It's my personal belief that in fact j = -i and nobody has noticed.Actually, j = ki.
My favourite set is the Russell set: A set which contains sets that do not have themselves as members.Well, isn't the whole point of formal set theory to make Russell's paradox impossible? So, for example the "set of sets that do not contain themselves" isn't possible. So for example:
Now the question remains is the Russell set a member of the Russell set? If it is a member of the Russell set then by definition it does not contain itself as a member and is therefore not a member of the Russell set.
In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms ... ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists.Which means, I think that there is no 'set of all sets that don't contain themselves'. Rather (I think) you can only look at some other pre-defined set and then say "of all of these, I want the ones that do not contain themselves"
...
In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.
A poet, a priest, and a lawyer are discussing whether it's better to have a wife or a mistress.--
The poet argues that it's better to have a mistress because love should be free and spontaneous.
The priest argues that it's better to have a wife because love should be sanctified by God.
The mathematician says, "I think it's better to have both. That way, when each of them thinks you're with the other, you can do some mathematics."
An engineer, a physicist and a mathematician are driving through the high country in Scotland. Atop a hill, they see a black sheep.--
The engineer says: "All sheep are black!" The physicist says: "No, no, some sheep are black." The mathematician: "At least one sheep is black on at least one side."
An engineer, a physicist, and a mathematician find themselves in an anecdote, indeed an anecdote quite similar to many that you have no doubt already heard. After some observations and rough calculations the engineer realizes the situation and starts laughing. A few minutes later the physicist understands too and chuckles to himself happily, as he now has enough experimental evidence to publish a paper. This leaves the mathematician somewhat perplexed, as he had observed right away that he was the subject of an anecdote and deduced quite rapidly the presence of humor from similar anecdotes, but considers this anecdote to be too trivial a corollary to be significant, let alone funny.Hmm... Most of the ones I find funny are of the "occupation X, Y and Z=mathematician variety
Among other things, Godel proved that all sufficiently complex formal systems necessarily yield true statements that cannot be proven within those systems (basically, any formal system worth its salt is incomplete--it can yield truths that aren't consistent with the formal axioms of the system. So, to put it really crudely, no formal system is powerful enough to explain all the truths it necessarily implies. Those necessary but incompatible truths, therefore, aren't products of the formal systems, but exist independently of them. The accepted interpretation of the proof is that formal systems describe an independent mathematical reality.
2) The claim that set theory is the foundation of all mathematics is an early-20th century view. The theory of categories is becoming increasingly important and can avoid a lot of the unpleasant consequences of set theory.
Mathematics is a method for describing patterns and modeling the consequences of their relationships. We created the mathematics we use, and we've quite often gone on to discover features of reality whose relationships are well described by some of the mathematics we create. This is the only connection that mathematics has with the world outside our minds. No minds -> no mathematics (they don't have to be our minds - those of any sufficiently sentient being would do).Well, congratulations on solving a millennia old philosophical problem so tidily. I guess it must be easy being so much smarter then Stephen Hawking and all.
This seems perfectly clear to me
which precisely my point, which is why the idea that there must exist some mysterious universal process responsible for "breathing fire into the equations" has always struck me as weird.Weird != false.
I think many of its practitioners forget that mathematics is map, not territory ... [primes are] map feature, and only occurs in the territory ... Facts are map entries we use to navigate within the territory we find ourselves occupying. ... facts an existence independent of the territory that they are facts about and independent of the minds ... My view is that mathematical objects are best considered as map features, not territory features. ... map entries that might possibly be made by some sentience at some time in response to its having noticed some features of the territory - then it still fails. ... no two objects are completely identical, and no experiment is absolutely and completely replicable. The territory is simply not that regular ...Confusing the map for the territory? I think you've confused an Analogy for Reality. Or metaphysics. All this reasoning about "Maps and territories" is also itself a "map" about something else, and therefore in your own model incapable of actually describing the "territory" of the interaction between mathematics and reality.
(If you want to find something I _do_ consider controversial in my belief system, it's that I think quantum entanglement actually does prove hidden variables. Spooky action at a distance is a _hack_, and making excuses that it doesn't send information is positively Ptolemaic.Google the Holographic Principle. There was a great video that someone linked too in some FPP talking that spent an hour explaining the entropy of black holes and how their surface contains a ton of information. So the speculation is that the universe exists as the information on some surface, and that gravity is a thermodynamic process, by which objects being together has a higher entropy then things being far apart. I can't find the damn thing now, though. Very annoying.
I'm using the "map" and "territory" analogies for the purpose of concise exposition. I can readily understand how my idiosyncratic use of those terms could result in a lack of clarity in communicating my position. If you know of clearer words I could use instead, please do share.Well "math" and "universe" work. The relationship between math and the universe is not identical to a map and a territory because, as I said, a physical hunk of land cannot create it's own map without outside help. And ordinarily they have not been created from another map, certainly not without intelligent help. But perhaps the universe was created on the basis of underlying math.
