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What Do You Mean By Undefined?

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Posted by Harvey on December 27, 2001 23:21:04 UTC

Paul,

I see that we may have a different concept of the term undefined as it is used in mathematics. My understanding is that mathematicians do not define certain terms (e.g., points, lines, groups, sets, and various other terms), however this does not mean that those terms lack total meaning. Rather the undefined terms lack formal meaning.

This is not to say that the undefined terms hold no meaning. Every math book I can think of uses undefined terms that would make absolutely no sense unless some meaning applied to those undefined terms.

Usually what is meant by undefined terms are terms that have no formal meaning within mathematics. For example, a set has no formal meaning within mathematics, but it obviously means a collection of members sharing some quality which makes them share a particular quality (thus making them part of a set), but unique in that they have other qualities which set them apart from other members of the set. In the philosophy of mathematics there is a great deal of discussion on the actual meaning of sets and set theory, but this is philosophy since mathematicians are only concerned with working within set theory without consideration of the philosophical meaning of sets as they apply to the world.

>>>In science, the ideas a - g are shunned while h - o are accepted. In mathematics, while idea k is a by-product, only l - o are accepted in the formal development, although math professors condescendingly use a limited amount of j in order to teach their students (just kidding).>>As with any theorem, Dick's result only makes an abstract statement of the form If A then B. In Dick's theorem, A is an undefined set of data and B is a gawdawful differential equation. Any correlation between A, B, and anything in the real world is something for science to make. Dick has discovered that solutions to B yield some familiar laws of physics, but he has been the only person working on solving that equation and he hasn't done any of that work for some time. I think the potential is there to find additional solutions to B that will not only describe new developments in physics, but possibly even suggest novel developments.

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