中文摘要：

蔡沉在其《洪范皇极内篇》中象与数对举，建立了一套范数学体系，从象与数的关系来说，象学体系中有数，数学体系中有象，二者不是截然相分的。象学体系与数学体系的区别在于，象学体系是一套以数字二为基数的推衍模式，数学体系是一套以数字三为基数的推衍模式。蔡沉范数学的目的是“因数以明理”，数只是一种明理的工具。数的变化关涉自然界阴阳消长的规律，数还能穷该万事万物，万事万物都能以数的形式进行表达。蔡沉极为重视处于洛书九个数中间位置的数字五，并由此阐发了他由“中”及“皇极”的理学思想。数是一种明理的方式，它穷该万物绝不是对事物的量化；蔡沉的数，哲学的意味重，科学的意味浅。

英文摘要：

Juxtaposing images and numbers, Cai Chen（1167-1230）established a numerological system with his Hongy＇an Huangji Neipian （Inner Chapters of the Supreme Principles That Rule the World Conceived in the Great Plan）. So far as the relationship between images and numbers is concerned, there are numbers in the imagery, and there are images in the numerology, which shows the inseparability of the two. The distinction between these two systems lies in that the imagery is a deductive mode based on the number 2, whereas the numerology is a deductive mode based on the number 3. Cai Chen＇s establishment of numerology aimed to reveal principles by numbers; thus numbers function only as a tool to this end. The changes of numbers are related to the principle of waxing and waning of yin and yang, and can be used to thoroughly comprehend the myriad things; in other words, the myriad things can be expressed in the form of numbers. Cai Chen attached great importance to the number five which is in the central position of the nine numbers in the Luo River Chart, based on which he expounded his Neo Confucian idea from centrality （zhong 中 ） to Supreme Principles （huang ji 皇极） that rule the world. Principles can be expressed in the form of numbers which do not quantify but can thoroughly reveal （the principle of） the myriad things. Cai＇s numbers are more philosophical than scientific.