At Multiverse Impasse, a New Theory of Scale
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simonsfoundation.org
On the other hand, due to the nature of HN, one has to choose between either spending a few days researching the comment and posting it with a very slim chance of being read or answered, or blurting out something not very polished.
In this case it's the latter. (How you define “objective”? “reality”? A rough example of how could the answer to the question “what reality is” roughly look like? Any links to existing research—people certainly attempted to ask a question of this nature before?) The resulting discussion may be worth it, though.
And I felt that the Planck-scale was somehow a proof for this.
We, as clever little beings found similarities and analogies between the universe and mathematics and leverage those to make conjecture regarding the form and nature of the universe. However, having a "complete" knowledge of either mathematics or the physical universe does not confer complete knowledge of the other.
I often use this line of thought to talk about God with open-minded believers. If there is a God that created our reality and acts on it, then the effects of these actions are real in our sense and thus measurable. It then follows that God isn't separate from the natural world and can be observed like many other natural phenomena we don't look at directly. Without a chain of interactions this wouldn't be possible.
What I mean is our reality is one of interactions. What is real to us, and what can be called "existing" is what interacts directly or indirectly with our senses and our minds. What we mean by "nature" on a large scale is ultimately internal structure. Once we hit a level where internal structure doesn't make sense anymore, I'm not sure what we are looking at and what the word "nature" can be made to encompass.
Any physical system that can be described by a set of mathematical laws is bound to have some axioms that simply cannot be proven within its context. Godel's incompleteness theorems prove that.
Hence, we will never get to the bottom of the reality of our universe.
This means that at some point, existence must have been started by something non-existent. Because we only comprehend cause-and-effect, we can't possibly find the origin in a provable way while researching from within existence.
Gödel's incompleteness theorems are about mathematical theories, not about physical theories. It may well be that for physical understanding a very limited arithmetic is sufficient. Also, there may be statements that are unprovable, but these might turn out to be irrelevant to sufficiently understanding the universe.
Also, an axiom, by definition, can not be proven. The things that are proven are theorems, statements, etc. An axiom is an assumption.
Eventually, it all depends on what level of understanding you want. But Gödel's theorems have little to do with that.
In fact, the theorem showing how to do so (Bayes' Theorem) is so utterly elementary we teach it in the first probability course everyone takes.
Where Gödel comes into the picture is that he gives us no right to believe that only those axioms are all there is.
If the reality is mathematics, Gödel justifies the Multiverse hypothesis: just because we are objects within some theory doesn't mean there don't exist other theories (perhaps superset of this one).
"Unable to prove" in the Gödel sense can be then equated with "unable to peer into another universe".
Gödel's sentence becomes analogous to some phenomenon or artifact of another universe which cannot exist in ours because it cannot be derived from its axioms.
I wish the word "reality" would stop being thrown around without justification. "Fundamental" might be a more appropriate term.
Only among high-energy physicists. Perhaps not coincidentally there's also a belief among high-energy physicists that only they count as physicists.
In reality, they're a small minority of physicists. The largest specialty in physics, accounting for about half of all research, is condensed matter physics. Then in no particular order you've got AMO (atomic/molecular/optical), astrophysics, geophysics, biophysics, and a bunch of other stuff sitting between or outside those categories. With the exception of just a few corners of astrophysics, none of these fields needs, or can make sure of, the stuff being done on fundamental interactions; regular old quantum mechanics plus relativity is good enough for describing everything we can actually measure.
Bardeen is the real deal. His papers are very interesting and feel a bit like reading Sidney Coleman's papers. If you're interested...
http://arxiv.org/find/hep-th/1/au:+Bardeen_W/0/1/0/all/0/1
Is nature scale invariant? So far the answer is absolutely NO but I strongly advise to wait and see. There are many topics that point to some breakdown in scale or reorganizing what we think of space and distance (dualities in string theory, conformal field theory).
OK, now the important thing I want everyone here to realize. You are living through a GOLDEN AGE of physics. You wouldn't think that based on what all the popular magazines tell you. Here's why...
1. Higgs particle - discovered!
2. Inflation - discovered! Denying this one is like denying the Big Bang itself. The evidence is overwhelming and in fact I would list this as the single greatest scientific discovery of all time. The concurrent discovery of gravity waves, quantum gravity and a real life example of a Hawking process only sweetens the deal.
