Care to provide a reference to a paper where the Copenhagen interpretation has been terminally discredited? Or one that proves that observation does not have a part to play in wave function collapse? This issue is by no means dead - one interpretation is more popular than others at any one time, but this is more of a result of what is fashionable, not of what is experimentally verifiable.
Don't confuse unfalsifiable interpretations with testable hypotheses.
Damn, one hundred years of quantum physics and it takes you only a few seconds to demolish it.
Seriously, you can't use your everyday "common sense" when doing quantum physics. It's a subtle subject that goes against much of our everyday experience. Quantum physics does seem to suggest that conscious observation does have a role to play in the collapse of wave functions.
Not quite. The quantum state of the first object can change the quantum state of this second object, but the wave function of the second object will still be in an undetermined state until you observe it. *You* then must observe this second object, and only then does the wave function of the second collapse, consequently collapsing the wave function of the first object. What you've described is simply a chain of interdependent quantum states - until you observe them, they'll be undetermined.
Yes, but what about the safety of those on the moon? When we build our moonbase there, and start mining the moon for nuclear fuel and dumping our nuclear waste there, a simple accident could cause a tremendous explosion. This explosion could knock the moon from orbit, sending the moonbase and its inhabitants into the far reaches of space.
They would then have to spend the rest of their days hoping to bump into an earth-like planet. It's just a fate too boring to contemplate.
You're kidding me, right? I mean, I like Led Zep and all, but it's painfully clear that most of their songs have riffs that are "borrowed" from blues artists and others, or are straight out covers.
Examples (list found on web somewhere):
1. Bert Jansch - Blackwater Side (3:46)Black Mountain Side 2. Bert Jansch - Go Your Way My Love - intro (0:26)Going To California 3. Bert Jansch - The Waggoner's Lad - intro (0:51)Bron-Y-Aur Stomp 4. Blind Willie Johnson - Nobody's Fault But Mine (3:11) 5. Bobby Parker - Watch Your Step (2:49)Moby Dick 6. Bukka White - Shake 'Em on Down (3:01)Hats Off To (Roy) Harper 7. Davey Graham - She Moved Through The Fair (3:07)White Summer 8. Howlin' Wolf - Killing Floor (2:51)The Lemon Song 9. Jake Holmes - Dazed and Confused (3:48) 10. Joan Baez - Babe, I'm Gonna Leave You (2:41) 11. Josh White - Jesus Gonna Make up My Dying Bed (3:05)In My Time Of Dying 12. Leadbelly - Gallis Pole (3:01)Gallows Pole 13. Little Richard - Keep A-Knockin' (2:15)Rock And Roll 14. Memphis Minnie - When The Levee Breaks (3:11) 15. Ritchie Valens - Ooh! My Head (1:48)Boogie With Stu 16. Robert Johnson - Travelling Riverside Blues (2:41) 17. Sonny Boy Williamson - Bring It On Home (2:36) 18. Spirit - Taurus (2:37)Stairway To Heaven 19. The Small Faces - You Need Lovin (3:57)Whole Lotta Love 20. The Yardbirds - Knowing That I'm Losing You (2:55)Tangerine
Led Zep took a tradition and gave it a unique sound. To claim that they were original is a bit of a stretch. It's no different from the acts of today. You are just experiencing - how to put this politely? - a bit of cultural chauvinism.
Of course, because mathematics is all about adding long lists of numbers. Higher mathematics is about adding even longer lists of numbers. (*rolls eyes*)
Oh, please! You're recommending Spivak to someone who hasn't grasped the basics of pre-calculus yet? Spivak is a gem - but only for those that already know calculus from a more basic text, or for exceptional students. I agree that Spivak's Calculus is a fantastic book. However, it's not for a struggling high-school level student - the poor soul would be totally lost after reaching Spivak's discussion on "least upper bounds" (and this level of rigor would be useless for someone who is finding pre-calculus hard).
Spivak should be more appropriately called "Introductory Real Analysis".
reminds me of "aunt hillary" in "godel, escher, bach" - the intelligent ant hill that explained how it was she could be smart even if the ants she was made from were as dumb as sin. even had the ant-eater "manage" her ants for her.
one passage described how a flood disrupted a "relative" of hers, and even though the individual ants survived, the "relative" was no more - instead, a different "relative" was born.
it's one heck of a book - recommended for everyone into ai.
Lots of people put forward this ftl argument to say that it just wouldn't be possible to travel interstellar distances.
