Talk:Spider diagram
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Link/Citation Needed Re "Several mathematicians have questioned the validity of this area of mathematical research." Reference please? Jaybee 08:53, 28 June 2007 (UTC)
I'm removing "Mathematicians within the Russell Group Universities do not study the body of knowledge associated with Spider Diagrams." since I am doubtful about the purpose of the statement. If it's there to question the validity of the research, this point is already made (and needs a citation). The academic community in the UK (since you're talking about the Russell Group) is wide -- why not make the point that other equally respectable universities *do* study Spider Diagrams? Also, the body of knowledge associated with Spider Diagrams could be argued to involve logical reasoning and the theories of Sets, Functions, groups etc, could it not?
- When I first found this article, there was no citations, and I think I wasn't sure if such a thing wasn't something invented one day. I found a few articles, and added a reference with a web-cite, although better articles exist I'm sure. But I didn't really disagree with the statement you requested a citation for, so I didn't remove it. Even so, I'd say feel free to further stubify the article, at least until someone with a more intimate and cited understanding comes along. Best, Smmurphy(Talk) 16:32, 5 July 2007 (UTC)
[edit] Some Comments
[NOTE : I am not entirely happy with the below comments - but if I did not post them now, I might never do....]
I agree that the statement “Mathematicians within the Russell Group Universities doe not student the body of knowledge associated with Spider Diagrams” is probably not as objectively framed as one would expect a mathematical criticism to be. The issue here is that spider diagrams can be widely pinpointed as plainly just not being mathematical in a sense which I attempt to outline below. Of course, there is no objective proof of this (beyond the fact that the subject might satisfy the criterion of Fringe Mathematics, if such a definition exists at this time – where Fringe refers to the number of researchers who focus on spider diagrams. ).
There are, however, vast swades of mathematical research and widely accepted mathematical material whose application to everyday life and everyday technology show that said mathematical research is naturally applicable - it arises in a scientifically economical process of problem solving. Equally well, such mathematical subjects/branches arise due to the fact that it is plain impossible to solve real world problems without resorting to the language and terminology of those mathematical subjects. This is in contrast to the descriptive capability of spider diagrams which cannot be economically utilised to describe even basic mathematical results (how would I use spider diagrams to state that 1+1=2?, or even the basic property of induction?*).
For example, linear algebra (which relates to fields and the linear maps which are isomorphic to matrices over those fields) can be found throughout nature – either for representing things like co-ordinate transforms, or within optimisation (say). It would be nearly impossible to understand vector quantities and their evolution in time without linear algebra – meaning that this branch of mathematics could not be said to a an unnatural aberration, but rather an integral linguistic tool for speaking about both mathematical and physical nature (which might be outside the purview of pure mathematics – though even the pure mathematician finds that their work is applicable to physical reality at times). Linear algebra could not be argued to be an unnatural artefact for conceptualising the world – as so many people would intuitively use it even if they had not be taught basic linear algebra concepts (a truth which could be easily verified by asking school children to write down the co-ordinates of an arbitrary point after rotation by a certain angle).
Of course, I have chosen one example (linear algebra). I could have chosen MANY others. These include geometry (without which it would not be possible to phrase geometrical problems and theories – let alone solve them/prove them), algebra (I've already mentioned linear algebra which I basically think of as “working with linear maps”, but group theory and so on deal with algebra), differential equations (which are important to predicting the behaviour of dynamical systems), galois theory (without which, it would be impossible to prove that the general quintic is insolvable), number theory (which finds application to computing via counting – that is, combinatorics – as well as searching algorithms used in computers and the like). The last subject seems to me to be the most important indicator of mathematics – number theory, even if it had NO application to everyday reality, would still be worthy of pursuing for mathematical reasons alone as it displays some kind of 'beauty' to mathematicians. Theorems such as Fermat's little theorem or the proof that there are infinitely many primes, are easy to prove due to their “natural foundations” or intuitive beauty – and, in fact, such theorems even have application to everyday reality (the distribution of primes is important in some very interesting computer algorithms, and the notion of mathematical induction crops up everywhere in mathematics as a tool for proving results and making definitions – the term “inductive definition” crops up often in mathematics).
