The limits of science

Imagine a fantasy universe in which both natural and supernatural events take place. There is a set of scientific laws running in the universe, and most events take place as a direct consequence of these scientific laws. Yet at the same time, on rare occasions, there are events which take place without any natural law causing them at all. They happen magically.

Now let’s imagine a group of scientists in this universe who insist that there are no supernatural events. There are merely natural events we do not yet understand. Faced with a supernatural event, they will look for an explanation in terms of natural law.

Suppose that in this universe, Joe is walking to school one day and throws an apple seed on the ground. Suddenly, the seed sprouts and grows into an apple tree in the space of 10 seconds. Will this convince the scientists that supernatural events happen? Surely not.

There are two cases.

First, perhaps the scientists will manage to come up with an explanation for the rapid growth of the tree in terms of natural events. People are clever. We can come up with all sorts of explanations for how things might have happened, whether or not they really happened that way, and certainly there are ways to explain the tree. (For example, we can simply claim that Joe hallucinated or that he fabricated his story.) Obviously, having come up with such an explanation, they will refuse to believe that anything supernatural happened.

Second, the scientists will fail to come up with an explanation in terms of natural events. Will they say, “I guess there are events which cannot be explained naturally”? No, of course not. What they will say is, “We don’t know why this event occurred, but we know things like this don’t happen without a natural cause of some kind. One day, scientists will be able to explain this.”

That’s a little troubling. Apparently, if someone believed that all things are caused by natural law, but lived in a universe in which some supernatural events took place, he would never be able to be persuaded by evidence that he was wrong. Science helps us distinguish between two different naturalistic explanations, and decide which is better, but it seems incapable of even recognizing events that are not the result of natural laws.

There are three reasons that I first became interested in philosophy. One of the three is that I want to understand the meaning and limits of the scientific method. If there is a God and there are miracles (as I believe there are), then what is the proper role of science?

Meno’s paradox

Below is something I just wrote up about Meno’s paradox.

PHL 102 Students: Note how it follows the basic structure your paper should follow: a) argument from the Meno, b) my argument in response (expanding on Meno’s), c) an objection that might be raised to my argument, and d) a response to the objection. With a little more care in the citation and some expansion of the last part of the paper, it’d make a good term paper.

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In Meno, the following exchange occurred:

MENO: And how will you enquire, Socrates, into that which you do not know? What will you put forth as the subject of enquiry? And if you find what you want, how will you ever know that this is the thing which you did not know?

SOCRATES: I know, Meno, what you mean; but just see what a tiresome dispute you are introducing. You argue that a man cannot enquire either about that which he knows, or about that which he does not know; for if he knows, he has no need to enquire; and if not, he cannot; for he does not know the very subject about which he is to enquire.

Meno wonders whether we can successfully enquire about anything if we don’t know what it is. He gives two related reasons for doubt.

Meno’s first argument is:

  1. Before we enquire about anything, we must determine what it is that we are enquiring about.
  2. We cannot determine that which we do not know.
  3. Therefore, until we know what something is, we cannot enquire about it.

Meno’s second argument is:

  1. To successfully enquire about anything, we must both discover it, and recognize that we have discovered it.
  2. We cannot recognize that we have discovered something unless we know what it is.
  3. Therefore, until we know what something is, we cannot successfully enquire about it.

Meno applies these arguments to virtue. If he and Socrates do not know what virtue is, then how can they even begin to enquire about it? And having discovered a definition of virtue, how will they know that it is the correct definition?

Socrates responds with the geometry demonstration. He asks Meno’s slave boy how to double the square. The slave answers in two different ways, but in both cases after Socrates questions him he recognizes that his answer does not work. Then Socrates asks him to consider a third procedure, and the boy recognizes, upon questioning, that this procedure will double the square.

So, as Socrates points out, the slave boy did not know the procedure for doubling the square, but his enquiry into how to do it was successful.

This counterexample seems to refute Meno’s conclusion, but what was wrong with his arguments? Socrates seems to say: Nothing, actually. Meno is right. One must know what doubling the square is in order to discover anything about it. Therefore, the slave boy must have known how to double the square in at least some sense. The boy knew enough to determine the subject of enquiry. He knew enough to reject false answers and recognize the true one. It wasn’t that he started not knowing and ended up knowing; rather, he started out knowing but not knowing that he knew, and ended up knowing that he knew.

Similarly, when it comes to virtue, Socrates hopes that they may discover they already have the essential knowledge of what virtue is within them, if they can only recollect it.

I think there is a simpler way to explain what goes wrong with Meno’s two arguments, which is to distinguish between the knowing a definition and giving an example.

