The Riemann Hypothesis as a Counterexample to Coherentism in Epistemology?

Coherentism is the main alternative account of justification to that of foundationalism. With the latter, justification is based on beliefs, much like with axioms in mathematics, which are held to be basic and self evidently true. From those beliefs a system of knowledge is built in linear fashion.

With coherentism, by contrast, what matters is how belief “coheres” with the wider system of knowledge. The image is of a web of knowledge as it were, or the spoke like arrangement of a wheel.The thicker the coherence with the wider system of knowledge the more justified are we in holding that a belief constitutes knowledge.

Although foundationalism takes its inspiration from Euclidean geometry, so thereby is heavily influenced by the manner in which proof is developed in mathematics, it is possible to think of mathematics in coherentist terms.

Many theorems of mathematics are built by first moving from theorems already proven. Of course, this fits into the linear foundationalist picture but it is also possible to imagine all the theorems of mathematics constituting a coherent web of knowledge. Now consider the Riemann hypothesis, perhaps the leading problem of mathematics, in the context of coherentist epistemology.

The Riemann zeta function takes the form, very crudely arrrrgh!!!!!

z(s)= 1/1^s + 1/2^s + 1/3^s …

S is a complex number, i.e s=x+iy. The Riemann hypothesis is the conjecture that all non trivial solutions of the zeta function that have a value of zero, that is all values of zero except when y=0, have a real part, that is x, of ½.

One of the most important reasons why the Riemann hypothesis is the leading problem of mathematics is because, if true, the Riemann hypothesis tells us that there is an underlying order to the distribution of prime numbers, the atoms of number theory. It is not hard to see how a proof of the Riemann hypothesis takes on considerable intellectual significance here.

Most popular discussion on the Riemann hypothesis and its significance focuses on the distribution of primes. However, many theorems of number theory have been developed by first assuming that the Riemann hypothesis is correct. The two considerations are not unrelated, of course, but the latter is interesting with respect to coherentist theories of knowledge.

The Riemann hypothesis strongly coheres with many theorems of number theory. Given this thick coherence are we then to say that, even though the hypothesis has not been proved, that we nonetheless are justified in saying that all non trivial zeros of the zeta function have a real part ½?

The matter of coherence becomes more stark when we consider that the Riemann hypothesis plays an important role in other areas of mathematics, that is not just number theory. Moreover, the Riemann hypothesis is also connected to physics by way of quantum mechanics for complex numbers play an important role in the theory.

That is thick coherence indeed.

Do we not “know” the Riemann hypothesis to be true, because of its thick coherence with number theory, even though we have not “proven” it to be true? A mathematician would say that this would be an odd way of asserting that we are justified in believing the Riemann hypothesis to be true. Justification can only come after a proof. If we side with the mathematicians on this point it seems we must say that coherentist epistemology is false.

That is, the Riemann hypothesis and the way it relates to number theory (but not just there) forms a counterexample to the coherentist theory of knowledge.

But any epistemologist worth her salt knows, with intended pun, that justification and knowledge might turn out to be two different things altogether. Of course, coherentism *is* offered as a theory of justification based on the standard conception of knowledge as *justified* true belief.

Such epistemological considerations are not important for number theorists as they subject Riemann to assault.

However, if the Riemann hypothesis should turn out to be undecidable? What then? Well, epistemological considerations will definitely loom large then!!! I suspect coherentism will get a boost should that be the case.

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