3. Supersymmetry has basically already been discovered IMHO. They aren't announcing anything at CERN and won't until they have so many sigmas under their belt but trust me, it's coming and truth be told, it isn't really so surprising. SUSY physics has always been rock solid from the beginning. The situation is very similar to that before offical Higgs announcement and before someone went knocking on Andre Linde's front door. Many were extremely confident in the Higgs particle a least a year before the official announcement. The BICEP 2 results were even more glaringly apparent than the Higgs results. Many people were walking around the Earth with 'secret knowledge' that inflation theory was correct even 2 to 3 years before the official announcement.
So you are living through EXTREMELY interesting times but you wouldn't know it with all the big science bashing being thrown around.
You've likely seen it but if not you might enjoy it.
-- Evan from TQHN.
Why does an absence of evidence imply that the arrival of evidence would be shocking? They're doing the experiment because they already believe things are this way and they're trying to get evidence to confirm it. If they disconfirmed it, that'd be surprising.
Hypothesizing is one thing, but to believe without evidence is just blind faith.
Even in the case under discussion, that someone might have faith that something is a particular way, and are doing an experiment to confirm it (more likely disprove an alternative), that doesn't mean it's 'blind' as in without reason. People can have a lot of good reasons for believing something is a particular way without having measured it directly yet.
That's not science, that's confirmation bias. In science, one would want to look for evidence that one's theory is false with as much vigor as a search for evidence that it's true. Science isn't law, and it's not religion.
> If they disconfirmed it, that'd be surprising.
If they falsified their theory while only seeking confirming evidence, yes, that would be surprising. This is why open-minded scientists try to avoid assuming what they should be proving (the real meaning of the expression "beg the question").
No, the expression "assuming what you should be proving" has a special semantic meaning -- it refers to a thought process that uses its conclusion to support its investigation, or takes the preferred outcome as a given from the start, without seriously considering alternative explanations.
Hypothesizing means taking existing theory and extrapolating new untested properties, then presumably investigating whether there is any evidence for the hypothesis.
> You don't really design experiments that can both confirm and refute a theory.
On the contrary, the best experiments have the chance to either confirm or refute a hypothesis. The Michelson & Morley ether experiment is a classic of its kind -- its outcome would either confirm or refute the ether as it was imagined to be.
http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experi...
> When an "open-minded" scientist attempts to perform an experiment to test theory, they aren't personally responsible for subsequent theories and experiments that could disprove their results.
Not so. An open-minded scientist wouldn't pass up the chance to uncover any positive or negative evidence for or against his theory -- both kinds of evidence contribute to our understanding of nature. Remember the story about Bell Labs engineers Penzias and Wilson cleaning bird droppings from their microwave dish? They did that so someone else wouldn't scoop them by discovering that they had been fooling themselves about the source of the noise in their antenna (which ultimately was identified as the cosmic background radiation, now standing as evidence for the Big Bang).
http://www.aps.org/programs/outreach/history/historicsites/p...
The bottom line? Science isn't law, it's not adversarial, there aren't two competing sides, and a responsible scientist maintains an open mind with respect to evidence both for and against his theories.
Regarding SUSY, strictly speaking no particles have been directly detected but anomalous currents abound and just about everything seen works perfectly with light SUSY and much better than with the standard model. The dilepton events alone are very compelling. http://www.science20.com/a_quantum_diaries_survivor/a_susy_e... SUSY is in plain view much the way the Higgs was in plain view for about a year before the official announcement. I have no special insider information, just a humble internet connection to download the important papers and a decent knowledge of particle physics. In hind sight, everyone will say 'of course' which always happens. From a theory point of view, SUSY is simply not an option unless there is some spectacularly new theory out there with novel concepts to replace QFT. The Coleman-Mandula theorem is extremely compelling. It basically tells us accept SUSY or find something to replace QFT. Since QFT works and there's no good reason to move on to something else (especially when something else doesn't exist presently and possibly never will), it's a safe bet SUSY must exist somewhere. It just so happens to be light SUSY and we're seeing it at the LHC right now. I think we're presently seeing the equivalent of a COBE picture of SUSY and next year it will sharpen up into a WMAP picture.
I'm just an engineer, so I'm not sure I see the utility in any of that--sorry to sound closed-minded, but am genuinely curious.
For the record, my father was far more amazed by rambling, not entirely coherent speculative physics ideas than my mother or either of my aunts.
Apologies for taking the conversation off on a tangent, but you seem passionate about this.
