[FOM] Re: Sharp mathematical distinction between potential and actual infinity? + Re: John Baez on David Corfield's book

[Aatu Koskensilta]

tom holden wrote:
 >
> Koskensilta's reply to Chow is precisely the kind of mathematical attempt at
> answering the philosophical question, which seems ill advised to me.
> Complexity measures all (?) come down to length of program and computation
> time on a suitable universal machine. And even if such machines do not have
> instructions as obviously vulnerable to scepticism as INC, their
> instructions have still been chosen to coincide with the ones which seem
> simple to us. Take the Turing machines one "merged" instruction of write
> value, move left or right and set new state. Why should the one instruction
> program "if in start state and current cell is blank, write one and move
> right, setting state back to the start state" write 1111111111111111111...
> rather than 1010101010... or 1101001010101010101010101011001... Our choice
> of the Turing machine's instruction rather than one of the rogue ones is
> based on our "subjective" considerations of its simplicity: the same
> considerations Koskensilta hoped to justify by appeal to Kolmogorov
> complexity and Turing machines.

I mentioned Kolmogorov complexity here because it's the obvious 
complexity metric that comes to mind. However, I explicitly said that 
Kolmogorov complexity in bare does not provide an adequate measure for 
deciding between competing rules, and that I don't know what an adequate 
metric would look like.

There is a sense in which you can't refute the skeptic's arguments: 
whatever justification you propose, he can claim to "not understand it", 
or understand it incorrectly. This doesn't mean that there is no 
interesting notions to be unearthed here. The bare fact exists that 
people do in fact learn to correctly follow rules, and hopefully it's 
not impossible to come up with an interesting epistemological notion 
explaining in general terms this ability. Similarly any skeptic can 
remain unconvinced of Chruch's thesis, but this doesn't mean that the 
notion of computability provided by the various formalisations is of no 
value.

There are some properties a complexity metric should have, e.g. it 
should make intuitively ad hoc rules and explanations more complex than 
intuitively more coherent and cohesive rules. Several other guiding 
principles can be formulated, but I won't go into them here, since they 
have no direct bearing to the skeptic's argument. I believe this line of 
research to be directly relevant to machine learning and also to more 
philosophical inquiries, although it doesn't touch the skeptic's 
argument directly.

It's also noteworthy that in the hypothetical Wittgensteininan dialogues 
with a skeptic (or an idiot who fails to grasp a simple rule), the idiot 
doesn't simply produce arbitary output after 10000 (or whatever), but 
starts to apply what seems to be another *rule*, i.e. add 2. The 
situation could thus be described as the student having failed to grasp 
the *correct* rule, not necessarily as him having not grasped any rule.

I also think it's fairly obvious that no human who is familiar with 
natural numbers and their properties will be *systematically* 
misinterpreting the operations carried on these numbers - i.e. that he's 
following a systematic rule, which however is not the correct rule -, on 
the basis that any such systematic misinterpretation is bound to be 
highly complex if it's to agree on all computations carried out (known 
to the person).

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus