Unresolved probability paradox that goes to the heart of scientific objectivity

I never heard about this. I never worried about this.

The idea that there is a rule that says that you must assume that the probability has a uniform distribution is ridiculous. The probability is never uniforms except in very special cases that are build on purpose.

In a similar problem, the first step is try to guess a good model for the distribution of the probabilities, and then refute experimentally all the alternatives.

The Philosophy professor is wrong. This is not a paradox.

Anyone applying the Principle of Indifference needs to make a valid argument about why it is reasonable to measure the outcomes in a specific way. You can't just arbitrarily choose what you'd like to be indifferent about.

For example, the video is about squares of wood, in which the edges are between 1 and 3 feet. The first possibility is measuring outcomes by edge length. To apply the Principle of Indifference here, one could make an argument such as "The wood is cut with a table saw. The set of customers wants squares with a variety of edge lengths. Therefore, the boss instructs the laborer to configure the table saw so that the scale on the fence's rail is set anywhere between 1 and 3 feet. This is a single physical adjustment that doesn't require the laborer to make any calculations. Therefore, it is very plausible to believe that the laborer uses the scale to achieve arbitrary values between 1 and 3 feet. Also, without further information, it is reasonable to guess that the values would be evenly distributed between 1 and 3."

It is much, much less plausible that the laborer makes a series of calculations in an effort to achieve an outcome in which the square's areas are uniformly distributed between 1 and 9 square feet. There is no common wood-cutting technology in which a laborer dials in an area, and a machine then cuts to the specified area. Also, there is no common wood-cutting technology in which a laborer sets a length on a nonlinear scale, e.g., if the length is set to the halfway point, the machine knows to cut more than halfway (so that small areas aren't over-represented). Without further information, it would be incredulous that the laborer surmises that uniformly distributed areas are desirable, AND either has unusual technology or has the mathematical skills to achieve this by hand.

Going further, it is completely implausible to believe that the laborer, when given the direction that edges must be between 1 and 3 feet, decides to choose lengths such that the generated sawdust incrementally contributes to the factory's indoor Air Quality Index by uniformly distributed amounts. The effect of wood cutting on the Air Quality Index has complex dependencies on the duration of a cut relative to the speed of air currents within the factory, the size of the wood relative to the volume of the factory's work space, etc. It is not reasonable to expect that the laborer knows how to calculate this, or (even if known) has a straightforward way to configure the saw to achieve that Air Quality Index distribution.

If a scientific study simply doesn't have a reasonable argument to justify how to apply the Principle of Indifference, then the investigators should state that statistical reasoning about their results is not necessarily useful.