I suppose you're referring to Lisi? Neither he nor E8 were even mentioned in the article. You're both OT and unnecessarily unpleasant.
of course. Natural thing when a field is dominated by orthodoxy that closes their eyes to everything that they don't want to see. Like yesterday "entangled photon imaging" where what really happens is that a beam modulated by an image heats/excites crystal (with that heating/excitement thus obviously modulated by the image) which generates another beam (thus that another beam is obviously also modulated by the image) which hits CCD - no miracle of entanglement here, yet Nature published it as such : https://news.ycombinator.com/item?id=8234221
as "entanglement" and "mutiverse" are very much in fashion this season and get you published.
With Higgs as a "mass" boson it was also non-starter because the theory of it failed to address gravitational and inertial mass equivalence. I mean i don't doubt that CERN found new particle of course, yet nowhere it was shown that it is the boson "generating mass". The article seems to suggest that finally the mainstream physics starts to seriously ponder whether the mass is a result of dynamic interaction - that has been obvious for decades to the "fringe" physicists, who couldn't just dismiss the above mentioned gravitational and inertial mass connection, a pretty fundamental fact that has to be at the center of anything called physics :)
I'll admit most physicists I've met are suspicious of wild-eyed theory. But I wouldn't say they're overly orthodox. I think it's more a healthy skepticism of anything which involves humans projecting science-ficiton wish-fulfilment into science.
I'd go further and say that "entanglement" and "multiverse" are not at all en vogue. I think most seasoned physicists realize that these were trendy fields some years ago (for solipsistic reasons - ie, make us feel special).
So I think the return to a fundamental assessment of phenomena is pretty natural thing. Calling out assumptions like scale. What's more, I think the more discerning will also be suspicious of the "post-multiverse" dialectic. It's just the process really.
...
>So I think the return to a fundamental assessment of phenomena is pretty natural thing. Calling out assumptions like scale.
i see big difference between "Calling out assumptions like scale" as a "return to a fundamental assessment of phenomena" and "Calling out assumptions like scale" in order to fit it into a wild-eyed theory like superstrings (basically to salvage the theory).
One of those is a theoretical idea for reducing the Solomonoff complexity of the universe with no clear experimental implications. The other is an integral component of how we predicted and explain experimentally confirmed phenomena like violations of Bell's inequalities, the Elitzur-Vaidman bomb tester, quantum cryptography, and quantum teleportation.
If we get quantum computers working at a non-trivial scale, would you consider that to be strong evidence for entanglement?
i still can't in details :) I so far have worked my way only until double-slit experiments - no superposition there :) For Bell violations i bet we have the same mis-interpretation of statistical aggregate description as individual realization (this "mis-interpreation" is the Copenghagen's superposition) as in double-slits. In case of Bell violations experiments it seems to be the same as "disjoint sampling loophole" http://en.wikipedia.org/wiki/Loopholes_in_Bell_test_experime...
>If we get quantum computers working at a non-trivial scale, would you consider that to be strong evidence for entanglement?
it would be strong evidence for superposition and thus for entanglement. Want these computers like anybody else, yet don't really see them happening :(
There's no magic: you're just sampling a joint distribution in the first place.
that is exactly my views too. QM people call it "hidden variable" theory.
>two separate distributions whose results are "forced" by "magic" to conform to the appropriate correlation after the "sampling"
and that is Copenhagen interpretation superposition wave function collapse based theory of entanglement.
>you're just sampling a joint distribution in the first place.
if somebody shows that Bell violations can happen here too the Nobel prize will be his/her. I though think that Bell violations can't happen at all. That would be another Nobel prize i guess :)
If the universe is actually very likely to occur in the state that it's in, then we don't need to use this "vast multiverse where our universe was the one that produced an observer" explanation. It could be a single universe in all of existence, where everything makes sense mathematically and no further posturing is required in that silly unscientific field of cosmic philosophy.
Or if it fails, and it remains that our universe is unexpectedly perfect, then we are forced to resort to statistical explanations within an impossible-to-observe external multiverse. Scientists don't very much like conclusions built on not being able to observe things.
Though to me, the multiverse theory has a sort of fractal appeal to it. It is surely applicable to our planet, after all. Why not our entire universe?
The whole http://en.m.wikipedia.org/wiki/Flatland Idea.
A "Multiverse" is a superset of that idea—it simply means a condition of many universes existing. They could have arisen in the same way ours did through a "big bang" in some larger space alongside our universe, they could be the many iterations of our one universe expanding and contracting over an infinite time, or they could be overlapping in the "many worlds" sense; we don't know.
