Talk:Mathematics/Archive 5

message: By definition, mathematics is abstract, and science is about gathering empirical knowledge (it can refer to the process, the people, or the knowledge itself). Surely something can't be both abstract and empirical?
Why do you think science must be empirical? this is only a point of neopositivists' view.
Notice the article I linked to in the heading? That article says it is empirical. Also, the common usage (the common usage I've noticed, anyway) indicates that it is empirical. The definition at Wiktionary does not indicate that it must be empirical, but that definition seems too broad.
If you know anything about neopositivism (I don't), you can start the article! Brianjd 04:17, 2005 Mar 6 (UTC)
Why is the definition at Wiktionary too broad? Does it not imply that Wikipedia is a "science"; that accounting is a "science"? Brianjd 04:18, 2005 Mar 6 (UTC)
:Mathematics is not a science. Mathematics is mathematics, it is ubiquitous, it is not a subcategory of something greater. We don't have to define it in terms of something else. --Tothebarricades.tk 05:46, 6 Mar 2005 (UTC)
replies: text: Calling it a science no more "defines it in terms of something else" than calling it one of the academic disciplines "defines it in terms of something else". Michael Hardy 02:03, 20 Mar 2005 (UTC)
thread_title: Mar 2005 "Not a [[science]], by definition"

message: The science article (for me anyway) clearly states that science is empirical. How can something be both empirical and abstract ("the science of abstraction")? Brianjd | Why restrict HTML? | 04:09, 2005 Apr 23 (UTC)
:To think in abstractions is the strategy of simplification of detail. You can select an empirical detail from a mass of other empirical detail, thus simplifying or abstracting. No contradiction. Ancheta Wis 08:27, 23 Apr 2005 (UTC)
thread_title: The mathematics article and the [[Science]] article contradict each other

message: *No. Empiricism means applying logical induction to the real world.  Mathematicians use logical induction to come up with conjectures, but pure mathematics is independent of physics. --Sean Kelly
*Sometimes it is, actually. Charles Matthews
*Empirical methods used to form hypotheses but logic used to prove theorems - so results are not contingent, and mathematics is not just empirical investigations. Gandalf61
*No Kevin Baastalk "Numbers exist independantly of the things they number." 'Nough said.
*I guess you mean "Does mathematics include taking measurements in the real world?" and then my answer would certainly have to be No. Instead from intuition you may suspect that a certain statement holds. This is your hypothesis. Then you can proceed to investigate and try to find a proof or a counterexample. These are your measurements. In this way you could say that mathematics is empirical. -MarSch 17:46, 23 Apr 2005 (UTC)
thread_title: Is Mathematics empirical?

message: *No.  Though m. is artist and beautiful, and metaphorically an art. --Sean Kelly
*No Charles Matthews
*No Kevin Baastalk If math was an art, it would be taught in art school.  Math does not express human emotion or irrational impressions in a relatively unrestrained form.  To summarize, there are a few basic subjects taught in middle school: art, science, math, history, english. give or take a few, like phy. ed., etc.  in any case, point made.  is art a language?  is history a science?
*Unsure. Mathematics requires great skill and ingenuity, which would also make engineering an art. Yes art is a language and music is a language, even mathematics is a language. History is surely a science. It is even certainly empirical. How else do we know about our evolution or dinosaurs? "Hey, when I dig here I get a lot of bones and broken crockery. I wonder how it got here. Hmmm" -MarSch 17:46, 23 Apr 2005 (UTC)
thread_title: Is Mathematics an art?

