just wasting our time, but having fun doing it
Poincaré, in several of his famous essays on the philosophy of science, characterized the difference between mathematics and physics as follows: In mathematics, if the quantity A is equal to the quantity B, and B is equal to C, then A is equal to C; that is, in modern terminology: mathematical equality is a transitive relation. But in the observable physical continuum "equal" means indistinguishable; and in this continuum, if A is equal to B, and B is equal to C, it by no means follows that A is equal to C. In the terminology of the psychologists Weber and Fechner, A may lie within the threshold of B, and B within the threshold of C, even though A does not lie within the threshold of C. "The raw result of experience," says Poincaré, "may be expressed by the relation
A = B, B = C, A < C,
which may be regarded as the formula for the physical continuum." That is to say, physical equality is not a transitive relation.
...
Instead of distinguishing between a transitive mathematical and an intransitive physical relation of equality, it thus seems much more hopeful to retain the transitive relation in mathematics and to introduce for the distinction of physical and physiological quantities a probability, that is, a number lying between 0 and 1.
Elaboration of this idea leads to the concept of a space in which a distribution function rather than a definite number is associated with every pair of elements. The number associated with two points of a metric space is called the distance between the two points. The distribution function associated with two elements of a statistical metric space might be said to give, for every x, the probability that the distance between the two points in question does not exceed x...
when i read things like this i want to give up on math. but why would i do that? things are just starting to get interesting.
A = B, B = C, A < C,
which may be regarded as the formula for the physical continuum." That is to say, physical equality is not a transitive relation.
...
Instead of distinguishing between a transitive mathematical and an intransitive physical relation of equality, it thus seems much more hopeful to retain the transitive relation in mathematics and to introduce for the distinction of physical and physiological quantities a probability, that is, a number lying between 0 and 1.
Elaboration of this idea leads to the concept of a space in which a distribution function rather than a definite number is associated with every pair of elements. The number associated with two points of a metric space is called the distance between the two points. The distribution function associated with two elements of a statistical metric space might be said to give, for every x, the probability that the distance between the two points in question does not exceed x...
when i read things like this i want to give up on math. but why would i do that? things are just starting to get interesting.