The point of a Multiverse set is to explain our universe's existence statistically rather than empirically (as in, with a satisfying observational explanation). It says "we know that the chance of our universe existing is extremely small, so we must be one in an infinite number that had the right combination to result in life."
This is sort of like a car crash, and saying: "whelp, it happens with a probability of 1 per 1000, so it was bound to happen to someone" after a car accident rather than finding out exactly why and what caused the event—"whelp, the brakes didn't work, we should probably look into that."
Just like that car crash, finding out exactly why our universe exists, and how likely it is to have the reality we enjoy, gives us far more information than a simple philosophical statistical conclusion. It doesn't mean that the statistical conclusion isn't correct—there still may be a Multiverse, we may never know—it just means we know more about it than just how often it happens.
But, knowing how likely our universe is to happen is a big step. If it's very likely, then we know a lot more about universes. If it's unlikely, then we could just chalk up our existence to chance and it would be perfectly valid. In the latter case, however, it would be extremely unsatisfying scientifically. We'd love to know more.
If mathematical logic is a property of our universe, then aren't we again entering the realm of circular logic? i.e: "Given the logical frameworks that are correct in our universe, our universe is very likely."
Unless math is something that somehow exists independently of our physical universe... but then "where" does it come from?
My hunch is that mathematics is more than universal; truths hold regardless of which medium of calculation is used to conclude them.
In any case, our conclusions about our own universe at least hold within our own universe, and are therefore still very useful for our own understanding thereof.
"First, for maximum sense-making, read my previous comments here, here, here, and here. I realize that's a lot, sorry. The Universe is complicated. Then, for what follows, recall that to a particle physicist such as myself, there is no difference between lengths and energies (or indeed, any other dimensional quantity). Length is inverse energy: very energetic phenomenon probe very small lengths ...."
https://www.metafilter.com/142211/Multiverse-No-More-a-New-T...
The question boils down to this: are we one of an infinite number of universes that popped into existence that just happened to have the right variables to be stable and result in life? (The anthropic principle, a statistical conclusion, and a philosophical mind-bender...)
Or are we in a single universe whose properties are mathematically aligned in such a way as to be very unlikely to result from chance, even if repeated an infinite number of times? The details of this are interesting, but extremely numerous and complex: basically, Physicists are trying everything.
The latter does not require the hand-wavey wishy-washy existence of an infinite number of non-observable extremely large things, and therefore garners more enthusiasm.
However, if observations conclude that the parameters of our universe are in fact paradoxically perfect, then the idea that we're an evolved observer of one chance universe out of many holds water.
Hence the intense research on how this whole universe thing works. Fun.
[0] http://www.amazon.com/Our-Mathematical-Universe-Ultimate-Rea...
That is naive. There is no way to prove or disprove it; it can be safely regarded as true without ever contradicting any possible observation we can ever make.
I do not think physics will ever settle the questions in such a way that the "multiverse ennui" is not invoked to cover the loose ends; in fact, it will become more and more obvious as the inescapable conclusion.
The deeper we dig into nature, either we keep finding more mathematics, or we encounter some absolute that can no longer be analyzed.
Once we know every physical law, and are certain nothing more is to be discovered, how do we distinguish our universe from a set of mathematical axioms? We have two choices then: to suspect there is some outer universe which implements the rules and axioms of this one. Or else admit that axioms do not require an implementation: they just are, and that's what makes their system exist (which means that any other axioms and rules we can imagine also exist just as much).
Either hypothesis is an unprovable "cop out"; but the latter of the two is more plausible. The first hypothesis still leads to an infinity of universes, and they have to be nested in each other in an infinitely regressing sequence. The second one has no such silly requirement; and it has the anthropic principle which plausibly explains everything that seems arbitrary or special.
http://www.wired.com/2014/08/multiverse#comment-1560643455
Edit: Never mind, http://rationalwiki.org/wiki/Mensur_Omerbashich
EDIT: Look -- if you don't understand physics, don't compound your ignorance by downvoting the posts of people who do. Instead, post a written objection, and I will explain why you're wrong.
I like to consider ∆S the result of a smoothing operation over energy differentials.
You can put work into a system to reestablish an energy differential. It just necessitates obtaining that other energy differential with which to do the work.
> I don't like that definition.
Physics isn't a popularity contest.
http://en.wikipedia.org/wiki/Entropy
> You can put work into a system to reestablish an energy differential.
