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Difference between revisions of "Practice Everyday"
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这也太简洁明了了吧！（比我小练笔简介🙃️）那怎么做到呢？我们需要强大的抽象能力，抓住重点，把其他一切有抛弃掉，就做到了。 | 这也太简洁明了了吧！（比我小练笔简介🙃️）那怎么做到呢？我们需要强大的抽象能力，抓住重点，把其他一切有抛弃掉，就做到了。 | ||
--- Joe version --- | |||
'''具象''' | |||
昨天大美妞给我看了一篇帖子，大意是：po主和的儿子一起学数学，po的儿子问：“负数加上负数，这个意思是欠的更多了，比如我欠你两个苹果就是-2，我再欠你三个苹果就是-3，两个加在一起就是-5，这个我能明白。可是负数乘以负数是什么意思呢？”这个问题把po主难倒了，于是发文求教。一时间好多人参与了讨论，各路神仙、各显神通。 | |||
关于一个概念，比如“负数”，是否需要具像化的去理解，因人而异。正如哲学里讨论的那样：观念源于事物，观念与事物的关系如果是正确的，那么观念就能够得以应用、发展和推广，但是如果观念不正确，就需要回归到事物，重新确立观念。也就是说确认无误的观念，不需要回归至事物层去理解。如果的确想回到事物层，去找寻观念的诞生，哪怕是被人类目前确认无误的观念，最好的办法是通过科学史的方法和视角去理解。那就需要回到“负数”这个概念诞生之初去理解，才能够真正的理解这个概念。哥廷根大学教授Felix Klein出版过的著作“Elementary Mathematics from a Higher Standpoint: Volume I: Arithmetic, Algebra, Analysis”，这套书有三卷，在第一卷里的P27页开始，就对负数这个概念从诞生到发展进行了清晰的介绍，并说明了负数相乘的意义。当然顺便在那里，作者做了如下的描述(P30)： | |||
If we now look critically at the way in which negative numbers are presented in | |||
the schools, we find frequently the error of trying to prove the logical necessity of the rule of signs, corresponding to the above noted efforts of the older mathemati- cians. One frequently likes to present the heuristic derivation of (-b)(-d)=+bd , from the formula (a-b)(c-d), as a proof, completely ignoring the fact that the validity of this formula depends on the inequalities a > b, c > d. Thus the proof is fraudulent, and the psychological consideration which would lead us to the rule by way of the principle of permanence is confounded with logically proving considerations. Of course the pupil, to whom it is thus presented for the first time, cannot possibly comprehend it, but in the end he must nevertheless believe it; and if, as it often happens, the repetition in a higher grade does not supply the corrective, the conviction may become lodged with some students that the whole thing is mysterious, incomprehensible. | |||
在同一页内，作者简单的说明了为什么数学只有发现，没有发明。为什么新概念的建立过程中，纯逻辑只起到规则或规范的作用，而不起唯一的指导原则的作用： | |||
In this connection one cannot repress that often recurring thought that things sometimes seem to be more sensible than human beings. Think of it: one of the greatest advances in mathematics, the introduction of negative numbers and of operations with them, was not created by the conscious logical reflection of an in- dividual. On the contrary, its slow organic growth developed as a result of intensive occupation with things, so that it almost seems as though men had learned from the letters. The rational reflection that one devised here something correct, compatible with strict logic, came at a much later time. And, after all, the function of pure logic, when it comes to setting up new concepts, is only to regulate and never to act as the sole guiding principle; for there will always be, of course, many other conceptual systems which satisfy the single demand of logic, namely, consistency. | |||
Felix Klein没有写所谓的“天才”，但是他含蓄的说：The simple explanation, which was brought out in the nineteenth century only, is that it is idle to talk of the logical necessity of the entire approach, in other words, the rule of signs is not susceptible of proof; rather, one can only be concerned with recognizing the logical permissibility of the rule, and, at the same time, that it is arbitrary, and regulated by considerations of convenience, such as that principle of permanence. | |||
在这里是否可以穿凿附会的理解，利用逻辑，通过对世界的认知，对事物进行抽象，找出或发现其中的规则。这些规则主要是考虑应用的便利性。满足上述条件之后，再进行诸如符号逻辑辅助下的证明。所以一开始就谈论那些规则、原则的逻辑必然性，会让我们失去对这个世界探知的能力，只能在前人发现的世界里活动。 | |||
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 | |||
model: "gpt-3.5-turbo" | |||
binary pattern stage Tenant Bond | |||
Binary Bond Binary Connect Code Meta Key | |||
== Day 71 == | == Day 71 == | ||