Talk:Mathematics/Archive 3

message: I think that it would be useful to have a wiki project for the proofs of mathematical theorems.  The goal would be to eventually prove all of the most important theorems(in basic math at least), using only basic axioms and other proofs in the wiki.--Todd Kloos 08:14, 30 May 2004 (UTC)
thread_title: Wiki Proofs?

message: Ongoing discussions of the structure of the page
===Page organisation===
I have a suggestion for a slight reorganization of the main page:
* "Methods" becomes "Foundations"
* "Miscellanea" becomes "History and Miscellanea"
* "History of Mathematics" (to be created) goes into "History and Miscellanea"
* "Special Functions" moves to "Change"
* "Fermat's little theorem" goes on a "Number theory" page (to be created under "Structure")
* Links to "Symbolic Logic" and "Set Theory" are placed under "Foundations and History" (Set Theory also remains under Finite Math)
* "Finite Math" moves down the list, after "Space"
* "Discrete Math" moves under "Finite Math"
What do you guys and gals think? --AxelBoldt
----
An alternative well thought out classification scheme for math is at http://www.math-atlas.org/index/beginners.html ----
As one of the original autors of Math entries here, like Group, Field, Linear algebra, Trigonometric Functions, I think that the new group of articles, are too high level for the average reader, particularly without examples. Many college grads get out of college with
a general math course. I think we have to begin to fill in the lower levels.
In regard to your placement of Finite Mathematics under Space, I do not agree. We also need people to draw graphics for some of these entries, not to mention a way to do matrices. RoseParks
===Arithmetic===
"Arithmetic does not count as a "foundation for mathematics"; it is part of elementary algebra"--AxelBoldt
Then why does the elementary algebra article begin: "For this introduction, knowledge of arithmetic (including the use of parentheses) is assumed." --BlackGriffen
No substantive knowledge, except of the English language common to an intelligent high school graduate, should be assumed--unless an exposition of the subject really does require such assumptions.  In this case, since we introduce school children to arithmetic all the time, I should think we need a simpler article about arithmetic...  In other words, Wikipedia's math articles should, while being maximally useful for mathematicians, also be very useful for non-mathematicians. --LMS
Yes, the elementary algebra article is incomplete and should start out very gently with introducing the order of operations, parentheses, commutativity and associativity etc. (or factor out to an arithmetic article and add a link to elementary algebra if that seems preferable). --AxelBoldt
Should there be a page for arithmetic?  There's a page for modular arithmetic, defining it as "a modified system of arithmetic", without any definition of that term.  Is it considered to be such a simple concept as to require only a dictionary definition? - Stuart Presnell
:See the above discussion. I think we should have a separate article on arithmetic, but I'm not really sure how to write elementary stuff like this, so I haven't attempted it. --Zundark, 2002 Jan 5
::Arithmetic is taught as a mixture of two subjects: algebra, and basic counting.  Most of arithmetic is concerned with counting.  This is just an applied practice, and nothing remotely abstract.  This is where you learn 4 + 4 = (1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) = 8.  This is where you learn the rules about carrying when you add up to more than 9, or when you subtract to get less than 0.  Addition and subtraction are taught as DIFFERENT operations, and you learn about negative numbers near the end of the subject.  You learn about the commutative and associative rules around that time.
::All in all, the teaching of arithmetic is a contentious point, and one of the most frequently cited subjects for much needed reform.  The teaching of arithmetic is brittle and requires a great deal of un-teaching for the rest of the students'  mathematical careers. —Daelin 04:18, 7 Dec 2004 (UTC)
thread_title:  Reorganization and rewriting 

message: "Arithmetic does not count as a "foundation for mathematics"; it is part of elementary algebra"--AxelBoldt
Then why does the elementary algebra article begin: "For this introduction, knowledge of arithmetic (including the use of parentheses) is assumed." --BlackGriffen
No substantive knowledge, except of the English language common to an intelligent high school graduate, should be assumed--unless an exposition of the subject really does require such assumptions.  In this case, since we introduce school children to arithmetic all the time, I should think we need a simpler article about arithmetic...  In other words, Wikipedia's math articles should, while being maximally useful for mathematicians, also be very useful for non-mathematicians. --LMS
Yes, the elementary algebra article is incomplete and should start out very gently with introducing the order of operations, parentheses, commutativity and associativity etc. (or factor out to an arithmetic article and add a link to elementary algebra if that seems preferable). --AxelBoldt
Should there be a page for arithmetic?  There's a page for modular arithmetic, defining it as "a modified system of arithmetic", without any definition of that term.  Is it considered to be such a simple concept as to require only a dictionary definition? - Stuart Presnell
:See the above discussion. I think we should have a separate article on arithmetic, but I'm not really sure how to write elementary stuff like this, so I haven't attempted it. --Zundark, 2002 Jan 5
::Arithmetic is taught as a mixture of two subjects: algebra, and basic counting.  Most of arithmetic is concerned with counting.  This is just an applied practice, and nothing remotely abstract.  This is where you learn 4 + 4 = (1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) = 8.  This is where you learn the rules about carrying when you add up to more than 9, or when you subtract to get less than 0.  Addition and subtraction are taught as DIFFERENT operations, and you learn about negative numbers near the end of the subject.  You learn about the commutative and associative rules around that time.
::All in all, the teaching of arithmetic is a contentious point, and one of the most frequently cited subjects for much needed reform.  The teaching of arithmetic is brittle and requires a great deal of un-teaching for the rest of the students'  mathematical careers. —Daelin 04:18, 7 Dec 2004 (UTC)
thread_title: Arithmetic

