What does it mean to say that mathematics can be regarded as a formal game devoid of intrinsic meaning? If this is the case, how can there be such a wealth of applications in the real world?

Mathematics is a formal game devoid of intrinsic meaning because it begins and ends with arbitrary human inventions. A mathematical investigation starts with formulas, numbers, variables and the like -- all of which mean absolutely nothing if not placed in a very specific context and subjected to intense scrutiny. This separates mathematics from other areas of study because its results, as well as its questions and processes, are not of any value alone -- they can only be used to explain or relate to other ideas. The "discovery" of a new mathematical theorem simply means that someone has designed a new way to play the game -- a new way to use an artificial system to produce artificial answers from artificial input.

This being said, there are certainly fields relating to and using mathematics which are quite valuable and meaningful. Mathematical formulas are valuable shortcuts in solving problems in various areas of science which do produce intrinsically significant results. Though the specific formulas used in designing jets are arbitrary, the fact that they can help create a flying machine is useful.

These applications exist because mathematics has been designed around them -- we made the rules, so we can play the game wherever we want. We fool ourselves into thinking that we understand the world because we have discovered the mathematical truths at its core when in reality we have simply attempted to mirror what was already there. The examples of natural patterns Lillian mentioned are indeed all mathematical. However, this does not mean that we "found mathematics" after careful study of nature.

The patterns are there no matter how we describe them. Mathematicians cannot claim to have "discovered" mathematics in nature -- the patterns were in nature; they designed the mathematical system around them. I could just as easily (and more directly and specifically) describe the same phenomena in words. Mathematics is necessary because complex situations would render the use of words far too cumbersome -- we streamline and stylize the situation by using our system. The complex problems of the modern world are greatly facilitated by mathematics -- just we might design a game to aid in our understanding of any occurrence. The game has no value in itself -- the points are meaningless, the rules are arbitrary, and the score is not an ultimate answer to any question. However, the game of mathematics can lead to greater understandings just as a simulation involving students orbiting around a room might serve as an analogy for the solar system.

The fact that there are additional patterns in nature which cannot be described by our mathematical system serves as definitive proof of the seperation between natural patterns and human mathematical representations. On some level, everything must be a pattern and occur for a reason -- the cycle and operation of life, the function of the basic forces of physics, the creation of the universe. Our mathematical system cannot mirror these truths because we cannot yet comprehend them. If we could create mathematical representations of these collassal patterns, it would mean that we would have come to understand everything -- but it wouldn't be mathematics that we were understanding.