Platonism and Number

 20 May 2023

Platonism and Number

“The starting-point of our investigation is: can number exist by itself, or must the two be observed in two things and the three likewise?”

(Plotinus, Ennead VI.6, On Numbers, translated by A. H. Armstrong, Plotinus: Ennead VI.6-6.9, Loeb Classical Library, Harvard University Press, Cambridge, Massachusetts, 1988, page 33, ISBN: 9780674995154)

1.  Throughout the history of Platonism there has been a strong connection between mathematics and Platonism.  Mathematicians have tended to be Platonists, though there are exceptions.  

2.  There is a video on youtube by a mathematician/philosopher by the name of Steve Patterson, called ‘Coming Around to Platonism’.  I think it is an interesting presentation about how one mathematician, who used to have anti-Platonist inclinations, decided that Platonism had a better explanatory range than other philosophical theories of numbers and mathematics.

3.  Platonism understands the study of numbers and their relationships, that is to say mathematics, as a means for experiencing the noetic.  Just as the experience of beauty can lead the beholder of beauty to the experience of beauty as such, and beauty as such is a noetic reality, so also the experience of number can lead the beholder of number to the experience of number as such, and number as such is a noetic reality.

4.  Generally speaking, and even though it was beauty that first gave me the experience of the noetic, I suspect that the experience of number has some advantages over beauty for bringing about the experience of the noetic.  First, beauty tends to induce grasping; we want to hold on to that which is beautiful and at times it is very difficult to let a beautiful object go.  This turns the experience of beauty into a primarily material experience with the result that with those kinds of experiences of beauty the fixation on materiality grows stronger.

In contrast, our experience of number does not normally lead to the grasping after number in the way that we often grasp on to beautiful things.  I mean that when we see three trees it does not stimulate in us a desire for three as such.  It may stimulate a desire for the tree(s), but not for ‘three’ as such.

And often our experience of number is abstract and absent specific content.  For example, I might recall that 3 + 2 = 5 without needing to instantiate material objects for the numbers.  In this way I am taking a small step in the direction of the noetic because noetic realities transcend material instantiations.  Numbers are ‘before’ any material instantiation.

As Plotinus writes, “But, if numbers were before beings, they were not beings.  Now number was in being, not as the number of being – for being was still one – but the power of number which had come to exist divided being and made it, so to speak, in labour to give birth to multiplicity.  For number will be either the substance or the actual activity of being, and the absolute living being is number, and Intellect is number.”

(Ibid, page 35.)

“Being, therefore, standing firm in multiplicity was number . . . “

(Ibid, page 37)

5.  The Platonist view of numbers and their relationships is that we discover numbers and their relationships rather than create numbers and their relationships.  We discover these in the realm of being, which is a noetic reality.  (As an aside, notice how Plotinus references the three primary facets of the noetic realm in the quote above which are being, life, and intellect.  Plotinus first mentions ‘the actual activity of being,’ then follows with ‘the absolute living being’, and then concludes that ‘Intellect [or Mind] is number.’  I’ve noticed that this is a pattern when Plotinus refers to the second hypostasis.)

6.  The focus on number in Platonism likely reveals the Pythagorean background of Platonic thought.  I see Platonism’s view of number as a blossoming of a perspective that had been planted centuries before and came to fruition in the Platonic tradition.

7.  The wonderful thing about numbers is that they are a good example of an abstract, non-material reality that people use every day with ease.  Numbers are a good way of explaining to people why Platonists believe that non-material realities actually exist.  It is more difficult to convince people that forms such as the Good, or the One, or the form of the Beautiful exists before material realities; that, for example, beautiful things depend on the Beautiful as such, the form of Beauty.  It is a bit easier, though I’m not saying it is easy, to point to numbers as form-based realities that are instantiated in material objects.  If nothing else, you can point out that many mathematicians have this view.

8.  The attraction of numbers to the mathematician is the attraction of transcendence.  Numbers can, and often do, lift consciousness from the material realm into a more restful and serene domain.

9.  Numbers are also a good example of the power of Platonic forms.  I mean by ‘power’ something like ‘efficacy.’  For example, numbers permeate time and space and are not confined by material borders.

10.  The beauty of numbers in the noetic realm offers the opportunity to go further, to step into the One and the Good itself.  Not very many mathematicians seem to do this.  I think that is because they don’t give much thought to what numbers themselves are dependent upon; that is to say what gives birth to numbers.  The One transcends number by being, analogically, a kind of ‘empty set’ – the set of numbers before any specific number is generated by noetic being.  Again, that’s just an analogy; it is meant to indicate that just as number exists before its material instantiations, the One exists before the instantiations of Being, Life, and Intellect that give rise to number in the noetic.