Ask a Thomist

Do Angels Know All Infinite Numbers?

How do certain infinite concepts relate to God and the Angels? An example would be Pi. To my knowledge, Pi is an irrational number with an infinite number of digits. However, it is a ratio of a circle’s diameter and it’s circumference: two natural concepts. Further, to my knowledge, Angels do not need to think and tease out knowledge of natural things. They simply know them in their entirety when they behold them. So my questions are: can an angel know all infinite digits of pi? Do angels know all infinite numbers (e, root 2, etc.)? Does God, being the Supreme Being, know more about Pi than only the digits? How can an angel know an infinite amount of digits and not take on the ‘quality’ of infiniteness? Thank you, – Andrew Fisher

So first, it is helpful to understand what is meant by infinite. In the first part of the Summa, St. Thomas interrogates that concept as it pertains to God. In the process though, he makes some helpful clarifications about infinity in the created world. Specifically, he entertains the question of whether there can be an actually existing infinity. He determines that absolute infinity pertains to God alone. Other things can only be relatively infinite. He speaks about created infinity in terms of magnitude and multitude. He goes on to say that there cannot exist an actually existing infinite magnitude as all created things are circumscribed by their form. For similar reasons, there cannot exist an actually existing multitude. But, St. Thomas describes, we can speak of potentially infinite magnitude and multitude. Extended bodies are potentially divisible ad infinitum. And, to any multitude, we can always add a further unit.

Now, one more clarification. We have to distinguish between different modes of knowledge. Human knowledge passes through the material world and develops in stages. Angelic knowledge is intuitive and immediate, but is still dependent upon a created participation in the divine light. Since they don’t abstract concepts from things like we do, they don’t reason and formulate new concepts in the way that men do. Rather, it’s more proper to say that their knowledge is given to them by God. Now, can they illumine each other and “learn” after a fashion? Yes, but we need to be careful not to just extend our mode of knowledge into the celestial sphere. Finally, God’s knowledge of creation is of an entirely different stripe. His knowledge is both exemplary and creative. Which is to say, God does not learn in any way, shape, or form. His knowledges of things is an implication of his knowledge of himself. In knowing himself, he knows how his nature can be participated in a variety of ways, and he conjoins his will to make what those types or ideas connote.

So, to speak directly to your questions, it seems to me that an angel cannot know all the infinite digits of pi. Those digits are a potential infinity, as described above. For every set of significant digits, a further significant digit can always be added. Thus, this potential infinity cannot be traversed in the strict sense. Angels cannot do the impossible. What the angels can know, and better than we, is the interrelation of the pertinent forms, namely that of radius and circumference. These mathematical notions, provided they are given to the angels to know by God, would be far more transparent to the angelic intellect than to the human.

Finally, with respect to the divine intellect, God himself furnishes the types or ideas of existing things. In order to answer whether God knows pi, we need to inquire whether it is something, or simply the relation of somethings. The latter seems more likely. If that’s the case, then it has a logical or notional existence in the abstract, but a real existence in concrete things only in the category of relation (accidental being). Therefore God knows pi as concreated with the material world from which the mathematical notions are abstracted, but not as a distinct existent. Furthermore, that relation is far more transparent to him as Creator, Provider, and Governor of all things, for he knows everything innermostly. Even the hairs (and interrelations of the hairs) on your head are counted.


Fr. Gregory Pine, O.P. serves as the Assistant Director for Campus Outreach for the Thomistic Institute. Born and raised near Philadelphia, PA, he later attended the Franciscan University of Steubenville, studying mathematics and humanities. Upon graduating, he entered the Dominican Province of St. Joseph in 2010 and was ordained in 2016. “It was St. Thomas Aquinas who first introduced me to the Order, and by his prayers that I grew in knowledge and love of its saving mission and ultimately came to find my happiness in Order of Friars Preachers.”

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