joigus

@MigL wasn't talking about your force on the ball. He was talking about gravity. (My emphasis.) It weighed. Now it doesn't. Where's the force?

!!!

That's why I didn't say concepts like "force" or "interaction" are sterile. I said your concept of "real" is:

Not necessarily. The set of integers modulo 7 are numbers. And very useful ones at that. And there's nothing in their nature that even remotely suggests a line. They are quite independent from the concept of a point. Some numbers can be assimilated to topological and geometric concepts. But they don't have to. A line has no zero point. What is the origin of a geometric line? Geometry is one thing Topology is another And algebra, yet another Mathematicians love to play with these things. I'm sure @studiot and @wtf can tell you volumes about it. They go like: Can I drop this property and still get something interesting?

Then, a unicorn is real, as it is a real literary artifact used in story-telling. Which leads me to think that you've expanded your concept of what's real to the point of rendering it completely sterile.

How do you know these laws are real, and not a part of your theory? Are meridians and parallels real, or just a cartographic artifact?

Forces are not observables in quantum mechanics. Little wonder. "Force" is a very derived concept. How much force does a W boson exert on a neutron when it decays? Etc...

I brought it up because I know the theme of so-called detrimental genes that can be a blessing in disguise for the group has been the subject of extensive study and discussion in medical science / biology. Examples I can remember are sickle-cell anemia --which in its mild version protects you from malaria--, schizophrenia --which in its mild version is thought to have played a role in shamanism--, etc. I'm sure there are others. It could be the case for autism. I honestly don't know. That's a very good point. The brain is nothing but --it could be argued-- an organ evolved to map the world in certain ways. How it does that is subject to many variations. Some people count by projecting imaginary sounds in their minds, others do it by picturing images. It's even possible that there were some kind of 'maximum common divisor', so to speak, that affects us all, neurotypical or not. In that case J.B.S. Haldane's words "Now, my own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose" would be totally spot on as to what's given to us to know.

You haven't specified if this index is to be applied to ancient civilisations or modern ones. I concur with doubts expressed by @exchemist @Genady, and @Peterkin. Any definition you come up with is bound to meet criticism from almost any direction conceivable. You could try to formulate some kind of multi-index that takes into account factors such as, technological/scientific (functionality) artistic/spiritual/religious (symbolism) altruistic collaboration (solidarity) profit-focused collaboration (trade) If you think in terms of aspects that have played a role in defining 'civilisation' by anthropologists, historians, etc, you'll probably see that they're somehow or other included there. Examples: writing, accounting (technology), monumental architecture (symbolism and technology), etc. But there's plenty of room for ambiguity --eg, are old cosmogonies science or religion? This is where @Genady's comment on societies evolving becomes very relevant indeed. Or, I'm more inclined to try to work out a ratio very much inspired in biological definitions --like, eg, primary productivity-- that gives you an overall idea of what's going on in that direction, and to which a concept that most would agree underlies any reasonable definition of 'civilisation.' Such an index could go something like, #C:=(spare time)/(individual)x(day) Spare time being the time per day left over after having subtracted the time necessary for survival IOW, if a society has amazing technological capabilitites, but not enough time is left for people to dedicate to other activities than those essential for survival, then it's not civilised. Not enough, according to my index.

LOL. Beta decay = "when the neutron gives up the ghost" I understand 'ghostly' as 'interacting very weakly.'

By "fundamental" I mean resting on minimal/general assumptions, although I know how slippery that concept can be. Yes, theory of the mind --understanding that others have minds probably like we feel we ourselves have minds, and acting accordingly-- is one of the most important adaptive pressures that shaped the evolution of the human brain. I don't have the sources at hand to assert this, by I know from the reasoned arguments of many scientists of human evolution I've sampled through the years. I am in no doubt that there must be a reason why Nature has kept the genes that give us autism, which at first glance looks like just a cognitive impairment. Look closer and more can be seen. From my experience, from what I know from you, and others like you: Genuinely caring individuals, who suffer when they see conflict, generally devoid of manipulative intentions. Never foul players, sincerely concerned about problems, both human and technical, their own, and those of others. Hard workers, obsessive in a way that can be very productive, given the proper outlook, focus, and advise about how letting go when the time comes. Sometimes we're discussing something and we're being petty and narrow-minded. And here comes Markus Hanke and shines his light. All the pettiness is dissolved. The problem is re-focused to what the problem is. I guess that's why you lot are here for. Don't look now, but you activate us in a direction that --always in my experience, mind you-- usually is a good one.

Thanks, Phi. Now, what do you wish to discuss concerning neutrinos, @Vette888?

Right, but not news.

I couldn't agree more. For every bundle of sensory input we receive, we build models in our minds, filled with switches and state variables, or whatever you like to call them. Right now, in my mind, I picture @Genady's opinions and ideas, criteria, etc as --somewhat loosely-- some kind of a list with states of opinion, philosophical tenets and so on, that are perfectly defined as a series of 0s and 1s, so to speak. And you me, I'm sure. We work under such assumptions, and assign states, probabilities, etc to all these things. There is no fundamental reason why this should be true. What's more, even if it happens to prove itself useful to a certain degree, there's no fundamental reason why this process could be continued till the last least little consequence for everything we experience.

Exactly. I agree 100% with your professor. I have no qualms about considering the refraction index as a complex number, and its real and imaginary parts as real numbers. What I try not to forget, ever, is that there is a corpus of theory undelying this idea. It could just be approximately right. A real number is a convenient tool that gives you room to accomodate infinite precision. What's not to like about this wonderful tool?

