Steve says: `Since mathematics is a formal system ...' is it? I gave a talk some years ago at the national college of the UK women's institute and the title I used was: Mathematics, a human activity. My point was that mathematics is done by mathematicians (amongst others). Until we find to the contrary, mathematicians are more often than not human (in the widest sense of the word!!!!!). The form and direction of mathematical investigation is determined by curiosity, and similar human emotions., (sometimes also by rivalry, hatred, envy , and other ones of less beauty). A (subjective view) good proof convinces the `reader' that the statement is true. The 'explanation' behind a proof by contradiction explains somewhere along the lines: the result is trapped, it cannot get away, therefore we have it. That is a human judgement and is sometimes accompanied by the sentiment of `but that argument leaves me dissatisfied as I do not see why'. (The level of belief in the use of contradiction is sometimes an issue but not always.) `Explanation' can be modelled by a worldview approach, but then you have the problem of the teaching situation where the teacher gives an explanation of some mathematical result, but has to say that the proof has to take a different route. In category theory, many proofs are transparent and of the form: what do we know about the situation, just one fact, so we have to use that.... it works. (I am thinking of classical Yoneda lemma type situations, since the only elements in hom-sets that we can be sure exist are the identities.) A thorough understanding of the proof does give an explanation of why the result holds. (The problem I have with the original request for examples is that explanation requires understanding of the situation so is dependent on the knowledge of the `codomain'/ reader!) Tim On 25/04/2011 14:17, ClemsonSteve wrote:
Quoted from Jean-Pierre Marquis email: "yes, of course, Salmon is certainly one of the important contributors to the field [scientific explanation]. In mathematics, Paolo Mancosu has been pushing the issue for the past 10 years or so, following the paths of Steiner, Resnik and a few others."
In science, the issue of explanation has been discussed for at least a century. Since mathematics is a formal system and not a physical system, we have to be more careful about what *explanation* means. This makes it a "worldview" problems? As a constructionist/computationalist I would say no constructive, computational proof then there is no explanation. Platonist have their own. Is their explanation useful to me? Don't know because if I can figure a constructive technique out from the plationic technique, I'm good.
steve stevenson clemson
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