Your usage of “imperfect” presupposes “perfect”, which is just another concept, designating some idealized form.
This “idealized form” is the thought object of the proof. Try to conceive of an “ultimate idealized form” instead of the word “greatest” or “perfection”. I have knowledge of the concept of “complete knowledge of all things” or “omniscience” only because I myself lack a complete fullness of knowledge. The concept exists and is understood by means of a negation. It is logically possible that some sort of being has this attribute that I do not have. This concept of “complete knowledge of all things” would also then by necessity include “complete knowledge of the knowledge of all things” and complete knowledge of the being itself that is capable of containing the concept. This “knowledge of” meta-realization presents itself as an infinite string, …ex knowledge of knowledge of knowledge, etc ad infinitum, and therefore must eventually be abbreviated.
When contemplating “existence” and more perfect forms of it, it may be helpful for you as a mathematician to picture a number line. Negative numbers technically do not exist, but we can conceive of them and understand the meaning. Just like there is always a “greater” number on the number line, my own existence is inferior, to the “greatest” mode of existence, which is no where to be found on the number line but is abbreviated with infinity. Because I came into existence or began to exist, I lack a FULLNESS of being, which is what I mean when I say “perfect existence” or “greatest existence”…this fullness and lack of absence. My existence is “greater” than the existence of my dog, in the respect that its duration is longer. It is possible to conceive of an mode of existence that neither began nor will cease. This “infinite existence” is the totality and fullness of being that I lack, and must exist out of necessity wedded to the “ultimate idealized form” that is not found as a finite number on the number line, and cannot not exist also in reality, but not not exist beyond reality and is reality itself since this mode of being must be omnipresent. Such is God.
Coincidentally, I am going on vacation also. Attached in this post is the section from Anselm’s work where the syllogism I posted earlier is derived.
CHAPTER III.
God cannot be conceived not to exist. --God is that, than which nothing greater can be conceived. --That which can be conceived not to exist is not God.
AND it assuredly exists so truly, that it cannot be conceived not to exist. For, it is possible to conceive of a being which cannot be conceived not to exist; and this is greater than one which can be conceived not to exist. Hence, if that, than which nothing greater can be conceived, can be conceived not to exist, it is not that, than which nothing greater can be conceived. But this is an irreconcilable contradiction. There is, then, so truly a being than which nothing greater can be conceived to exist, that it cannot even be conceived not to exist;. and this being you are, O Lord, our God.
So truly, therefore, do you exist, O Lord, my God, that you can not be conceived not to exist; and rightly. For, if a mind could conceive of a being better than you, the creature would rise above the Creator; and this is most absurd. And, indeed, whatever else there is, except you alone, can be conceived not to exist. To you alone, therefore, it belongs to exist more truly than all other beings, and hence in a higher degree than all others. For, whatever else exists does not exist so truly, and hence in a less degree it belongs to it to exist. Why, then, has the fool said in his heart, there is no God (Psalms xiv. 1), since it is so evident, to a rational mind, that you do exist in the highest degree of all? Why, except that he is dull and a fool?
CHAPTER IV.
How the fool has said in his heart what cannot be conceived. --A thing may be conceived in two ways: (1) when the word signifying it is conceived; (2) when the thing itself is understood As far as the word goes, God can be conceived not to exist; in reality he cannot.
BUT how has the fool said in his heart what he could not conceive; or how is it that he could not conceive what he said in his heart? since it is the same to say in the heart, and to conceive.
But, if really, nay, since really, he both conceived, because he said in his heart; and did not say in his heart, because he could not conceive; there is more than one way in which a thing is said in the heart or conceived. For, in one sense, an object is conceived, when the word signifying it is conceived; and in another, when the very entity, which the object is, is understood.
In the former sense, then, God can be conceived not to exist; but in the latter, not at all. For no one who understands what fire and water are can conceive fire to be water, in accordance with the nature of the facts themselves, although this is possible according to the words. So, then, no one who understands what God is can conceive that God does not exist; although he says these words in his heart, either without any or with some foreign, signification. For, God is that than which a greater cannot be conceived. And he who thoroughly understands this, assuredly understands that this being so truly exists, that not even in concept can it be non-existent. Therefore, he who understands that God so exists, cannot conceive that he does not exist.
I thank you, gracious Lord, I thank you; because what I formerly believed by your bounty, I now so understand by your illumination, that if I were unwilling to believe that you do exist, I should not be able not to understand this to be true.