In the mathematics of the physics, gravity is a scalar field. We don't really know what gravity is, we just have various descriptions that seem to be useful at various levels of detail. So when folks talk about gravity waves, in the maths it is a pulse traversing a scalar field, and we don't really know what is happening to the space or whatever it is that makes the pulse possible. Is space actually stretchy like elastic? Unknown.
But it means that if there is no other potential field that can "bend" a moving gravitational wave's trajectory then gravitational waves will travel in absolutely straight lines.
Light, on the other hand, moves through the potentials generated by the gravity field and is governed by its energy, sort of, to follow geodesics of the gravity field. So in GR light takes the curvy path around objects with gravitational potentials around them. The article doesn't mention if there were any influential masses along the journey.
1.7 secs is about 320,000 miles. Maybe the curves?
On the connected question of why human societies accelerated their development over time (recently posted on HN) a couple of comments are pertinent (see [0] and [1]).
If one values the continued acceleration of human development (in all its expanding pluralities) then there is a strong case for something like Z-lib to exist for a lot of factual or science-generated information being available for everyone. For such texts, I would guess that what the library contains could lag behind the commercial publication front by 5-10 years without appreciable profit loss to the publishing companies, nor significant losses to the estates of individuals who created them. And taking away some of their back catalogue might actually reduce publishers' running costs... (Ask: Can anyone else on HN put hard numbers to this?)
The debate seems similar to the one regarding taxation vs. inflation rate rises. Governments are currently choosing to raise inflation rates because that doesn't impede trade flows as much as some potential friction coming from moving money through a government bureaucracy for some cause. The particular cause matters, but the choice is about efficiency in essence. The same with Z-lib, as I see it. A Z-lib type free digital library could raise global efficiency in all sorts of areas, but the argument will only pass if various forms of extortive-because-looking-very-obsolete-now old capitalist inefficiences can be demonstrated to be such. Lots of publishing companies are probably holding on to old titles that for them are already digital landfill in their archives. Maybe they just need to see that clearly.
The other catch is that the "for everyone" phrase implies a representative global stakeholder is needed for it to "get legal". UN.org/z-lib anyone?
Efficient proprioception is useless if key behavioural incentives are misaligned with key aspects of the environment.
Humans are quite good at creating collective organisations that are delusional, even though some of those organisations have excellent information about the state of their environment, because the invented incentive structures for action favour simplistic short term "maximisation" of fetishised behaviours instead of optimisation for (even) the medium term healthy future. Resource allocation is still looking frighteningly suboptimal right now, and simply empowering that with a "smart" central AI bot following the same patterns probably won't improve much.
I'm skeptical about the human race getting past this. Humans want a lot of stuff, and are prepared to form huge organisations to get that stuff, without really thinking through whether they really need it, what the other impacts will be, or why they are even doing what they do.
What is the human race trying to be?
Why?
How does the pattern of incentives get aligned?
It seems like we only have the weakest responses to these questions right now.
Imho, RRO only works well if some additional rules are added, to hold from the outset and throughout the process. As a prototype offering:
1. There must be sufficient representative diversity on the committee.
2. No agent on the committee should be allowed to buy/bribe other agents to form a persistent (all powerful) majority bloc.
3. To combat issues with (2) all such blocs should be (re)considered as reduced to single agents with a single vote.
Imho, the situation (2) seeks to combat can happen in the real world, with clear evidence of agents no longer being independent in any meaningful sense; so additional strong requirements for transparency surrounding a process are needed, maybe enforced in the committee charter.
I agree with the comments that the article is misdirecting fluff. Cutting through...
All organised systems (anything that we would describe as a system as opposed to random or chaos) require additional energy inputs to maintain organisation over and above those energy inputs required for performing input-output function. Maintenance requires work.
All life on Earth depends on the thermodynamic gradient supplied by the Sun being appropriately steep enough for that additional organisational margin to be provided within tolerances, but not too steep to make everything excessively random, or too shallow to stop stuff moving.
Failing to provide that extra margin of energy for organisation for any part of the functionality of a system results in breakdown and halting.
E.g. Quite a lot of the extra margin of human organisation energy is provided via being omnivores _at the top_ of the food chain. If all the plankton and birds and bees die -> Humans don't get enough food -> Disorganisation -> Dead humans.
It is possible to build forms of organisation that are inherently unstable outside of limited tolerances. Stepping outside of those tolerances exponentially raises the extra costs required to maintain organisation.
