关于生物学中数学模型的评论

Swarthmore College 生物系的 S. Gilbert 在 Progress in Biophysics and Molecular Biology 发表观点文章[1]，对生物中的数学模型提出 点 caveats， 分别是：

1. Mathematical models are limited by the science known at the time.

2. Mathematical models can tell what can happen, not what does or did happen.

3. Real-world models can provide a better explanation than the mathematical model.

4. In abstracting reality, the things left behind can be very important.

5. Mathematics models can be Platonic rather than evolutionary.

5 点 caveats 有一定道理，对很多数学模型来说，确实有这样的问题，在我们解释或者应用数学模型的结果时是需要注意的。但是如果认为所有的数学模型有这些问题，那就有些以偏概全了。因为这些内容有关于数学模型方面的诸多误解，我在这里一一评述。下面内容都是我个人观点，有不当之处，欢迎指出并进行讨论。

Gilbert 文的摘要中写道：“However, there are certain dangers associated with mathematical modeling and knowledge of these pitfalls should also be part of a biologist’s training in this set of techniques” 如果作者把 mathematical modeling 仅仅看成是 a set of techniques, 我想这是未免是很片面的看法了。数学在研究中的作用从来就不是以一种技术出现的。数学对于科学研究的作用应该主要体现在两个方面：1. 数学体现的是逻辑思维方式(mathematical thinking)，对于生物学（其他学科也是一样）来说，是一种能够帮助人们理解实验事实背后的逻辑关系的一种能力；2. 数学是科学的语言(formulation)，是准确地表达人们对自然规律理解的最佳方式。对数学模型的理解和在生物学中的应用应该要放到这样的框架下去思考才可以看到更加本质的内涵。

1. Critique 1. Mathematical models are limited by the science known at the time.

2. Critique 2. Mathematical models can tell what can happen, not what does or did happen.

3. Critique 3. Real-world models can provide a better explanation than the mathematical model.

Mathematical models are limited by our understanding of the science known at the time.

4. Critique 4In abstracting reality, the things left behind can be very important.

5. Critique 5. Mathematical models can be Platonic rather than evolutionary.

General themes of applied mathematics -- From handwriting of C.C. Lin

1. Gilbert, S. F. Achilles and the tortoise: Some caveats to mathematical modeling in biology. Progress in Biophysics and Molecular Biology 137, 37–45 (2018).

2. HODGKIN, A. L. & HUXLEY, A. F. A quantitative description of membrane current and its application to conduction and excitation in nerve. Bull Math Biol 52, 25–71; discussion 5–23 (1990).

3. Lander, A. D., Nie, Q. & Wan, F. Y. M. Do morphogen gradients arise by diffusion? Dev Cell 2, 785–796 (2002).

4. Morelli, L. G. L., Uriu, K. K., Ares, S. S. & Oates, A. C. A. Computational approaches to developmental patterning. Science 336, 187–191 (2012).

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