It is possible to know how to do one of these two things without the other. Meno gave examples of virtue, even though he could not define it. The slave boy on the other hand, knew what the definition of “a procedure to double the square” was – he had no difficulty grasping what Socrates was asking him to produce – but he was unable (at first) to discover an example of such a procedure.

Meno’s two arguments blur this distinction. The first premise of the first argument states:

Before we enquire about anything, we must determine what it is that we are enquiring about.

Meno’s point is that we can’t look for something if we don’t know what we are looking for. We can’t look for virtue if we don’t even know what virtue is. We can’t look for a way to double the square if we don’t even know what doubling the square is. Therefore, when he says “enquire about” he actually refers to finding a specific instance of it. We cannot do that unless we know “what it is” – that is, unless we know the definition of it.

Therefore, Meno’s principle would be more precisely stated this way:

Before we seek for an instance of X, we must know the definition of X.

I’m not convinced this is true, but it is at least plausible. It seems reasonable to suggest that before we can look for examples of virtue, we must know the definition of virtue, or that before we can look for a way to double the square we must know what it means to double the square.

Meno’s second argument relies on this principle:

We cannot recognize that we have discovered something unless we know what it is.

Again, the first phrase – to “discover” something – should be interpreted as referring to a specific instance, while “what it is” refers to a definition. Meno is saying we cannot know if we have found an example of virtue if we don’t know what the definition of virtue is. We cannot know if we have found a way to double the square if we don’t know the meaning of doubling the square. Thus the principle is more precisely stated this way:

We cannot recognize that we have discovered an instance of anything unless we know a definition of it.

Again, I am not convinced this is true, but it is at least plausible.

In light of these rephrased arguments, the geometry demonstration makes perfect sense. Meno’s first argument is now:

  1. Before we seek for an instance of X, we must determine a definition of X.
  2. We cannot determine that which we do not know.
  3. Therefore, until we know a definition of X, we cannot seek for an instance of X.

In terms of the geometry demonstration, the first premise means simply that before the boy could seek for a specific way to double the square, he had to understand what Socrates meant by doubling the square – which he did. Therefore, we conclude that he could not seek for a way to double the square until he knew what was meant by “a procedure to double the square”.

Not only is the geometry demonstration not a counter-example to this revised argument, it is actually a nice illustration of it.

Similarly, the second argument is now:

  1. To successfully seek for an instance of X, we must both discover an instance of X, and recognize that we have discovered it.
  2. We cannot recognize that we have discovered an instance of anything unless we know a definition of it.
  3. Therefore, until we know a definition of something, we cannot successfully seek for an instance of it.

In terms of the geometry demonstration, this argument merely points out that the boy, after being led to consider a procedure that actually works, was able to recognize that it was a correct answer. He was able to do this because he already knew what it meant for a procedure to double the square. Again, the demonstration does not provide a counterexample to the argument but rather illustrates it.

What about the virtue case? This is trickier. Whereas Meno’s slave sought a specific instance of a square-doubling procedure, Socrates and Meno were not seeking a specific instance of virtue. They were seeking a definition of it. Yet, as Meno proposed definition after definition, they were able to critique these definitions and see that they were lacking. That is because Socrates knew, and Meno, with Socrates’ guidance, knew what a good definition of virtue would be like.

The key observation is that to know what a good definition of virtue would be like is to know a definition of a definition of virtue.

With this idea in mind, we can see that the two Meno arguments again pose no obstacle.

The arguments imply that Meno and Socrates would have had a hard time seeking for a definition of virtue if they hadn’t known what a good definition would look like. That is why, before looking for the right definition, Socrates had to help Meno understand what was wrong with the old ones. One Meno understood the criteria of an adequate definition, he was ready to seek for one, and to recognize it if were discovered. This is exactly what Meno’s revised arguments predict.

One might be tempted to object that Meno’s arguments still contain a flaw. For they imply that until we have a definition of virtue, we cannot seek for specific virtues or recognize them as such when they are found. Yet both Meno and Socrates agreed that justice was a specific instance of virtue. If they had no definition of virtue, how did they know that justice was one?

Even worse, one of the criteria of a good definition of virtue was that it should cover all the specific cases. How can that be if none of the cases are recognizable until after the definition has been found?

Technically, Meno’s arguments do not say that we cannot already know that something is a virtue without a general definition, but only that we cannot seek and discover new instances. Therefore I propose that we accept, with Meno and Socrates, that there are some virtues that we already know are virtues, without needing a definition. Meno’s question is not to show where all knowledge comes from, but to show how we can gain knowledge we did not already possess.

For both Meno and Socrates, it is possible in theory to start with some knowledge of certain virtues already given, use these cases to work out a general definition of virtue, and then use that definition to identify new, previously unsuspected cases.

Thus, when due attention is paid to the difference between giving an example of something and giving a definition of it, Meno’s paradox is easily resolved.