Of course, supposing that number theory has no application to everyday reality hides the fact that any language (“formal system”) capable of expressing arithmetic is also capable of expressing large bodies of mathematics (for example, the use of induction is usually first comprehended as being an argument used when dealing with natural numbers – though it occurs as a common argument throughout the whole of mathematics, including, say, linear algebra).
Spider diagrams do not seem to sit well with any of this (as I hope to describe systematically below).
So – what was the point of the above diatribe? Well, spider diagrams do not seem to pass the following tests that are required for being called “good mathematics” (yes, there was a paper by Terence Tao which discusses what good mathematics is – and it is probably worth referring to at some point, but anyhow...) :
1)Problem Solving. “Good mathematics” provides the linguistic and problem-solving (computational?) tools for solving problems which would otherwise be impossible (or very hard) to solve without using the tools provided by that “Good mathematics”. Spider diagrams do not seem to have been created to solve any particular problems – they do not naturally lend themselves to satisfying any kind of need (linear algebra is necessary to understand vector quantities like force and displacement – but most mathematicians and physicists would be hard pressed to point out spider diagram-like quantities). 2)Some type of universality. I mentioned above that, even if a sufficiently intelligent person had NOT been taught anything about linear algebra – they would still end up using linear algebra terms/ideas and might create their own terminology and language to speak about things like “isomorphisms” and “endomorphism”, “linear maps” and “determinants”, etc... But it would be clear (probably from their use of diagrams) that they were speaking about the same concepts in the same way as someone who had been taught such results. That is, mathematical ideas occur naturally and universally, even to individuals who do not come from a mathematical background. This is perhaps (statistically) the strongest argument – ideas (and, in fact, whole families of theorems and proofs) which it would be statistically improbable for others to generate to solve a particular problem would in fact be generated in order to address particular mathematical issues.
Spider diagrams do not seem to have occurred naturally to other mathematicians (but then, neither did the proof to Fermat's Last Theorem – so this does not constitute a valid point in and of itself). Spider diagrams are probably not universal (due to the fact that they have not been created to address a particular mathematical problem – spider diagrams would likely not be reproduced by others to diagrammatically represent information as seems to be indicated in the External Links section of the article).
3)Economy. Of course, I will let others determine the issue of the actual economics of spider diagrams. Here, I do not refer to economy of wealth – but rather the economy of information. There is a school of thought which would describe the habits of mathematicians and physicists as nothing more than “enhanced information compression/processing devices”. Mathematicians and physicists seek either to solve mathematical problems or phrase theories in a way which captures all of the information concerning a given system but in such a way that the information is so well structured that only a handful of physical laws (for physicists) or mathematical axioms (for mathematicians) is required to describe the behaviour of the whole system/structure/branch. Thus, the reason behind why so much of mathematics and physics find application to so many different areas of life is that they can compress into textbooks the workings of profoundly complex systems (like the world in which we live, or the theories needed to solve the problems we which to solve). Mathematics and Physics succinctly describe reality not only qualitatively – but, most convincingly of all, quantitatively as well. And the level of succinctness to faithful qualitative and quantitative description is mind-bogglingly profound – this “information compressing” capability of mathematics in applied hands (physics) enables scientists to explain and then PREDICT the behaviour of the world in which we live.
Spider diagrams do not possess this information economy – there is no problem I ever met in mathematics that could be more succinctly described using spider diagrams than by using an already well-established branch of mathematics. Spider diagrams do not seem to add anything 'new' and provide no intuitive insight to the solution of mathematical problems (the article does not indicate how spider diagrams would aid problem solving – one of the major goals of mathematics). If an article is linked which can counter this point, I would be interested in it.
4)Harmony (arguably a certain type of consistency, but I will not get into that here...). Mathematical subjects exist in harmony with one another – and there is often some synergy/synthesis of idea which occurs from knowing of several branches of mathematics. For example, Ideas within group theory can find application to polynomial computation (I have already mentioned that Galois Theory states that the general quintic is insolvable). Searching for last year's email on your hard disk requires that some poor soul implement an efficient combinatorial search algorithm which relies on some abstracted number theoretical result (showing how computer science links to combinatorics/number theory – obviously this is a bad example). Liouville's theorem (the phase space one) which states that the velocity field in phase space can be thought of as an incompressible fluid/other fluid phenomena (providing a link between a general result about Hamiltonian dynamics and fluid dynamics – showing how phase space can be thought of as being fluidic, in a suitable sense).
