Kant, when he considers mathematics, is concerned with the assumptions of applied mathematics, of those sciences which, though mathematical, make statements about existing objects, and in which the old distinction between understanding and perception which was based on the difference in the objects of these two faculties breaks down. The sciences which Kant is investigating imply that principles which are clearly not derived from mere observation are yet the basis on which we order and arrange what we observe. Now, if we held that the objects of mathematics were independent entities quite separate from the things we perceive, it would be impossible to explain how we might assume that the things we perceive would be subject to the rules of mathematics. If, on the other hand, we held that in mathematics we were simply concerned with the various objects of the senses, it would be impossible to explain how mathematics can have a generality and necessity which no statements can have which rest on observation of the various things we see. The existence of applied mathematics implies firstly that understanding and perception are distinct, and that neither of them can be reduced to the other, for that would mean that we should have to give up either the element of observation and experiment or the element of necessity and a priority, and secondly, that understanding and perception are combined, and must be combined for any advance in science. Now, Kant finds his answer to the problem he has raised by concentrating his attention on the fact that, while understanding and perception are distinct, they are both present in all knowledge. His argument is that we are necessarily in a difficulty if we think of understanding and perception as having each its separate objects, and then try to explain their combination. If we begin with their combination, we may see that the reference of principles of thought to objects of sense is not an accident, but that these principles of thought or of understanding, as Kant calls them, are only concerned with objects of sense, and have no other meaning. If we object, But how can principles of thought be universal if they are concerned with the many and varying objects of sense? Kant’s answer is that they are not concerned directly with these objects, but with the conditions under which these objects can be understood. They are therefore not statements about objects, but statements of the conditions of possible experience. If we find out that all perceiving and thinking imply certain conditions, then we can affirm the validity of principles based upon these conditions, so long as we do not try to apply the principles beyond our perceiving.