What do mathematicians teach us about the World--论文代写范文精选

他们经常晚上醒来解决问题,可在早上发现不是一个合适的解决方案，这是令人沮丧的。他们常常让枯竭的精神努力追求数学研究。他们喜欢它,他们认为这是一个领域,可以做的很出色。下面的paper代写范文进行详述。

Abstract
Aristotle was a thorough-paced scientific man such as we see nowadays, except for this, that he ranged over all knowledge. As a man of scientific instinct, he classed metaphysics, in which I doubt not he included logic, as a matter of course, among the sciences, - sciences in our sense, I mean, what he called theoretical sciences, - along with Mathematics and Natural Science, - natural science embracing what we call the Physical Sciences and the Psychical Sciences, generally. This theoretical science was for him one thing, animated by one spirit and having knowledge of theory as its ultimate end and aim  (Peirce 1992 [1898] : 107)

I did not train as a mathematician, I trained as a Social Scientist. I had chosen however Mathematics as my main subject when at the Athénée , the equivalent of High School in Belgium, the country where I was born and where I was raised up to graduate level. At the Free University of Brussels, I learnt mathematics for economics as part of the curriculum for sociology undergraduates. As a Graduate student I had the privilege of being one of Georges Théophile Guilbaud's students at his seminar called Mathematics for Social Scientists hosted by the Ecole des Hautes Etudes en Sciences Sociales in Paris. Next followed many years of conversations and joint informal work with Professor Sir Edmund Leach, at Cambridge University, a small portion of which has recently found its way to the publisher (Jorion 1993). Then I went on learning mathematics on the job, first as an anthropologist, then as an Artificial Intelligence researcher and more recently in finance. This makes me a long-term Applied Mathematician which - I have realized over the years - is a profession very different from Pure Mathematician.

Let me tell you how I see these two experiences as being different. In my conversations with Pure Mathematician friends I have discovered people who are performing a demanding task which despite the intellectual hardship they very much like and enjoy. Mathematicians tell you about the torment of trying to establish something like demonstrating a theorem or designing a complex mathematical object, and often failing to do so after weeks, months or even years of effort. They tell you about being woken up at night by the solution to a problem which may depressingly turn out in the morning not to be a proper solution after all. They mention being often left drained by the mental effort which the pursuit of mathematical research requires. But despite the hardships they are on the whole happy with mathematics : they like it and they believe it is a field which is doing a good job in the world of science and in the world at large.

My own experience with mathematics - essentially as a customer of mathematics which have been produced by others - is of a different nature. No doubt what justifies the honor I am being awarded of being here today at University of California, Irvine has got much more to do with the circumstances when I have been a satisfied customer of mathematics than when I have been a frustrated one - the latter not leading to any noteworthy conclusion ! I have been very fortunate some twenty-eight years ago to encounter permutation groups as part of a full-bodied theory and being able then to find original ways for applying permutation groups to the systematic exploration of genealogies, under the guidance first of Guilbaud, then of Leach. At times in my dealings with mathematics I have managed to do somewhat better than being a passive consumer. This was possible when the question I was trying to address was close enough to some mathematics I was familiar with that I could customize to my own demands an existing object such as the dual of a directed graph. But on the whole I have been very much a frustrated customer of mathematics who hardly ever found in the mathematical toolbox what he was looking for. This frustration has led me over the years to ponder about the production of mathematics by mathematicians as a part of our culture, and in this quality can be studied in an anthropological perspective. Today I shall try, from an anthropological point of view, to shed some light on the question What can mathematics tell us about the world ? 

Conclusion
I have examined the activity of mathematicians in what I claimed was an anthropological perspective. By the latter I meant that, to rephrase Malinowski's apocryphal words, I was more interested in what mathematicians do, than in what they claim they do. Also, as opposed to the philosopher who ponders on mathematics' foundational issues, epitomized in the distinction between realist platonism and anti-realist constructivism, the focus of my attention has been, in metaphorical terms, practice rather than dogma or liturgy. My unmentioned assumption has been that examining the task effectively performed may reveal an agenda which may not be professed either by the orthodox mathematician, or by the heterodox.

Freud's metapsychology introduced the methodological principle that actors are poor judges of their own motives. Derrida's deconstructivism imported the same principle at the cultural level for cultural actors. I have attempted to show - in the brief time which a lecture allows - that, independently of their own representation of their task, mathematicians produce in actuality a virtual physics.

I have proceeded is the following way. I have introduced the principles of demonstrative proof as described and assessed by Aristotle. Modern authors could not have been cited instead as however accurate their cataloguing of methods of proof, they always refrain from grading these methods . Thus was shown the latitude in demonstrative methodology open to mathematicians, being able to resort to modes of proof ranging from the compelling to the poor.

Then I have shown that even such leeway in the matter of proof has been felt at times as an intolerable constraint. The proof by reductio ad absurdum, more appropriately called by its traditional name of per impossibile, was shown to be by-passable and effectively by-passed by mathematicians. The arising of an impossible conclusion used to signal a flaw on the path leading to it. An epistemological coup allowing the by-passing of such impossibilities consisted in defining the demonstrative process as unassailable, and shifting the property of impossibility into a positive attribute of the conclusion : undecidability, although distinct from impossibility, belongs to this type.

This would be starting a different subject, but a similar practice has characterized the development of the part of physics known as quantum mechanics. A particle is here or there. Sometimes it is possible to say whether it is here or there, sometimes it is not. You can blame your ignorance : there is something inadequate in your method for locating a particle. Or, you can say - as has been said - that it all lies in the nature of things themselves : particles have a third possible state of being neither here or there or of being simultaneously here and there.

How can you tell which view is justified ? You cannot. The only thing which is clear is that if you blame your ignorance you are currently stuck in your research, and this for an unknown length of time. On the contrary if you assume that having an indeterminate location is one potential attribute of particles, you are entitled to move on. The peril lying here though is that, one day, the mathematical object allowing to envisage once again particles in terms of their being exclusively here or there may be forthcoming. In which case any theorizing in the meantime on the basis of indeterminacy will turn out to have been building a house of cards.

I have quoted Morris Kline as saying both that the calculus was the most original and most fruitful concept in all of mathematics and that it had been plagued by its lack of mathematical rigor. The reason for this, we have seen, is that the world in its very build forced the calculus to be what it became.

The mathematician enters the world of mathematics armed with his intuition of how the world at large operates. This he imports within mathematics and, quite automatically, designs mathematical objects with an in-built virtually physical plausibility. The culture around him is impatient with mathematics which do not find their way to providing models. A double system of constraints, both inner and outer, contribute at making mathematics a virtual physics. The price to be paid is often high : unjustified discarding of terms is a sore, cancellation of errors is putrescence.

Sometimes the mathematician needs to bear blinkers, sometimes even a blindfold, sometimes he needs to grab an iron from the fire and blind himself purposely. This is the burden the universe imposes on mathematical pursuit. The anthropologist puts the mathematician under his anthropological microscope, and scrutinizing his works exclaims : What extraordinary achievement !

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