Hi Harv,
>>>I see that we may have a different concept of the term undefined as it is used in mathematics.>I think our disagreement hinges upon what you mean by mean.>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.>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>>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.> Terms must mean something otherwise mathematicians would be unable to distinguish a group from a set, or a set from a point, or a point from a line, etc.>>On the other hand, they lack formal meaning. That is, there is no definition of say a derivative which talks about limits, Δ x → 0, etc. This is how I understand the use of undefined terms in mathematics (i.e., not mathematically defined).As for mathematics, I don't think you can construct mathematics without some meaning to those 'undefined' terms.>That is, if a term has absolutely no meaning then how can you construe meaning? Without meaning you cannot construct something meaningful.> The reason that math is possible is because we can use the meaning of abstract terms and from that we can construct abstract definitions, equations, theorems, etc.However, in both in his premises and his conclusions he seeks to discuss philosophical issues. That's a no-no if you are strictly a mathematician.>>P.: 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). H.: I won't quibble about these issues. However, I can't resist in commenting that you have (g) being shunned by science which is in fact the most significant aspect of science (observation). Other than that, I won't say any more. |