Friday, September 15, 2006


There is a debate, as far as I can tell, in the world of mathematics between those who believe that the human mind casts a net of meaning over the Cosmos by inventing mathematical ideas that enshroud it, and those who believe the mathematical ideas are already out there in the Cosmos and that all that human consciousness does is discover them. The paper from which the following fragment comes is far beyond my ken. The whole paper by Max Tegmark appears in Science and Ultimate Reality: from quantum to Cosmos, honoring John Wheeler’s 90th birthday and published by Cambridge University Press (2003). The ultimate argument of Tegmark’s long paper is that, indeed, there are an infinity of realities out there. The concept of alternative universes is beyond my imagination to imagine with enough force to excite me. All I can do is state it like a school boy reciting his multiplication tables.

Anyhow, math is no longer numbers. It transcends them. I guess?????????????

“Many of us think of mathematics as a bag of tricks that we learned in school for manipulating numbers. Yet most mathematicians have a very different view of their field. They study more abstract objects such as functions, sets, spaces and operators and try to prove theorems about the relations between them. Indeed, some modern mathematics papers are so abstract that the only numbers you will find in them are the page numbers! 'What does a dodecahedron have in common with a set of complex numbers? Despite the plethora of mathematical structures with intimidating names like orbifolds and Killing fields, a striking underlying unity' that has emerged in the last century: all mathematical structures are just special cases of one and the same thing: so—called formal systems. A formal system consists of abstract symbols and rules for manipulating them, specifying how new strings of symbols referred to as theorems can be derived from given ones referred to as axioms. This historical development represented a form of deeonstructionism, since it stripped away all meaning and interpretation that had traditionally been given to mathematical structures and distilled out only the abstract relations capturing their very essence. As a result, computers can now prove theorems about geometry without having any physical intuition whatsoever about what space is like.

“Figure 8 shows some of the most basic mathematical structures and their interrelations. Although this family tree probably extends indefinitely, it illustrates that there is nothing fuzzy about mathematical structures. They are "out there" in the sense that mathematicians discover them rather than create them, and that contemplative alien civilizations would find the same structures (a theorem is true regardless of whether it is proven by a human, a computer or an alien)."

Figure 8. Relationships between various basic mathematical structures (Tegmark 1998). The arrows generally indicate addition of new symbols and/or axioms. Arrows that meet indicate the combination of structures—for instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold. The full tree is probably infinite in extent—the figure shows merely a small sample near the bottom.

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