However, we're only scratching the surface when it comes to solving the general relativity equation. For instance, one solution of the GR equation consists of a wormhole that allows one to exit this universe into... who knows? One possible solution is extensible, allowing a hypothetical space traveller to travel into a completely different universe, or potentially a different region of our own universe. Some solutions are so bizarre that causality is violated, e.g. the solution of the spinning charged black hole, which has a ring singularity and a region where "time" becomes circular and loops back on itself. Admittedly, some of these solutions are considered "unphysical", but it's not clear yet why they are unphysical, what prevents them from happening. It's not even clear that they are impossible yet. All we know is that there is some truly bizarre stuff out there and we don't understand it all.
Sadly, though, if a traveller did ever manage to survive going through such a wormhole, the GR solutions appear to indicate that there is no way back... but as you can imagine, none of this is certain as yet. Maybe there are terms in the GR eqn that are missing. Maybe a theory of quantum gravity would prevent any of this from happening. Who knows?
Nonsense. You're obviously unaware how rare aperiodic tilings are. There is a vast universe of tile sets, almost all of which give periodic tilings when they tile at all. Aperiodic tilings are like the proverbial needle in a haystack, which is why it took us so long to discover them (let alone to show they even exist).
It's not possible to "design" the pattern for aesthetic reasons without some understanding of the mathematics behind it. It's a very unique pattern. It's like solving Rubik's cube - someone can't just solve it "by accident". I'm not saying that they necessarily knew the connection Penrose Tiles have with condensed matter physics, but they must have understood what made them unique, the precise angles required, possibly the connection between the angles and the "golden ratio", etc.
It must be hard for Europeans to accept that Persians are more than barbaric "towelheads".
During the dark ages, most of the texts of classical antiquity had been lost. The European rediscovery of their Classical past came not directly via Latin and Greek texts - but via translations of arabic translations of classical texts.
"Early translations [into latin] included works by Avicenna, Al-Ghazali, Avicebron, etc.; books on astronomy, astrology, and medicine; and the works of some of the Ancient Greek philosophers, especially Aristotle, who unlike Plato had been unknown or at least largely ignored in medieval Christendom. The latter were accompanied by the commentaries of Al-Ghazali, Avicenna, and Averroes, to the point of there being an identifiable Averroist school of philosophy in Christian Europe."
As much as it pains Europeans, the fact is that during the middle ages Europe was a cultural backwater, a land of barbarians. At the time, Islam kept the flames of civilisation alive. One of the effects of Islamic expansion in Europe was to allow Europeans to rediscover their own classical civilisation. So, for instance, greek philosophers were rediscovered in Europe not in their original greek, but in latin translations of arabic translations of the original greek.
No - aperiodic crystals aren't a just a pretty shape. They're very difficult to discover - what makes them interesting is that tilings don't repeat themselves. It's a bit like saying that the Greeks knew about the number 3.14159.. but didn't necessarily know what it stood for - it's pretty clear that if the only way they could find that number is if they knew what it represented.
There is a universe of potential tiles, the vast majority of which always give periodic tilings - to come up with a set that makes aperiodic tilings is nothing short of astonishing.
I think you mean ancient Rome. Rome had a Senate, "Greece" did not. In fact, Greece was made up of independent city states with different government systems. Athens had what we know call a classic democratic system - one citizen, one vote - but the citizen assembly was not checked by a senate. And citizenship was restricted to men with land as property. Your (the US) government was more heavily based on the Roman model than the Athenian model.
The issue people have with Cryptonomicon is that there is no denouement following the climax, so the ending appears abrupt. It's almost certainly deliberately done by Stephenson - the effectiveness of this is up to the individual reader to decide. IMO, there is nothing wrong with the climax itself.
But real numbers aren't fundamental either, they are a mathematical convenience to assign a number to every point in a number line. They are formally defined as Dedekind Cuts, which is based on (infinite) sets of rational numbers.
Oh, and rational numbers aren't fundamental either, they are a mathematical convenience defined as an equivalence class of a ordered pairs of integers.
"God created the integers... all else is the work of man".
But integers can also be defined as equivalence classes of sets, ala Principia Mathematica, so they're not fundamental either...
Alternatively, you can dispense with all this nonsense about complex numbers being somehow unknowable. They are as knowable as any other mathematical structure. Claiming that complex numbers are more mysterious or unknowable than reals is utterly meaningless. Ask any quantum physicist or electrical engineer.
Ptolemy once asked Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry.
That was true then, and it is true now.
You will find a mountain of contradictory advice. I can only offer what has worked for me (I have a BSc in CompSci/Mathematics, and an BE in Elec Eng).
1. Learn to abstract - from concrete objects, to numbers in a number system, to algebraic symbols in an algebraic structure. Almost all of mathematics is abstraction from simpler concepts. 2. Learn to reason - learn how to prove things, directly and via reductio ad absurdum. 3. Do the exercises. Some people have a knack for understanding things straight away - the rest of us need to work hard.
