Monday, July 26, 2010

Proofs in metaphysics


On Tues, Jul 20, 2010 at 07:29:34
Vaidyanathan asked this question:

I am a newcomer to philosophy, and metaphysics in particular. I would like to know about the method of analysing and proving statements in metaphysics. Being a student of mathematics I am familiar with the axiomatic method. Is there any systematic method of proving statements in metaphysics?

Vaidyanathan has taken me on a trip down memory lane. How I wrangled with this!

Spinoza in his Ethics (1677) is probably the best example of a philosopher who explicitly uses Descartes' 'geometric method' for proving propositions in metaphysics. But you'd be totally wrong to think that Spinoza is any different from the majority of metaphysicians who eschew Spinoza's barbaric apparatus of axioms, definitions, propositions, scholia etc. etc.

You've got to start somewhere. Descartes starts with the 'Cogito'. Whatever proof you offer (and we'll get on to the question how there can possibly be 'proofs' in metaphysics in a minute) you need to assume something; or have you?

Thirty years ago, I showed this to my harassed thesis supervisor John McDowell. He was predictably nonplussed:

1. I begin with nothing: only an unspecified commitment, a pure question mark, a certain mental attitude. I want to tell the truth in a true way, before I even have any truths to tell; to grasp the nature of ultimate reality while reality itself presents no point of entry to its innermost circle; to forget all that I have learned and begin this time without a beginning, empty-handed and empty-headed.

It gets worse:

2. The dialectic is pure impulse to movement; and it is omnivorous. Everything serves as raw material, including its own self. When pure movement feeds upon pure movement, something may indeed arise out of nothing: the dialectic becomes conscious of itself and begins to construct its net.

Finally:

3. In metaphysics, the truth wholly ceases to be true when told in a false way. The activities of the misguided thinker issue, neither in partial truth nor partial falsehood. They have no issue. From the point of view of ultimate reality the activities remain confined within themselves; they fail to acquire an external reference...

(I hope you're following this.)

...For to 'begin with nothing' means rejecting the 'matter in hand' and the 'common purpose'. Metaphysics is not a 'subject' concerning which there may be partial agreement or disagreement. One simply refuses to understand 'results' which the dialectic cannot be made to generate entirely through its own resources.

It's a wonder that I ever succeeded in writing my D.Phil thesis. Needless to say, this version of Chapter 1 didn't make it into the final draft.

The thing is, Vaidyanathan, I know exactly what it was I was trying to do. I really thought this was possible. You start without any assumptions. Ground zero. Nada. Then you spin the dialectic, say 'Abracadabra' and 'something' emerges out of 'nothing'.

(I've tried this exercise of retracing my steps before, in the Glass House Philosopher, Notebook II, page 9. Disappointingly, the attempt fizzles out after a few pages.)

However, there's nothing wrong with starting again. What was I trying to do?

Let's get back on course. You're a mathematician, so you're familiar with mathematical proofs. Here's a famous proof invented by the Ancient Greeks long before anyone thought of axiomatizing arithmetic.

To prove: The square root of 2 is irrational.

1. Assume (for the sake of reductio) that the square root of 2 is rational.

2. Therefore the square root of 2 can be expressed as the fraction m/n, where m and n have no common factor.

3. Squaring both sides of the equation, 2 = m2/n2.

4. Therefore m2 = 2 n2.

5. Therefore m2 is even.

6. Therefore m is even.

7. If m is even, then m is 2k for some number k.

8. Therefore (substituting 2k for m) 2 = 4k2/n2.

9. Therefore 2n2 = 4k2.

10. Therefore n2 = 2k2.

11. Therefore n2 is even.

12. Therefore n is even.

13. If m and n are both even, then they have a common factor, viz. 2.

14. But this is a contradiction because we assumed that in m/n, m and n have no common factor.

15. Therefore the square root of 2 cannot be expressed as m/n where m and n have no common factor.

16. Therefore the square root of 2 is irrational.

As I said, this proof is familiar to any mathematician. But what I want you to try to do is picture what is going on in your mind, as someone who doesn't know that the square root of 2 is irrational, as you work through the proof step by step. Remember the first time you learned this proof (and imagine what you would have thought if you hadn't been taught in school that the square root of 2 is an irrational number). Or picture the (unknown) Greek mathematician who discovered it.