message: OK, while I think it's pretty obvious that math is not a science, I guess enough people have argued about it for long enough that I can't just walk in and say what I want.  But I do think something has to be said about it, it does have to be addressed, because it's a (mis)conception that many people may have, and may come to wikipedia wanting to learn more about that idea.  Maybe a heavily qualified statement like: "many people consider that math is not a science for the following reasons, while many others think math should be counted among the sciences for these reasons".  Right? -Lethe | Talk 06:04, July 11, 2005 (UTC)
:The best we have so far is kind of a compromise in the subheading titled "Queen of the Sciences":
::Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions, a phrase first used by Carl Friedrich Gauss. If one considers science to be strictly empirical, then mathematics itself is not a science. That is, mathematical knowledge exists separate from the physical world.
::Mathematics shares much in common with the sciences. Experimentation plays a large role in the formulation of reasonable conjectures, and therefore is not by any means excluded from use by research mathematicians. However, theorems are only accepted if proofs have been found for them.
:Though I must admit the organization of this article is a bit flimsy. Perhaps this could be copied (though not moved) to the "what mathematics is not" section.
:Actually, a lot of people hold the opinion that there shouldn't be a "what mathematics is not" section, based on the arguments above.  I'm beginning to concur.  —Sean κ. + 06:11, 11 July 2005 (UTC)
thread_title:  latecomer to the debate 

message: : Albert Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions, a phrase first used by Carl Friedrich Gauss.  If one considers science to be strictly empirical, then mathematics itself is not a science.  That is, mathematical knowledge exists separate from the physical world.
If mathematical knowledge exists separate from the physical world, the second sentence does not follow, unless it can be shown that knowledge of non-physical things cannot be empirical.  Ontological materialists, of course, would hold that knowledge of non-physical things cannot be empirical, but fundamentally that's because they deny the existence of non-physical things, and they must therefore deny also that mathematical knowledge exists separate from the physical world. --Trovatore 20:59, 17 July 2005 (UTC)
thread_title: "Queen of science" section problematic

message: Here are some reasons to delete the section on "Is mathematics a science?":
* It starts semantic arguments (see above).
* It's about the philosophy of mathematics. Pretty much any answer you give will make one controversial claim or another.
* It distracts from presenting an overview of the subject suitable for a non-expert.
* The article is too long anyway, and there are many more-important things that should be kept instead.
Are there any reasons to keep it?
If there are no strong protests, I'll delete it. --Ben Kovitz 06:33, 2 October 2005 (UTC)
replies: text: Someone will re-open the question. Charles Matthews 12:50, 2 October 2005 (UTC)
thread_title: Proposal: delete this section

message: ===Orionix's proposal===
Practically it is possible to devide mathematics into 15 main divisions: history and foundations, number theory, arithmetic, algebra, analysis, geometry and trigonometry, combinatorics; game theory; numerical analysis; optimization; set theory; probability theory and statistics. Noteworthy is that analysis is the largest branch of mathematics.
A new organization would be very helpful and useful, especially in the advanced areas of physics such as relativity and quantum mechanics. -- Orionix 04:22, 13 Mar 2005 (UTC)
replies: text: Though I feel that the current scheme is a bit whimsical, I'm not sure I would agree with your divisions.  I feel most of these can be combined or split depending on who is reading them.  History and foundations doesn't seem to be a division of mathematics, rather a division of history about mathematics.  I would ask why arithmetic split off from number theory, and why geometry and trig split off from analysis.  And why is analysis the largest division?  Why not group theory or algebra?  I personally find the most useful scheme of divisions to be: algebra, analysis, logic, and other.  --Sean Kelly 07:38, 15 Mar 2005 (UTC)

text: Geometry is a tough one, though.  Classical geometry is clearly a unique field, but it's also very old, and I'm not sure how many geometers are left.  Modern geometry seems to be its bastard child.  Once we abandon the preoccupation with Euclidean geometry, then the study of spaces in general falls under analysis and topology.  Algebraic geometry is really just utilizing what we learned back in classical geometry to solve tough problems in algebra.  So we can't say that geometry is analysis, can't say it's algebra.  Thoughts?  --Sean Kelly
The fields of algebraic geometry and differential geometry are just more advanced from Euclidean geometry. Euclidean geometry forms the basics to all other non-Euclidean geometries, the most important one is Riemannian geometry.  
Now most of the modern 'geometries' are basically part of abstract algebra combined with modern topics in analysis, such as fourier analysis which is widely used in astrophysics and quantum cosmology.
Algebraic geometry has its roots in analytic geometry and differential geometry has its roots in calculus. K-theory (or cohomology theory), groups shemes, lie algebras and noncommutative geometry are also active areas of research, especially in mathematical physics. -- Orionix 16:37, 23 Mar 2005 (UTC)

