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This is interesting, but I'm not sure I agree with the conclusions.
Difference of a factor ... it's all subjective.
I'm aware...
It's not really accurate to say it's subjective. Some of the math really is cleaner. The unit circle is probably the most shining example. It's nicer that 1/4 the way around the circle is 1/4 the circle constant.
It's comparable to units of measure, I think. Some metrics are nicer than others because they're cleaner, more intuitive, etc.
I just don't think it's worth the effort. Writing two everywhere is okay with me.
Agreed. However, cleaner is subjective. Appearance doesn't change the math.
Where the imperial system uses the King's foot size as the measure of a foot, the metric system uses water and rare metals to define various units. How is that any more ideal than the King's foot? Shouldn't everything be based on hydrogen since that's the universe's most basic element? No, what about based on masses of subatomic particles? No, quarks! No, strings!
In the days before the computer, the metric system proved to be convenient and easy for conversion. What does the metric system have to show for itself now? Absolutely nothing.
It's all subjective and relative. "Convenient conversion" is just a subjective reason for the metric system over the imperial units for example.
I do believe the world ought to use a single metric system. I just don't really care which because it doesn't really matter.
Of course it doesn't change the math. Appearance isn't all that it changes, though. It makes some math a lot more intuitive, and that's something more than appearance.
Again -- I don't give a shit. I think it's interesting, but definitely not worth more than passive pondering.
I don't agree with your analysis of the metric system. Sure, what it's based on is pretty arbitrary and not incredibly helpful, but how all of the units relate definitely is helpful. A bunch of different units that seem to have been developed independently is a lot more icky and nebulous (and difficult to deal with) than conversion factors of 1.
Sure, this benefit is completely lost when it's convenient to apply modern technology, but that's not the only time one encounters units of measure.
The benefit for using $\tau$ instead of $\pi$ is even smaller. I just thought it was an interesting read.
I agree with everything you've said. However, your reasoning for both issues are purely opinion (which I happen to share).
Which number system do you prefer? Well, I grew up using decimal so I prefer that best. Computers use binary and many ancient civilizations used quinary (5), vigesimal (20), sexagesimal (60) (wikipedia was helpful!) as well as other forms of number systems. Many ancient civilizations didn't arrive at the intuitive decimal. So much for 10 fingers and all that junk about 10 being more intuitive (even though the brain is suspiciously good at doubling quantities).
Just making a comical point. It's all convention.
Sure, that's a good point. The metric system is only as nice as it is if you assume a decimal numbering system.
However, granted we do express numbers in base-10, we might as well have a system of measurement that exploits that.
I'll concede that most of this crap is pretty trivial and meaningless, but I'm not sure it's entirely fair to call it subjective as long as you're able to accept something along the lines of "simpler is better" axiomatically.
There is nothing to concede.
Luckily, the number 10 is invariant to base. In any base, you can always count on 10*n to append a 0 digit to the number n. The metric system would generalize in funny ways. Imagine base 13, a kilo would have the value 2197.
I actually think this discussion is profound. We're discussing the conventions used to describe the intangible.
Heh, sure -- if you think of "10" in a sort of abstract way (i.e., "10" is 1b1 + 0b0, for a number system with base b).
I suppose powers of (decimal) 10 are rather arbitrary. There's no reason a powers of 10b measurement system would be any better or worse than our powers of 1010 measurement system, as long as b isn't too big as to disallow a reasonable level of resolution.
I doubt it's an accident that most number systems in history have multiples-of-ten bases. Too many number systems have this property. Sure, there are exceptions, but if there weren't some biological/neurological/physical influence, I think we'd probably see number systems with "stranger" bases.
I think the claim that we've arrived at a base-10 number system because we have 10 fingers is probably crap, but it's interesting nonetheless.
In terms of units, you're not necessarily required to do computation with units themselves. You can always transform complicated physical problems to equivalent dimensionless problems. This is probably the easiest approach metric system or not. It's amusing that you can describe reality with math that is completely detached from reality.
Maybe one can view the optimal numerical base as a solution to some sort of compression problem.
Define the energy function:
[latex]
E(b) = \text{Number of digits to encode } L + \text{ Number of distinct symbols}
[/latex]
where [latex]b[/latex] is the numerical base and [latex]L[/latex] is what you consider a
large number that is
frequently encountered. Mathematically, the energy is given as:
[latex]
E(b) = \left \lfloor \log_b L \right \rfloor + 1 + b
[/latex]
We can make it smooth by removing the floor
[latex]
E(b) = \log_b L + 1 + b
[/latex]
Now solve:
[latex]
E'(b) = - \frac{1}{b (\log b)^2} \log L + 1 = 0
[/latex]
This gives:
[latex]
-\frac{1}{b (\log b)^2} = -\frac{1}{\log L} \\
b (\log b)^2 = \log L \\
\sqrt{b} \log b = \sqrt{\log L} \\
\sqrt{b} \log \sqrt{b} = \frac{1}{2} \sqrt{\log L} \\
e^{\log \sqrt{b}} \log \sqrt{b} = \frac{1}{2} \sqrt{\log L} \\
\log \sqrt{b} = W \left ( \frac{1}{2} \sqrt{\log L} \right ) \\
b = e^{2 W \left ( \frac{1}{2} \sqrt{\log L} \right )}
[/latex]
Where [latex]W(\cdot)[/latex] is the
Lambert W functionIf [latex]L = 10^9[/latex] then [latex]b \approx 6[/latex]
(computation). So base 6 or 7 would be
optimal.
We could generalize the energy to be
[latex]
E(b) = \log_b L + 1 + \lambda b
[/latex]
where [latex]\lambda \geq 0[/latex] describes how much
weight you give the number of distinct symbols. In other words, if you don't care how many distinct symbols you have to remember, then [latex]\lambda = 0[/latex].
By similar derivation, this optimal [latex]b[/latex] is given as
[latex]
b = e^{2 W \left ( \frac{1}{2} \sqrt{\frac{\log L}{\lambda}} \right ) }
[/latex]