Interactions, class of observers to class of observers transformations, thermal equilibrium. All of these are notions that require some kind of idealisation or another that is not real in any strict sense. Interactions: Consider an electron. You want to study it in detail in the ultraviolet regime, you need to shoot something at it and, pretty soon, you excite the virtual degrees of freedom, and you have to worry about virtual photons and pairs that seem to come out of this --initially at least-- well-defined thing. Lorentz transformations: You consider classes of observers that are at rest wrt each other and fill up all of space and time. Thermal equilibrium: In order to rigorously define thermal equilibrium you need to go to the so-called thermodynamic limit. That is, all the extensive --additive-- variables are infinite. So what's interesting to me is that, in order to study reality --whatever that is-- you absolutely must to take some distance from it, go to a theoretical framework that is not real in any meaningful way --it's just a convenient abstraction--, and draw your conclusions from there.

Numbers (real numbers) are not atoms, they're not beads on a string, they're not pieces of code on a DNA strand, they're not characters on an alphanumeric string. They're abstractions. That's, I think, at the root of why you're finding it so hard to wrap your head around the ideas that define them. Same reason why people I will describe in anecdote a1) at the end of this post, completely missed the point too. So you're not alone. Axiom of completeness: (1) Semi-intuitive notion: Real numbers have no gaps or holes (2) Rigorous notion: Every nonempty subset \( X \) of \( \mathbb{R} \) that is bounded above has a least upper bound. That is, \( \sup X \) exists and is a real number. Consider this sequence --not made up of atoms, nor made up of beads, nor made of little LED lights; made up of numbers--, \[ X=\left\{ x\in\mathbb{Q}/x=1-\frac{1}{n},n\in\mathbb{N}\right\} \] In other words, \[ X=\left\{ 0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8},\cdots\right\} \] Now consider the following facts: A) Every number in \( X \) is less than 1. B) For every \( r \) such that every number in \( X \) is less than \( r \), \( r\geq1 \). That means that 1 is not only an upper bound to \( X \); it is, actually, the best upper bound we can find. That is, 1 is the least upper bound. Challenge: Find a number in \( X \) that gives you exactly 1. If you get to understand how you will fail to find that number, say \( x_{n}=1-\frac{1}{n} \), you will be a step closer to understanding this logical frustration. Let me finish with a couple of anecdotes: a1) I was once sitting at a Calculus class and the professor told us about the axiom of completenes in the form, "every monotonically increasing sequence in the real numbers possesses a least upper bound." A couple of students in front of me started giggling and went something like, "Doh! Why of course." Needless to say, they'd completely missed the point. a2) Our Electricity and Magnetism professor said. "Imagine an R, RC, or LRC circuit with an amperimeter. You measure the current and it gives 1.3 Amperes. What kind of number is that? Is it real, rational, imaginary? After a prolongued murmur somebody uttered: It's a real number. The professor said: "No, it's neither one of them. It's a measured number." Don't let the adjective 'real' deceive you. They're real alright. In some sense.

The \( f^{0} \) comes from the 0 component of 4-momentum by differentiating wrt coordinate time. It's not to do with the usual gamma factor in Lorentz transformations. It's to do with a time-dependent gamma factor. The energy of a particle is \( mc^{2}\gamma \), but this \( \gamma \) is a function of time. The power gain/loss for the particle is the time derivative of its energy, so that, \[ \frac{d}{dt}\left(mc^{2}\gamma\right)=qu^{\nu}\left.F_{\nu}\right.^{0}=q\left(c\gamma F_{00}+\gamma v_{k}F_{k0}\right)=q\left(c\gamma F_{00}+\gamma\boldsymbol{v}\cdot\boldsymbol{E}\right) \] \( F_{00} \) is zero, as F is a 2-rank antisymmetric tensor. Its non-diagonal elements are the electric and magnetic field. The zero component of the time derivative of 4-momentum is thereby the power. It's all beautifully wrapped up in space-time formalism. I've re-instated the gamma factors, but I might be missing some c factor and perhaps my initial definition of F (the electromagnetic tensor) had the wrong sign. I'm sorry that I'm missing the main point in relation to physics and reality at the moment. Please, let me come back tomorrow and try to catch up with the finer points.

The Lorentz force does have a time component, though. When you write down the complete --covariant-- form of the equation it gives, \[ f^{\mu}=qu^{\nu}\left.F_{\nu}\right.^{\mu} \] as a 4-vector equation. Where \( \left.F_{\nu}\right.^{\mu} \) produces all the components of \( \boldsymbol{E} \) and \( \boldsymbol{B} \). These equations decouple into, \[ f^{0}=q\boldsymbol{v}\cdot\boldsymbol{E} \] \[ f^{k}=q\left(\boldsymbol{E}+\boldsymbol{v}\times\boldsymbol{B}\right) \] and where I think I may be missing a gamma factor. The 0-th Lorentz equation gives you the power gain or loss, and the spatial equation is the conventional Lorentz equation.

Very small curvature in both cases, never mind Riemann tensor components or curvature scalar (some kind of average of all Riemann components). It can be estimated by means of Newtonian gravity.

It certainly doesn't cramp my style.

Equivalence relations are at the basis of categorical thinking, or Aristotelian categories. We're hardwired to think in terms of categorical thinking. When we can't place the category, when that category does not close in mathematical terms, it's loosely defined, we feel confused. I think it was Wittgenstein that worried a lot about that problem.

Correct. https://en.wikipedia.org/wiki/Self-energy Is it just a manner of speaking due to electrons not being isolated 'things' in any meaningful sense?

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel One has to be careful even with the natural numbers. The ocean of natural numbers admits arbitrarily many more drops! The real numbers are even more counterintuitive.

Just stretching/shrinking space without time being involved... I see this difficult to reconcile with known physics. Physics with matter --massive-- is not invariant under scale transformations. Time-dependent scale transformations would make this even worse. I don't see how you could save conservation of charge, for example, if space is actually shrinking at small scales... Exactly!