joigus

Agreed. There is such a thing as punctuated evolution, and the concept can be applied to some extent to scientific progress too, I think. Abrupt changes appear out of complex scenarios that could not have been predicted in any analytic way with the tools at hand. AAMOF, they are to be expected somewhere along the way. Alexander Fleming's discovery of penicillin is a perfect example. We are in the middle of a massive data-gathering phase now, measuring things we could barely have dreamt of only decades ago. That's why something new is to be expected. That's also probably why, I think, no continuous model like the law of diminishing returns really applies when it comes to predicting this kind of abrupt changes.

In the words of George Costanza (Season 3, episode 9, The Nose Job), "You can't stop modern science. You can't stop it. You can't stop it. Can't stop science. Can't be stopped. No way, no how, science just marches..."

So do I. There are silent, progressive revolutions, and you understood my point perfectly.

Topological insulators Quantum computing High-precision tests of the standard model (not science-spectacular, but extremely important) Neutrino physics (flavour-changing neutrinos) Observational cosmology (gravitational waves, supermassive BHs, accelerated expansion, exoplanets, etc.) Negative tests of proton decay (negative-result test are extremely important) Non-linear optics And the list goes on... As to other sciences, Ancient DNA Gene therapy Stem-cell therapy Ribozymes Cloning techniques And the list goes on... I think it's the other way around: It's very hard to keep up, really. It's because we're piggybacking on the shoulders of giant breakthroughs that it's so hard to tell how fast we're going --relativistic metaphor-- and even harder to relate the information in order to get a glimpse of any kind of big picture. I think there hasn't been a major change of paradigm, and that's easy to be misinterpreted as no advance. Whether these major advances will coalesce into a paradigm shift is neither certain, nor necessarily the case to be expected.

Maths is the proper language to describe/ascertain uniqueness and/or complexity. What makes you think there is a better language? Approximate calculations without maths? And how would that go?

It is my opinion that words themselves are worthless without the world of meaning behind them. It's what it means what's been, I'm sure, essential in human evolution. Long-distance trade, collaboration, etc. would have been impossible without the phatic function of language. Having others know the communication line is open even if you didn't completely understand the full import that they're trying to get across is a priceless function of language. I learnt that word, as usual, by carefully listening to others. I wouldn't have understood it by just reading a book. Thanks for appreciating...

I personally don't parse every sentence I hear or read through propositional logic. That's all I meant. Language has a phatic function too, you know.

There's nothing wrong with seeing with your mind's eye what your heart feels. It's not science, that's all. And it certainly isn't a scientific speculation.

You have no basis to assert this. All we have is endocasts and certain genetic sequences. How do you know?

Try with different definitions of existence and see if you can make some progress in your understanding. Maybe everything is fleeting, under the proper time perspective. You didn't mention what your definition is, BTW.

You're probably right. We got lost in geometry. I suspect there's something about time that's not entirely geometric. To me, it has the unmistakable flavour of abstract algebra. Suggestions from QFT are clear. Deeply involved in microcausality, operator-ordering questions, and the CPT theorem.

I don't know either.

Why not? Minkowski is R4 with signature (-+++), while R4 with signature (++++) is 4-dimensional Euclidean space. S1xR3 with signature (-+++) would be a Minkowskian cilinder, while with signature (++++) would be Euclidean. Metric signature and topology are quite independent.

What about a 4=1+3 cilinder? A cilinder is flat, it's S1xR3, but it's not R4.

Do you mean provide another example of tautology, which we would only escape by proposing further relations in the initial tautology? My example was classical mechanics, which would lead us too far from OP's goal at almost any amount of detail. But I think it's plausible that, if we were to make any progress in the problem of time, new concepts would have to appear, being circular in their initial formulation not constituting a difficulty impossible to overcome. If I were to try, I would take inspiration from similarly groundbreaking advances. In the case of Newton, F=ma seems to be both a definition of both mass and force, which at first sight doesn't look as much of a step towards progress, does it?... Until you formulate a law of force F(x,v), the concept of inertial frame as somewhere far removed from sources of interaction where systems satisfy the 1st law, etc. The problem with time is it's so difficult to conceive of intuitions that would lead the way as, in the case of Newton, isolated systems, force, and mass.

