Talk:Theory (mathematics)

Thomas, if my use of the word "consequential" is wrong, the phrase "The Greek (axiomatic) tradition treats a theory as a tower of more consequential facts called theorems" has to be changed, too. How to change it? Or should I say "consequent" (in both cases)? Or "derived"? Or "derivative"? --Boris Tsirelson 01:39, 31 March 2011 (EDT)


 * How would you define "consequential"? I think the word means something has consequence (that is, significance), but maybe it has a mathematical usage that I'm not familiar with. Maybe you could use "consequential" but explain what it means in parentheses?—Tom Larsen (talk) 19:46, 16 April 2011 (EDT)


 * That is a problem of my poor English. No, it is not a math usage, just common English. I did not mean "something has consequence", rather, "something that is a consequence of something else". Is there an apt word? Again, what about "consequent", or "derived", or "derivative"? --Boris Tsirelson 01:34, 17 April 2011 (EDT)


 * Ah, I think I understand what you mean. How about:


 * They answer differently the question of whether or not some mathematical facts require less derivation, and are therefore more "fundamental", than others.


 * Is that kind of what you intend?—Tom Larsen (talk) 02:09, 17 April 2011 (EDT)


 * Maybe. But axioms require no derivation at all. --Boris Tsirelson 03:08, 17 April 2011 (EDT)

"Mathematical truth is sharp, not fuzzy. Every mathematical statement is assumed to be either true or false"
I would question whether that statement is true universally. It certainly is true from the perspective of classical mathematics, by which I mean the mathematical approach that starts with classical propositional and predicate calculus, which it uses to build ZF set theory (or ZFC, or whatever), which in turn is used to construct classical real analysis, and so on we can go. But, what about non-classical approaches, such as intuitionistic or paraconsistent logic? Intuitionism/constructivism denies the law of the excluded middle (everything is either true or false, with no in-between) - it is willing to accept mathematical statements which are neither true nor false. Paraconsistency denies the law of non-contradiction (nothing is both true and false). Both form parts of mathematics, since they are parts of mathematical (or formal/symbolic) logic, which is part of mathematics; and both can be used as a foundation for further development of mathematics - a fair amount of work has been done developing the major mathematical disciplines (e.g. real analysis) on an intuitionistic/constructivist base; I have seen some work done on a paraconsistent base too, although I don't think that project has ever reached as far. So, this statement is an adequate definition of the most significant (in terms of influence, adherents, practicioners, etc.) approach to mathematics, but not of mathematics as a whole. Zachary Martin 02:49, 10 June 2011 (EDT)


 * Yes, I agree. It is worth to note somewhere in the article that all said holds for the mainstream math. Also it is worth to add (probably at the end) sections about non-mainstream approaches. --Boris Tsirelson 03:12, 10 June 2011 (EDT)