Yes, but that doesn't change the definition of entropy -- it just borrows energy from somewhere else.
Why link me to the Wikipedia article or insinuate I'm pulling this out of--to put it more politely--ex culo? Given my use of notation, one might assume I am at least a little familiar with how the thermodynamic entropy functions are mathematically derived.
To be pedantic for a bit, Entropy is an abstract mathematical state function useful for the prediction of thermodynamic systems. Much like the Gibbs and Helmholtz terms, entropy has no direct qualitative basis; there are many open-ended interpretations seeking to justify what it means, which is not unlike looking for God or faeries or what have you.
Mathematically speaking, I think you got your equality / definition wrong. Which energy function do you refer to? If you are still convinced about an error on my part, you can demonstrate in classical thermodynamics that,
S = "the inverse of available energy to do anything"
What's my incentive to continue a discussion with someone who has abandoned the topic? In a closed system, the amount of exploitable energy, and the quantity called entropy, move in opposite directions.
> ... or insinuate I'm pulling this out of--to put it more politely--ex culo?
Point to where I said this. If you can't, then you have abandoned the topic.
> Much like the Gibbs and Helmholtz terms, entropy has no direct qualitative basis; there are many open-ended interpretations seeking to justify what it means, which is not unlike looking for God or faeries or what have you.
Nonsense. Entropy is clearly understood, it's not a philosophical puzzle as you suggest. It's easily quantifiable. It's science, not philosophy.
> Given my use of notation, one might assume I am at least a little familiar with how the thermodynamic entropy functions are mathematically derived.
Anyone can adopt the notations used in science. This doesn't tell me anything about their knowledge of the topics under discussion. You simultaneously apply standard terminology, but you go on to say that explaining entropy is "not unlike looking for God or faeries or what have you", which means you don't understand the significance of the symbols you used.
Entropy = k log(n)
k = Boltzmann's constant.
n = number of states the system can be in.
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.ht...
Quote: "Temperature is expressed as the inverse of the rate of change of entropy with internal energy, with volume V and number of particles N held constant. This is certainly not as intuitive as molecular kinetic energy, but in thermodynamic applications it is more reliable and more general."
Yeah, but physics aren't a la carte. Either prove it wrong (and get a nobel) or accept it.
Okay, here's a commonly used explanation. Entropy and time are bound together -- as time passes, overall entropy increases (with local violations like us). This is one idea about why time "moves" in the direction it does -- if time were to reverse, we could tell because some classic entropy results would run in reverse, violating common sense.
Imagine that there's a room, and in one corner of the room there's a perfume bottle. At time zero, the cork is removed from the perfume bottle. Because of entropy, the tendency of isolated systems to move from order to disorder, the perfume disperses through the room.
Ask yourself what the probability is for the perfume to spontaneously recombine in the bottle, versus the probability that it will disperse through the room. That probability differential is a measure of entropy.
> So is there somewhere in universe some energy that is not available to do anything at all?
Sure -- any isolated system can eventually get to a point where no useful work can be performed. Imagine an engine without any temperature differentials anywhere, or any way to acquire a temperature differential from an external source. Such an engine cannot do useful work.
A steam engine requires a heat source and a heat sink. Without a temperature differential, the engine cannot function. Same with a gas engine.
> Is it too diffuse such that the energy needed to pull it together to perform a unit of work is greater than the energy available ?
But without tapping an external energy source, the energy in an isolated system won't concentrate itself or spontaneously create a temperature differential suitable for exploitation. In an isolated system, the level of disorder always increases over time.
> If so how did it get to that state?
Simple physics. For two masses having a temperature difference, one with temperature "a", the other "b", and a temperature conductivity of "k", they will eventually reach the same temperature this way:
Δ = (b-a) e^(-t k)
Δ (delta) = temperature difference at time t
a = temperature a
b = temperature b
t = time
k = energy transfer coefficient
As an aside, this is my own pet theory as to why some actions of living beings seem to resist scientific prediction. Observed within its own boundaries, a living being appears to be a system that locally decreases entropy; therefore within those boundaries it does not appear to operate causally.
So for example we can predict a human being who is deprived of food will die (basic energy balance), but we can't use science to predict what color shirt they will choose to wear next Tuesday.
(caveat: I know very little of real physics)
That's a layman's definition of entropy, but we're a long way from that state.