text: It is really difficult to partition mathematics. Instead all these proposals are coverings. Thus it does no harm to provide several different "orthogonal" coverings. -MarSch 17:53, 23 Apr 2005 (UTC)

text: Here are some suggestions to bridge the period till we discover a theory of everything. I agree that the "topics" section, and also the subsequent "tools" section, is not very informative, and I think they should ideally be removed. Thanks to Charles' excellent work, most of the "topics" section (specifically, all except Theorem & Conjectures and World of Mathematicians) can in my opinion already be deleted. We might need a section on the activity of doing mathematical research, with links as Fields medal and Fermat's last theorem. I think there should also be a bit more on applications. But it is very hard to write an article on the whole of mathematics (which is of course why I am only making suggestions instead of editing the page directly!) -- Jitse Niesen 12:02, 5 Apr 2005 (UTC)
replies: text: I think we should move the topics section to something like list of mathematical topics, perhaps mathematical topics or fields of mathematics. Tools section can be deleted. -MarSch 18:02, 23 Apr 2005 (UTC)
thread_title: How should the "Topics" section be structured?

message: Practically it is possible to devide mathematics into 15 main divisions: history and foundations, number theory, arithmetic, algebra, analysis, geometry and trigonometry, combinatorics; game theory; numerical analysis; optimization; set theory; probability theory and statistics. Noteworthy is that analysis is the largest branch of mathematics.
A new organization would be very helpful and useful, especially in the advanced areas of physics such as relativity and quantum mechanics. -- Orionix 04:22, 13 Mar 2005 (UTC)
replies: text: Though I feel that the current scheme is a bit whimsical, I'm not sure I would agree with your divisions.  I feel most of these can be combined or split depending on who is reading them.  History and foundations doesn't seem to be a division of mathematics, rather a division of history about mathematics.  I would ask why arithmetic split off from number theory, and why geometry and trig split off from analysis.  And why is analysis the largest division?  Why not group theory or algebra?  I personally find the most useful scheme of divisions to be: algebra, analysis, logic, and other.  --Sean Kelly 07:38, 15 Mar 2005 (UTC)

text: Geometry is a tough one, though.  Classical geometry is clearly a unique field, but it's also very old, and I'm not sure how many geometers are left.  Modern geometry seems to be its bastard child.  Once we abandon the preoccupation with Euclidean geometry, then the study of spaces in general falls under analysis and topology.  Algebraic geometry is really just utilizing what we learned back in classical geometry to solve tough problems in algebra.  So we can't say that geometry is analysis, can't say it's algebra.  Thoughts?  --Sean Kelly
The fields of algebraic geometry and differential geometry are just more advanced from Euclidean geometry. Euclidean geometry forms the basics to all other non-Euclidean geometries, the most important one is Riemannian geometry.  
Now most of the modern 'geometries' are basically part of abstract algebra combined with modern topics in analysis, such as fourier analysis which is widely used in astrophysics and quantum cosmology.
Algebraic geometry has its roots in analytic geometry and differential geometry has its roots in calculus. K-theory (or cohomology theory), groups shemes, lie algebras and noncommutative geometry are also active areas of research, especially in mathematical physics. -- Orionix 16:37, 23 Mar 2005 (UTC)
thread_title: Orionix's proposal

message: 1. History, Foundations & Philosophy. 
2. Arithmetic & Number theory.
3. Algebra & Combinatorics. 
4. Geometry & Trigonometry. 
5. Analysis: Calculus & Real analysis, Complex analysis, Differential equations, Theory of functions & Modern analysis. Analysis also includes Numerical analysis & Optimization. Other areas are Global analysis, Constructive analysis & Non-standard analysis.  
6. Probability and statistics
7. Set theory & Logic
8. Modern geometries, Modern algebra and Topology
Another thing of incalculable importance in modern physics is group theory and the study of lie groups, algebraic groups and topological groups. See also quantum gravity and superstring theory 
In the future, mathematics will become more idealized and abstract of its subject matter. --  Orionix 01:16, 4 Apr 2005 (UTC)
replies: text: It is really difficult to partition mathematics. Instead all these proposals are coverings. Thus it does no harm to provide several different "orthogonal" coverings. -MarSch 17:53, 23 Apr 2005 (UTC)
thread_title:  Orionix's other proposal 

