Throughout my life, or more precisely, ever since I’ve attained wisdom, there have been individuals (“Mutants”) who have, through their sheer intellect and brilliance, have managed to influence, impress upon and shape my thoughts . This familiar story of a single individual seeking and marking down a list of men who have provided him inspiration is not new. For example, Alexander Grothendieck (one of the “mutants” in my list), listed his own set of mutants in his Notes pour la Clef des Songes. His list contains eighteen names.
My list contains a modest six.
- Leonardo Da Vinci
- Otto Weininger
- Ludwig Wittgenstien
- Jorge Luis Borges
- Alexander Grothendieck
- Vladimir Voevodsky
These “Mutants” are human beings who are ahead of their time, precursors of a coming “New Age”. They are distinguished by internal freedom, insight into the nature of humanity and by the depth of Platonic genius inherent in their work. Inspection of their lives reveals periods in which each was tortured by their own mind, as if the weight of their genius was unbearable to them. All of these men (possibly with the exception of Da Vinci), at some point in their lives, were struck by melancholy, and learning how they dealt with their melancholia is greatly enlightening.
I myself have the tendency to slip into depression often; and during one of these manic depressive episodes, I happened to recall a line of Borges’, where he talked about writing a poem as “working [his sadness] out of his system and making something out of his experience”. So, I decided that I should do something similar myself: work the sadness out of my system and squeeze out something positive from it. It was then that I played around with an idea in my head, just for fun and to see where it would take me.
The main idea is derived from Jorge Luis Borges’ short story “Tlön, Uqbar and Orbis Tertius“, wherein Borges describes a universe which has completely adopted Berkelyean Idealism without a God. In a nutshell this means that while Berkeley has posited that only minds and mental constructs exist and thus the world exists because it is the mental construct of a God, Borges describes a Berkeleyan universe without a God, so all that exists is only that which people imagine in that particular instant, and the world is a series of such instants.Borges then describes various features of this curious universe, including its grammar, literature and so on.I liked to imagine this universe as an infinitely dark room with people having attached flashlights to their heads. If a person’s flashlight falls on something, it would mean that he is imagining that thing. Structures which are imagined by the people in the room are only as vivid as the amount of light falling on them, and are capable of being extinguished completely if there is no light around them. This is precisely what Borges describes at the very end of the text.
I was interested in the imaginary mathematics of such a universe.
At first, I thought it would it be sensible to say that the imaginary mathematics of such a universe would be equivalent to the Ultrafinitism of our own mathematics.
A few words about Ultrafinitism first. Ultrafinitism is a branch of Constructivism as a Philosophy of Mathematics. Constructivists believe that it is necessary to “find” or “construct” a mathematical object in order to prove it’s existence. So for example, exists because we have
On the other hand, something shown to exist by proof of contradiction is something which the constructivists don’t allow to be labelled “existing”. Because you haven’t explicitly constructed it.
Ultrafinitists take it a step further. Ultrafinitists deny the existence of the set of naturals , because it can never be completed. Here is an example of a conversation (taken from Harvey M. Friedman “Philosophical Problems in Logic”) with a well known Ultrafinitist, Alexander Esenin-Volpin, who sketched a program to prove the consistency of Zermelo-Frankael Set Theory with the Axiom of Choice in Ultrafinite Mathematics.
I have seen some ultrafinitists go so far as to challenge the existence of 2^{100} as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 2^{1}, 2^{2}, 2^{3}, … , 2^{100} do we stop having “Platonistic reality”? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Esenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2^{1} and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2^{2}, and he again said yes, but with a perceptible delay. Then 2^{3}, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2^{100} times as long to answer yes to 2^{100} then he would to answering 2^{1}. There is no way that I could get very far with this.
On the other hand, Intuitionistic Logic and Primitive Recursive Arithmetic are agreed to be foundations for Constructivism and Finitism respectively. The appropriate foundations for Ultrafinite mathematics is still an open question.
Now coming to mathematics in Tlön, Borges actually writes a couple of lines about how the mathematics and geometry of Tlön is. He write that in Tlön, the “very act of counting, changes the number being counted”.
My initial hunch about the equivalence of Tlön arithmetic and Ultrafinite arithmetic was based on the fact that our minds can only “picture” small numbers. We surely cannot picture 2^100 trees without being unsure of the number being pictured. To make this intuition rigorous, I can think of the following steps.
- Pin down axioms for mathematics in Tlön: One can start this by looking at analogues of Peano Arithmetic in Tlön.
- Pin down axioms for Ultrafinite mathematics: This may be a problem because in the preliminary reading that I have done, I have learnt that there are no formal foundations for Ultrafinitism and this is one of the main problems of this field. If not, maybe one can start with axioms for Constructivism (Intuitionistic Logic) or Finitism (Primitive Recursive Arithmetic)
- Show that they are equivalent: Show that the axioms imply each other. A cool way of doing this would be through Coq, the proof assistant.