William M. Briggs

Statistician to the Stars!

I Was Wrong About Axioms: Day One Teaching

Mistakes can happen

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.

Teaching

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.

More Stove

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.

15 Comments

  1. I have noted several times on Dr. Curry’s site that ‘Climate Science’ is the only science that is based on an axiom rather than theory.

    The Central Axiom of Climate Science is: ‘The Temperature of the Earth (TOE) is controlled by the amount of CO2 in the atmosphere. Human civilization, as a byproduct of producing energy, is injecting massive quantities of CO2 into the atmosphere, raising the concentration of atmospheric CO2. This injection of CO2 is causing the TOE to rise at a historically unprecedented rate. The results of this rise in the TOE will prove catastrophic unless coordinated government action is taken to drastically reduce (90+%) the production of anthropogenic CO2. The reduction can be achieved most efficiently by taxing and regulating every human activity that produces a ‘carbon signature’.’

    You will note that Climate Science, writ large, does NOT treat the above as a theory, subject to confirmation or rejection by observations. It, all of it, including the mitigation prescription, is treated as an axiom, as defined by Wikipedia: “An axiom, or postulate, is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.”

    Any nominal scientist, no matter his credentials, who questions any part of the axiom, however mildly, instantly loses his status as ‘Climate Scientist’. Ask Dr. Curry, Dr. Rossiter, and Dr. Bengtsson, to cite three recent examples.

    All other sciences propose a theory, collect and analyze related data, and, based on the observations, either confirm, adjust, or reject the theory. Climate Science collects data and adjusts the data as necessary to confirm the axiom. Or simply declares that ALL observations (too hot, too cold, too wet, too dry, too much snow, too little snow, more tropical storms, fewer tropical storms, more Antarctic ice, less Arctic ice, etc.) confirm the axiom. Only axiomatic science is ‘settled’.

  2. This is an excellent post. The pithy ‘there is nothing in the intellect that was not first in the senses’ alone is worth the price of admission. Every once in a while Kant comes up in class, and while I don’t quite have a Randian aversion to the man (‘the first hippie’), I have a strong aversion to his overly ‘a priori’ way of looking at knowledge. The above pithy aphorism, and I’m willing to admit I may be rushing to judgment , seems to put the dagger into Kant’s ever-beating heart. I hope I don’t have this completely wrong.

    I get what you say about logic, though. I’m still not sure that, somehow, that knowledge still had to come to us first through the senses. ‘Somehow’ is my wiggle word. Thank you!

  3. Don’t be too hard on yourself Briggs; the axiom par excellence, the Priciple of Non-Contradiction, can only be known by intuition, and any other axiom that we know to be true is indirectly (at least usually) derived from it by reductio ad absurdum.

    To use the Peano axiom as an example, to deny it implies that, in at least in some instance, a natural number x may be equal to some distinct natural number y, but y not be equal to x. But, since we have already agreed that x=y, that means that y must have the same value, and thus, be the same value of x; to say otherwise goes against the definition of what is meant by x=y. Thus, to deny Peano’s axiom implies that both y=x always and that y=x not always. But this violates the PNC, therefore, Peano’s axiom is true. Note however, that without knowledge that the PNC is true, we could affirm that y=x always and not always, so the PNC is absolutely critical to establishing the axiom.

    So, we start with our knowledge of the PNC, which we cannot prove but know to be undeniably true, and from there, gain knowledge of other axioms by gaining knowledge of the natures of things, definitions, by experience, and then can gain knowledge of other axioms by seeing the truths that are implied by the definitions.

    To go back to the example, we see that the definitions of “natural number”, “x”, “y”, and “=”, combined together, imply that “y=x” always. And since the PNC is true, we can conclude that the axiom is true.

    To wrap up my (rather long-winded) comment, in one sense, we only know all of the axioms because of our intuition; because our knowledge of them is based on our intuitive knowledge of the PNC. But in a different sense, we can say that they require no more intuition than the PNC, and thus, it’s fallacious to conclude that they need any MORE intuition than the PNC, or that ALL axioms are impossible to prove in at least in some sense.

