi i}}} \right) = {\left( { - 1} \right)^{ - i}}[/math] Which is [math]{e^\pi } = {\left( { - 1} \right)^{ - i}}[/math] Which is an irrational number raised to an irrational power expressed as a rational fraction of two rational numbers.

The Navier Stokes equations are readily derivable equations connecting perfectly well defined mathematical entities. However they are non linear and as such have only few known particular solutions in simple cases. We do not have a general solution, although the existence and uniqueness theorem suggests there must be one. That is why there is a prize offered for a general solution. Well I can't go with this definition, since it rules out my arch. But it is an interesting proposal and admittedly narrow. @joigus and I seem to be dancing around one characteristic that you have named 'novel'. Thank you for your explanation of that use of novel. I agree but suggest that by itself novel is too brief as others have read it differently. Your Nature article also acknowledges the problem of including a time factor in any definition. The one characteristic is the idea of a large number of parts/components/elements combining in some way to produce the emergent effect as a result. The Nature articles implies this can only happen with an infinite count. I hold that the number need not even be large, let alone infinite, although it could be. Going back to my arch consider the following thought experiment (which is actually possible in practice). Take a bunch of wooden bricks (I say wooden because ceramic ones could break during the experiment). Toss them up into the air to land in a jumbled heap. Sooner or later a few of these will land in the configuration of an imperfect arch. Further trials will bring better and better arches. So the 'emergence' of an arch configuration has a statistical dependance. The probability of a perfect arch is very small, one in a very large (perhaps infinite) number. Chemical reaction dynamics has a similar statistical dependence. The rate of a reaction product being formed depends upon the probabilities of the right configuration of the right molecules occurring. Unlike the arch, we do not know the right configuration or the right molecules for the formation of Life, so we cannot substantiate the hypothesis that Life is an emergent phenomenon that occurred this way. It is worth noting that in the case of the arch we know both the right elements and the right configuration so we can deliberately construct an arch, ie change the probabilities to approach certainty. So I would like to propose a modification to @Eise proposal that instead of saying it does not matter what the elements the phenomenon emerges from, to certain formative elements must be present and correctly configured, but it does not matter how they themselves arise or arrive.

Are you referring to euler's identity ? [math]{e^{\pi i}} + 1 \equiv 0[/math]

Developing the concept of necessary and sufficient sounds good to me. +1 A work in progress sounds good to me. +1 Good question worth exploring +1 irreducible ? out goes Joigus' reference to elements. Nor does my arch example fit. Unpredictable ? people have built weak arches but theya re predictable. Novel ? Arches have been around thousands of years. But perhaps arch action is not emergent. As I said it is difficult and I am not offerering a definition, just trying to help.

So by this definition a pile of broken egg or humpty dumpty shells or perhaps the ashes of a burnt letter are emergent phenomena. I think that tying down the concept is really difficult and that there are as many definitions as there are definers. Nevertheles it can be a useful concept. Thank you again for introducing the subject +1

I agree but my point is that I don't think your definition is enough to make a phenomenon special enogh to be called emergent. I agree that you may or may not know where the phenomenon arose or came from. My example with the light is just such. I see light. It could have come all the way from the furtherst star in the universe or the candle by my bedside, it doesn't matter. I don't atually need to know its source. But I don't call it an emergent phenomenon, just light. Yet it conforms to your definition quoted above. That is why I say your definition is too broad. Please also note it is a horse and cart or a horse and carriage ( both precursors of the abbreviation car) Sorry to hear about the covid problems in the Netherlands. We didn't make it to Leiden this season as we didn't last season either. My nephew and niece made it back there, don't know when or how they will be able to get back (to work) in the England. They did get a luxury BA flight for £20 though as no one wanted to go to Schipol before Christmas. On another tack entirely, I can udnerstand why people think perhaps Schrodinger leads to an emergent phenomenon. When we solve the equation we use (introduce) suitable boundary conditions and then find quantum numbers 'emerging' from the solutions in the form of values where the solution can be set to zero in a periodic fashion. I prefer to observe that this is common with many equations and nothing special at all. Consider the following much simpler equation x2 -4 = 0 therefore x is 2 (yes or minus 2) Now change this to x2 - 4 = 5 Now x is 3. Nothing special at all. Wouldanyone say the numbers 2 or 3 emerge from this ?

There is no paradox. As an engineer have you considered the work done moving the magnet ?

Whilst I agree with you about the QM and programming, there are many sorts of computer whose working do not depend upon QM. Some of these are purely mechanical from astrolabes and orrerys to slide rules, some are electrical, some elctromechanical, some fluidic. De Morgan can also be demonstrated with a variety of devices. I shan't tell him if you don't But no, I mean that I can tell when there is day and when there is night around me. I don't need to see or know the source of any light but I don't consider any light I do see to be 'emergent'. Pretty well everything in this universe had a beginning, a middle and an end but I don't consider that because everything came from something that went before it, it is therefore emergent.

The quality (not qualification) or characterisitc of being emergent applies to a phenomenon, not to an equation. However I think that since emergence only occurs in special circumstances, these special circumstances need be incorporated in any description or specification of an emergent phenomenon. As a for instance, my earlier example of arching action. Despite the antiquity of the arch, we do not have an 'equation' to this day that properly describes arch action, although we understand it. Furthermore arch action only occurs in particular circumstances.

Well I think it is neither unreasonable, nor unexplained. 'Unsuprising' would be theonly un-word I would choose. The whole issue reminds me of (I think it was Strange who first said it here) of the story of the puddle. The puddle who woke up sentient one morning and said to himself. "Wow look at that, this hole fits me so well, it must have been made for me!"

The existence and uniqueness theorems.

