Self Evidence and Self Reference

The notion of self evidence as a justification for knowledge has fallen from its two-and-a-half millennia rule as the beginning point of all science and philosophy to the low point where today it can be casually dismissed as Peter Suber does here:

Self-evidence seems to be a byproduct of culturation or paradigm-influenced perception, not a theory-free anchor by which to judge theories. New discoveries have forced us to unlearn the self-evidence of the commensurability of all numbers, the motionlessness of the earth, the parallel postulate of Euclid, and most recently the naive definition of a set as any collection of any elements.

What I found striking about this list is that although it is very typical of arguments against self evidence, not one of the items in it strongly supports the point. I’ll go over each example except for the one about Euclid which I put in a separate post.

A proposition is said to be self-evident if when the proposition is present to your mind, you can simply tell, by some rational intuition, whether it is true or false. For example, consider the proposition that two parallel lines never intersect. If you have reasonably good geometric intuition, you can visualize two parallel lines, try to imagine what it would look like for them to intersect, and realize that such a thing simply can’t happen.

In fairness to Mr. Suber, let me make three points: first, though three of his four examples involve mathematics, what he may be primarily thinking of, given the first sentence, is the use of self evidence in moral and metaphysical reasoning. Second, the document is unfinished, so he may have changed these examples if he had finished, or justified them better. Third, he follows these examples with another criticism of self-evidence that is more compelling, which I will not answer (please read the document to see it).

My purpose here is not to defend self evidence, but to counter what I see as a very wide-spread and very unconvincing argument against self-evidence in the domain of mathematics and logic. Suber claims that mathematics has mostly dropped the notion of self evidence, but I disagree. Yes, if put on the spot, many mathematicians today will say that they are only following the consequences of their premises, but if you pay attention to what they are doing, you will find that they think their premises are true. For example, practically all mathematicians assume set theory is true, though set theory has no other foundation than self evidence. I think a better way to express today’s mathematical attitude towards self evidence is that mathematicians trust it, but don’t want to bother thinking about it or defending it because they aren’t philosophers.

So, with that:

The commensurability of all numbers

Two numbers, x and y, are said to be commensurable if they are related by a ratio of integers a/b. that is,

x = (a/b)y

Some numbers are not commensurable; in fact, no irrational number such as pi or the square root of 2 is commensurable with any rational number.

Presumably, Mr. Suber’s idea is that it was once considered self-evident that any two numbers are commensurable, but I doubt that it was ever widely considered self-evident that all numbers are commensurable because it was known to be false before the time of Plato, and as far as I know, the first explicit explanation of the idea of self evidence (that we know of) is found in Aristotle, who followed Plato.

But would the Classical Greeks have considered it self-evident that all numbers are commensurable? I doubt it, because they had a very geometric idea of numbers including angles, lengths, areas, and volumes. They had lots of proofs about the relations of these magnitudes; for example: if you have a right triangle, the area of the square that you can make from using hypotenuse as a side of a square is the sum of the areas of the two squares that you can make from the two other sides of the triangle.

If numbers are all commensurable, then this means (for length) that any two lengths you can construct by any process whatsoever, such as the side and the diagonal of a square, or the circumference and the diameter of a circle, can be related by a ratio of integers. How is that self-evident? How can you hold in your mind the notion of every possible way of constructing two lengths from geometric figures, and somehow just see that however you construct them, there is some ratio of integers relating them? If mathematicians ever assumed that all numbers are commensurable, it was just that–an assumption–and not self evidence.

The motionlessness of the earth

Again, I doubt that the motionless of the earth was ever widely viewed as self-evident. The Greeks discussed the possibility that the earth moves, and there were arguments constructed to prove that it does not; you don’t construct arguments for something that is self-evident.

Between Copernicus and Newton, there were many arguments against the motion of the earth including Bible passages, the lack of wind from the earth’s motion, and the lack of observable stellar parallax (that is, if the earth were moving, you would expect to see the stars change position over the year just like your finger seems to change position if you hold it in front of your face and close one eye, then rapidly close the other eye and open the first).

I won’t go into the quality of these arguments, but I will note that they were arguments about whether the earth moves, and since people were giving arguments, that means that they did not consider it self-evident that it does not.