message: After two dozen edits, I'm now fairly happy with the page, with the exception of the topics sections. Those seem to be somewhat arbitrary, and certainly not well explained. In fact, a bit old-fashioned in WP terms. What to do with those?
Charles Matthews 10:12, 4 Apr 2005 (UTC)
replies: text: Here are some suggestions to bridge the period till we discover a theory of everything. I agree that the "topics" section, and also the subsequent "tools" section, is not very informative, and I think they should ideally be removed. Thanks to Charles' excellent work, most of the "topics" section (specifically, all except Theorem & Conjectures and World of Mathematicians) can in my opinion already be deleted. We might need a section on the activity of doing mathematical research, with links as Fields medal and Fermat's last theorem. I think there should also be a bit more on applications. But it is very hard to write an article on the whole of mathematics (which is of course why I am only making suggestions instead of editing the page directly!) -- Jitse Niesen 12:02, 5 Apr 2005 (UTC)
replies: text: I think we should move the topics section to something like list of mathematical topics, perhaps mathematical topics or fields of mathematics. Tools section can be deleted. -MarSch 18:02, 23 Apr 2005 (UTC)
thread_title: More discussion

message: Ive been working on a couple templates for organizing basic math stuff. :Template:numbers is being worked on now, but could use some checking/reorganizing. Working on Wikipedia:Access, :Template:Access, :Template:Unsolved - more offline. -SV|t 20:37, 2 May 2005 (UTC)
replies: text: Are you aware of the discussion of the similar :Template:Calculus at Wikipedia talk:WikiProject Mathematics#Template:Calculus -- is that needed?, where some people question its usefulness? I think that the numbers template also takes a lot of room and the reason why it is created is not quite clear to me. But perhaps it is time to discuss the wider issue on how the reader is expected to navigate through all the maths pages. -- Jitse Niesen 21:15, 2 May 2005 (UTC)
thread_title: Templates

message: I've rearranged the article slightly to make the definitions easier to find, but I still don't understand what "the study of abstraction" means. It sounds like it means "study of how to remove unnecessary detail from things" but that doesn't correlate with how I've heard the word "mathematics" being used or anything else in the article (including the formal definition). Brianjd | Why restrict HTML? | 09:33, 2005 May 8 (UTC)
:Definition from Abstraction (mathematics): Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications.. In my view, defining mathematics as "the study of abstraction" is valid, but rather ... errm ... abstract, and so not very helpful in isolation. Gandalf61 13:27, May 8, 2005 (UTC)
::It is relevant in that mathematicians tend to be Platonists, and Platonism accentuates the abstraction from the real world to an ideal level. It's not much good as a definition; it doesn't have much explanatory value, and it is quite possible to be a kind of nominalist mathematician who rejects all that (Haskell Curry comes to mind). It is just not a lot of fun thinking of mathematical reasoning that way. Charles Matthews 13:35, 8 May 2005 (UTC)
thread_title: Dodgy definition

message: I'm still trying to wrap my mind around this paragraph...
:Since the result of mathematics inspired by mathematics is often pure mathematics and thus has no applications outside of mathematics yet, the only value it has is in its aesthetics. Surprisingly often, it has happened that pure mathematics, which was considered only of interest to mathematics, has become applied mathematics because of some new insight, as if it anticipated later needs.
Unfortunately, I don't have time to go through all of MarSch's changes, but IMO a partial revert is in order. —Sean κ. ⇔ 18:17, 18 May 2005 (UTC)
replies: text: I understand that you find my formulation suboptimal. . Perhaps you should take a look at what was in the old version before saying you want to revert. -MarSch 12:35, 19 May 2005 (UTC)
replies: text: While the formulation may indeed well be "suboptimal" in many places, it is not worse than before. However, I do not understand why the section Overview of fields of mathematics was moved to history of mathematics. In fact, I found this the best section in the whole article. As the heading already indicates, it is not about the history, though it does sometimes mention the fields in historical order because that happens to be the most logical order. Therefore, I undid the move. -- Jitse Niesen 23:42, 9 Jun 2005 (UTC)
replies: text: Fine by me. I didn't move it, but merged it, since it was duplicated at history of. --MarSch 09:19, 10 Jun 2005 (UTC)
thread_title:  Wading through the recent changes 

