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Old Jan 25th 2017, 08:09 AM   #1
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Approximation

Just wondering!

I believe every consideration, every non-mathematical
aspect of all of physics is approximate. Put otherwise, every
discussion, all proofs, even base ideas of physics begin with
assumption (some made explicit others implicit or unknown).
Seems physics always begins with some "model" that is
approximate. Approximate in gets approximate out!

I wonder about approximation in mathematics. I don't mean a
solution in math that results as a series. The series is exact.
Truncation of the series is convenience, not approximation.
I guess math is exact because it has no "models" (at least in
the little math I know).

JP
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Old Jan 25th 2017, 10:08 AM   #2
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I would state it slightly differently: in physics everything is an approximation because it is impossible to measure anything in nature to infinite precision. Hence 2 feet plus 2 feet is not precisely 4 feet; instead what we mean is something like this:

$\displaystyle 2.00 \pm 0.005 \ ft \ + 2.00 \pm 0.005 \ ft \ = \ 4.00 \pm 0.01 \ ft$

In mathematics typically we use values that are infinitely precise: when we say that 2 + 2 = 4 we mean the result is exactly 4.0000... (i.e. infinitely precise).

One area where we may have an argument is statistics - if we can agree that statistics is a branch of mathematics then the uncertainty of results, expressed by probabilities, is imprecise.
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Old Jan 25th 2017, 10:47 AM   #3
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I have spent quite a number of years working with mathematical models of one kind or another.
They broadly fit into two camps "empirical" and "analytical"
Empirical models just use a mathematical function that has been observed to fit the observed behaviour of the system being modelled.
Analytical models actually apply the known relationships between the physical features of the system.
Thus the relationship: F=ma allows the Acceleration to be modelled knowing the Force and the Mass.

Theoretically, with sufficient care and effort, we can determine the mass and the force to any degree of accuracy we might care to, and hence the acceleration will be known to the same degree of accuracy.

However the above statement assumes that F=ma is the complete and completely accurate description of the system.
However it is not.
For a start there will inevitably be interfering influences from outside the system being modelled.
I guess that these can be reduced to negligible proportions via the care and effort used to set up the system.

But even then can we really assume the F=ma is the final word in describing this relationship?
What happens as we move toward relativistic speeds?
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Old Jan 25th 2017, 10:53 AM   #4
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Just to be pedantic

Shouldn't your sum be :

$\displaystyle 2.00 \pm 0.005 \ ft \ + 2.00 \pm 0.005 \ ft \ = \ 4.00 \pm 0.007 \ ft$

If I remember correctly the error tolerance when adding two quantities is not just the sum of the individual tolerances,
rather it is the root of the sum of the squares of the tolerances.
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Old Jan 25th 2017, 11:41 AM   #5
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You remember incorrectly.

If $\displaystyle x= 2.00\pm 0.005$ then x can be as low as 1.995 and a high as 2.005.
If $\displaystyle y= 2.00\pm 0.005$ then y can be as low as 1.995 and a high as 2.005.

x+ y could be as low as 1.995+ 1.995= 3.99 or as high as 2.005+ 2.005= 4.01.

That can be written as $\displaystyle x+ y= 4.00\pm 0.01$.
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Old Jan 25th 2017, 11:46 AM   #6
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Originally Posted by Woody View Post
Shouldn't your sum be :

$\displaystyle 2.00 \pm 0.005 \ ft \ + 2.00 \pm 0.005 \ ft \ = \ 4.00 \pm 0.007 \ ft$

If I remember correctly the error tolerance when adding two quantities is not just the sum of the individual tolerances,
rather it is the root of the sum of the squares of the tolerances.
Perhaps if the plus-or-minus values means the 1 sigma (or 2 , or 3 sigma) values, then yes, the new sigma would be as you describe. But in engineering the plus-or-minus values are actual limits that are acceptable. If I ask a vendor to give me a piece of metal that is 2 feet plus or minus 0.005 feet, they may provide pieces that range from 1.995 to 2.005 feet. So its quite possible that they send me two pieces that are both 2.005 feet long, so their sum would be 2.01 ft.
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Old Jan 26th 2017, 07:35 AM   #7
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Hi,

Interesting comments, thank you! I guess statistics is mathematics of sets of
entities. Thanks, Jim
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