message: It's a course in universities but it seems I couldn't find an article about it in wikipedia. So is there anyone with experience would start Engineering Mathematics
Roscoe x 04:17, 9 Oct 2004 (UTC)
thread_title:  Engineering Mathematics 

message: The attention message has been removed. Is the template ready to go on yet?? 66.245.106.126 23:49, 1 Nov 2004 (UTC)
thread_title:  Template 

message: ===Definitions For Math Aren't Dictionary Definitions===
There is a fundamental problem in trying to arrange mathematical knowledge in a dictionary or encyclopedia and it is made worse by hypertext links. This is that mathematical definitions are not definitions of words or short phrases.  For example, there is no mathematical definition of words such as "limit" or "point".
A mathematical definition may contain the word "limit" but it involves explaining a relation between complete statements, not trying to equate a single word with some statements.  We don't define "limit".  We give a statement, such as "The limit of the function f(x) as x approaches a is equal to L"  Then we say this is equivalent (or "means") another statement such as "for each epsilon greater than 0 there exists a delta greater than 0 such that ...".  And all this must take place in a context where we state or imply by notation conventions that f is a real valued function and epsilon and delta are real numbers. 
This seems tedious and even abstract mathematical textbooks make concessions to the ordinary use of language.  For example, after a long definition is given for "f is a measure on the sigma algebra S of subsets of the set Omega", the text may say "A probability measure" is "a measure such that f(Omega) = 1".  This would appear to be a definition that says one phrase is another phrase rather than saying one statement means another statement. The books assumes the reader will recognize that it amounts to saying that the statement "f is a probability measure on the ..." is equivalent to the statement  "f is a measure on the ...  and  f(Omega) = 1".
In the case of "point", this word is usually taken as an undefined term. A fact  that is disturbing to non-mathematicians is that undefined terms must be a central feature of any mathematical system. A layman's idea of a definition is an explanation that gives him an intuitive understanding.  The trouble with such a definition in mathematics is that an intuitive idea can tempt us to introduce properties that mathematics does not assume.  For example, if the teacher says "Let f(x) be a function such that f(3) = 7", a student may object: "That can't happen. The point 3 is infinitely small, so f couldn't do anything there because there is no time or room. And f couldn't find it exactly because it's too small to see."  We regard this as a confusion originating from intutive ideas.  The student misinterprets a function as a matching-up process that must have some physical realization.  His idea of a point is something that has no temporal or spatial extent.
In the case of words that mathematics takes as undefined, I suppose an encyclopedia is obligated to give the reader intuitive guidance.  In such a case it would be wise to mention the fact that what is being offered is an intuitive idea that explains the motivation for creating certain mathematical models. And that some intutive ideas about the term may not be reflected in the mathematical assumptions used in the model.
In the case of words like "limit" we have a term that is embedded in various mathematical definitions. (For example the definition for "f(x) is the uniform limit of the sequence of functions f[i](x) as i approaches infinity"  is different than the definition given above.)   I would prefer to see mathematical definitions gven in the form of one statement being defined as equivalent to another statement.  This doesn't preclude giving the intuitive and cultural background for the isolated word.  But a clear distinction should be made between the formal and informal discussions.
I have a friend who gets upset at the imprecise use of words in cultural and political discussions.  He asks, with exasperation, "When will people learn that words have meanings?".  My reply is that "Words don't have meanings.  Only statements have meanings."  This summarizes the cultural conflict between the dictionary and the mathematics textbook.
Stephen Tashiro
I can't agree with all of that. There seem to be different cases. With some things, like the integral sign, you do have an 'incomplete symbol', and the Frege-style analysis of the meaning as recovered from complete propositions is a good match to what is going on. In other cases such as 'limit', there may be difference usages, implied in different contexts, but there is a perfectly good definition (although you can also treat 'lim' as an incomplete symbol, you don't really have to). And in the case of 'point', it is effectively undefined in standard pure mathematical usage, but reflects the way we say or understand: point is to space as element is to set. 
Charles Matthews 08:35, 11 Sep 2003 (UTC)
replies: text: Do automated theorem proving, law of excluded middle and axiom of choice provide what you want ? If not, then be bold. Gandalf61 14:10, Nov 10, 2004 (UTC)
I've begun to take a crack at rewriting the intro into something interesting and understandable.  It's at  my page.  Input always appreciated.  -- Sean Kelly 09:09, 9 Dec 2004 (UTC)
thread_title:  On the Introduction and Definition of Mathematics 

