It can be asked how we can know or prove what we can't know or prove. Epistemologists, for example, have come at this question from different angles.

Take
David Lewis. He had a notion of “proper
ignoring” (shown in his paper 'Elusive
Knowledge' – 1996) in which he more or less says that some
propositions either can't be known to be true (or false) or must be ignored in
order to get on with those propositions we do need to know at a
particular juncture. Or, in Wittgenstein's case, those things we know
we can't know (or at least question) act as hinges on which the
rest of our questioning and endeavours hang. (Having said that, this
is more a question of not attempting to know rather than one of the
impossibility of knowing.)

**Kurt Godel**

Kurt
Godel did something similar with his theorems. However, instead of
knowledge, we can now talk of proof.

Godel
proved that certain statements within a system can't be proven even
though they can still be taken to be true. Godel knew that we
couldn't prove these statements. Still, as I said, such statements
are still taken to be true; just not

*provably*true.
Nonetheless,
Godel's proof of a

*lack of proof*did go against what certain mathematicians – or even the majority – believed at the time (e.g., the mathematical constructivists).
More
recently, John Horgan quotes Gregory Chaitin (a mathematician and
computer scientist at IBM)
saying the following:

“

**Normally you assume that if people think something is true, it's true for a reason. In mathematics a reason is called a proof, and the job of a mathematician is to find the proof, the reasons, deductions from axioms or accepted principles.”****(230)**
However,
Chaitin goes on to say:

“

**Now what if I found is mathematical truths that are true for no reason at all. They are true accidentally or at random....”**
The
constructivists certainly
believed that truth - as well as numbers themselves - came along (as it were) with proof. A statement
becomes true only when it's proved to be true. It can't be true for
any other reason. So, as far as the constructivists were concerned,
statements such as “mathematical truths that are true for no reason
at all” simply don't make sense. It's almost equivalent to
saying: “I think that Santa exists; though I can give no reason for
believing that he exists.”

The
same can be said of mathematical statements being true “accidentally
or at random” - on a certain picture of maths, such things are
quite literally impossible (or even meaningless).

Yet,
of course, Godel proved otherwise. He proved that certain true
statements can't be proved to be true; yet they're still true. The
question remains:

*How did Godel know that these statements were true?*
More
specifically, is it possible for someone who doesn't know the maths
to get his head around the the notion of unprovable - yet true -
mathematical statements/equations? (It's here that Roger Penrose
talks of 'seeing' their truth. Others speak of 'intuition'.)

**Alan Turing**

Whereas
Godel was concerned with unprovable truths, Alan Turing discovered
unsolvable problems. Ray Kurzweil (mentioned in the last post)
writes:

“

**These are problems that are well defined with unique answers that can be shown to exist, but that we we can also prove can never be computed by any Turing machine – that is to say, by***any*machine...” (187)
Admittedly,
this is a rough explanation of what Turing advanced.

First
of all, it's hard to get a grip of what's meant by 'problems' here. I
assume they're mathematical problems which the computer fails to
prove. So here again computers are backing-up what Godel had already
discovered in his “incompleteness theorem” of 1931. (Having said that,
the Turing machine, at least in the Turing test, isn't all about mathematical questions or solutions.)

The
words “unique answers that can be shown to exist, but that we can
also prove can never be computed by any Turing machine” again show
the Godelian angle. Put simply, it's about proving that

*X*or (*X*'s) can't be proved. You have a proof that*X*can't be prove.
Despite
saying that, I still have a problem with the opening clause which
says that “unique answers that can be shown to exist”. So those
unique answers are known, though we also know that they can never be
computed by any Turing machine. Can we know of unique answers if they
haven't been computed? If the answer is 'yes', then how is that the
case? Perhaps “showing” the existence of “unique answers”
isn't the same as

*knowing*(not even proving) the unique answers. Indeed the word “showing” (rather than “knowing” or “proving”) seems to hint at intuition or “mathematical insight” again.
Following
on from some of the things which have just been said, it's also hard
to understand what's meant by saying that “Turing showed that there
as many unsolvable problems as solvable ones”. One's immediate
reaction to a statement like that is:

*How could Turing have possibly known that?*The only answer I give is the mathematical truism that because all infinities are equal (in that they're all, well, infinite), then if solvable mathematical problems are at least potentially infinite in number, then the same must also apply to unsolvable mathematical problems.**********) Of course, according to Georg Cantor and others, infinite sets aren't all “equal” (i.e., they're "larger" or "smaller" than each other) even though they're all, well, infinite – they're only equal in, as it were,*infiniteness*.
## No comments:

## Post a comment