Abstract
Why is mathematics able to describe the phenomena of the natural world? Evidence that it can do so
with a remarkable degree of success is all around us. Engineers use mathematics to design bridges, dams,
roads, skyscrapers, computers, cell phones, aircraft, coffee makers, and, indeed, the greater part of all
devices and structures we see and use every day. The laws of physics are expressed in mathematical
equations. The other sciences also employ mathematics to one degree or another to describe and explain
the phenomena they study (albeit few of the other sciences use mathematics to the degree and extent it is
used in physics).
Yet mathematics is indisputably the product of human minds whereas the natural world is not. Further
more, you can search the world over and you will never find a single sensible experience in which you
have any direct physical encounter with any object of pure mathematics. A mathematical point, line, or
circle is nowhere to be found; nor is a transcendental number, a mathematical hyperspace, or any other
denizen of what we will call "the mathematical world." The only places where we find any immediate
connection between the mathematical world and the world of physical nature are in the minds of human
beings whose understandings of natural phenomena call upon supersensible mathematical objects to
describe and explain them. By "supersensible object" I mean an object that can never be experienced by
our senses or by any instrument capable of extending our senses (e.g., a microscope that allows us to see
bacteria invisible to the naked eye or a telescope that allows us to view the rings of Saturn). It seems that
mathematical objects exist outside of physical nature. Yet, if this is so, how is it possible for us to
successfully understand the latter by means of the former? Is this not a fundamental paradox?
This treatise is written around the theme that, although many find a paradox here, it is not a fundamental
paradox. It has a resolution, and this resolution is found by understanding more fully what mathematics is,
what science endeavors to do with it, and what practical connections we make between the two. We will
examine relationships between math and science and, by doing so, cultivate a practical understanding of
how to invent mathematical objects, bring them into understanding nature, and use them to improve
human lives and conditions.