There's no royal road - even those who have a knack have developed this knack through practice. Most people have a psychological block regarding mathematics, because it is probably the most poorly taught subject, starting from kindergarten. Practice and develop confidence - it then becomes a "virtuous cycle" where confidence encourages you to practice more, the extra practice turns into ability, and the ability into even more confidence.
There are people like Feynman that are like magicians - they produce brilliant results without giving any indication of how they came to produce such brilliance. But even they have to work at it. Feynman was a master showman, and loved to confound people with seeming flashes of brilliance that really stemmed from very simple ideas (eg, see "Surely you must be joking", the anecdote where he competes against the abacus operator by mental calculations - using very simple tricks, he must have appeared to the abacus operator to be a walking computer). The point here is that Feynman was a brilliant man, but not above a little showmanship. So don't be intimidated by showmanship in mathematics - sometimes things are simpler than they appear, and not necessarily beyond your abilities - it's just a question of working out the trick these magicians use...
Oh, and finally: real mathematicians can't do arithmetic. They may be able to do tensor calculus on multivariate curved manifolds, but they will struggle with basic numeric arithmetic like most people. It's a badge of honour for mathematicians to get arithmetic wrong occasionally, because arithmetic is "mindless"
As has been explained, that was not the claim. Of course that nuclei determine what element an atom is. But that's different to saying that its chemicals are determined by their nuclei. Chemical properties are determined by the outermost electron shell, and nothing else.
Many scientifically literate people I know get infuriated by "new ageism" and pseudo-sciences like astrology, parapsychology, etc, etc. Most of them are concerned, and rightfully so, about the effect that these fads have on children, the uneducated, the naive.
How is this mumbo-jumbo different? How many of you are naive enough to think that this is anything but pseudo-scientific quackery? Not many, I hope, but just think of the damage this causes among the scientifically illiterate nerd crowd.
Many of you may disagree, but I feel that these sort of stories, like stories on astrology, numerology and clairvoyance, simply don't deserve to be given further publicity.
Where is your hard evidence? All your post seems to me to be is a prejudiced rant rationalising your ageism.
For every stubborn elderly person I know, I know a stubborn youngster. Where is your *hard* evidence? Not anecdotes, not "I know people who..." but hard numbers, statistics, etc, etc.
Actually, that quote is an old one, and goes back to the days when the *Soviets* led the US in space, not the other way around. In reality, it said that the Russian Nazis were better than the American ones.
This was before Apollo, when the Soviets were much better at this sort of thing than the Americans
Slashdot Mirror
Care to provide a reference to a paper where the Copenhagen interpretation has been terminally discredited? Or one that proves that observation does not have a part to play in wave function collapse? This issue is by no means dead - one interpretation is more popular than others at any one time, but this is more of a result of what is fashionable, not of what is experimentally verifiable.
Don't confuse unfalsifiable interpretations with testable hypotheses.
Damn, one hundred years of quantum physics and it takes you only a few seconds to demolish it.
Seriously, you can't use your everyday "common sense" when doing quantum physics. It's a subtle subject that goes against much of our everyday experience. Quantum physics does seem to suggest that conscious observation does have a role to play in the collapse of wave functions.
Not quite. The quantum state of the first object can change the quantum state of this second object, but the wave function of the second object will still be in an undetermined state until you observe it. *You* then must observe this second object, and only then does the wave function of the second collapse, consequently collapsing the wave function of the first object. What you've described is simply a chain of interdependent quantum states - until you observe them, they'll be undetermined.
Yes, but what about the safety of those on the moon? When we build our moonbase there, and start mining the moon for nuclear fuel and dumping our nuclear waste there, a simple accident could cause a tremendous explosion. This explosion could knock the moon from orbit, sending the moonbase and its inhabitants into the far reaches of space.
They would then have to spend the rest of their days hoping to bump into an earth-like planet. It's just a fate too boring to contemplate.
Denied!
You're kidding me, right? I mean, I like Led Zep and all, but it's painfully clear that most of their songs have riffs that are "borrowed" from blues artists and others, or are straight out covers.