Now, I'm going to show you another proof. A direct quote this time. (I've just interpolated numbered steps.)

Let us imagine the following case.

1. I want to keep a diary about the recurrence of a certain sensation. To this end I associate it with the sign 'S' and write this sign in a calender for every day on which I have the sensation.

2. — I will remark first of all that a definition of the sign cannot be formulated.

3. But still I can give myself a kind of ostensive definition.

4. — How? Can I point to the sensation?

5. Not in the ordinary sense.

6. But I speak, or write the sign down, and at the same time I concentrate my attention on the sensation — and so, as it were, point to it inwardly.

7. — But what is this ceremony for? For that is all it seems to be! A definition surely serves to establish the meaning of a sign.

8. — Well, that is done precisely by the concentration of my attention; for in this way I impress on myself the connexion between the sign and the sensation.

9. — But 'I impress it on myself' can only mean: this process brings it about that I remember the connexion right in the future.

10. But in the present case I have no criterion of correctness.

11. One would like to say: whatever is going to seem right to me is right.

12. And that only means that here we can't talk about 'right'.

L. Wittgenstein Philosophical Investigations para. 258

On the face of it, there are notable differences between Wittgenstein's reductio ad absurdum of the notion of a 'private object' and the proof by the unknown Greek mathematician. But I would argue that these are superficial. In his famous para. 258, as well as the paragraphs leading up to and following it, Wittgenstein uses all his rhetorical gifts to get inside the head of someone who thinks that the notion of a 'private language' is possible. If you take all the extra trappings away, you get a 'proof' that is two, or at most three lines long. Even so, exactly the same thing is happening as in the arithmetical case.

My aim is: to teach you to pass from a piece of disguised nonsense to something that is patent nonsense.

L. Wittgenstein Philosophical Investigations para. 464

I can write down, 'The square root of 2 = m/n' but what I am writing down is impossible. It cannot be true. However, it takes a proof to see this. Prior to discovering the proof, you don't see it. The nonsense is disguised. Maybe (as the Greeks probably did), you spend hours and hours looking for a fraction which correctly represents the square root of 2, not realizing that all the time you were chasing a chimera. That is exactly what Wittgenstein says philosophers are doing, who put forward theories of the mind according to which feelings and sensations are 'private objects'.

Thousands of miles of ink have been spilled expounding, or defending, or attacking Wittgenstein's 'private language argument'. I'm not going to add anything to that now, except to say that I believe (as I believed 30 years ago) that this is the most important argument in the whole of metaphysics.

As I once described it, the argument is like a 'metaphysical wall'. You see a wall, blocking the path of your thoughts. You imagine that there must be some way round the wall, or under it, or through it. But there is not. 'You reach the wall, only to find you are facing the other way.'

Just as we can tease out the assumptions, the 'axioms' behind the theorems of arithmetic (as Peano attempted to do) so I'm sure that there are plenty of assumptions or axioms to tease out if you want to formalize your intuitive grasp of what metaphysics is, that is to say, what it is to seek a 'definition of reality' (or 'Being qua being' in Aristotle's sense). But it would be a mistake to think that the results of metaphysics therefore 'derive from axioms'. They do not. The arise through the determined attempt to think whatever is thinkable, to the ultimate extent and wherever that may take us.

Metaphysics progresses by demonstrating what is not thinkable. Metaphysics shows us that the things we thought were thinkable are not thinkable. It seeks to 'make the nonsense manifest'. It is tempting to assume (and here's a possible 'metaphysical axiom' if you want it) that whatever emerges from this exercise unscathed is thinkable. But we can never know this for sure. And so, just like mathematics, there is no end to the discovery of the 'truths of metaphysics'.

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