Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof
Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof
复制标题
无穷小分析的自由变量公理基础：具有有限一致性证明的片段
DOI：
10.2307/2275512
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发表时间：
1995
影响因子：
0.6
通讯作者：
P. Suppes
中科院分区：
文献类型：
--
作者：
R. Chuaqui;P. Suppes
在理论物理的论文或高级教科书中，很明显，数学的使用方式与数学书籍中发现的非常不同。比如，分析方面的书，一方面，和量子力学的经典教科书，比如，Schiff, b[11]，或者最近的书，比如Ryder， b[10]，关于量子场论，没有紧密的联系。理论物理的书并不是用纯数学特有的定义-定理-证明风格来写的，这种差异比这一事实要深刻得多。尽管在物理学的书中证明了许多命题，但几乎无一例外地没有存在证明，因此也就没有真正严肃系统地使用量词。另一个重要特征是无限小的自由使用。事实上，从物理学家的角度来看，大多数结果不会失去任何东西，只要让它们保持近似形式，也就是说，不使用严格的等式或不等式，只使用无穷小的等式或不等式。理论物理学中通常的数学处理方式与任何标准的现代观点中从基础观点出发建立数学的方式之间的差异，提出了一个问题，即是否有可能直接建立一个严格的基础，反映理论物理学中这种标准实践的重要部分。物理学中标准实践的其他部分，例如，使用物理直觉但非严格的论证，在我们的系统中不存在。
In treatises or advanced textbooks on theoretical physics, it is apparent that the way mathematics is used is very different from what is to be found in books of mathematics. There is, for example, no close connection between books on analysis, on the one hand, and any classical textbook in quantum mechanics, for example, Schiff, [11], or quite recent books, for example Ryder, [10], on quantum field theory. The differences run a good deal deeper than the fact that the books on theoretical physics are not written in the definition-theorem-proof style characteristic of pure mathematics. Although a good many propositions are proved in the books on physics, there are almost with exception no existential proofs, and consequently there is no really serious systematic use of quantifiers. Another important characteristic is the free use of infinitesimals. In fact, most results would not lose anything, from a physicist's point of view, by leaving them in approximate form, i.e., instead of strict equalities or inequalities, using equalities or inequalities only up to an infinitesimal. The discrepancy between the way mathematics is ordinarily done in theoretical physics and the way it is built up from a foundational standpoint in any of the standard modern views raises the question of whether it might be possible to construct quite directly a rigorous foundation that reflects a significant part of this standard practice in theoretical physics. Other parts of standard practice in physics, for example, the use of physically intuitive but nonrigorous arguments, are not present in our system.