message: Seems to have coined the phrase "queen of the sciences", but he is not a scientist and no Einstein nor Gauss either. Further he says that theology is the queen of sciences. This information does not seem relevant to me, but I also hate to see theology mentioned in this way in this article. What's your POV? --MarSch 12:02, 8 Jun 2005 (UTC)
thread_title:  Thomas Aquinas 

message: I think that all mathematics can be reduced to geometry. We have:
1. Precalculus: Elementary Algebra and Trigonometry ----> Linear Algebra 
2. Calculus & Multivariable Calculus ----> Topology and Differential Geometry
An alternative approach is:
1. Precalculus ----> Linear & Abstract Algebra ----> Algebraic Geometry & Clifford Algebra
 
2. Calculus & Multivariable Calculus ----> ODEs + PDEs ----> Topology & Differential Topology 
3. Real and Complex analysis ----> Functional analysis
-- Orionix 01:06, 9 Jun 2005 (UTC)
replies: text: Where does set theory of any kind fit into this? Brianjd | Why restrict HTML? | 09:33, 17 July 2005 (UTC)
thread_title:  Mathematics and Geometry 

message: Try ranking, in order of what they manage to communicate about mathematics, these pages:
:(A) axiomatic system (B) Bourbaki, (C) :Category:Mathematics, (D) formalism, (E) history of mathematics (F) mathematics, (G) philosophy of mathematics.
An exercise to try to define where effort is needed. Charles Matthews 10:20, 28 September 2005 (UTC)
replies: text: Eigenvalue, eigenvector, and eigenspace
which was COTW but hasn't still reached the FAC state but maybe is close to be.  It needs thorough copyedit and systematic review Vb 16:25, 28 September 2005 (UTC)
thread_title: Time out

message: I note that an anon editor has removed the dab to the hip-hop producer. Now, I didn't like it either. But I do wonder how people will find that article without the notice. (I also wonder why people would want to find it, but that's not my judgment to make.) --Trovatore 18:03, 27 September 2005 (UTC)
replies: text: Well, you can alwas try. I suspect that the AfD will fail; however, this raises an issue about the bar for "notability" in WP.  Recently, CH made a case for what constitutes a  "notable" mathematician; I suspect the bar he set is higher than it is in other areas of WP. (The point being that there are others who are more notable than this producer, and yet they've already been ruled as "not notable enough"). linas 22:38, 27 September 2005 (UTC)

text: For other uses, see mathematics (disambiaguation)
but I am truly not sure if it is worth putting it just for the sake of the hip hop producer. Oleg Alexandrov 00:35, 28 September 2005 (UTC)

text: As a general rule, if there is only one other use of a term, even if it's vastly less common than the main use, then a direct link should appear from the top of the main use's page. If this producer is noteworthy enough to have an article, then the link must be restored. – Smyth\talk 16:42, 28 September 2005 (UTC)
thread_title: [[Mathematics (producer)]]

message: So I'm still not convinced we need a history paragraph in the intro section, but I think I can live with the one that's there. I do think we need to be very disciplined about making sure it doesn't grow. I propose as a rule of thumb that a reader using 800x600 resolution and without perfect eyes (let's say he uses "large fonts" in Windows) should still be able to see the top of the TOC. This is partly as a practical matter to accomodate such a user, and partly because I just think that's about how long intros should be.
As far as history yea or nay, how about a poll? --Trovatore 17:02, 1 October 2005 (UTC)
replies: text: No, I don't believe in having a length rule like that.  Let's just keep searching for ways to make the article better, without predefining what we're going to find.  --Ben Kovitz 12:14, 3 October 2005 (UTC)
replies: text: Listen to Charles, In my experience, he knows (generally) whereof he speaks. If we must have a rule of thumb about length, then the above seems about as good as any. Paul August ☎ 17:59, 3 October 2005 (UTC)
thread_title: History in the intro section

message: Keep or remove the history paragraph above the TOC?
*weak remove --Trovatore 17:02, 1 October 2005 (UTC)
thread_title: Poll-history in intro