The point here is that many branches of mathematics exist in harmony with one another – results in group theory can be applied to geometry, and ideas within Partial Differential Equations theory can be applied to Dynamical Systems. Probabilistic and Statistical ideas occur in Statistical Physics and so on. The point here is that many mathematical ideas which may have once been derided have proven their worth through their applied utility or through their predictive power within science through their ability to fit in with other branches of mathematics/physics (which is what the mentioned examples are meant to signify). It often appears to a mathematician that it is possible to view a particular problem within one branch of mathematics as being (for example) a specific instance of another somehow equivalent problem within a different branch of mathematics. Such “intuitive insights” show how large bodies of mathematical work sit in harmony with one another (problems can be translated from the language of one branch into another).
In fact, that such ideas could be so well integrated with one another, and find application within each other's fields by chance alone is so astronomically improbable that one would be right to state that there must be some underlying structure (which, in some sense, is physical) from which mathematical insights are obtained. But this latter observation would seem to be quite a deviation from facts, etc...
However, it seems clear to me that spider diagrams are “profoundly unnatural” – there are no real problems of mathematical interest (which I can think of) that they solve! How would one state the riemann zeta hypothesis using spider diagrams? What about Sylow's theorems? The latter 2 ideas are “real” ideas in the sense that they could (in a realistic way) relate to problems of practical interest within physics, etc... However, spider diagrams do not seem to harmonise well with other branches of mathematics. If they did, surely this should be stated clearly in the article - ? If spider diagrams do harmonise well with other branches of mathematics – how do they do so? If there were some way of showing how spider diagrams express non-trivial results in combinatorics/mathematical logic (say) – then this should be indicated and technical examples should be provided.
I accept that the criticism that spider diagrams seem to be bad mathematics because they are not taught in Russell Group universities is an unworthy one – but one which outlines some cause/effect relationships (individuals interested in "good" academic work, might focus on 'traditional' and well proven maths!). However, I could (and should if I find time) explain/outline how large bodies of mathematical theory satisfy the various above criteria in a way which spider diagrams do not – hopefully anyone who gets round to reading this will find such an observation so profoundly obvious that they will be gob smacked that I felt the need to post up this text.
Some individuals might state that mathematical results such as Sylow's theorem satisfy the above criteria in a way which is, frankly, more illuminating than the way in which spider diagrams do so. I have tried avoiding such arguments, and (instead) going for some of the qualitative properties mentioned within Terence Tao's “what is Good Mathematics” article (which can be found on Arxhiv). Note that merely providing these qualitative descriptions (though providing plenty in the way of criticism) does not go far enough – ideally, I would provide examples of mathematical theorems and the complexity associated with such theorems – showing in some quantitative way why those ideas are more beautiful/complex and adept to problem solving than are spider diagrams. Of course, in a sense, all of the above is obvious – though stating that spider diagrams are “obviously” not useful to mathematical investigation would not have provided a sufficiently proper argument against the utility of spider diagrams for the purposes of wikipedia.
Hopefully these comments don't ruffle any feathers – and hopefully there will be some out there (aside from, perhaps, the researchers who study spider diagrams) who will be thinking carefully about whether spider diagrams are good research. It appears novel – but is it of any use? The answer to this last question may very well depend upon who is answering.
Also note, my arguments may not be critical of spider diagrams themselves (or of pure mathematics/mathematics without application in general where I have criticised on grounds of applicability), if the article can introduce or address those points which have brought up above (which, incidentally, would lead to a better article).
P.S - Apologies for the long comment.
- - Also note that the article does not seem to state that spider diagrams have as their purpose "visual reasoning" (if any purpose at all). Perhaps the article should describe why so few individuals in the mathematical community might know of >this< particular type of spider diagram (rather than the whole "mind-mapping" type). I don't think that I ever met a *mathematically useful* system of reasoning which would not allow for it to be possible to state that 1+1=2.
Spideee diagrams wiki (talk) 01:19, 8 February 2008 (UTC)
