"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude." (Gottfried Leibniz, [letter to Christiaan Huygens] 1670)
"A problem was posed to me about an island in the city of Königsberg, surrounded by a river spanned by seven bridges, and I was asked whether someone could traverse the separate bridges in a connected walk in such a way that each bridge is crossed only once. I was informed that hitherto no-one had demonstrated the possibility of doing this, or shown that it is impossible. This question is so banal, but seemed to me worthy of attention in that not geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind, with any number of bridges in any arrangement, whether or not such a round trip is possible […]" (Leonard Euler, [letter to Giovanni Marinoni] 1736)
"At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied." (Timothy Gowers, "The two cultures of mathematics", 2000)
"Euler's proof that in Königsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all the bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Königsberg graph had four such nodes, one could not find the desired path." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws [...]. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"The important graphs are the ones where some things are not connected to some other things. When the unenlightened ones try to be profound, they draw endless verbal comparisons between this topic, and that topic, which is like this, which is like that; until their graph is fully connected and also totally useless." (Eliezer S Yudkowsky, "Mysterious Answers to Mysterious Questions", 2007)
"The mathematical structure known as a graph has the valuable feature of helping us to visualize, to analyze, to generalize a situation or problem we may encounter and, in many cases, assisting us to understand it better and possibly find a solution." (Arthur Benjamin, "The fascinating world of graph theory", 2015)
"The theory of graphs is the fundamental study of relations in their purest, non-trivial form: binary connections between abstract points. And as so often in combinatorics, this simple assemblage of trivial objects results in a dazzlingly rich theory of seemingly endless depths." (Felix Reidl, "Structural Sparseness and Complex Networks", 2015)
"The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation." (James J Sylvester)
No comments:
Post a Comment