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Author Topic:   Physics contradicts maths - how is this possible?
Modulous
Member
Posts: 7801
From: Manchester, UK
Joined: 05-01-2005


Message 46 of 69 (442726)
12-22-2007 1:15 PM
Reply to: Message 45 by sinequanon
12-22-2007 1:05 PM


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.

This message is a reply to:
 Message 45 by sinequanon, posted 12-22-2007 1:05 PM sinequanon has not replied

  
Chiroptera
Inactive Member


Message 47 of 69 (442753)
12-22-2007 3:45 PM
Reply to: Message 43 by sinequanon
12-22-2007 10:17 AM


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

This message is a reply to:
 Message 43 by sinequanon, posted 12-22-2007 10:17 AM sinequanon has replied

Replies to this message:
 Message 48 by sinequanon, posted 12-22-2007 3:52 PM Chiroptera has replied

  
sinequanon
Member (Idle past 2864 days)
Posts: 331
Joined: 12-17-2007


Message 48 of 69 (442755)
12-22-2007 3:52 PM
Reply to: Message 47 by Chiroptera
12-22-2007 3:45 PM


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.

This message is a reply to:
 Message 47 by Chiroptera, posted 12-22-2007 3:45 PM Chiroptera has replied

Replies to this message:
 Message 49 by Chiroptera, posted 12-22-2007 4:11 PM sinequanon has replied

  
Chiroptera
Inactive Member


Message 49 of 69 (442766)
12-22-2007 4:11 PM
Reply to: Message 48 by sinequanon
12-22-2007 3:52 PM


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

This message is a reply to:
 Message 48 by sinequanon, posted 12-22-2007 3:52 PM sinequanon has replied

Replies to this message:
 Message 50 by sinequanon, posted 12-22-2007 4:24 PM Chiroptera has replied

  
sinequanon
Member (Idle past 2864 days)
Posts: 331
Joined: 12-17-2007


Message 50 of 69 (442773)
12-22-2007 4:24 PM
Reply to: Message 49 by Chiroptera
12-22-2007 4:11 PM


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?

This message is a reply to:
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Replies to this message:
 Message 51 by Chiroptera, posted 12-22-2007 4:33 PM sinequanon has replied

  
Chiroptera
Inactive Member


Message 51 of 69 (442778)
12-22-2007 4:33 PM
Reply to: Message 50 by sinequanon
12-22-2007 4:24 PM


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

This message is a reply to:
 Message 50 by sinequanon, posted 12-22-2007 4:24 PM sinequanon has replied

Replies to this message:
 Message 52 by sinequanon, posted 12-22-2007 4:44 PM Chiroptera has replied

  
sinequanon
Member (Idle past 2864 days)
Posts: 331
Joined: 12-17-2007


Message 52 of 69 (442782)
12-22-2007 4:44 PM
Reply to: Message 51 by Chiroptera
12-22-2007 4:33 PM


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.

This message is a reply to:
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Replies to this message:
 Message 53 by Chiroptera, posted 12-22-2007 4:51 PM sinequanon has not replied

  
Chiroptera
Inactive Member


Message 53 of 69 (442783)
12-22-2007 4:51 PM
Reply to: Message 52 by sinequanon
12-22-2007 4:44 PM


*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


Message 54 of 69 (442793)
12-22-2007 5:17 PM
Reply to: Message 41 by Chiroptera
12-22-2007 9:11 AM


Das Kontinuum
(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.

This message is a reply to:
 Message 41 by Chiroptera, posted 12-22-2007 9:11 AM Chiroptera has replied

Replies to this message:
 Message 55 by Chiroptera, posted 12-22-2007 5:54 PM Son Goku has replied
 Message 57 by sinequanon, posted 12-22-2007 5:58 PM Son Goku has not replied

  
Chiroptera
Inactive Member


Message 55 of 69 (442810)
12-22-2007 5:54 PM
Reply to: Message 54 by Son Goku
12-22-2007 5:17 PM


Re: Das Kontinuum
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

This message is a reply to:
 Message 54 by Son Goku, posted 12-22-2007 5:17 PM Son Goku has replied

Replies to this message:
 Message 58 by Son Goku, posted 12-22-2007 6:19 PM Chiroptera has not replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 56 of 69 (442813)
12-22-2007 5:58 PM
Reply to: Message 42 by RAZD
12-22-2007 10:04 AM


Re: Math models reality with abstract constructions
RAZD writes:
quote:
Because the math can be perfect and still wrong.
No, that's a contradiction. Instead, the math can be perfect but not applicable to the situation at hand.
quote:
Math is based entirely on assumption
Since everything in science eventually comes back to math, that means everything is an assumption.
quote:
As such there is no real tie between any mathematical computation and any object of the world of objective reality.
On the contrary, nothing can exist without mathematics for it is the very nature of existence. But, we've had this conversation before.
quote:
Math can be used to model reality, but the model is only as good as the assumptions used in the maths
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:
Bees fly. An engineer calculated that it couldn't, but it was the assumptions that the aerodynamics of man-made plane wings to the flight with bee wings that was in error.
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.

This message is a reply to:
 Message 42 by RAZD, posted 12-22-2007 10:04 AM RAZD has replied

Replies to this message:
 Message 60 by RAZD, posted 12-23-2007 10:58 AM Rrhain has replied

  
sinequanon
Member (Idle past 2864 days)
Posts: 331
Joined: 12-17-2007


Message 57 of 69 (442814)
12-22-2007 5:58 PM
Reply to: Message 54 by Son Goku
12-22-2007 5:17 PM


Re: Das Kontinuum
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.

This message is a reply to:
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Son Goku
Inactive Member


Message 58 of 69 (442821)
12-22-2007 6:19 PM
Reply to: Message 55 by Chiroptera
12-22-2007 5:54 PM


Re: Das Kontinuum
Chiroptera writes:
Bwahahaha! Are there still constructionalists?
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.
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:
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.
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.

This message is a reply to:
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Buzsaw
Inactive Member


Message 59 of 69 (442855)
12-22-2007 8:07 PM


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.

Replies to this message:
 Message 61 by Taz, posted 12-23-2007 11:18 AM Buzsaw has replied

  
RAZD
Member (Idle past 1405 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 60 of 69 (442986)
12-23-2007 10:58 AM
Reply to: Message 56 by Rrhain
12-22-2007 5:58 PM


Re: Math models reality with abstract constructions
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.

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This message is a reply to:
 Message 56 by Rrhain, posted 12-22-2007 5:58 PM Rrhain has replied

Replies to this message:
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