Who breaks a butterfly upon the wheel?

Imagine that you are writing out a complicated mathematical proof. Most philosophers would say that it would be irrational to believe you are justified in thinking that you have conducted a proper derivation without first checking over each short step in the proof. It’s obvious that you might have made a mistake.

One should always lower the level of confidence assigned to the conclusion of any argument, according to the probability that one has made a mistake in the argument. However, David Hume points out that this estimate in probability may itself be in error. We ought to seek out an estimate for this estimate, which may itself be in error … leading to an infinite series of corrections. This leads to systematic doubt over the truth of synthetic *a posteriori* statements. The very tools we use to check the truth of our fallible beliefs are themselves fallible.

Now, assume that all the lemmas of this mathematical proof are produced without error. Imagine a computer program is conducting the proof. This computer program cannot err in its derivations of the logical consequences from the axioms. *But are the axioms true?*

For instance, if Euclidian geometry is to describe anything outside a coherent mathematical system, we must require that the parallel postulate not just be assumed, but be *true*. When mathematicians such as Gauss, Bolyai, and Lobachevsky attempted to show the parallel postulate false in a *reductio ad absurdum*, it turned out that the parallel postulate could be false without resulting in a contradiction.

We’ve moved from certain *a priori* truths to coherent *a priori* truths. Both systems that assume the parallel postulate and systems that reject its truth cannot both be true at the same time without violating some very fundamental intuitions about truth. What we’ve really shown is that even intuitively true *a priori* statements are not necessarily true; even *a priori* statements are doubtable. Numerous coherent systems of belief can each be true according to their own assumed axioms. If any system should run up against a contradiction, one need only follow St. Thomas’s advice that “When you meet a contradiction, make a distinction,” and then one is out of the trap again. When caught in a corner, you can always wiggle your way out.

We now know from Einstein’s general relativity that Euclid’s geometry is only an approximation to the actual geometry of physical space. Yet, it may be true that overall, Euclid’s geometry is a better approximation than hyperbolic geometry when describing the physical universe. This judgement about the structure of space is prone to err, for it is a synthetic *a posteriori* judgement. Once you try to bridge the gap between *a priori* systems and *a posteriori* facts, we run into a problem: we cannot know with certainty if either of two coherent *a priori* systems are true.

We can’t go about saying that “all triangles have angles that add up to 180 degrees”, for this isn’t true anymore. We would have to amend the following to the statement: “… within Euclidian geometry”.

The problem for *a priori* truths is formulated as follows:

- To know an
*a priori*statement is true and not just part of a coherent system, we must appeal to some synthetic*a posteriori*(SAP) statements. - To know which SAP statements are true, we need a good procedure …

And then we’re back to spinning on the wheel.

Take the problem of induction as a case-study; it’s meant to solve a wider problem: *how do we learn from experience?* As Popper points out, the answer includes a logical solution and a psychological solution. Both solutions, he thinks, do not resort to inductive inference, but use the process of trial and error. Once we reevaluate the problem of induction in light of the larger problem it is intended to solve, we realize that induction was never part of theory-formation or theory-validation, since it was not part of our psychological learning-processes and could never tell us if theories were true.

If we disregard the particulars and focus on the overall issue at hand, I think that it is clear that, like the problem of induction, the problem of the criterion is intended to solve a larger problem.

Without reevaluating your presuppositions, you won’t abandon the search to square a circle, even though it is a problem that will never bear fruit. The answer ‘Choose justificationism!’ is intended to solve *the problem of rationality*, yet justificationism runs into the problem of the criterion. The *diallelus* or ‘wheel’ argument is a problem for those that require immediate good reasons for belief in order for one to be rational. I conjecture that the wheel argument is then a pseudoproblem: it is impossible to solve. This does not mean the problem of rationality is unsolvable, only that the intended solution is inadequate.

The problem of rationality might be expressed as follows: “What process should we use to make correct decisions?” One answer is that we set out to justify all of our beliefs; however, this program cannot be successful, since (1) their very foundations must be taken as an act of faith, or (2) one cannot rationally choose between incompatible but coherent systems.

Once we reevaluate the problem of justification in light of the larger problem of rationality, we realize that one cannot have justified true beliefs. Yet we’re not discussing the real problem — that of rationality, but a subsidiary pseudo-problem about a theory of rationality that is impossible to adopt.

As a scheme of things is modified by inroads from outlying existence, it loses authority, is less able to banish dread; its adherents fall away. Eventually it fades, exists only in history, becomes quaint or primitive, becomes, finally, a myth. What we know as legends were once blueprints of reality. The Church was right to stop Galileo; activities such as his import into the regnant scheme of things new being which will eventually destroy the scheme. (Allen Wheelis,

The Scheme of Things)

My intention is to help sow the seed, *the rejection of the justificationist program*, yet not abandon what made its rationalism, its focus on rigorous argumentation, so valuable. The truth is that there are no first principles. Plato’s ‘divided line’ where darkness of mere belief and where the sunshine of truth illuminates the world around us is a myth, nothing more.