Some, including Morris Kline, in a book dealing with uncertainty in the basis of mathematics,
indicates that eachh definition in mathematics references some other or other definitions;
but that this leads backwards to undefined terms, which must be there. Perhaps there are other problems, that whatever you are defining may not even exist.
Given the historical roots of math, it is not so clear. That math ought to be based also on other disciplines, counting, land measurement and so on.
Hence that defenitions may better hinge on a descriptive intuitive account that earn us the work of looking up every other definition made in mathematics, This is a great barrier in understanding a lot of work these days, the huge number of definitions one must absorb.
This would best guide our intuition to see if the results are logical. But the comon practice these days are far from it, strait into having to prove everything, and know everything about very narrow specialties, meaningless to most outsiders.
What is your opinion? Should we reform this?
Yes, I agree with you. The Mathematics now needs a radical change to the basic concepts, with a logic that brings them out of the silence of their interconnectedness.
Dear Juan Weisz,
I suggest you to see links on topic.
https://www.goodreads.com/book/show/748807.Mathematics
https://fr.scribd.com/document/391891945/Morris-Kline-Mathematics-the-loss-of-certainty
https://www.maa.org/sites/default/files/pdf/upload_library/22/Evans/september_2005_13.pdf
https://www.nytimes.com/1992/06/11/nyregion/morris-kline-84-math-professor-and-critic-of-math-teaching-dies.html
Best regards
Dear Dr Juan Weisz
Definitions in mathematics are developed hierarchically, so we find that one definition is generalized by another definition, but this does not mean that they are overlapping with each other. Algebraic structures have been developed since the basics of mathematics were discovered. For example, the natural number system was developed to define the integer system until we reached the complex number system. Each system has its own definition, but each system is partially obliterated by the next system. In my opinion, there is no problem in getting to know mathematics
Borrowing from Whitehead and Russel (Principia Mathematica, Vol. I):
"...the definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous."
Mohamed
Your first reference is the book I have a Spanish translation of
The references are interesting.
Anton
I dont see how you can do with no definitions, you must at least have concepts.
Perhaps they are talking more about logic, that simply relates different propositions.
Maged
If you generalize you must partly overlap.
No one denies that perhaps research math. can continue on its very protective
isolating course for specialists, but what is the profit for the rest of society?
What efforts are made to break out of this?
Perhaps the style of exposure is more important than discussing origins or foundations. No agreement on foundations yet, a subject matter still trying to define itself.
Should it further integrate with other subjects?
Children are able to count, and gradually they are able to measure the length of a rod. From there it is not hard to gradually become accustomed to the natural numbers and the reals. In what sense these numbers exist, is an ongoing debate between philosophers of mathematics, and these questions ar very much related to the ZF(C) axioms, and the representation of the natural numbers in terms of sets. I myself take the existence of the natural numbers, also as a set , for granted, and that can safely be done, although, admittedly, whether ZF is consistent, that question has not been answered until so far. The offspring of these seemingly futile discussions are present in the theory of computing everywhere, and that can be considered as very useful! Mathematics is, what mathematicians do, and you only become a mathematician by practicing a lot, and that for years! And when you would like to become a physicist, that also takes you years. And it seems to be impossible to really get a good idea about mathematics from the outside! Just step in, and join the club, or else, do something , that is more to your liking.
JW
You are right Im a bit of a butter in. I think I earned the Physics title,
or applied math. Not so pure math. It would take me too long perhaps
to prove things like mathematical consistency.
It is style of presentation and too many definitions that impeds me access.
Also I dont like formal proofs as much as mathematicians. Applied math seems mostly well consolidated for our purposes.
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