Help in understanding a paragraph from 'What is Mathematics'? "Whatever our philosophical standpoint may be, for all purposes of scientific observation an object exhausts itself in the totality of possible relations to the perceiving subject or instrument. Of course, mere perception does not constitute knowledge and insight; it must be coordinated and interpreted in reference to some underlying entity, a "thing in itself," which is not an object of direct physical observations, but belongs to metaphysics."
What does the author mean by "underlying entity"? - Is it what the object is made off? What is inside the object?
 A: Think about the formula for the volume of a sphere. Which sphere(s) does the formula apply to? There are concrete physical objects that are (roughly) spherical … billiard balls, soap bubbles, planets (roughly), or apples (even more roughly). But we don’t think of the volume formula as something that applies to planets or apples; it applies to all spheres, whether they exist physically or not. The sphere is the “underlying entity” for all of these physical objects.
I don’t know if this will help you, but I’d say that these ideas are related to Plato’s theory of forms. Plato would argue that there is an abstract non-physical thing called a sphere, and that physical objects like planets and apples and even billiard balls are just imitations (approximations) of the sphere “form”.
A: What the quote is saying can be rephrased as follows. When we study a physical  object, we use some instruments (rulers, etc.) and compare it with other objects, which give us some properties of the object. These do not give us complete information about the object because there is always something ("thing in itself") about the object which cannot be measured by the instruments (here should be a reference to Kant).
This "thing in itself" is of course a philosophy concept but it eventually lead to very concrete statements from math such as Goedel's theorems.
