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Author | Topic: Physics contradicts maths - how is this possible? | |||||||||||||||||||||||
Modulous Member Posts: 7801 From: Manchester, UK Joined: |
Yes, that might be the case, which is why I said I eagerly await any replies by Agobot to see where things go from here.
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Chiroptera Inactive Member |
Convergence would normally be proved by showing the series tends to one. In this case assuming convergence would be assuming the thing you are trying to prove. No, I would first show that the sequence of partial sums is monotonic and bounded (this would be pretty easy) -- convergence then follows trivially, and then Modulus' method suffices to prove the value is indeed 1. It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi
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sinequanon Member (Idle past 2895 days) Posts: 331 Joined: |
Not the normal method. Much easier and more direct to write down the sum to n terms of a geometric series and examine its difference with 1 as a function of n.
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Chiroptera Inactive Member |
Actually, there are a lot of series and sequences where the actual limit isn't obvious from the form of terms. In that case, it's a pretty standard technique to first show that it converges, and then use algebra tricks like Modulus' to find the actual limit. These sorts of things especially common on exams; in real life (that is, real mathematics papers), the actual value of the limit is often of little interest -- what is of interest usually just convergence, and showing convergence is often a lot easier than figuring out the actual value of the limit.
Anyways, since you just mentioned that it's a geometric series, the easiest thing to do is just use the geometric series test to show that it converges, and then the geometric series formula to evaluate the limit. No need to even discuss limits. It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi
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sinequanon Member (Idle past 2895 days) Posts: 331 Joined: |
Values of limits of great interest in "real mathematics" papers. Every time you solve a differential equation or an integral, you are evaluating a limit.
Applying the formulae you mentioned would not be taken as proof in a "real mathematics" exam. More like a demonstration. Have you ever had the displeasure of marking maths undergraduate exam papers?
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Chiroptera Inactive Member |
Applying the formulae you mentioned would not be taken as proof in a "real mathematics" exam. More like a demonstration. What? A geometric series? This is basic Calculus II. Or showing that a sequence is monotonic and bounded? This is undergraduate elementary analysis. -
Have you ever had the displeasure of marking maths undergraduate exam papers? As a matter of fact, that turns out to be my job. Do you know anything about mathematics? "The guilty one is not he who commits the sin, but the one who causes the darkness." Clearly, he had his own strange way of judging things. I suspect that he acquired it from the Gospels. -- Victor Hugo
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sinequanon Member (Idle past 2895 days) Posts: 331 Joined: |
Do you know anything about mathematics? Me too. Majored at Cambridge University here in the UK and went on to research in non-deterministic fluid dynamic modelling of mid-latitude weather systems. Would have got a fail in my first year exams if I'd submitted a proof depending on the two formulae you supplied. You'd be expected to do it more from first principles.
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Chiroptera Inactive Member |
*shrug*
I dunno. Maybe they were trying to force you to learn the basic principles. Me, I'm not going to try to second guess other peoples' teaching methods. "The guilty one is not he who commits the sin, but the one who causes the darkness." Clearly, he had his own strange way of judging things. I suspect that he acquired it from the Gospels. -- Victor Hugo
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Son Goku Inactive Member |
(Yeah, a (very) rough sketch of the proof is that any given algorithm is specified by a certain number of symbols which can be "Gdel Numbered". That is you can assign a natural number to any given algorithm. Since the cardinality of the Natural numbers is Aleph-Zero, then the cardinality of the computables is also Aleph-Zero. As you know the cardinality of the Reals is greater than Aleph-Zero, hence most reals are uncomputable.)
This is a major bone of contention with constructionists. As you know a continuum like the Real numbers is needed to do analysis and calculus. This is because of properties like convergence e.t.c. that Modulous and sinequanon have been talking about. However if most of the Reals are numbers that can never be described, labelled, reached, e.t.c. (even in principle) then you get this picture of the Real numbers as being like the night sky. All the numbers we use are just pin point stars in the vast black of uncomputables.However even though we never use all that black, never even speak of it, you need it (formally) for calculus. Certain people are uncomfortable with most of the Reals being simply "formal junk" that's only generated to do calculus. This caused a program at the start of the twentieth century which attempted to see how much of the reals you can remove and still have calculus work. It turns out that removing the uncomputables means the fundamental theorem of calculus no longer holds.
