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The second principle, from mathematics, is to make all
assumptions explicit.^{2} This is
taken to an extreme in contemporary set theory where the only
fundamental entities are the empty set and the relationship of set
membership. The only property of the empty set is that it contains
no members. This is an explicit axiom. Properties are created in
more complex sets by the axioms used in constructing them. The
integers are a good example. Zero is the empty set. One is the set
containing the empty set. Two is the set containing zero and one.
Any positive integer N is the set containing all positive integers
less than N including zero.

Mathematicians did this in part because they were embarrassed by a history of making self evident assumptions that proved to be false. The most notorious was the postulate that parallel lines never meet. It is true of Euclidean Geometry, but not of the geometry of the surface of a sphere like our planet.

Relying only on these two principles avoids the inherent circularity of self evident or intuitively obvious assumptions. What is self evident depends on evolutionary, cultural and personal history. These are, in part, a result of the fundamental principles being sought. An example of the difficulties this can lead to is Strawson's argument about extension in physical space.

Suppose someone -- I will call him pseudo-Boscovich, at the risk of offending historians of science -- proposes that all ultimates, all real, concrete ultimates, are, in truth, wholly unextended entities: that this is the truth about their being; that there isIt may be absurd but it is true in both mathematics and conscious experience and may be true in physics. Mathematics builds all continuous structures as sets of discrete points, starting ultimately with the empty set. The human eye has discrete receptors creating an image composed of an array of dots. Each receptor produces a neural output centered at a spatial point reflecting the average brightness over the area of the receptor. Neural networks in the brain `connect the dots' to create the illusion of continuous extended objects. Many leading physicists have come to suspect that time and space are ultimately composed of discrete points. These include Albert Einstein, `I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures.[20, 467]', Richard Feynman[14, 57] and the 1999 Nobel prize winner, Gerard 't Hooft[28]. This may or may not be true of physical reality but it is a possibility fully compatible with mathematics and science.nosense in which they themselves are extended; that they are real concrete entities, but are none the less true-mathematical point entities. And suppose pseudo-Boscovich goes on to say that when collections of these entities stand in certain (real, concrete, natural) relations, they give rise to or constitute truly, genuinely extended concrete entities; real, concrete extension being in this sense anemergent propertyof phenomena that are, although by hypothesis real and concrete, wholly unextended. Well, I think this suggestion should be rejected as absurd.[26, 15].

Our intuitive sense of spatial extension starts with the way our brain is wired to visualize the world. Such intuitions are generally valid in the context in which they evolved. The history of physics, especially quantum mechanics, ought to make it obvious how wrong applying such intuition to fundamental questions of physics, mathematics or philosophy can be.

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