Needless to say Nature doesn't keep many such systems around, if any, particularly when under significant stress. They get weeded out fast.
The extra kicker in human economic organisation is money as a mediator for controlling energy allocation.
Organisations retain function if they can afford to support the additional cost of being organised.
Within monetary systems, profits have to remain high enough to pay off loan interest and support organisation.
But...
If people without enough jobs or money starve -> The latent labour pool dies -> No support for growth, no maintenance -> breakdown.
So the labour force loses a margin of value-add generated in order to support organisation.
The hidden labour force that is Nature loses out once human economies/populations grow too fast or too large.
So modern humans do this "supporting organisation" without considering Nature as part of the organisation, and frequently fail to support the labour pool properly.
Obsolete monetary/value system designs are a big problem.
Globally, monetary systems to support sustainable societies should provide for: equilibration against destructive monopolies - especially at the ForEx level; enough liquidity - supplied through limited liability loan failures under capitalism - to meet aggregate interest on loans and organisational costs AND latent labour pool support; maintenance of key functional inputs, like food, sunlight,...; and zero cumulative pollution;
Why shouldn't every viable economic or social unit operate monetarily locally according to Kirchhoff's current law, if money is an adequate proxy for energy?
Why do we not price in the extra energy input Nature requires to remain coherent and still balance supply and demand?
Humans previously kept building inherently unstable civilisation structures with inherently unstable monetary systems that cannot cope with externally imposed constraints arising.
They kept failing to preserve and maintain the inputs sensibly, and they designed monetary systems that are typically destined to create and exarcerbate those failures.
They could collectively: stop breeding like rabbits; design better monetary systems; stop shitting on the environment.
All of it is technologically feasible already.
How much of that is politically feasible right now?
People are holding people back by being people. Wicked problem. Difficult to handle kindly.
In the first Open Problem, maybe the generator tweak for random Hamiltonian paths is that two grids are needed: one for the eventual H-path; and one of points that are the centroids of the first which is used to draw a random tree starting from some point, subject to some rules about distances to unconnected nearest neighbours being acceptable (hunch e.g. of length >= sqrt(2)). Then draw around that tree on the first grid to get the H-path. (Grid graph duality use.)
I'd guess this can be generalised to 3D by drawing the H-path on each 2D plane slice, and some simple rules about choosing the degree of interconnection between those planes.
It seems like all the tools for tuning the shape of the initial random trees already exist in MJ, based on the examples shown.
Maybe someone familiar with the graph theory math can help me.
I don't understand why these properties aren't numerically accessible. Why is the estimation necessary?
E.g. Assume N nodes, and edges E can be created at random with redundancy (picking same edge e twice just keeps the prior edge) by picking pairs of points.
Assume N random edge picks are made, then the probability of e.g. a Hamiltonian cycle appearing in those N picks can be calculated.
(My quick back of envelope sketch says P(H-cycle) = N! / N^N , but please consult the expert literature.)
But one can also calculate the probability of a H-cycle given N+1 random picks, N+2 random picks, ... and that appears as
= P(H-cycle) * (1 + extra factors that account for increases in other edges not redundantly in the H-cycle)
(Again, my back of envelope sketch for N + 2 picks gives:
= P(H-cycle) * ( 1 + 2/(N-1)[N - 3 - 1/N]) , but please consult the expert literature.)
These probabilities would seem to tell a user that given a graph with N nodes and E edges, that if e.g. in the H-cycle case, E > N the user can get an explicit probability for the likelihood of a H-cycle being present.
Are there graph properties that prevent this approach being viable?
Why CO2 from air vs. extraction from seawater? I remember seeing some papers (e.g. [0]) suggesting the from seawater route was way more efficient per m^3 processed with a lot less environmental disturbance.
I am not a lawyer, but... a public interest clause might not be the right tool, since public interest can be interpreted as restricted to information that would affect a person's voting preferences and nothing broader.
In the UK TV broadcasters get their licences and general charter terms from the government, based on standards previously contested in law and, I believe, broadly enshrined in statute and/or regulations. One of those terms is a crucial clause to the effect that broadcasters are not allowed to broadcast anything that would make ordinary people throw up over their TV dinner or grannies die of apoplexy, etc.