Examples (list found on web somewhere):
1. Bert Jansch - Blackwater Side (3:46)Black Mountain Side
2. Bert Jansch - Go Your Way My Love - intro (0:26)Going To California
3. Bert Jansch - The Waggoner's Lad - intro (0:51)Bron-Y-Aur Stomp
4. Blind Willie Johnson - Nobody's Fault But Mine (3:11)
5. Bobby Parker - Watch Your Step (2:49)Moby Dick
6. Bukka White - Shake 'Em on Down (3:01)Hats Off To (Roy) Harper
7. Davey Graham - She Moved Through The Fair (3:07)White Summer
8. Howlin' Wolf - Killing Floor (2:51)The Lemon Song
9. Jake Holmes - Dazed and Confused (3:48)
10. Joan Baez - Babe, I'm Gonna Leave You (2:41)
11. Josh White - Jesus Gonna Make up My Dying Bed (3:05)In My Time Of Dying
12. Leadbelly - Gallis Pole (3:01)Gallows Pole
13. Little Richard - Keep A-Knockin' (2:15)Rock And Roll
14. Memphis Minnie - When The Levee Breaks (3:11)
15. Ritchie Valens - Ooh! My Head (1:48)Boogie With Stu
16. Robert Johnson - Travelling Riverside Blues (2:41)
17. Sonny Boy Williamson - Bring It On Home (2:36)
18. Spirit - Taurus (2:37)Stairway To Heaven
19. The Small Faces - You Need Lovin (3:57)Whole Lotta Love
20. The Yardbirds - Knowing That I'm Losing You (2:55)Tangerine
Led Zep took a tradition and gave it a unique sound. To claim that they were original is a bit of a stretch. It's no different from the acts of today. You are just experiencing - how to put this politely? - a bit of cultural chauvinism.
Of course, because mathematics is all about adding long lists of numbers. Higher mathematics is about adding even longer lists of numbers. (*rolls eyes*)
Oh, please! You're recommending Spivak to someone who hasn't grasped the basics of pre-calculus yet? Spivak is a gem - but only for those that already know calculus from a more basic text, or for exceptional students. I agree that Spivak's Calculus is a fantastic book. However, it's not for a struggling high-school level student - the poor soul would be totally lost after reaching Spivak's discussion on "least upper bounds" (and this level of rigor would be useless for someone who is finding pre-calculus hard).
Spivak should be more appropriately called "Introductory Real Analysis".
one passage described how a flood disrupted a "relative" of hers, and even though the individual ants survived, the "relative" was no more - instead, a different "relative" was born.
it's one heck of a book - recommended for everyone into ai.
However, we're only scratching the surface when it comes to solving the general relativity equation. For instance, one solution of the GR equation consists of a wormhole that allows one to exit this universe into... who knows? One possible solution is extensible, allowing a hypothetical space traveller to travel into a completely different universe, or potentially a different region of our own universe. Some solutions are so bizarre that causality is violated, e.g. the solution of the spinning charged black hole, which has a ring singularity and a region where "time" becomes circular and loops back on itself. Admittedly, some of these solutions are considered "unphysical", but it's not clear yet why they are unphysical, what prevents them from happening. It's not even clear that they are impossible yet. All we know is that there is some truly bizarre stuff out there and we don't understand it all.
Sadly, though, if a traveller did ever manage to survive going through such a wormhole, the GR solutions appear to indicate that there is no way back... but as you can imagine, none of this is certain as yet. Maybe there are terms in the GR eqn that are missing. Maybe a theory of quantum gravity would prevent any of this from happening. Who knows?
It's not possible to "design" the pattern for aesthetic reasons without some understanding of the mathematics behind it. It's a very unique pattern. It's like solving Rubik's cube - someone can't just solve it "by accident". I'm not saying that they necessarily knew the connection Penrose Tiles have with condensed matter physics, but they must have understood what made them unique, the precise angles required, possibly the connection between the angles and the "golden ratio", etc.
It must be hard for Europeans to accept that Persians are more than barbaric "towelheads".
See, for instance, the case of Aristotle, http://en.wikipedia.org/wiki/Arabist
"Early translations [into latin] included works by Avicenna, Al-Ghazali, Avicebron, etc.; books on astronomy, astrology, and medicine; and the works of some of the Ancient Greek philosophers, especially Aristotle, who unlike Plato had been unknown or at least largely ignored in medieval Christendom. The latter were accompanied by the commentaries of Al-Ghazali, Avicenna, and Averroes, to the point of there being an identifiable Averroist school of philosophy in Christian Europe."
As much as it pains Europeans, the fact is that during the middle ages Europe was a cultural backwater, a land of barbarians. At the time, Islam kept the flames of civilisation alive. One of the effects of Islamic expansion in Europe was to allow Europeans to rediscover their own classical civilisation. So, for instance, greek philosophers were rediscovered in Europe not in their original greek, but in latin translations of arabic translations of the original greek.
There is a universe of potential tiles, the vast majority of which always give periodic tilings - to come up with a set that makes aperiodic tilings is nothing short of astonishing.
