[FOM] predicative foundations
aatu.koskensilta at xortec.fi
Fri Feb 17 20:29:01 EST 2006
On Feb 17, 2006, at 10:51 AM, Eray Ozkural wrote:
> However, Turing's formalization and conception of computation, by
> 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
> 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
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
More information about the FOM