Evolution, Induction and Rational Intuition
Intro
Rational intuition: There are a number of things which can be called rational intuition[1]. Here I simply mean the fairly immediate/spontaneous/off the cuff inclination to say that a certain sentence is true without, say, looking a proof or learning this by testimony. So, for example when you immediately accept or feel that you can directly ‘see’ the truth of a statement or when you go through one example (or fewer) and then are inclined to think, without proof, that all other examples will work out the same way these would all be examples of rational intuition in the sense that I have in mind.
We take rational intuition to very frequently lead us to true conclusions though it is not infallible.
There is probably an evolutionary explanation for how such intuitions can so frequently get things right. In this section I will consider some philosophical problems for such an account.
One note before we get started though:
Often we have rational intuitions about propositions which can be proved from accepted axioms so one might think that all such intuitions are a matter of subconsciously working out a proof. This doesn’t seem likely for a number of reasons. First off we can have apparently quite similar intuitions about statements like the axiom of choice which can’t be proved from accepted axioms. Secondly accepted axioms often seem intuitive themselves in a very similar way as propositions which can be proved in terms of them – rather than feeling somehow especially trivial. This is, of course, consistent with the idea that the mechanism which produces rational intuitions effectively tries to prove the statement in question from our accepted axioms and gives us the feeling that such statements must be true when such a subconscious proof can be given (in the case of the axioms these proofs will just be proofs of one line). However, this hypothesis that cultural acceptance of certain axioms precedes mathematical intuition seems very unlikely. It is surely more plausible to think that intuition precedes axiom choice: that a number of related statements all seem intuitively likely and we choose some of them as axioms in such a way as to entail as many of the others as possible.
Thus for these two reasons the fact that many things which are intuitive can be proved should not be taken to mean that mathematical intuition is a matter of unconsciously grasping a proof (though, of course one thing we can have a mathematical intuition about is the claim that something can be proved).
Impossibility
argument
If scientific induction is completely unreliable in the realm of necessary truths then it is surprising that evolution would lead people to accept true axioms and inference procedures.
For, if there is a humanly detectable kind of inferential procedure such that the fact that a number of instances of a procedure of this kind don’t lead from truth to falsehood makes it likely that the inference procedure is infact truth preserving, then we will get both a justification for induction about math+logic and an evolutionary explanation for how we could evolve a faculty of rational intuition. Otherwise we get neither.
I think that we should accept both the evolutionary account of rational intuition and the idea that in some cases induction can justify us in believing necessary logico-mathematical truths.
A strong Platonist might object that what we evolve is a faculty that literally detects the forms. But if you don’t accept such causal powers then it is hard to see what the sub-personal mechanisms evolved could do to get reliability that conscious induction can’t.
Here are a few points to soften the blow of accepting that we can have mathematical knowledge by induction
- calculator example
- could learn that everyone in group A qualifies for insurance plan 5 inductively
- obviously some mathematical predicates aren’t very inductable but neither are some empirical ones… all we need is that there is some subset of mathematical claims which humans can distinguish which are inductible
- in some cases we are inclined to use knowledge to mean possession of a canonical proof as distinct from reliable belief: so if you are inclined to say that you cant *know* mathematical truths by scientific induction in the special sense which is normally relevant to mathematical statements this does not entail that induction can’t lead you to reliable true beliefs.
Specific problems for
coming up with an evolutionary story about rationality:
What is it for a creature to be able to infer from ‘P v Q and ~Q’ to ‘P’ – need to associate these two logically equivalent propositions with different mental states such that some creature could be evolutionarally disadvantaged by not connecting these states.
[This is a variant of the problem of logical omniscience: it is tempting to think of a creature’s mental states in terms of the set of possible worlds which are actual for all we know.]
In order to get an evolutionary grip we would need to have separate behavioral states associated with the two necessary beliefs, which there could then be some evolutionary value to evading.
Suppose:
Mice can detect vixen urine.
Mice can detect foxes visually, and go into fox evasion procedure when they do.
If a mouse smells the vixen urine but doesn’t start the fox evasion procedures we might say that it knows that there is a fox but not that that there is a vixen.
In this way there could be selection for either
a) mice with brains that automatically connected these two states
b) mice with brains that would end up connecting these states if they were frequently enough activated next to each other (i.e. brains that treated these predicates as inductible)
In this way, if we think that it makes sense to attribute logical abilities to pre-lingusitic animals we can make sense of evolution giving them these linguistic abilities.
On the other hand if you don’t think it makes sense to attribute logico-mathematical abilities to animals then the story is even easier to tell once language is in place
Suppose:
People can recognize vixens by seeing them or by hearing others say vixen
People can recognize foxes by seeing them or by hearing others say fox, and they have fox hunting/evasion procedures
There would be survival value to going to get your fox spear directly when someone says vixen rather than waiting for someone to say that is a fox too, or waiting for it to come into sight.
Since language chances so quickly it’s unlikely that there would be benefit in making a brain that ‘automatically’ believes that vixens are foxes.
But there would be a benefit to build a brain which is likely to connect these kinds of states (one that treats these states as inductive).
We would be evolved to have a sense of the ‘right’ kinds of inferential procedures to accept as universally true after relatively few confirmatory experiences, just as we are evolved to have a sense of the ‘right’ kind of generalizations about the empirical world (pots that look like this will crack when fired, treating bees like this will make them aggressive) to believe on the basis of very limited experience.
In this way our intuition that the pigeon-hole principle is true is like our intuition that you can’t cut a banana with a telephone wire (we are evolved to quickly, subconsciously, make certain kinds of generalizations on the basis of very limited observation)
The only difference is that in the latter case we can form certain kinds of pictures of mere physical impossibilities but not metaphysical/mathematical impossibilities.
But these canonical methods of picturing are just formed by a) evolution and b) custom in such a way that everything which is actual/physically possible turns out to be picturable. But there are no further constraints: whether we say that a given description of a physically possible state of affairs is or is not metaphysically possible is just a matter of chance and convention.