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Chiroptera Inactive Member |
This is a major bone of contention with constructionists. Bwahahaha! Are there still constructionalists? I thought they all died off with the people who don't accept the Axiom of Choice. Oh, wait a minute. I know someone who doesn't accept the Axiom of Choice. Never mind. "The guilty one is not he who commits the sin, but the one who causes the darkness." Clearly, he had his own strange way of judging things. I suspect that he acquired it from the Gospels. -- Victor Hugo
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Rrhain Member Posts: 6351 From: San Diego, CA, USA Joined: |
RAZD writes:
quote: No, that's a contradiction. Instead, the math can be perfect but not applicable to the situation at hand.
quote: Since everything in science eventually comes back to math, that means everything is an assumption.
quote: On the contrary, nothing can exist without mathematics for it is the very nature of existence. But, we've had this conversation before.
quote: No, the model is only as good as the assumptions used in the model. The math will always be correct. But if you have left something out that your model requires in order to be accurate, it isn't the problem of the math but rather of the model. If you're going to make bread, you mustn't forget the salt. If you do, the bread won't taste very good. If you don't add the salt, it isn't the fault of cooking. The cooking process can only work with what it has. If you've forgotten something, then that's your problem. Mathematical models can only work with the information that you provide them. If you've neglected to account for certain variables, then that is your problem, not the problem of math. Now, science is an observational process, so we will never know if we have accounted for all the variables. And the equations involved can be so complex that we don't know how to untangle them. But just because we don't know how to do it doesn't mean it can't be done. Obviously, things happen despite our models. That's a problem of the model, not the math. Take the difference between linear and relativistic mechanics. The mathematical model is perfect...it just isn't applicable to the world in which we live. If the universe were linear, then linear mechanics would be accurate. It isn't that there's something in the math that makes the universe non-linear. The universe follows its own mathematics. Part of the point of science is to discover what it is.
quote: And thus you prove the point. The problem wasn't the math. After all, the mathematics of rigid-wing aerodynamics is accurate since airplanes fly. It's just not applicable to flexible-wing aerodynamics. The problem is not the math but the model. Rrhain Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.
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sinequanon Member (Idle past 2895 days) Posts: 331 Joined: |
The sweeping wonders of set theory, eh? It allows you to talk of things for which no representation exists. Then the axiom of choice allows you to select elements from this 'soup' of abstraction.
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Son Goku Inactive Member |
Chiroptera writes:
Yep, sure there are now people even more extreme, the ultra-finitists. They don't even agree that big numbers (e.g., 10^google) are sensical. Bwahahaha! Are there still constructionalists? To be fair though the people who don't like the uncomputables are constructionists-lite. They don't share the views of other constructionists, they're just a bit unsettled by analysis being based on numbers you can't talk about.
sinequanon writes:
I love the axiom of choice. It implies stuff that is totally crazy and you intuitively feel like rejecting and yet if you get rid of it you lose stuff that is totally obvious and necessary from every area of maths.
The sweeping wonders of set theory, eh? It allows you to talk of things for which no representation exists. Then the axiom of choice allows you to select elements from this 'soup' of abstraction.
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Buzsaw Inactive Member |
I've been reading and thinking about my message signature relative to this discussion and whether it has any application to this discussion relative to the Buzsaw hypothisis that the universe is unbounded in area/space and having no beginning and incapable of ending.
The immeasurable present eternally extends the infinite past and infinitely consumes the eternal future. Put differently, the immeasurable present eternally converges the infinite future into the eternal past and infinitely diverges the eternal past out of the infinite future. BUZSAW B 4 U 2 C Y BUZ SAW. The immeasurable present eternally extends the infinite past and infinitely consumes the eternal future.
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RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
But, we've had this conversation before. We have, and I can understand your point of view, where you are coming from.
It's just not applicable to flexible-wing aerodynamics. The problem is not the math but the model. Which was mathematical.
Instead, the math can be perfect but not applicable to the situation at hand. A wrong use\application of math\model still means the math is wrong for the situation, no matter how perfect it is for other use\applications. We'll just have to disagree. Enjoy. we are limited in our ability to understand by our ability to understand RebelAAmericanOZen[Deist ... to learn ... to think ... to live ... to laugh ... to share.
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