It's all about the deep questions.
Category Archives: Math and Logic
A familiar problem for the philosophy of mathematics is the ontology of mathematics. One view, platonism, maintains that the things that mathematicians talk about, ultimately sets, actually exist as mind-independent abstract objects. Here’s one problem for the view that I’ve come up with, although being no expert in the field, it could have some conceptual mistake.
1. If platonism is true, there is a fact of the matter as to what cardinalities sets can be.
2. If there is a fact of the matter as to what cardinalities sets can be, we can take the union of one set for every cardinality that there is.
3. The resulting set A must have the same cardinality as one of its subsets A’, since by hypothesis there exists subsets of A with every cardinality.
4. The resulting set A must have at least the same cardinality as the powerset of A’, since the powerset of A’ is a distinct cardinality and there must be a subset of A with the cardinality of the powerset of A’.
5. Therefore, the resulting set A must have a cardinality both equal to and not equal to (greater than) A’.
6. Therefore, platonism is false.
The most well-known argument against the existence of God is the problem of evil, which basically asks why there is so much pointless suffering and evil when the world is really governed by an all loving, all powerful, all knowing being. Most people think this is a problem since God wouldn’t make a world with pointless suffering. However, I think one can go further. Not only does a perfect God have to make a world without pointless suffering, he would also have to make a perfect world. But is a perfect world even possible? Here, I would like to defend two propositions:
1) If God exists, then no possible world would be better than the actual world.
2) There is a possible world that is better than the actual world.
From 1 and 2, it would follow that God doesn’t exist.
Note: Here I am using “better” in a moral sense. For example, a world with a total of 100 people who all were loving Christians would be “better” than a world with a total of 100 people who were all arrogant, sinful atheists where all other factors were constant between the two worlds.
God is supposed to be a perfect being of which no greater can be conceived. But, if God created a world that was worse than some other possible world that he could have created, isn’t it pretty easy to conceive of a greater being? Namely, one who would have actually created that better world! As far as I see it, this premise is hard to deny. God wouldn’t just draw a world from the “all possible worlds” hat and make it the actual world; he would carefully choose which world to create. Furhtermore, if he were asked why he picked that world as opposed to any other world, he would be able to give reasons why he considered the actual world “better” than any other one. And, since he is all-knowing, he would be right in choosing the world that he did. I would think that the vast majority of theists would agree with these observations.
Again, I think this is a fairly uncontroversial premise. In facing the problem of evil, theists usually deny the existence of pointless suffering by countering with various theodicies, or explanations why evil occurs (For example, God permitted the holocaust to let Hitler exercise his free will). To deny this premise, the theist would have to take a much stronger stance, however. Denying this premise is stronger since asserting that this world is perfect among all possible worlds implies that there is no pointless suffering, but asserting that there is no pointless suffering does not entail that this world is perfect among all possible worlds. So, first of all, in support of this premise I would like to drag all of the observations that atheists have made over the centuries in support of the existence of pointless suffering since they would add support to this premise as well. Additionally, however I would like to argue for the impossibility of a perfect world.
In arguing that that there is no such thing as a morally perfect world, it will be useful to introduce a sort of rough-and-ready “scale” of moral goodness. “0” would be a sort of morally neutral world (for example a world that just consists of empty space with no living things and no moral goods or moral evils). The more positive it gets, the better the world, and the more negative it gets, the worse the world. One reasonalbe observation about moral goodness is that it is additive. For example, if a universe X had a rating of 3 and another universe Y had a rating of 3, then if God were to create a multiverse containing universe X and universe Y, the rating of the multiverse would be strictly greater than 3. This assumption that moral goodness is additive is backed by strong intuitions. Which one is better: sponsoring 1 child in Africa or sponsoring 10 children in Africa? From here, the problem is straightforward. No matter what our rating on the scale is, God could always have decided to make our universe a multiverse with one additional universe with a positive rating. The resulting multiverse would be a possible world that is better than the actual world.
*Technical Observation Involving Math
One possible objection would be to claim that the actual world really is a multiverse consisting of infinitely many pocket universes each with a positive rating. This would result in a rating of infinite, and hence no multiverse could have a higher rating. This objection doesn’t succeed however. First of all, only theists who are perfectly comfortable with the actual infinite can make this objection (I’m looking at you Kalam Cosmological Argument supporters). Second of all, there are infinitely many different sizes of infinity, and there is no such thing as an infinity that is larger than all other infinities according to set theory. Therefore, say that the multiverse God created had a number of universes equal to an infinite cardinality X. According to set theory, the powerset of a set with a cardinality of X has a cardinality that is strictly larger. So, this objection fails.
In this paradox Zeno argues that all motion is impossible, which is a pretty substantial conclusion. What kind of reasoning could possibly lead someone to this completely counterintuitive conclusion? Well here is the gist of the story that Zeno spells out:
Imagine Achilles wanted to complete a race, lets say it was one mile long for convenience. It’s clear that if he wants to complete the race, he will eventually have to make it to the half way point. After he finishes the task of traversing the first half mile, he must then reach the half way point between 1/2 a mile and 1 mile. So, he must now finish the task of getting to the 3/4 mile point. However, continuing along this line of thought, Achilles has infinitely many tasks to complete! He has to traverse 1/2 a mile, 3/4 a mile, 7/8 a mile, 15/16 a mile, etc. Each task will also take a finite number of time! Since it is logically impossible to complete a series of infinitely many tasks (each requiring a finite amount of time), Zeno tells us that Achilles can never finish the race. Furthermore, this argument can easily be generalized to anything moving from point A to point B where A and B are distinct.
The Racetrack Argument
1. If something is to move from A to B, where A and B are distinct, then it would have to complete an infinite number of tasks.
2. It is impossible for anything to complete an infinite number of tasks.
3. Therefore, nothing can move from A to B, where A and B are distinct.
So, now we should take a hard look at the premises. Read more of this post