I think you mean ancient Rome. Rome had a Senate, "Greece" did not. In fact, Greece was made up of independent city states with different government systems. Athens had what we know call a classic democratic system - one citizen, one vote - but the citizen assembly was not checked by a senate. And citizenship was restricted to men with land as property. Your (the US) government was more heavily based on the Roman model than the Athenian model.
http://en.wikipedia.org/wiki/Denouement
As the article states, lack of a denuouement is a stylistic device, used for instance in Lord of the Flies.
Oh, and rational numbers aren't fundamental either, they are a mathematical convenience defined as an equivalence class of a ordered pairs of integers.
"God created the integers... all else is the work of man".
But integers can also be defined as equivalence classes of sets, ala Principia Mathematica, so they're not fundamental either...
Alternatively, you can dispense with all this nonsense about complex numbers being somehow unknowable. They are as knowable as any other mathematical structure. Claiming that complex numbers are more mysterious or unknowable than reals is utterly meaningless. Ask any quantum physicist or electrical engineer.
What would *he* know? He didn't have a brain.
Ptolemy once asked Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there was no royal road to geometry.
That was true then, and it is true now.
You will find a mountain of contradictory advice. I can only offer what has worked for me (I have a BSc in CompSci/Mathematics, and an BE in Elec Eng).
1. Learn to abstract - from concrete objects, to numbers in a number system, to algebraic symbols in an algebraic structure. Almost all of mathematics is abstraction from simpler concepts.
2. Learn to reason - learn how to prove things, directly and via reductio ad absurdum.
3. Do the exercises. Some people have a knack for understanding things straight away - the rest of us need to work hard.
There's no royal road - even those who have a knack have developed this knack through practice. Most people have a psychological block regarding mathematics, because it is probably the most poorly taught subject, starting from kindergarten. Practice and develop confidence - it then becomes a "virtuous cycle" where confidence encourages you to practice more, the extra practice turns into ability, and the ability into even more confidence.
There are people like Feynman that are like magicians - they produce brilliant results without giving any indication of how they came to produce such brilliance. But even they have to work at it. Feynman was a master showman, and loved to confound people with seeming flashes of brilliance that really stemmed from very simple ideas (eg, see "Surely you must be joking", the anecdote where he competes against the abacus operator by mental calculations - using very simple tricks, he must have appeared to the abacus operator to be a walking computer). The point here is that Feynman was a brilliant man, but not above a little showmanship. So don't be intimidated by showmanship in mathematics - sometimes things are simpler than they appear, and not necessarily beyond your abilities - it's just a question of working out the trick these magicians use...
Oh, and finally: real mathematicians can't do arithmetic. They may be able to do tensor calculus on multivariate curved manifolds, but they will struggle with basic numeric arithmetic like most people. It's a badge of honour for mathematicians to get arithmetic wrong occasionally, because arithmetic is "mindless"
As has been explained, that was not the claim. Of course that nuclei determine what element an atom is. But that's different to saying that its chemicals are determined by their nuclei. Chemical properties are determined by the outermost electron shell, and nothing else.
Tomb raider doesn't sound like a porno,
but womb raider certainly does.
Cool. I'm glad to hear I'm in good company.
Many scientifically literate people I know get
infuriated by "new ageism" and pseudo-sciences
like astrology, parapsychology, etc, etc. Most
of them are concerned, and rightfully so, about
the effect that these fads have on children,
the uneducated, the naive.
How is this mumbo-jumbo different? How many
of you are naive enough to think that this is
anything but pseudo-scientific quackery?
Not many, I hope, but just think of the damage
this causes among the scientifically illiterate
nerd crowd.
Many of you may disagree, but I feel that these
sort of stories, like stories on astrology,
numerology and clairvoyance, simply don't deserve
to be given further publicity.
"It is human nature to become set in our ways..."
Poppycock.
Where is your hard evidence? All your post seems to me to be is a prejudiced rant rationalising your ageism.
For every stubborn elderly person I know, I know a stubborn youngster. Where is your *hard* evidence? Not anecdotes, not "I know people who..." but hard numbers, statistics, etc, etc.
And continuing on this thread, why did the US
embark on an expensive space program when it
had a substantial part of its population living
in poverty?
(Also a rhetorical question - I don't dispute
the value of space exploration, I only wish to
point out the bias of the original post)
Actually, that quote is an old one, and goes back
to the days when the *Soviets* led the US in
space, not the other way around. In reality, it
said that the Russian Nazis were better than
the American ones.
This was before Apollo, when the Soviets were
much better at this sort of thing than the
Americans