message: There is a fundamental problem in trying to arrange mathematical knowledge in a dictionary or encyclopedia and it is made worse by hypertext links. This is that mathematical definitions are not definitions of words or short phrases.  For example, there is no mathematical definition of words such as "limit" or "point".
A mathematical definition may contain the word "limit" but it involves explaining a relation between complete statements, not trying to equate a single word with some statements.  We don't define "limit".  We give a statement, such as "The limit of the function f(x) as x approaches a is equal to L"  Then we say this is equivalent (or "means") another statement such as "for each epsilon greater than 0 there exists a delta greater than 0 such that ...".  And all this must take place in a context where we state or imply by notation conventions that f is a real valued function and epsilon and delta are real numbers. 
This seems tedious and even abstract mathematical textbooks make concessions to the ordinary use of language.  For example, after a long definition is given for "f is a measure on the sigma algebra S of subsets of the set Omega", the text may say "A probability measure" is "a measure such that f(Omega) = 1".  This would appear to be a definition that says one phrase is another phrase rather than saying one statement means another statement. The books assumes the reader will recognize that it amounts to saying that the statement "f is a probability measure on the ..." is equivalent to the statement  "f is a measure on the ...  and  f(Omega) = 1".
In the case of "point", this word is usually taken as an undefined term. A fact  that is disturbing to non-mathematicians is that undefined terms must be a central feature of any mathematical system. A layman's idea of a definition is an explanation that gives him an intuitive understanding.  The trouble with such a definition in mathematics is that an intuitive idea can tempt us to introduce properties that mathematics does not assume.  For example, if the teacher says "Let f(x) be a function such that f(3) = 7", a student may object: "That can't happen. The point 3 is infinitely small, so f couldn't do anything there because there is no time or room. And f couldn't find it exactly because it's too small to see."  We regard this as a confusion originating from intutive ideas.  The student misinterprets a function as a matching-up process that must have some physical realization.  His idea of a point is something that has no temporal or spatial extent.
In the case of words that mathematics takes as undefined, I suppose an encyclopedia is obligated to give the reader intuitive guidance.  In such a case it would be wise to mention the fact that what is being offered is an intuitive idea that explains the motivation for creating certain mathematical models. And that some intutive ideas about the term may not be reflected in the mathematical assumptions used in the model.
In the case of words like "limit" we have a term that is embedded in various mathematical definitions. (For example the definition for "f(x) is the uniform limit of the sequence of functions f[i](x) as i approaches infinity"  is different than the definition given above.)   I would prefer to see mathematical definitions gven in the form of one statement being defined as equivalent to another statement.  This doesn't preclude giving the intuitive and cultural background for the isolated word.  But a clear distinction should be made between the formal and informal discussions.
I have a friend who gets upset at the imprecise use of words in cultural and political discussions.  He asks, with exasperation, "When will people learn that words have meanings?".  My reply is that "Words don't have meanings.  Only statements have meanings."  This summarizes the cultural conflict between the dictionary and the mathematics textbook.
Stephen Tashiro
I can't agree with all of that. There seem to be different cases. With some things, like the integral sign, you do have an 'incomplete symbol', and the Frege-style analysis of the meaning as recovered from complete propositions is a good match to what is going on. In other cases such as 'limit', there may be difference usages, implied in different contexts, but there is a perfectly good definition (although you can also treat 'lim' as an incomplete symbol, you don't really have to). And in the case of 'point', it is effectively undefined in standard pure mathematical usage, but reflects the way we say or understand: point is to space as element is to set. 
Charles Matthews 08:35, 11 Sep 2003 (UTC)
thread_title: Definitions For Math Aren't Dictionary Definitions