  4. Sander van der Wal

    16 June 2014 at 11:40 am

    An axiom is just an axiom. All man are mortal is one. And because all man upto now have died, it is reasonable to add the proposition: ‘the axiom “all men are mortal” is true in this world.’

    But an axiom, any axiom, is not *automatically* a true proposition of this world. Take the Geometries, Euclidian, Hyperbolic and Spherical. All three geometries have axioms about parallel lines, and they are all different. But that doesn’t mean that they are all valid as a description of the geometry of this universe, or parts of it. On a sphere, like Earth, Parallel lines at the Equator cross at the Poles. The large scale universe is either Euclidian, or Hyperbolic.

  5. Faith is something that cometh from reading too, so I’m not sure why you’d put that with intuition. If by faith you mean “trust,” as a child has for a parent, then that is both learned and innate.

    When we observe “wild people” we see that we see they do not develop a recognizable epistemology and so receive experience in a way that we simply can’t understand. Understanding experience, beyond “fire hurts” to “Tom is the kind of guy you’d like to see in higher office,” requires a social assimilation, and the development of critical abstract intelligence so as form understandings of unexperienced phenomena.

    JMJ

  6. van der Wal,
    “On a sphere, like Earth, Parallel lines at the Equator cross at the Poles.”

    This is incorrect. There are no parallel straight lines (geodesics) on the surface of a sphere although you can have curved parallel lines, e.g. lines of latitude. You should be referring to Euclid’s fifth postulate and not parallel lines per se.

  7. I think the last post that I made on this site, your response to me was that you know your right “a priori”. I pretty much washed my hands of the discussion at that point.

    You might want to consider the full implications and correlations of discarding the mystical notion of a priori knowledge.

  8. Brandon Gates

    17 June 2014 at 12:34 am

    Briggs,

    Speaking of homework:

    http://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/093570275X

    When I was in college, my mother gave me a poster-sized print of the book cover. I framed it and hung it on my wall for years. Thanks for the fond memories.

  9. Jim’s right, too. There’s no such thing as a priori knwoledge.

    JMJ

  10. Sander van der Wal

    17 June 2014 at 1:05 am

    @Scotian

    On the equator, the lines are parallel, as the angles are 90 degrees.

    But you’r missing the point. You can define three different geometries by changing one axiom. All of these geometries are internally consistent. What differs is the truthfullness of the propositions you make about the geometries and their relation to the real world, or parts of it.

  11. Brandon Gates

    17 June 2014 at 2:05 am

    JMJ,

    There’s no such thing as a priori knowledge.

    I’ll go as far as to say that it gets abused. But doesn’t exist? Does knowledge itself exist? [big hint] Do explain.

  12. van der Wal,
    “On the equator, the lines are parallel, as the angles are 90 degrees.”

    That would be true in Euclidian space. The definition of parallel, as lines that never meet, remains. It is the fifth postulate that changes.

  13. @Brandon,
    It might come down to a difference of definitions…. but a priori knowledge is fully formed, conceptual knowledge that “appears” some how in the mind – bypassing observation. This is not logical inference or deductive reasoning, nor is it a predisposition to learn grammar/language etc.

    I’m not sure what your cryptic post is meant to communicate. Do explain.

  14. I’ve come to this late, but there is one point that seems to be neglected by most philosophers and cognitive scientists…. The human mind develops. I would doubt that a two year old or probably a four year old would do syllogisms…the question then is when do we develop this “innate” logical facility?
    See, for example, the work of Piaget and Rochaut
    http://psychology.emory.edu/cognition/rochat/Five%20levels%20.pdf

  15. Brandon Gates

    18 June 2014 at 11:15 pm

    Jim S, there’s was this little fragment running around from Sunday’s Aquinas post (I think) which I can’t find now. Anyway, my caution to JMJ is that discarding any and all a priori knowledge may get him into a bind if he doesn’t watch out because it leaves open the possibility that God could have defined a different value for pi.

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