Fractal figures are of a length, area, or volume which cannot be pinned down to exactly one value. Depending on when one stops "zooming in," so to speak, on the figure, different values can be ascribed to the figure, none of them deserving the right of being the quantity of the figure "itself."
What does this mean? I think that one consequence is that measurement is a decision, an act of our minds combined with natural phenomena. The crucial thing here is that it is literally our minds that "determine" the length, area, of volume of a given fractal figure.
This challenges two common conceptions. First, it challenges the concept of measure as a simple representation of something which exists independently of the mind. Second, on a more broader scale, it also challenges the concept that natural properties exist determinately independently of the mind. The second lesson might be particularly counter-intuitive for Lockeans who still believe that "primary properties" exist "out there."
The role which the mind plays in determining nature - this is something which needs to be spelled out. Not in terms of a simple-minded Kantianism, but neither in a way which merely reduces this question to that of "how the brain works when a human body interacts with nature or thinks."