They don't really work for me, because they don't emphasize the one-way representational relationship between territory and map.But those are the actual names of the actual things. And the whole point is: we don't actually know what the relationship is. You've decided it's a certain way, and then are arguing based on an analogy to your own idea. But all the flailing about "maps" and "territories" has nothing to do with the actual things. That's why you use the real words, rather then ones that (you feel) are analogous.
I don't think of them that way. I think photons are map features. I will go so far as to allow that photon emission and absorption/detection events are probably territory features.See this? This is what I'm talking about. The "Map/Territory" stuff only exists in your head. But more importantly they don't necessarily exist in anyone Else's head. That's the important thing. If you want to thing that way, feel free, but no one else is obligated to think the same way just because you do. It's not at all obvious.
Really, all we know is that when we set up an experiment thus and so, we see correlations we would not expect to see in the light of present models. And all that means is that some of our assumptions about the entities and relationships involved are are unjustified.What? If you're talking about quantum entanglement and stuff, then no, the models predict it perfectly. The experiments about those correlations were done to prove QM, not open up gaps in it.
Here's a testable prediction: if we do ever come to understand how to model these cases properlyAgain. You're just wrong. We can model these things just fine using the standard model. The only "Gaps" in quantum mechanics are things like we don't understand where gravity comes from and things like the Hierarchy problem, where experimental results are different then what theory suggests but if you just plug those constants in then you get a working system that predicts everything perfectly.
and I have never attempted to suggest that anybody else is obligated to think the same way I do. I am simply pointing out that this view is availableThat's not true at all, here's what you said at the start of the thread:
This seems perfectly clear to me, which is why I find myself getting a little exasperated when well-regarded thinkers like Stephen Hawking and Paul Davies go on about what "breathes fire into the equations" as if mathematics were in fact somehow prior to existence. Apparently there's money to be made in deliberately mistaking the map for the territory and torturing the idea of "existence" until it gives up and means anything you want it to.See, you said they were deliberately getting it wrong. In order for something to be deliberate, it has to be done knowingly, which means that you apparently thought that these people do know of your idea, and know it's right, and are either lying about it or being "exasperatingly" pig-headed by not agreeing with you. Like it was so obviously true
That's not a bug, that's a feature.
the problem with your whole argument is that you're taking this "map/territory" thing as a given and reasoning from there.
I agree with this, except such a symmetry is expressly denied by QM.Since you clearly don't know anything about quantum mechanics, why are you talking about it?
I agree with this, except such a symmetry is expressly denied by QM.... Maybe. Measurements that show a violation of Bell's inequality are usually interpreted as ruling out "local hidden variables." Whether a global symmetry is subject to the same constraints is a question I don't know the answer to (not for want of trying).
Also, how is a symmetry different from a hidden variable?That's a question with a long answer, maybe for another day.
See, this is why I hate the words "observe" and "measure". Me putting the output of two spatially distributed detectors into a spreadsheet, does not represent a flow of information between two detectors.You're right, it doesn't. It's when you compare the results of the detectors that you have information flowing between them.
I've been thinking about situations where probability distributions are inverted, i.e. you have 100% to 0% in one direction, and 0% to 100% in the other. That would, over a large number of samples, agree with test data -- it's something of a midpoint. Is this something akin to what you're thinking?No, I don't think it is. Here's what I was thinking.
"all physical laws still working?- Keep continuing physical laws operating... Checking States... Measuring Size. If size is over limit begin compaction. If size = Smaller than limit... continue expansion. Particles operating on Wave/particle manner. Observers Present. Sol System>begin operation quantum state/non-locality. "Without this "Near instant communication" between "spatially distant" "elements" of the Universe, it wouldn't exist. It is how "Universal Physical laws" are enforced... and it happens with things that we are not creating in Labs also.
"Shadows live in a simple world. They glide effortlessly across any sort of surface, oblivious to the higher dimension of space in which 3-D bodies move, collide and sometimes block the paths of rays of light.
Shadows have no idea how important that third dimension is, and how objects in it endow those very shadows with their quasi-physical existence. Indeed, the laws of shadow physics all depend on the third dimension’s presence. And just as the clueless inhabitants of the shadow world require an extra dimension to explain how they exist and interact, reality for humans may also depend on an invisible dimension or dimensions unknown."
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"String math describing such ripples stems from an idea called the holographic principle, used by string theorists to describe certain kinds of black holes. A black hole’s entropy depends on its surface area — as though all the information in its three-dimensional interior is stored on its two-dimensional surface. (The “holographic” label is an allusion to ordinary holograms, where 3-D images are coated on a 2-D surface, like an emblem on a credit card.) The holographic principle has value because in some cases the math for a complex 3-D system (neglecting time) can be too hard to solve, but the equivalent 4-D math provides simpler equations to describe the same phenomena."
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A: "Banach-Tarski Banach-Tarski"
posted by Plutor at 6:28 AM on March 17, 2010 [63 favorites]