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If I were to try to find pi using a ruler and a compass, I would first try to find out how many rational line segments of the diameter I could fit around the interior circumference and then continue to refine until I began converging on some number. Before the foundations of Non-Euclidean geometry were made, people would not have even questioned that the euclidean pi had a unique nature.

Why not think of the standard model parameters in the same light? Just simply as ratios that might change with the geometry of space or some something similar?

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The number $\pi$ is a mathematical constant that can be calculated. And indeed, as expected from naturalness, it is of order one, so it is clearly not fine-tuned. So we wouldn't be talking about fine-tuning even if it were just a physical parameter and not a mathematical constant.

Fine-tuning only occurs when a theory requires at least 2 parameters with the same units whose numerical value differs by a big ratio. That's not the case of $\pi$. It is the case of the Higgs mass in the Planck units which is $10^{-15}$, and especially cosmological constant in Planck units which is $10^{-123}$. There are many other less extreme examples.

But if your suggestion is that all such constants should finally be as calculable as $\pi$, that's indeed a classic goal of theoretical physics, one that string theory has the capacity to resolve because everything in string theory is calculable. At the current state of progress, however, there exists a very high - but finite or countable - set of possible vacua where the results of the calculations are different. That's why some people defend the anthropic reasoning.

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Fine tuning and fitting are powerful instruments in describing experimental data. They make it possible to describe good data with a bad theory. If you lack free parameters to fit well the data, expand your coupling and masses in power series and consider each term as a free fitting parameter. Doing so one is bound to describe the data. $\pi$ is also possible to fit, if necessary - the theory must describe data so every means is good nowadays. – Vladimir Kalitvianski May 26 '11 at 8:21

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