Various entities, people or single issue political parties for instance, have attempted to get broadcasters to show films containing grisly stuff, and when they meet refusal have attempted to sue broadcasters on free speech grounds. However, the right to free speech does not allow a legal entity or person to compel some other party to shout out noxious views on their behalf. So broadcasters in the UK can exercise their rights against such compulsion and filter out extremal horrors if they wish. The tests they use are designed to maintain the same two or three standard deviations of a normal distribution curve that maximises their customer base and matches the same standards civil law requires to maintain order on the civil scale anyway.
In that context, what are social media companies if they are not media intermediaries between legal entities? Thus they can not be compelled to relay anyone's content to anywhere else if they don't want to. Their T&Cs typically say so anyway.
Folks worried about censorship of their partisan or minority views on such platforms don't seem to get what rights those intermediary companies have. They play nice relaying messages because they like mass market incomes.
So really the issue might be purely about how many messages could MetaFB/Twitter afford to kill off and still be popular enough to survive?
Incidentally, I just drew some diagrams and with the assumptions that the Earth is not at the centre of expansion, Earth orbit has extremes (for parallax), and all light only takes straight routes, my second question has a negative answer.
But if one puts a gravitational lens on the view of one of the two stars then (based on a fast doodle though) it looks like maybe expansion could be detected.
The executive summary of my paper is that provably there are fatal inconsistencies in Cantor's Diagonal Argument (CDA).
They take a few forms:
(1) Application of basic classical analysis tools reveals that the contradiction sought by CDA does not hold if those tools are used.
E.g.
Assume we have the table of all unique infinite length binary strings {0,1}* (where * represents the supremum of the natural numbers, e.g. the first ordinal infinity in set theoretic language).
Assume that those strings represent the fractional part of binary represetations of real numbers in the continuum interval [0,1].
Assume they start big endian, so for some string s, the value v(k) of some bit at the k-th position of s is v(k) = s(k)/2^k, for k = 1,2,3....
Let f: \N x \N -> {0,1}, be a function that accesses bits in the binary strings viewed as a matrix M(i,j).
Create the binary number z as z(k) = 1 - f(k,k).
Now deploy basic analysis.
Note that k -> \infty implies v(k) -> 0.
So no matter what the value of z(k) is, k -> \infty implies |z - f(k)| -> 0, for some suitable k.
Let f be the limit of f(k) as k -> \infty.
Then in the limit k -> \infty the number z = f, and (for all practical purposes) the diagonal z coincides with a binary number in the table f within a countable limit.
This is independent of any ordering of the table rows of M(i,j).
So the contradiction of CDA does not hold convincingly when these basic tools of analysis are used.
(2) Corollary: Platonists would need to prove that infinite digit existence somehow invalidates basic analysis in order to cling to CDA. Moreover, they would have to prove that "higher" infinite digit existence does not invalidate useful concepts of numbers and limits at all.
As far as I know, no one has done that yet.
(3) Using the same basic tools of analysis, key arguments about powersets having cardinalities that are "higher" infinities do not go through, and moreover aspects of "higher" cardinal arithmetic, as presently defined, can be shown to be self-inconsistent.
Therefore:
A) When the tools of basic classical analysis are so useful, why would any pragmatic user of mathematics throw them out at some key point within CDA (and CDA only) to endorse a notion of "uncountability" that creates manifold inconsistencies further on?
B) Why should any pragmatic mathematician allow the extra unconvincing machinery of higher infinities into maths?
In the absence of decent answers to those questions I am compelled towards the recommendation is that mathematicians should throw CDA out.
Lawvere's paper appears to suggest that if I refute one diagonal argument then I refute them all. Would you consider that to be an accurate description?
I don't understand the notation f(n)(n). Is it related to f_{nn} in LaTeX notation? Your later text suggests maybe it was aiming at f(n,n) so I will assume that.
I recognise a form of this argument and I might have tackled it in the supplementary materials I created that are referenced in the article. Let me know.
> However, z(k) = 1 - f(k)(k). Yet f(k) = z, so z(k) = 1 - z(k).
I'm assuming this was intended to be:
z(k) = 1 - f(k). Yet f(k) = z, so z(k) = 1 - z(k).
The mileage of others may vary, but it has been my experience that there is no cogent solid proof of uncountability that can withstand concerted critique.[0]
Being charitable one might argue that the meanings of terminology had been lost in translation over time and that perhaps Cantor was trying to create non-standard analysis, but then the diagonal argument seems to represent nothing more than the truism that finite numbers are smaller than transfinite ones.
Hence I worry about people who are still worrying about this issue, and I worry for the future of science and AI in particular if folks can't get clear of it.
Gravity, arguably, does not experience gravity as a curved surface.