message: Leaving aside the content of the introduction, how long should it ideally be? As the length of the current version may change during the poll, please compare to this version. Suggested answers: much shorter, shorter, about the same, longer, much longer, wrong question.
*shorter It isn't massively too long at the moment, but I worry that the length may start creeping up. ---Trovatore 19:25, 1 October 2005 (UTC)
replies: text: No, I don't believe in having a length rule like that.  Let's just keep searching for ways to make the article better, without predefining what we're going to find.  --Ben Kovitz 12:14, 3 October 2005 (UTC)
replies: text: Listen to Charles, In my experience, he knows (generally) whereof he speaks. If we must have a rule of thumb about length, then the above seems about as good as any. Paul August ☎ 17:59, 3 October 2005 (UTC)
thread_title: Poll-length of intro

message: "Even so, in the past it sometimes happened that something which had supposedly been proved turned out to be false."
This sounds good -- appropriately 'umble, hat in hand, no stuck up mathematicians in hear, no suh!  Further, I am certain it must be true, knowing what I know about human falibility.  But I can't think of a single important example.
Newton originally got the "product rule" wrong, but he scratched it out, he didn't publish it.
I can think of lots of cases where a published proof turned out to be incorrect, from Fermat's marginal scribble to the errors in Wyles original paper, plus holes in Euclid's proofs, the incorrect proof of the Dehn Lemma, many incorrect proofs of the Four Color Theorem.
But in every case I can think of, a better proof has come along.
I'm sure one of the mathematicians on wiki will enlighten me with an example of something supposedly proved that turned out to be false. Rick Norwood 21:45, 6 October 2005 (UTC)
thread_title:  false 

message: : for recreation—discovering and cataloging patterns for their beauty, without regard for practical application
Is "recreation" the only way to describe mathematics engaged in the sake of beauty rather than utility?  This might seem a bit like describing philosophy as "discovering and cataloging truths for beauty, without regard for practical application." Moreover, many of the Greeks (and many others, surely) saw mathematics as a kind of divine activitity, and  much of the brilliance of the mathematic field comes from people striving not necessarily towards utility but the beauty and awe of knowledge...certainly not something reducable to "recreation," I venture. --Dpr 01:50, 13 October 2005 (UTC)
replies: text: Recreation (compare with leisure, both in terms of definition and categorisation - it makes no sense) seems to be something done for enjoyment, and that seems to be the correct term here. Brian Jason Drake 05:08, 16 October 2005 (UTC)
replies: text: Yes, well, with that attitude, all of politics, music, art, literature, history, science and computer programming are "recreation".   But that would be a silly over-broad use of the term. Professional math, like professional sports, is not exactly "recreation" even though what football players do in the stadium vaguely resembles what you might do at the beach. linas 19:01, 16 October 2005 (UTC)
replies: text: I think "recreational math" is maybe better defined as math done without regard for application even within mathematics. For example, stuff that works only in base 10. I've modified the passage accordingly; see what you think. --Trovatore 19:27, 16 October 2005 (UTC)

text: I'm going to take a shot at that passage.  It seems clear to me that there is a difference between pure mathematics and recreational mathematics.  Euclid did not write The Elements because he thought it would be of use to carpenters, but neither did he write The Elements just for fun.  Is the Mona Lisa "recrational art"? Rick Norwood 23:10, 16 October 2005 (UTC)
thread_title: Beauty in mathematics

message: I have rewritten the much-disputed intro sentence as follows:
:Mathematics is the field of study which investigates the exact formal relations between numbers and other abstract entities, and the methods of deduction, reasoning and proof based on the relationships between these objects. Applied mathematics is the study of spatial and quantitative relations in practical situations.
This is a loose rephrasing in modern language of the public domain 1913 Webster's Dictionary definition, which, in its original form, is as follows:
:[Mathematics is] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.
I hope this is acceptable: please feel free to improve it. -- The Anome 14:10, 18 October 2005 (UTC)
replies: text: Well, of the basic three things, analysis, algebra, geometry, it slants towards analysis; as one would expect in 1913, given the history. Charles Matthews 10:58, 19 October 2005 (UTC)
thread_title:  Definition of "mathematics"