Euclid gave us a gloriously simple proof that there are an infinite number of primes. A prime number, of course, is a positive number that can be evenly divided only by itself or one.
Here’s Euclid’s proof. Don’t worry if you can’t follow along; it’s only important that you understand that the statement, “There are an infinite number of primes” is true given the information provided in the proof.
Assume there are only a finite number of primes; order them from smallest to largest. Multiply them all together and then add to that product one. For ease, call that product-plus-one, P.
P is clearly larger than the largest prime we know of, because P is the product of the largest prime and all the primes smaller than it. But also, if we take all the primes we know and divide any of them into P, we will have a remainder of one.
That means that the prime that divides evenly into P must be larger than the prime we thought was the largest. And since you can keep doing this procedure each time you discover a new “largest” prime, the number of primes must be infinite.
Euclid’s is not the only proof that the number of primes are infinite, but it’s the simplest among all the proofs I know of.
But suppose you find another proof easier to comprehend than Euclid’s; perhaps Dirichlet’s demonstration. And let’s also imagine that your cousin has discovered an entirely new proof.
One day, the three of us meet. We all agree on the truth of the statement “There are an infinite number of primes”, but each of us believe this because of different evidence. We discuss the differences in the evidence, but are unable to come to the conclusion that our three sets of evidence are equivalent.
By “equivalent”, I mean something like we mean when we say a sentence has identical meaning in a different language. It could be that my evidence is “The ball is blue” and that yours is “L’objet circulaire est en bleu“, but since I don’t know French and you don’t know English, we cannot decide whether the sentences are equivalent, that they directly translate.
But even though I don’t understand your evidence, I might agree with its truth; that is, I might accept its internal coherence. Or I may simply accept it for the sake of the argument. Or again, I may not accept its truth at all. Finally, I could understand it as well as I understand my own evidence, but I simply prefer mine to yours.
Thus far, we are on solid ground. But let’s move towards the beach and consider your cousin’s discovery. Your cousin does not prefer either of our arguments, and claims that the third offering allows deeper understanding of truth of the statement “There are an infinite number of primes”.
Now, we might accept—and here finally arrives the crucial pronoun—her evidence as true, yet feel our evidence is easier to comprehend, or more readily adaptable to new arguments.
Or we might not be convinced of the soundness of her evidence, yet we cannot offer outside proof that her evidence is unsound. Finally, we might claim that her evidence is in fact unsound. Yet we all agree that “There are an infinite number of primes” is true.
Because of their inherent, incontrovertible genetic differences, women think differently than men. This claim is the basis of “feminist epistemology.” Note very carefully that “differently” in no way implies “superior to” or “inferior to.”
Let’s accept the feminist claim as true (I think it is true, but that’s irrelevant). But in doing so, to what have we acceded? Merely that some people think differently than others? And that, thinking differently, people might, as our example attests, weigh evidence asymmetrically, even though they agree on a truth of a proposition?
None of this in the least controversial. But what if your cousin offers evidence which she claims proves “There are an infinite number of primes” is false? Further, she insists that her way of thinking allows her to understand her evidence in ways that you, being male, cannot.
You might try to prove her evidence is unsound with respect to exterior information, but she might counter those arguments with similar ones about how that exterior information is viewed differently by females.
All evidence, and all argument, form a linked web, the strands of which eventually hang on the single thread of the a priori, the unproved and unprovable truths from our intuitions. So, she may claim that, being female, her intuition about axioms is just different.
She might be right. That is, there is no way to prove she is wrong. The only fundamental counter we have that feminist epistemology is no different than male, or even sentient, epistemology is our belief (provided by our intuitions) that there is only one set of base truths; that every statement is either true or it is false (or nonsensical), but that it cannot be both.