Date: Jan 7, 2013 4:46 PM Author: Kaba Subject: Re: The Reason Why Tau Is Fundamental And Why Pi Is Not 7.1.2013 22:00, Jesse F. Hughes wrote:

>> That said, in my opinion the article makes a good case in favor of

>> adopting tau instead of pi. The mathematics is anyway objective, even if

>> described subjectively.

>

> It's all a matter of convention. Whether, as it happens, it's somewhat

> more convenient to use 2pi as a constant, rather than pi, surely is one

> of the least interesting mathematical points one can make, barely better

> than insisting that the numeral seven ought to have a crossbar to better

> distinguish it from one.

Essentially, you are making the claim that syntax does not matter.

<rant>

To see the significance of syntax, you have to pick a field where the

effects of bad syntax are amplified so much that making progress beyond

a given point becomes impossible. This then gives perspective also for

the smaller scale (mathematics). That field is software engineering.

Software engineers have coined the terms accidental complexity and

essential complexity. Accidental complexity is caused by choosing

abstractions incorrectly, causing needless branching in the logic, to

match the hidden underlying correct abstraction. Essential complexity is

the minimum amount of complexity that needs to be introduced to make the

thing work as it should (the ideal).

An essential problem in software engineering, and indeed in every other

field including the composition of abstractions, is that these special

cases combine combinatorially (and they occur multiplicatively, in the

sense that it affects every point of use). Unless actively dealt with,

in a very short time this makes the program incomphrehensible to the

programmers themselves. What separates good programmers from the bad is

that the former can keep the accidental complexity in control by

actively making choices which minimize special cases (and repairing such

errors in retrospect).

This underlines an important idea in controlling accidental complexity:

in an abstraction, the uniformity of the corner cases with respect to

the base case is the most important part of the abstraction. Perhaps

mathematicians could resonate with the mental image that an abstraction

should be continuous on the boundary points, _unless_ that discontinuity

is an essential (and not accidental) part of the abstraction.

Up to now, I have described the great monster of bad syntax, that

arising from the composition of bad abstractions, a kind of

asymptotically bad complexity if you will. But part of accidental

complexity are also smaller monsters, which cause only a constant amount

of excessive work. Examples of such are reading this word sdrawkcab,

presenting linear systems of equations by elements rather than matrices,

or indeed the tau versus pi notation. Such constant-penalty complexity

requires additional brain processing for nothing. It slows you down:

consider the time it took to read the word backwards above.

It's only syntax, some people say, but I say they haven't actually been

exposed enough to the monsters to recognize the problem, or never lived

monster-free. In a fantasy world of mine, everyone strives to minimize

accidental complexity, whether it be constant or asymptotic.

</rant>

--

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