Why do I need perfection? I merely need something better (less imperfect) to compare. Can you smell as well as a Bloodhound? Can you see as well as an Eagle? Obviously, given that their senses are better than ours then our senses must be imperfect.
Human language describes the mental models we all hold inside our brains. Because of our imperfect senses those internal models are also imperfect and can differ from person to person. Human language does not describe the actual world. It describes a set of common shared internal models which are imperfect reflections of the real world.
Some dogs do not bark and some do not chew bones. Any general model, like “dog” will miss specific nuances of specific individual dogs. “Humans have five fingers” is often true, but it is not an absolute. There are humans with more than five and less than five digits.
Mathematics is very far from an objectively real system, it is a logical system based on axioms. As long as the axioms are consistent then they can form the basis of a mathematical system. The classic example is the parallel axiom. There are three different inconsistent versions, each of which leads to a different geometry:
- Given any straight line and a point not on it, there exists no straight line which passes through that point and never intersects the first line:
- Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the first line.
- Given any straight line and a point not on it, there exist at least two straight lines which pass through that point and never intersect the first line
1 gives Spherical geometry, 2 gives Euclidian geometry and 3 gives Hyperbolic (or Lobachevskian) geometry.
Mathematics is an axiomatic system, and it changes if you change the axioms.
Of course. Is “unicorn” attached to anything existing? It is attached to a mental model inside people’s heads, not to any real-world entity.
rossum