IMO, we shouldn't be too afraid of tautologies, as long as we have an external hypothesis to get us out of it. In fact, if we're ever gonna find a way to understand time, I think it's very likely that we have to do it by formulating some kind of tautology, and then ponder what the external assumption must be if we're to make it into a predicting machine. An outstanding example is Newtonian mechanics. The bare formulation is as tautological as can be. What is force? Mass times acceleration. But hang on. What is mass? Oh, that's easy: It's the ratio between force and acceleration in any direction. We wouldn't get anywhere from just that. But there's a hidden assumption: Whatever we want mass to be, it must be the same in every direction. And then there's the amazingly consequencial assumption that, under different simple circumstances, force depends on position in some particularly simple way. Then we're in business, because we can predict. Something that, with the sheer tautology, was impossible.

It is a scalar field that's needed in inflationary models of cosmology to provide a mechanism for the vacuum to go through different phases of expansion. A scalar field is a field represented by a number at every point of space and every instant of time, so that it doesn't change at all under rotations. In a way, it's very much like the Higgs field. It affords you to "imprint" needed properties on the other fields and space-time itself to account for observed properties that don't quite fit the model without it. It's considered by some to be somewhat ad hoc --convenient, but not very well understood or logically compelling.

Exactly. It's always a why from a how, or a how from a further how. And an initial how --what to assume-- is always susceptible of a further why. That's why there's always a how in the last analysis. How about that?

Interesting...

Yes, I don't know what all of this has to do with interpretations of QM, but composition of collinear boosts is not linear in velocities, although it is in rapidities[?*] (which are essentially hyperbolic-angle arguments.) That's probably what @Markus Hanke meant. The part where I get lost, as I say, is what any of this has to do with QM and its interpretations. I do not wish to keep talking about this, as it has no bearing on the particular aspects I'm more interested in, but suffice it to say that linear or non-linear depends on the arguments on which the object is considered to be acting. Eg, and totally analogous to collinear boosts, rotations about the same axis are linear in the angle, but not in the trig functions the rotation depends on. * Is that the word?

Thank you, Markus. +1. As usual, you helped to take the discussion back on its track. And that's right. They are linear. Thank you for your attentive reading. +1 I must confess I must go back to @Genady and @Lorentz Jr's parallel discussion and understand the point. The argument has spilt over into different aspects of physics, and it's hard to keep up.

@Lorentz Jr, you'd be well advised to refrain from your usual hissy fits, stop using neg-reps as some kind of covering fire for your arguments, and stick to the topic. If I didn't understand your point, say it nicely and we can all get back to business. I can assure you I, and many others, think you make a valuable contribution to the forums. Lorentz transformations don't act on the v's, they're parametrised by them. That's whay I got confused. I'm sorry for having ruffled your feathers. Jeez! I've just reverted my red point.

Why do you act so childishly? Saying they are non-linear is not the standard way of referring to them, and certainly not the appropriate way. It's at best confusing. If you wish to say they are non-linear in velocities, you should specify that. That's not the way we refer to them, among other things because once you are in the domain of SR, 3-velocity is not a vector quantity on which to define linear / non-linear transformations. It's just a set of 3 parameters.

They are linear.

Absolutely no rush. This thing about topological Lagrangians giving rise to systems (quantum or not) entailing some kind of rigidity has kept me wondering for years. For trajectories of point particles, it's clear that you get the system frozen to a point. But for field theories it's not so clear. If you allow for non-commutativity (quantum) the question is even more involved. To me, it both is interesting and makes sense. But for some reason people who study topological field theories haven't put it on the front burner, for years and years. You only see it mentioned as some kind of quirk. Yeah, OK. I must confess I have to think about this harder, and read more stuff with as much attention as possible. Some of these conditions coming from constraints do look a lot like continuity equations. It's peculiar to me that conditions like the Lorentz gauge-fixing condition are formally identical to local conservation laws for a Noether charge, although they're just constraints. And in the case of gauge fixing, they seem quite arbitrary. In the case of these topological theories, they seem to appear more "naturally." At least the way I saw Lee Smolin handle them in those lectures. That's all I can say for now in the way of a reflection. It's just peculiar to me.