### Knowledge of the Reasoned Fact

#### Posted by David Corfield

In a comment I raised the question of what to make of our expectation that behind different manifestations of an entity there is one base account, of which these manifestations are consequences.

If I point out to you three manifestations of the normal distribution - central limit theorem; maximum entropy distribution with fixed first two moments; approached by distribution which is the projection onto 1 dimension of a uniform distribution over the $n$-sphere of radius $\sqrt{n}$ as $n$ increases - it’s hard not to imagine that there’s a unified story behind the scenes.

Perhaps it would encourage a discussion if I put things in a contextual setting.

On page 51 of Mathematical Kinds, or Being Kind to Mathematics, I mention the idea of Aronson, Harré and Way (Realism Rescued: How Scientific Progress is Possible, Duckworth, 1994), that one of the crucial functions of sciences is to organise entities into a hierarchy of kinds. One important idea here, found in other writings of Rom Harré, is that the use of first-order logic by logical positivists and then logical empiricists to analyse scientific reasoning has been a disaster. What these authors realised was that something important had been lost from earlier Aristotelian conceptions.

In the Posterior Analytics, Aristotle claims that are four kinds of thing we want to find out:

- Whether a thing has a property
- Why a thing has a property
- Whether something exists
- What kind of thing it is

As you can see, these four types come in two pairs, the second of the pair asking a deeper question than the first. Indeed, the second and fourth questions ask about the cause of something and its properties, cause being taken in Aristotle’s broad way. In fact, this is broad enough that he has no qualm in discussing examples from astronomy (‘Why do planets not twinkle? Because they are near.’), mathematics (‘Why is the angle in a semicircle a right-angle?’) and everyday life (‘Why does one take a walk after supper? For the sake of one’s health. Why does a house exist? For the preservation of one’s goods.’). This talk of cause in mathematics continued for many centuries. For example, as we learn from Mancosu’s book, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (OUP, 1996), mathematicians considered that giving a proof of a result by a *reductio* argument was not to give its ‘cause’.

The salient distinction here is that of Aristotle between ‘knowledge of the fact’ and ‘knowledge of the *reasoned* fact’. Aristotle gives this example to illustrate the difference:

Planets are near celestial bodies.

Near celestial bodies don’t twinkle.

Therefore, planets don’t twinkle.

Planets don’t twinkle.

Celestial bodies which don’t twinkle are near.

Therefore, planets are near.

To give an easy example of Aristotle’s distinction in mathematics:

$n$ is an even number.

Even numbers expressed in base 10 end in 0,2,4,6 or 8.

$n$ ends in 0,2,4,6 or 8.

$n$ expressed in base 10 ends in 0,2,4,6 or 8.

Numbers expressed in base 10 which end in 0,2,4,6 or 8 are even.

$n$ is even.

It’s not that the second syllogism is wrong, but rather that it hasn’t got the ‘causal’ ordering correct. Explanation is about tapping into the proper hierarchical organisation of entities. Your ability to do this is what needs to be tested, as here in this account of an MIT mathematics oral examination (in particular, the Symplectic Topology section).

The idea that mathematics has a causal/conceptual ordering seems to be yet more radically lost to us than the counterpart idea in philosophy of science. It’s at stake in the example I give in my paper Dynamics of Mathematical Reason (pp. 11-13), where singular cohomology goes from its being a cohomology theory which happens to satisfy the excision and other axioms, to its being a cohomology theory because it satisfies the excision and other axioms.

Now, should we expect convergence to a single ordering?

## Re: Knowledge of the Reasoned Fact

So only constructive proofs give causes?

This makes sense if we recall the computational content of constructive proofs; they can be turned systematically into computer programs. For example, a constructive proof of the infinitude of primes automatically (at least if you write it out formally in type theory) gives an algorithm that, given a finite list of natural numbers (including being given its length), calculates a prime number that is not on the list. More interestingly, a constructive proof of the uncountability of the continuum, applied to a (computable) enumeration of the algebraic numbers, automatically computes a (computable) transcendental number. So the “reason” that transcendental numbers exist is that we have a way to construct them.

However, I don’t believe that this can be all that there is to it. Consider a reductio proof of Lagrange’s Theorem that every natural number may be decomposed as the sum of four squares. This proof can easily be made constructive by checking that there are only finitely many possibilities (given any specific number) for such a decomposition (since the summands can never be larger than the desired sum). However, the algorithm that results from this step is terribly inefficient; it simply searches through all possible summands until a solution is found! Surely the “cause” (whatever that means) of this theorem must be something more than that, if you search through the possibilities, you eventually succeed. You still want to ask

whyyou will succeed, and the reductio, even constructivised, gives no help.Much more satisfying is a proof (the usual one is already constructive, not relying on reductio) using unique factorisation in the quaternions. (How close is this to Lagrange’s original proof? I don’t know.) I’m not sure that this really deserves to be called “cause”, but it strikes me as much more of an “explanation” at least!

(Personally, I don’t believe in causation. I’ve never been convinced by any philosopher, from Aristoteles to David Lewis, that the subject belongs in metaphysics (as opposed to epistemology) at all. I believe in a universe described by relativity theory with a low-entropy big bang, but the consequences of this fall far short of most people’s understanding of cause and effect; it certainly doesn’t apply to the theorem above!)