I was wrong about how belief in axioms are held. I was not wrong about the beliefs themselves, which is to say, the axioms which we all know and love are still true.
But I used to say, and say wrongly, that we knew axioms were true based on “no evidence” save our intuition, or on faith. This is wrong and bone-headed, as I should have recognized. Actually, we know that axioms are true the same way we know any universal is true: via the evidence of observations.
This dawned on me when I was reading Ed Feser’s new book Scholastic Metaphysics when I was reminded of the Aristotelian and Scholastic dictum “there is nothing in the intellect that was not first in the senses”. If that’s true (and it is) then the sort of a priori knowledge I had in mind, which was the fully formed idea being present to us (somehow), must be wrong.
I was originally convinced of that kind of a priori knowledge by a beauty of an argument by David Stove which appeared in his The Rationality of Induction. The argument is still right if only a few words in it are changed. It purports to prove, and does prove, that our knowledge of logic cannot be had empirically. And that’s so; at least, that is true if we mean all our knowledge. His argument went something like this.
In order to know via experience the validity of (say) the schema A = “For all x, all F, all G, either ‘x is F and all F are G’ is false, or ‘x is G’ is true”, we could make observations like O1 = “David is bald and David is a person now in this room and all persons in this room are bald.” But in order to get from O1 to A; that is, to know A is necessarily true, we have to already know that O2 = “O1 confirms A”, and that is to have non-empirical logical knowledge. Or you could insist at O2 was learnt by experience, but that would require knowing some other logical knowledge, call it O3, which somehow confirms O2. And then there would have to be some O4 which somehow confirms O3, and so on. There cannot be an infinite regress—the series must stop somewhere, at a point where we just know (my guess is O2)—so we must possess some innate logical knowledge.
Perhaps this sketch isn’t fully convincing, but it is in the context of the rest of the chapter, which contains several other proofs that our knowledge of logic cannot be (fully) empirical. (I have a more thorough summary coming in my book—which is still in the works; see more examples below.)
Anyway, the conclusion is still true. Consider the major premise in the old standby “All men are mortal”, which is also unobserved, but still true. Our senses confirm instances, and we extrapolate to the universal. Consider my favorite mathematical axiom that “For all natural numbers x, y, if x = y then y = x”. Again, our senses confirm instances, and we extrapolate to the universal, which we call an “axiom.”
The Scholastic approach solves problems the rationalist a priori raises. If there is rationalistic knowledge we “just know”, then does everybody know all of this knowledge or just a few? And if not everybody knows everything, who or what decides who gets what knowledge and who doesn’t? Too confusing. I should have thought of that.
Much more satisfying is the Scholastic approach. We observe, do we not, that not everybody knows Peano, to pick just one. But they can be brought to its knowledge, and brought pretty quickly, by giving them a few experiences where x = y and y = x. This explains the difficulties over the Axiom of Choice, too (look it up); agreement is harder to come by. Solves the “problem” of induction, too.
Of course I speak of the Aristotelian and not Platonic universals, about which there isn’t space to disagree here. Except to insist that because we are sometimes wrong in our judgement of universals/axioms, it does not imply always wrong. Intuition is a tool that can be misused like any other.
Last of course: The above is not a proof that we do not (all or those non-defective) come pre-wired according to some scheme so that we have the ability to come to universals. Let him that readth understand.
Very little thus has to change in what I wrote before about axioms, but there must still be changes. It is not “no evidence save our intuition or faith” but “the evidence of our senses and our intuition (or faith).” I’ll put this up in the Classic Posts page as a warning. Still unsure whether the language of faith and universals is best. I’m leaning on “yes.” More later.
My plan was to write posts about the class discussions of the day before. There is no day before yet. But I talk about axioms on the first day, so this is a natural place.
We can learn from observation the following argument is invalid: “‘All men are mortal and David is mortal’ therefore ‘David is a man” if perchance we see David is not a man (maybe he’s a puppy). And we can learn from observation the invalidity of “‘All men are mortal and Peter is mortal’ therefore `Peter is a man” only if we see Peter is not a man (maybe he’s a cow). But we cannot learn the invalidity of “‘All men are mortal and X is mortal’ therefore ‘X is a man” through observation because we would have to measure every imaginable X, and that’s not possible. If we believe “‘All men are mortal and X is mortal’ therefore `X is a man” is unsound, and it surely is, this belief can be informed by experience but it cannot be solely because of it that we have knowledge of it. Another universal is born, though of a more complicated form.
Stove himself: “If an argument from P to Q is invalid, then its invalidity can be learnt from experience if, but also only if, P is true and Q is false in fact, and the conjunction P-and-not-Q, as well as being true, is observational. This has the consequence, first, that only singular judgments of invalidity can be learnt from experience; and second, that very few even of them can be so learnt.” And here’s the kicker: “If the premise P should happen to be false; or the conclusion Q should be true; or if the conjunction P-and-not-Q is not observational but entails some metaphysical proposition, or some scientific-theoretical one, or even a mere universal contingent like ‘All men are mortal': then it will not be possible to learn, by experience, the invalidity of even this particular argument” (pp. 155–156). The key is that :scarcely any of the vast fund of knowledge of invalidity which every normal human being possesses can have been acquired from experience.” Can we allow the hilarious pun universal universals?
Examples? The invalidity of the argument “Given ‘The moon is made of cheese’ therefore ‘Cats do not understand French'” cannot be learned from experience. Neither can “Given ‘Men can breathe underwater unaided’ therefore ‘The atmosphere is largely transparent to sunlight'”. In neither can we can ever observe the conjunct P-and-not-Q.
Homework: your turn for examples.