[FOM] predicative foundations

[Aatu Koskensilta]

On Feb 17, 2006, at 10:51 AM, Eray Ozkural wrote:

> However, Turing's formalization and conception of computation, by 
> talking
> about possible marks, is perfectly admissible, and explains everything
> about computation.

The theory of Turing machines or computability in general is indeed 
perfectly acceptable and can be motivated by considerations about 
possible marks on paper. Number theory, higher set theory and so forth 
are also perfectly admissible, although not likely to be thought of as 
having anything to do with marks on paper. Whether these theories 
"explain everything" is obviously a whole another matter.

> On the other hand,  you said that a similar account
> cannot explain numbers, because it seems according to you a number has 
> to
> be a unique, universal, independent entity.

I have no such requirements and am not even sure what such requirements 
actually amount to. I merely pointed out that talk about possible marks 
is in general no more and no less dubious than talk about natural 
numbers - however these are conceived -, and does not provide a 
"metaphysically uncontroversial" way of explaining N. Later you explain 
that the signs 0,1,2,3... as squiggles on paper have no independent 
meaning and are interpreted as numbers by humans. No such interpreting 
is actually going to take place for such "possible marks" as correspond 
to numbers in the range of 2^2^2^2^2^2^2 to 2^2^2^2^2^2^2^2^2^2^2, say. 
Sure, we can imagine squiggles corresponding to these numbers written 
down in some sense and then interpreted by some imaginary humans 
similar to us in some respect - an image which would "justify" for 
example the law of the excluded middle applied to decidable or at least 
feasible properties. Similarly, we can imagine digital computers with 
arbitrarily large amount of memory and reason about them.

These images, however, in no way serve to justify the mathematical 
principles and theories they motivate. This is because the assumptions 
embedded in the images - that is, presumed by our thinking the images 
are coherent in the first place - are in the philosophically relevant 
sense just the same as those expressed by the mathematical principles. 
It is possible that this is not always so. For example, it is possible 
that someone comes up with an intuitively compelling and coherent 
picture that motivates the strange set theory NF. This would be much 
more interesting than trying to justify the coherence of the idea of N 
by appeal to possible marks on paper, since in case of NF the various 
mathematical set existence principles don't appear to correspond to 
intuitive ideas in the sense the axioms governing the successor 
function correspond trivially to the picture of adding one more stroke 
to a ("possible") tally mark.

As Joe Shipman mentions in his posting, it's a rather remarkable fact 
that the various images or stances give rise, sometimes - perhaps 
often? - after tortorous analysis, to natural mathematical theorems 
(and formal theories) of exactly the same strength. Perhaps there is an 
illuminating explanation for this, perhaps it's just a brute empirical 
fact.

As to the relevance of a computational theory of mind to these musings 
I have nothing particularly insightful to say except to note that we 
are not in possession of such a theory and it seems rather pointless to 
try to guess what such a theory might imply. In particular, it is not 
in any way clear that a computational theory of mind implies that 
arbitrary natural numbers can be conceived by our "finite minds" while 
measurable cardinals or real numbers can't.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

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