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This paper emerges from the concern that the world of mathematics may hold insights of critical relevance to wider society, but that these insights are effectively withheld because of the nature of that knowledge, the values and dynamics of mathematicians, and their preferences in ordering those insights.
The point is often made that mathematics has many highly specialized branches and few of the people associated with any particular one have any interest in other branches or in mathematics as a whole. Pure mathematicians are proud of the irrelevance of their discoveries to wider society -- although ironically it is also the case that it is the US Department of Defense that employs the majority of professional mathematicians in that country.
This paper is therefore necessarily a naive exploration of a vast terrain to discover whether it holds any insightful answers to questions that maybe of critical importance to wider society.
Implications related to this exploration have subsequently been explored separately in:
Mathematicians may be described as being concerned with certain kinds of 'problems' to which they endeavour to discover 'solutions'. Periodically they produce papers that identify 'unsolved problems'. A good point of departure is therefore to understand better what constitutes a mathematical problem -- a problem for mathematicians. Why is it a problem? How does it acquire that status?
A problem for a mathematician seems to have something to do with identification of a relatively complex pattern for which there is no explanation in simpler terms. Problems, like puzzles, conceal the way of seeing a pattern of relationships -- or being certain of that pattern. Mathematicians experience a sense of irritation when faced with such inexplicable patterns -- especially when, from the seeming relative simplicity of the pattern, it appears that an explanation should be easily forthcoming. Like mountaineers, they may then explore the problem 'because it is there'.
To the external observer it then appears that mathematicians select problems that are 'interesting' and offer a chance of being 'soluble'. How are these problems selected? What is 'interesting'?Again to the outside observer, mathematicians seem to select problems in a somewhat unsystematic way, possibly in an area that to which they are attracted. What can be said, in terms meaningful to a mathematician, about the attraction of a mathematician to one area rather than to another?
As with mountaineers the problems are then chosen because they are challenging and/or accessible. Strangely however, once conquered by the first to do so, they remain a challenge to other mathematicians. Like climbing routes, later generations of mathematicians can attempt the same proof -- or pioneer alternative, and better, routes. These routes may be distinguished by the special skills they require or by the brute force nature of the enterprise required for success. As with mountaineers, there may be concern at those mathematicians who favour heavy use of (computer) technology over solutions relying primarily on personal skill. As with stages in team efforts to climb mountains like Everest, major problems may call for a staged array of provisional steps to solve intermediary problems, .
(*** Check web mountaineering mathematics)
As with mountaineering again, the community of mathematicians is fond of associating the names of its pioneers with particular problems or their solutions. Within that community, there is much pressure to be a pioneer and problems may well be chosen because of the fame to which they lead. 'Trivial' problems are disparaged. 'Important' problems are a focus of collective attention. Some are seen as 'too difficult' for present expertise. But even partial success with them may well be appreciated.
The world of mathematics is typically described in terms of its 'branches'. Is there a 'tree' of mathematical knowledge resulting from the explorations described above? Can these explorations be seen as somewhat like mathematicians climbing along particular branches searching for fruit on the tree?
The question that is the focus of this paper is whether this body of knowledge has any structure that emerges from the mathematical insights obtained. Or, alternatively, in its entirety is it only to be understood as a tree -- one of the simplest structures in mathematical terms -- of some value only to librarians of mathematical institutes. To what extent are such librarians acquiring responsibility for the pattern of hyperlinks extending from particular papers, especially to other branches?
Any solution of a problem acquires considerable additional significance to the extent that it opens connections to other branches of mathematics. Such 'connections' are most interesting when they break the tree pattern. The solution becomes a new kind of nexus. But what is the order that then emerges? Presumably, as with citation analysis, this order can be described with the tools of graph theory -- connectedness becomes a measure of importance.
But are there more interesting ways of describing the order of the mathematical universe? Does each 'branch' of mathematics potentially offer insights into alternative orderings of the mathematical universe? In which case with what framework can these alternative orderings most insightfully be related? How is this framework to be described and understood?
As with mountaineering, one of the intriguing features of mathematical problems is that the capacity to solve them can be effectively lost. A pioneer may climb the mountain, or claim to have climbed it, but others may not be immediately able to replicate this or determine what was actually achieved. This is the case with Mallory on Everest and with Fermat's claim with regard to his Last Theorem.