If we were approaching any "universal energy" limitations, it'd stand to reason that our technology would be very much different than our technology now -- perhaps universe sized experiments would be feasible by that point, at which time we may hit a limit imposed on us not by anything other than the length of time since the last universe formation event, and our poor luck in meeting that point in technology at a time where the energy no longer exists to fuel it.
Unlikely, and I think your idea of life being threatened before science is probably much more rational, but it's fun to think about. Plus, if we were using that much energy for a single experiment, one would wonder about that experiments' safety.
http://www.multivax.com/last_question.html
I recommend you read the whole thing, but if you're looking for a plot summary: http://en.wikipedia.org/wiki/The_Last_Question#Plot_summary
Of course, what you say is even more immediate.
Unless by "heat death" he meant a cold death, current thinking has it that the universe will gradually cool off, and ultimately freeze, because of the endless expansion provided by Dark Energy.
The fact that the universe is expanding exponentially, i.e. with a positive second derivative, was the first shock. The second came when the implications of this observation began to sink in.
I had a scheme, once, for everyone to be remembered. It started with one nearly impossible scheme, producing a biography for every person ever (starting with the living, with some effort made to recover as many of the deceased as are still remembered today). The problem with that is that being recorded is not the same as being remembered. Being remembered means that someone living knows about you and thinks about you. They occasionally ask themselves what you would think or do.
So you would need a system for everyone's life to be read from the recordings periodically, by at least one person. For the near future -- say a few thousand years -- this could work. For the first few generations, each living person could remember a small handful of the deceased, and all of the deceased would be held in living memory. Once the number of people to be remembered grew too great for that, you could still cycle them in from time to time. Being remembered by one person every thousand years could still be pretty nice.
But five billion years, that dog don't hunt. At current generation lengths, there'd be 200 million generations to be remembered. Unless you are willing to postulate either constant, exponential population growth, dramatic lengthening of generations, or superhuman capacity to remember, you couldn't remember any one person more than once every several million years. It still beats being forgotten, if that's your fear, but not by much.
http://www.technologyreview.com/view/413483/could-all-partic...
Isn't the word for that fractal?
"That shit would be too much," Gupta said. "It'd be like that Dr. Seuss book Horton Hears A Who and shit. I read that when I was, like, six, and it totally weirded me out."
Nailed it.
http://www.simonsfoundation.org/quanta/20140818-at-multivers...
That is, half of the article is trash and the other doesn't represent the science correctly.
So two things standing next to each other are the same size as one thing alone?
Damn hard to imagine. I would think "size" does not really "exist" but is a man made concept. And by its definition, two things next to each other form a "bigger" thing.
But, to address your example: "So two things standing next to each other are the same size as one thing alone? Damn hard to imagine".
That might not hold for size, but a very similar thing does occur (and is proven) for speed.
If you are in a car that goes at 0.8 miles per minute, and you throw a dart that goes at 0.3 miles/minute, its speed is 1.1 miles/minute (the sum).
But if you travel at the 0.8x the speed of light and you throw something forward at 0.3x the speed of light, its speed is not 1.1x the speed of light, but c. That is, trying to add velocity doesn't get you faster than the speed of light.
Not really. Velocities near c don't sum that way. And your reference doesn't support your claim.
If you add two velocities that are less than c, you can never get c, you always get something smaller.
0.8c + 0.3 c ~= 0.887 c (It's smaller than c)
At low speeds the effect still exists, but it's negligible. There is an important difference between "it doesn't exist" and "it's so small that if we forget it we will be fine".
It's so small that you must use floating point numbers with too many digits to calculate it. It's more useful to calculate the difference between the "relativistic" sum and the "classical" sum. Another way is to aproximate the formula with a Taylor expansion
dif = (X+Y) / (1+(X * Y/c^2)) - (X+Y) ~= - (X + Y) (X * Y/c^2)
0.8mph + 0.3 mph = 11mph - 5.9 * 10^-39 mph
The correction is veeeeery small.
Indeed it is. The only place I can think where such a small correction matters is in the GPS system. The accuracy of the GPS system depends on tracking down and eliminating every possible source of error, and to maximize position accuracy, two relativistic effects are accounted for:
1. The time dilation caused by the satellites' velocity, from Special Relativity.
2. The rate of time passage at orbital altitude compared to that at the surface, which reflects the gravitational well effect of General Relativity.
The two effects move in opposite directions, but they don't cancel out. The applied correction (to the orbiting clocks) is very small indeed, but enough to avoid a gradual but serious decline in positional accuracy over time.