正是数学的抽象性使它大有用处。
问题一开始出在数学的抽象教学上。
It often starts with the problem of teaching math in the abstract.
他断言，在理解冗长且完全抽象的一系列观点方面，他的能力非常有限，因此他深信自己在数学上永远不会成功。
He asserted that his power to follow a long and purely abstract train of thought was very limited, for which reason he felt certain that he never could have succeeded with mathematics.
长期以来，舍伦贝格一直质疑科学研究所支持的这一说法：音乐教育提高儿童的抽象推理、数学或语言能力。
Schellenberg had long been skeptical of the science supporting claims that music education enhances children's abstract reasoning, math, or language skills.
显然，这些是一些抽象的数学概念。
Obviously, this is an abstraction of these mathematical concepts.
由于已经习惯于使用我们熟悉的数学抽象进行工作，我们常常会忘记这一点。
Being so familiar with the mathematical abstractions we are used working with, we tend to forget this.
不过涉及到抽象数学量的计算时，可能超出这些值的范围。
However, when you do calculations on abstract mathematical quantities, it is possible to exceed these values.
数学研究数和空间的抽象对象。
Mathematics is concerned with the abstractions of time and space.
如果数学思想在某处等待着被发现，那么即使人类从未构想，一个纯粹抽象的概念也必须存在。
If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it.
他建议，比如将孩子们集合到一个团队中就某一问题展开辩论，可以让他们更轻松地掌握数学或哲学中的抽象问题。
He suggests, for example, that children may have an easier time learning abstract topics in mathematics or physics if they are put into a group and allowed to reason through a problem together.
另外一些——比如说宇宙学——则对远离日常生活的一些事情做出假设，然后通过很高级的数学技巧加以抽象和理想化。
Others — cosmology, for example — make hypotheses that are more remote from everyday causes, and involve a high level of mathematical abstraction and idealization.
数学上的抽象还是感性的东西，虽然是没有特定存在的抽象的感性东西。
But these are still the object of sense, although the sensible is abstract and idealised.
不管数学的任一分支是多么抽象，总有一天会应用在这实际世界上。
No matter how any branch of mathematics abstraction, will one day be used in the actual world.
我虽然不会计算简单的算术题，但是通过她，我感到抽象的数学与音乐非常地相似。
I could not do simple arithmetic but through her I sensed that abstract mathematics was very much like music.
因此，人类学中的“文化”的概念，就像数学中“集”的概念一样，是一个抽象概念，它使得大量的具体研究和认识成为可能。
Thus，the anthropological concept of “culture” ，like the concept of “set” in mathematics，is an abstract concept which makes possible immense amounts of concrete research and understanding。
Derive是使用Lisp语言编写的，Lisp是一种公共编程语言，尤其适合基于规则的处理，该处理是将一个抽象数学表达式变换为另一个表达式所必需的。
Derive is written in Lisp, a common programming language that is especially suited for the kind of rule-based processing necessary for transforming one abstract mathematical expression into another.
数学涉及对抽象概念的理解。
Mathematics is concerned with understanding abstract concepts.
数学既是一个知识体系，又是一个理性的思辩体系，它是经过高度抽象和严格的逻辑推理形成的系统化理论知识总和。
Maths is not only a knowledge system but also a rational thinking and argument system. It is a systematism theory knowledge summation by height abstract and strict logic ratiocination.
这些数学对象是抽象的，不易掌握的外行人士。
These mathematical subjects are abstract, not easy to grasp for uninitiated persons.
这并不是一个证明，这里不存在一个抽象的证明，你不可以同证明一个逻辑上的或数学上的声明一样证明月亮不是绿色奶酪做的。
This is not a proof, there is no metaphysical proof, like you can proof a statement in logic or math that the moon is not made of green cheese.
因此，人类学中“文化”概念就像数学中“集”的概念一样，是一个抽象概念，它使大量的具体研究和认识成为可能。
Thus, the anthropological concept of"culture, "like the concept of"set"in mathematics, is an abstract concept which makes possible immense amounts of concrete research and understanding.
数学是一项工具，特别适合于处理任何一类抽象概念，而且，它在这方面的作用是无止境的。
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
数学是搞抽象概念的，而抽象概念则来自实物的属性。
Mathematics deals with abstract concepts, which are derived from properties of physical bodies.
更重要的是，直到约翰海因里希冯屠能的离体国家在1826年，经济学家没有制订明确的和抽象的行为模式，以应用数学工具。
More importantly, until Johann Heinrich von Thünen's The Isolated State in 1826, economists did not develop explicit and abstract models for behavior in order to apply the tools of mathematics.
更重要的是，直到约翰海因里希冯屠能的离体国家在1826年，经济学家没有制订明确的和抽象的行为模式，以应用数学工具。
More importantly, until Johann Heinrich von Thünen's The Isolated State in 1826, economists did not develop explicit and abstract models for behavior in order to apply the tools of mathematics.
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