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.
It is evident to anyone with eyes to see that men and women are wired differently, and that is by design. But, as you said, truth is truth, and we know that women and men can either accept truth or reject it. I find the “feminist” notion that “men do not have the ability to understand me” to be a bit silly.
The difference in “wiring” between the sexes, it seems to me, shows itself quite a bit in the generally more emotional mindset of a woman and the more analytical mindset of a man. Men and women approach problems differently. Of course, each person, regardless of sex can approach any given situation any number of ways. I’m not talking rocket science here, obviously.
Men and women are counterparts intended to get together in pairs to form a complete union–complete once God is involved, that is. Each person brings features to the relationship unique to his gender.
The marriage of a man and a woman, in a “perfect” world, is a very supportive and efficient thing, indeed. I know this is kind of a rabbit trail away from your article, but it’s just a few thoughts.
I like your aritcle Mr. Briggs.
Beautiful explanation! WXRGina states a common plaint: That men don’t understand women. Looking at the problem in the aggregate, it is impossible to untangle.
But bring it down to the most fundamental matter of arriving at a logical conclusion based upon a shared truth or conflicting truths, and the problem can be solved. Or not. Maybe you discover an entirely new question: Whose truth can stand the test of logic–but then epistemological differences rears its head again…
This is like looking in a mirror with a mirror behind you…
Ahhhh! My head hurts! Thanks for kickstarting my brain, Mr. Briggs.
Is epistemology now a synonym for hormones?
‘E piste m’ology? Well, since we’re piste-ing around, my wife saying “He doesn’t understand me” shows a deeper understanding she has for me than I for her.
“The difference in “wiring†between the sexes”
The wiring differences goes deeper then the balance between rational and emotion sides of the brain.
My daughter did some work at brown(unpublished) on visual perception.
They had originally started out trying to determine if people were biased to horizontal or vertical.
So they set up a quite simple experiment drawing checkerboards with vertical and horizontal lines of varying widths and distances from each other and asked a simple question.
Are the vertical lines closer or farther apart or the same as the horizontal lines?
The answers the females gave(the original test subjects, her friends) were in almost total agreement, so they thought they had determined a human bias towards vertical. Being fairly intelligent she realized her sample group may have similar biases. So she sent me the test. I had a horizontal bias. I’m color blind, so maybe my vision impairment was the reason for a different bias. Then they randomly sampled the boys on campus, they all had a horizontal bias. Then they randomly sampled the girls, vertical bias.
Of course a study that concludes the differences between male and female extend as far as visual perception will never see the light of day.
Harrywr2, I have horizontal bias myself. It’s extremely acute in the morning. My wife usually gets up before I do. Obvious vertical bias on her part.
Whatever could visual vertical/horizontal bias mean, actually? A tendency toward seeing all lines as vertical/horizontal? Seeing vertical/horizontal patterns in otherwise patternless features? It’s not clear that even if a visual difference exists that it indicates an actual thinking difference.
“Note very carefully that “differently†in no way implies “superior to†or “inferior to.†” Would that be better as “Note that “differently†need not imply “superior to†or “inferior to.†“?
dearieme, Brigg’s has fallen into that pc- neologism: different but somehow equal. It think “independent but otherwise equal” is a better concept.
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OT: I finally discovered how to change the Foxfire spell checker language. Why it doesn’t use the Firefox language setting is beyond me. For whatever reason, it had concluded I live in Hong Kong.
Obviously, the now inescapable (!) Kantian discursively constructed gender reality is more than linguistic. The Heraclitean version of historicism rejects totalized meta-narratives. Meaning holism entails dis-unified Marxist standpoint theory which excludes certain possibilities, i.e. social and psychological phenomena in which gender is implicated.
I could go on, but my old lady just made me lunch and she gets whiney if I don’t eat it hot.
Let me play Devils advocate.
1. It doesn’t follow from the assumption of different epistemologies that there is no commonality. ie Feminists might agree with all Mathematics but disagree on say Biology or more likely Sociology.
2. Think back to Plato’s cave. There is absolute truth and there is perception of truth. That is to say there is an absolute truth but as humans we may be unable to comprehend all truths. We are in the cave and whilst we can get closer to the entrance we can never get out. Thus our theories about the world are only simplified models of reality. Our models might be “good enough” to be better than no model but they will never be reality. [That’s my post half a bottle of wine précis of Plato so be forgiving]
If we accept that then by inference we might have two people approaching the cave mouth and coming up with competing models. [eg. light is a wave vs light is a particle.] We might think that men and women, being physiologically different, are more likely to come up with different models of reality (than say two members of the same sex). Assume further that it is impossible to reconcile the two models – thus we have two epistemologies.
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I think this is bollocks BTW but I doubt any feminist would openly disagree with Euclid. They might waffle around the possibility – perhaps point out the long history of exclusion preventing Feminist Maths from gaining a footing.
I don’t think I wrote the second point sufficiently clearly. Let me try again:
“there is an absolute truth but as limited humans we may be unable to fully comprehend it.”
Epistemologists slaloming through ontologies hazard the trap of uncontrolled equivocation. Just ask the Yrmo.
http://tinyurl.com/2cvpy7a
A woman buys her husband a pair of ties for his birthday. They go out for a celebration dinner that evening. When he gets ready the husband puts on one of the ties. When he presents himself for approval before they leave she says, “What’s wrong with the other one?”
Most women I’ve told this joke to tell me it’s not funny, it’s a sensible question. Most men just laugh.
If a man is alone in a forest with no woman present, and he says something, is he still wrong?
“Putting the ‘pissed’ in epistemology”.
(That’s the UK ‘pissed’ not the American one).
Polylogism.