In my view no wave equation qualifies as emergent, as you have to know the wave variable to have the equation in the first place. I think you condition is too broad by itself, though it is an interesting approach. +1 I don't agree that the Gas Laws show any emergent features. I do strongly agree with your second example of 'clusters'. I expect you have done a lot of simulation theory. I am not sure about 'free will' and like you I don't know about time as emergent phenomenon. Back to your condition. I look around and sometimes see light and sometimes see darkness. I fail to class either of these as emergent, but by tour condition they would be. I walk along the street and see a fish lying there. I can't know how it got there so that would make it an emergent phenomenon. (This example is inspired by today's local news about a seal pup that walked into a bar in Bristol)

Yes probability and probability density are different. I have already defined probability. Probability is a non negative number less than or equal to 1. 'Probability density' is introduced to overcome the problem of division by zero, as with all densities. Probability density is a function.

What an excellent post. +1 And isn't 'regularity' a superb word since not only does it apply to so many different branches of Mathematics in some wya, but it is a general word that has not been hijacked by any discipline. Regularity appears algebra in Group theory, in geometry in polygons and other figures, in the solution of differential equations many of which are important in Physics such as Schroedinger, the Wave eqaution. May I suggest that electrons would be a better example than water molecules since H2O and D2O have some different physical properties ?

Since several people liked my comment, (Thank you all) , I will expand a little. The issue here reaches many parts of theoretical Mathematics, but one part goes to the heart of applications in Science. This is in probability theory and therefore in statistics and quantum mechanics. The probability of an event P(E) is defined as the limit of the relative frequency of that event as the number of trials tends to infinity. For instance consider rolling an n sided die. As the number of sides increases the number of different possible outcomes increases. As the number of possible outcomes increases so the probability of any given outcome (ie an event) decreases. So as n tends to infinity P(E) tends to zero. So we have the apparent paradox to resolve of how can we have a probability when we know that the die must end up showing one face or another, yet the probability of showing any one face is zero. In QM we resolve this by taking the probability between x and (x + δx) and taking a limit as δx tends to zero. In 'shut up and calculate ' mode we don't think about this we just do it and get 'the right answer'

This Zeno paradox is deeper than any of the others and was not properly answered for 150+ years after the others. The other Zeno paradoxes rely on sequences of integers and their reciprocals. This one relies on something deeper. The solution came after it became necessary to integrate many functions that could not be integrated by the Riemann integral, commonly taught in high school today. As you likely know, the Riemann integral is the sum of lots of small rectangles that make up the area under a curve. In fact it is the limit as the width of these rectangles ten to zero. But Zeno's question is what happens when that limit is reached ie the width is zero? The generalisation the the Riemann integral was introduced by Lebesgue (1875 - 1941) adn this ushered in what today is known as measure theory. https://en.wikipedia.org/wiki/Henri_Lebesgue The other approach to this issue was also developed in the first half of the 29th century by Paul Dirac and is known as the Dirac Delta function.

No, an engine will not run without it. An assembly as described without friction or any load will need at least an initial input of work (energy) to start. Thereafter it will continue in its state of motion as you describe.

I don't quite agree. Things are slightly more complicated than this. An ideal flywheel, crank and piston assembly by itself is a closed system. Yes. So it will continue its state of motion or rest indefinitely. But if you want to supply shaft work you require to input that energy somehow. And since some of the shaft work output of an IC engine goes to run absolutely necessary auxiliary devices, continuous energy input is required.

Another well balanced post. +1

You are mixing up thermodynamics and mechanics. No you do not need to put energy into a mass to start it moving. Stand under my window where the flower pots are and let me push one off the ledge. When it hits your head tell me how much energy I put into it. No the engine is not a closed system. Mass in the form of air/fuel mixture enters and echaust exists. It is known as a constant flow system or pseudo-closed since the same amount of mass exits as enters. But that entering mass brings (chemical) energy with it. So it is not an isolated system.The rising piston does work compressing the gas. The expanding gas then does work on the piston. Although work and energy have the same units and are different aspects of the same phenomen, there are subtle differences I suggest you look up. https://www.google.co.uk/search?q=difference+between+work+and+energy&source=hp&ei=3YPVYdboC43KgQbT-6qYCw&iflsig=ALs-wAMAAAAAYdWR7R2QBD6hUnxptAl9631h-LuTuAZn&ved=0ahUKEwiWy9faw5r1AhUNZcAKHdO9CrMQ4dUDCAg&uact=5&oq=difference+between+work+and+energy&gs_lcp=Cgdnd3Mtd2l6EAMyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEMgUIABCABDIFCAAQgAQyBQgAEIAEOgsIABCABBCxAxCDAToOCC4QgAQQsQMQxwEQowI6CAgAEIAEELEDOg4ILhCABBCxAxDHARDRAzoFCC4QgAQ6CAguELEDEIMBOgUIABCxAzoICC4QgAQQsQNQAFj8LmCSMmgAcAB4AYAB2ASIAdI6kgELNi45LjguMy41LjGYAQCgAQE&sclient=gws-wiz But +1 for accepting that you were not exactly correct before.

Yes that is true. So 1) There are numbers of no special consequence in Physics. That is there are parts of Mathematics that have no special meaning in Physics. 2) There are facts (numbers) in Physics which have special meaning that have no special meaning in Mathematics. Both leading to the conclusion that there is incomplete overlap between Mathematics and Physics. Is that not rational thinking ? OK I consider myself driven off the forum.

So What ?

What behaviour ? I said there are mathematically definable consequences. Would these consequences not be different if the value was ten times different ? Put into what ? Does the 'standard model ' predict the values of such numbers ? If so why bother to have them ? Why not just use the numbers themselves ?