I was teaching a logic class yesterday and the subject of infinity came up. It’s a problem, whenever this happens, because I still don’t have a good definition for infinity. And this is a problem because you can’t debate something if you can’t define it. If we’re debating how many pieces of fruit are on a plate, and you don’t think tomatoes are a fruit, but I do, who is right? Well, we’re both wrong until we agree on a definition for “fruit” (whether to use the scientific or culinary one).

In the same way then, to discuss infinite logically, we have to agree on a definition. Sometimes infinity is just thought to mean a really big number–like a googolplex, but bigger. Mathematically, it is handled as an abstraction (not representative of a real number) and can be divided in different ways. For example, infinity can be either countable or uncountable. Infinity is uncountable if its cardinal number is larger than that of the set of all natural numbers (if it includes negative integers, for example).

This is a useful concept, because in Tibetan Buddhism infinity seems to have different meanings. It is said for example, that living beings are infinite. But living peoples are things, and things can be counted, which would mean that we are talking about a real number (just very big). However, time is also infinite, but can be thought of in discrete (moments) or indiscrete (a non-broken line made up of infinitesimally small points) terms. Time can also be thought of in non-natural terms (the past as negative moments in time going backwards). So what are we to make of the seven step cause and effect method, which tells us that everyone has been our mother, because we have all been reborn over and over through infinite time? Because theoretically, if there are an infinite number of beings, and in each life you have one mother, then infinity minus one still equals infinity. No amount of infinite time will allow you to have been born of an infinite number of beings; in each life into which you have been born there will still be an infinity of beings who have not been your mother.

Mathematics doesn’t give us much help here, the solution to infinity (beings) minus infinity (time) is indeterminate: it could be infinity, zero, or negative infinity (here’s an equation proving the answer is -2), because infinity is defined as a concept and not a number. (I found a nice analogy: what is the value of an apple divided by a cat?)

This is also a problem with regard to getting out of sansara (suffering life). We’re told that everyone gets out of sansara eventually, given infinite time. But logically, this doesn’t make sense: there are an infinite number of sansaric results (rebirth as a hell being, hungry ghost, animal, human, or in a god realm) so again and infinite number of possible results means you could keep spinning in sansara infinitely (one possible result of infinity minus infinity is still infinity).

So let’s look at the first problem as a syllogism:

Consider every living being,

They have been my mother,

Because I have had infinite lives in the past.

True or false? Run the tests. Test #1: is it true that I have had infinite lives in the past in which to experience every living being? Buddhism would say yes. Test #2: if I have had infinite past lives, does that mean that I must have been everyone’s mother?

So I would argue no here, for the reasons above. An infinite amount of past lives would still leave an infinite amount of living beings who have not been your mother. Unless “infinite” here just means a really big number, in which case infinite time could be an infinitely larger number than the infinite number of living beings. In which case all living beings have been your mother an infinite number of times. But the converse could also be true: if there are infinitely more living beings than the infinite amount of your past lives then there will always be an infinite number of people who have not been your mother.

So I’m going to say this syllogism is false. What do you think?