Re: Division by zero

Am Montag000003, 03.02.2025 um 16:51 schrieb Mikko:

On 2025-02-03 07:56:53 +0000, Thomas Heger said:
 

Am Sonntag000002, 02.02.2025 um 10:30 schrieb Mikko:
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Hi NG
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I'm actually not really certain, but found an error in Einstein's 'On the electrodynamics of moving bodies' which is quite serious.
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See page six, roughly in the middle:
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There we find an equation, which says this:
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∂τ/∂y= 0

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Do you mean on page 899 (9th page of the article) in §3?
The operation is not division but a partial derivative.

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You should answer this question. It is not useful to talk without telling
what you are talking about.
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I'm referring to the English translation, which can be found here
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https://www.fourmilab.ch/etexts/einstein/specrel/www/
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The English pdf version has other page numbers than the original article.
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But in a way, these original page numbers are also possible as reference.
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But unfortunately I have here only the English version (the German I have on a different computer).
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So I have to tell you the page from the English version or make the meant part available to you by other means.
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So, § 3 was meant and roughly the middle, which can be found on page 6 of the English pdf version.
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And you are absolutely right, that a partial derivative was meant.
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The problem was: of which function was a partial derivative meant?

 He obviously means the function needed to determine τ. It does not matter
whether he means the function from x, y, z, t or x', y, z, t as ∂/∂y is
the same in both cases.

There ain't no thing as 'obviously'.
If an author doesn't write, what he has in mind, the reader is requirred to guess. And the result of such a process is by no means 'obvious'.
Einstein used τ for three different types of mathematical objects:
a value τ (meaning: time in k)
a function τ (a coordinate transformation between K and k)
as function value τ of that function τ.
Therefor it would requirre some brains to find out, which one was actually meant.
Correct would have been to make the type explicit, e.g. by different fonts.
But REALLY bad would be 'switching' between different uses of the same symbol τ.
This is so, because it is absolutely NOT obvious, which meaning τ has, if three different meanings are used.

Einstein didn't define the used variables and simply assumed, the reader would know anyhow, what he had in mind.

 Variables are clearly defined. For example, x, y, z, and t are defined as
the coordinates of the system K.

Possibly we disagree about the meaning of the term 'definition'.
For instance t is not really a coordinate in K, because K is an Euclidean coordinate system, hence 'timeless'.
The construct actually meant is called 'frame of reference' today, which could be understood as a coordinate system plus time measure.
Those FoRs have coordinates, too, but are not Euclidean spaces (which Einstein wanted to use).

But that wasn't particularly easy, because Einstein used the symbol τ for three different types of objects.
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a) the time values of clocks in system k were named τ
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b) a function τ was derived, which should serve as coordinate transformation between system K and system k

 Although modern mathematicians don't consider that correct, it is common
to use the same symbol for a quantity and for a function that computes
that quantity. It is obvious from the context which is meant: function
name is used with arguments, the quantity name without.

It is really bad, to use the same symbol for different types of objects and do not tell, which meaning was meant.
A funtion f(x), for instance, has an argument x and produces some output f(x)=y.
But 'f' is a name and belongs to the 'machinery' of the function and not to the output, hence f != f(x).
...
TH