Are all self-evident truths necessarily redundant?

Question

Every justificationist theory of knowledge has axioms and premises that it begins with. This fact has led skeptics to criticize the possibility of knowledge by noting the infinite regress within any attempt at a proof. But this skeptical problem has been offered an answer in Aristotelian and medieval theories of knowledge; there are self-evident truths. It is here that the explanatory regress of verification stops.

Now this answer is often criticized for being either 'dogmatic' or for creating a distinction between two sorts of statements, a distinction that might seem contradictory to what can be said of our statements in their proper context. That is, there is made a distinction between statements that require verification and statements that don't, between statements known 'in virtue of their meaning' and statements known in virtue of some experience.

Overlooking the discussion that has raged on about what such a distinction would imply and whether it actually exists, one might focus on another question: would these self-evident principles really not bear any information whatsoever, being only meaningless tautologies?

Answer 1

It depends on your definition of information. If you use the information theoretic definition, self-evident truths do not bear information because they say nothing about the universe that was not already presumed to be known (no possible states were ruled out by the statement). That being said, it is admittedly a circular reasoning for information theory requires some basic axioms before it can be written down in a form which permits calculation of information content.

On the other hand, self-evident truths are often used as a shortcut for a long line of reasoning ("it can be shown that..."). Those self-evident truths would have quite a lot of information content, though it may not be obvious to the person using it!

On the (third) hand, a search for the smallest truth which one must call "self evident" for a theory to hold true has interested quite a large number of scholars. The school of reverse mathematics is basically an exercise in how little one can assume when proving a point.

As an addendum, you might be very interested in the Aggripan trilemma (aka Münchhausen trilemma) which is the basis for a strong skeptic argument along the lines you are exploring.

Answer 2

If all self-evident truths are tautologies or are somehow redundant versions of the same thing, then Mathematics suddenly becomes a vacuous and pointless domain.

The elements of Peano's arithmetic are self-evident -- to over-simplify them a bit: we understand equality (of discrete and finite things) correctly, we can always get another number, equals added to equals net equals, and complete induction leads to meaningful generalizations.

Yet there is a great deal of content to arithmetic, not just silly redundancy, and even a bit of controversy. (E.g. we can always get bigger numbers, but we believe in infinity. So how many infinities are there, and can we identify them all?)

Some things are apparent because they make communication possible, and it is taking place. Basic math falls in that heap. Whether those are 'ultimate' truths that in any way transcend human language, or whether they are just genetic biases that come naturally to most healthy and complete human brains is another issue altogether. But the whole of math is not empty, redundant or tautologous.

Answer 3

I would point out that in mathematics, an axiom does not need to be "self-evident". Indeed, new forms of geometry were invented by altering the "self-evident" Euclidian axiom that "only one line connects two points". Whether the Riemann or Lobatchevski versions are "self-evident" is a matter of personal opinion (relative) and many people might agree that they are less "self-evident" than the Euclidian one.

In essence, an axiom is a basic proposition in a model. It seems that its property of being "self-evident" is not necessary; but merely useful in order to build a successful theory that could have practical applications.

One of the least "self-evident" axioms that I can think of, is those that found geometries in dimensions over 3.

Answer 4

Not necessarily, self-evident truths simply do not require a proof, e.g. see Russells comment upon his completion of the proof for "1 + 1 = 2"

As for redundancy, sure "2+2=4" can be considered redundat, but it is a very different statement than "4". Likewise with "dividends require financing" - the statement is obvious to those understanding finance but may be revelatory to those who do not, and, in the case of either reader, the sentence is meaningful beyond the utterance of "dividends" wholly unconcerned with the logistical considerations of the actual financing.