An already solved mathematical problem may be repeatedly presented as exercises to student mathematicians (or those from another branch of mathematics) who may or may not be able to solve it without assistance. If the paper reporting the solution is lost, or the mathematician who understood it dies, the problem may effectively revert to its initial status of being unsolved. It may even disappear from the awareness of the mathematical community (cf the Diophantine ***). A variant of this is experienced in the life of every mathematician when, tragically, they age to the point of losing their skills to solve problems or follow papers reporting on their solution -- including those they themselves pioneered. Like mountaineers, a significant number of mathematicians fall over edges into some form of insanity -- carrying with them insights into what they have explored but been unable to publish.
With the vast numbers of solved problems presented in mathematical papers each year -- far beyond the capacity of any mathematician to digest -- the question is to what extent the body of mathematical knowledge can actually be carried from generation to generation. Whilst solutions may be published, understanding those solutions may be a problem in its own right, whether or not they can be replicated. It is one thing to know that a mountain has been climbed, or even to know the route, it is quite another to be able to follow that route successfully. Even knowledge of the existence of the mountain may also be lost.
Also intriguing is that the proof that a problem is solved may be so complex that it may require an inaccessible level of expertise to validate it (cf the case of of the proof of Fermat's Last Theorem). In the absence of that expertise the proof may be considered worthless by those unable to appreciate it and who question whether the problem may be soluble. Mountains may be climbed without it being possible to prove that they have been climbed, or that they existed, or even that they were worth climbing (cf attitudes to Cantor's work on transfinite numbers)
For a mountaineer, climbing the mountain is the problem which is resolved by reaching the summit -- the solution. Like the mountaineer, the mathematician can usually see the objective -- the summit -- without necessarily knowing at first how to get there from the present level of knowledge. For both, once the objective is achieved a new vista may open up. The relationship of the mountain to other mountains becomes evident, just as the relation of the solution to other solutions becomes evident for a mathematician
The question of whether a problem is rated 'trivial' or 'important' may change over time with the fashions of the mathematical community.
Can the body of mathematical knowledge as a whole be imagined to have any structure, shape or dimensionality -- other than that implied by a branching 'tree' structure? Where 'area' is preferred to 'branch' as the appropriate metaphor, what can be said of the set of such areas, whether as a volume or a terrain -- ranges of mountains??
What might then be some of the questions and criteria to be considered in envisaging this structure?
Is it reasonable to ask how many problems there are in the universe of mathematics? Can anything be said about the number of such problems in relation to the properties of the space in which they are encountered -- or the perspective of the explorer in encountering them? Can problems be usefully thought of as points whose relationship to a contextual array is determined by the solution?
If the problems cannot be understood as mathematical objects, can the solutions? Is the number of problems/solutions constrained or characterized in any way? Does the notion of a 'branch' of mathematics lend itself to any kind of formal definition which might constrain the numbers of problems/solutions to be found within it?
How are 'interesting' or significant problems/solutions to be distinguished in this global ordering -- notably in relation to 'trivial' problems/solutions? What makes a problem fundamental within that framework? Does such importance emerge from characteristic formal properties?
Is there anything characteristic of the way that significant connections emerge between distant problems/solutions? How can these best be represented and understood? Web hyperlinks would be one way to hold these links, irrespective of the ways in which papers are ordered by mathematical librarians. This was done for physicists in the earliest development of document hyperlinking at CERN.
Can the ordering of mathematical knowledge be approached and/or achieved in different ways? Or is there only one sequence through which understanding of it can be achieved? Is there anything useful that might be said about the properties of the global ordering of mathematical knowledge? What are the characteristics of a Theory of Everything?
Does the ordering in any way predict the stages in which its degrees of order can be understood? Expressed otherwise, is it to be expected that the body of mathematics as a whole will be understood differently in 1,000, 10,000, or 100,000 years? Or again, how does it provide for partial glimpses which tantalizingly suggest the existence of an organized whole? Or again, with what partial insights does a mathematical neophyte approach this ordering -- and is this relevant to the process of mathematical education? At what points in this exploration do 'vistas' open up to sustain further exploration by any neophyte?
Suppose there were only say 10 mathematical problems. With what priority would problems be attributed to that number? And if there were only 50, 100, 500, 1,000, 10,000, etc? Can problems be ordered meaningfully in any way?
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