message: It's a shame the mathematics page still lacks a good general article about mathematics. With so many mathematicians about, you'd think a general discussion and characterization of mathematics would be forthcoming. --LMS
I strongly agree with the above remark. The lack is striking but perhaps people feel this would become too much a subject of dispute?
BozMo(talk)
I doubt that's it. It's just typical of wiki work, that when things are active in the 'specialist' areas, that's where the blood flows.
Charles Matthews 11:44, 6 Apr 2004 (UTC)
replies: text: Do automated theorem proving, law of excluded middle and axiom of choice provide what you want ? If not, then be bold. Gandalf61 14:10, Nov 10, 2004 (UTC)
I've begun to take a crack at rewriting the intro into something interesting and understandable.  It's at  my page.  Input always appreciated.  -- Sean Kelly 09:09, 9 Dec 2004 (UTC)
thread_title: General intro

message: This is sure to be an ongoing discussion --Sean Kelly 00:16, 21 Dec 2004 (UTC)
thread_title:  Is Mathematics a Science? 

message: I know what mathematics is and how it is used. I use it when I'm doing science, and I explore it as if it were a science. Methodical, logical, observational, structured, repeatable and recorded. 
If you are going to remove mathematics from the realm of science, you should make sure all your definitions here on Wikipedia share that point. It is quite humorous to look through the pages of Wikipedia see contradictions among your definitions and the arguments put forth here on the subject of whether mathematics fits the bill of science or not. 
Axioms can be either empirical and dreams. Mathematics started with only empirical axioms. 'Equals' is the observed quality of being the same. Addition is the observed quality of collecting things.
Scientific methods are many. Experimental methods are but one area. Observation and recording is the main method common to all sciences. 
Geometry has as axioms four that were descriptions of physical constructs. The fifth axiom was added because the observations didn't match the world until a parallel line axiom was added. The page on geometry makes it look as if geometry is empirical by nature, and therefore consistent with being both a science and mathematics. 
Science has definitions rooted in the word 'knowledge'. Until you remove that root, with such a statement that the word's root is from knowledge, but it has lost that definition and is now limited to the area of experimental observations, with falsification. 
Falsification however, is only one tool of science and to limit all of science to that classification would severely hamper the valid understanding of the world and leave many powerful sciences looking for a new home. Such sciences for example would be field botany or field biology, where the sole goal is to merely observe and record, with no experimentation or prediction intended. 
The reason why empiricism is so prevalent in science is that it is a way of gathering knowledge about the world rapidly. It is not the only method used to produce knowledge. 
Mathematics alone has pushed forth the desire to do empirical work in a large number of areas, such as antimatter, and time paradoxes. Mathematics is at the very root of empirical work. Without mathematics, you can do little, if any, observations. 
Please, understand that mathematicians do work in both empirical areas, and in pure deductive exploration areas. Both areas require the methodologies and critiques that any good science uses. Nothing strikes me as more scientific than mathematics. Empirical knowledge gathering is just as scientific.
Physics is a repository for all the other sciences. It is a combination of electronics, chemistry, mechanics, material science, astronomy, ....etc... It is also an area of study all on its own. The theory of relativity is a good example. 
No one area of science is basic to any other area. All areas are valid pursuits of knowledge. 
I say define mathematics as it was originally defined, the first science and let is stand. Try to show there is a fuzzy difference and non-complete separation between empirical and pure sciences. Show the overlaps. Show the differences. Call it a science. To call it a non-science indicates some skepticism is valid against it. To call it a science allows one's skepticism to explore it in valid ways as is true of all sciences. Mathematics is valid knowledge useful for understanding other areas of knowledge. Science is about the validity of knowledge. Mathematics validates all science, including itself. (EDN 2005-3-22)
"Mathematics is not a science, in the proper connotation of the word. Some thinkers see mathematicians as scientists, regarding physical experiments as inessential or mathematical proofs as equivalent to experiments. But they themselves do not see mathematics as a science, since it does not require experimental test of its theories and hypotheses. In either case, the fact that mathematics is such a useful tool in describing the universe is a central issue in its philosophy. But in any case, mathematics is essential to science, being its most important function the role it plays in the expression of scientific models."
Mathematics is not a science? Bahh, I haven't read such idiotism before. Nor the statement neither its justification is acceptable for someone who is not a philosophical positivist. I think this statement is a POV, and should be deleted (I will delete it in a few days time if someone can't explain it to me, what the author was thinking about). I think if someone think somethink, someone not, we shouldn't accept it as a NPOV statement. Well, mathematics is not surely an empiric science, but there are a lot of sciences what aren't empiric. Gubbubu 21:25, 1 Sep 2004 (UTC)
thread_title:  Not a science ?:-)