The naive definition of a set as any collection of any elements

Presumably this is a reference to Russel’s Paradox and related antinomies. It is not well-worded, because any collection of any elements does in fact form a set, so let me rephrase this as

(1) For any coherent description, there is a set of objects that satisfy the description.

A coherent description is a noun phrase such as “all the utensils in that drawer” or “all the odd numbers less than 7” which pick out objects in the sense that some objects satisfy the description and others do not. Keep in mind that even if there is no object that satisfies the description, there is still a set of objects that do–the empty set.

Now, (1) is a great candidate for a self-evident proposition. If you have a description–any description at all, so long as it makes sense–then you can imagine an abstract collection of all of the things that satisfy the description. What could be more self-evident than that?

Now consider this description:

(2) All sets that do not have themselves as a member.

Since (2) is a description, then according to (1), there must be a set of all objects that satisfy (2). Let’s call this set S2. Now, is S2 a member of S2? If it is, then it isn’t, and if it isn’t, then it is. This problem is called Russel’s Paradox.

Does Russel’s Paradox show that (1) is false? Eh, kind of, but not really. The problem is that (1) doesn’t say what a set is. If you have a set S, must it be the case that for every object in the universe, there is a determinate answer of whether that object is in the set or not? If so, then that’s another fact that is needed to form a paradox from Russel’s set, so Russel’s Paradox doesn’t show that self-evident statement (1) is false, it shows that the combination of (1) and the determinacy of membership in sets can’t both be true. And actually, it’s worse than that, because we left a lot open about the nature of coherent descriptions.

What Russel’s Paradox really showed was that a particular, rather complex theory of sets proposed by Gottlob Frege was inconsistent–that is, all of his axioms could not be true. I don’t think anyone would ever have claimed that they could hold Frege’s entire theory in their mind at once to check it for self evidence as I described above.

On the other hand, the point of an axiomatic approach such as Frege’s was to produce axioms in small-enough chunks that each of them, taken independently, can be considered self-evident, so I can’t argue, as I did for the previous two examples, that Frege’s failure was not a genuine blow against the concept of self evidence, but it was a somewhat deflected blow, because we already knew about problems of this type from Classical Greece. Here is another proposition that might seem self-evident:

(3) Every coherent declarative sentence is either true or false.

And consider this counter-example:

(4) This sentence is false.

Apparently, (4) is true if it is false and false if it is true. So more than 2,000 years before Frege and Russel, we knew that there are situations involving self-reference that subvert our expectations. Notice that self-reference is a special sort of case; there are other counter-examples to (3) such as

(5) Ice cream tastes good. (subjective)

(6) Violets are blue. (a judgement call that depends on your color palette)

Sentences (5) and (6) also are arguably not true or false, but in a much less surprising way than (4). In the cases of (5) and (6), we would argue that (3) expresses an idealization of truth in which there is nothing subjective or imprecise, just like geometry expresses an idealization of space in which there are no imprecise lengths.

By contrast, (4) is not just pointing out examples that don’t fit the idealization; it is pointing out an example that doesn’t work even within the idealized spirit of (3). It seems to reveal a genuine hole in our ability to evaluate the self evidence of abstract situations: it can seem to us that we can clearly see all possibilities, even though we are not considering possibilities that involve self-reference.

And that is the problem, not that we can’t analyze the situation and tell whether some proposition is true or false from a rational intuition, but that something escaped our notice–we didn’t consider self reference or its effects. Once we consider self reference, our intuition changes as we thing about its implications. What this tells us is not that self evidence is not a reliable guide to truth, but that it is not a full-proof guide. It needs to be handled like any other sort of evidence: it needs to be tested to make sure that we are not missing something; it does not eliminate the possibility of error, but in the end, it is what we necessarily rely on for most of mathematics and logic.

To prove from history that self evidence cannot be a reliable source of knowledge, a few examples, all related to the same basic correctable problem, is not enough; you need a list of several historical examples of propositions that were widely believed, and widely believed because they were self-evident, and were wrong, and were wrong for various unpredictable reasons, not just a few specific reasons that can be corrected for. I don’t think such a list can be constructed for self evidence, but on the other hand it certainly is possible to construct such a list for science–a very extensive list. Given this, it is odd how many philosophers (and I don’t if Suber is among them) have a profound distrust of self evidence along with a profound faith in science.