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Extending mathematics

If infinity is a potential and never a completed reality then infinite sets do not exist. Mathematicians try to define the most general infinite structures imaginable because that seems to give the most bang for the buck. If no infinite sets exist this would be the construction of a fantasy.

Mathematics should be directly connected to properties of nondeterministic programs in a potentially infinite universe. This would limit extensions to a fragment of the countable ordinals and the sets that can be constructed from them. Of course the previous sentence only has meaning in the philosophical context of contemporary mathematics. In the philosophical approach we are advocating it is illusion to think of the countable ordinals as having an objective existence.

The objects definable within a formal mathematical system no matter what axioms of infinity it includes are countable (they can be mapped onto the integers). This result is called the Lowenheim-Skolem Theorem. The idea of the proof is that a formal system can be interpreted as a computer program for generating theorems. Such a program can output all of the names of the objects or sets definable with the system. These names and thus the collection of all objects they refer to are countable. They can be mapped onto the integers.

All real numbers and for that matter larger cardinals that can ever be defined in any mathematical system that finite creatures create will be countable. They will not be countable from within the system. Cantor`s proof is correct as a proof about formal systems. If real numbers do not exist Cantor's proof is about the structure of formal systems and not some greater metaphysical reality.

This suggests that the theory of cardinals is an illusion. It is talking indirectly about ways of extending mathematics that are countable and reducible to properties of computer programs. The set of reals definable within a formal system is a countable set in a more powerful formal system. In the more powerful system there is a countable ordinal that characterizes this set. Because we are talking about names generated by a computer program there is a connection between this countable ordinal and properties of computer programs.

There is nothing inconsistent in believing the illusion of the completed infinite. But perhaps this fantasy gets in the way of extending mathematics to the full degree that the human mind aided with computer simulations is capable of.

Of course there is a limit to what we are capable of understanding as a species. It is only by continuing evolution in a nondeterministic way following an ever increasing number of divergent paths that we can avoid creative stagnation in mathematics and everything else.

But we are far from understanding all that we are capable of. Using the enormous power of computers to leverage our intuition and intellect has led to great strides in science. It is ironic that research in the foundations of mathematics is still largely conducted with pencil and paper. As long as the focus is on the most abstract and powerful notions of the infinite then computer experiments seem irrelevant. That alone suggests this approach to mathematical truth is the contemporary version of historical failures. There is a tendency in mathematics to postulate axioms that are too strong. Mathematicians have perhaps learned to avoid absolute contradictions but not the folly of Icarus. By attempting to fly too high foundations research in mathematics has stagnated.


Completed second draft of this book

PDF version of this book
next up previous contents
Next: Is the cardinality of Up: Creative mathematics Previous: Trees of trees   Contents


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