On May 24, 2012, at 9:55 PM, Ed Barkmeyer wrote:
I am really feeling like my thread has been hijacked by
people who like to read their own writing:> conjecture. I have
purposefully avoided quoting any one person--but you know who you are.
Sitting on the sidelines, it seems to me that the thread has been
hijacked by people asking two rather important questions:
1) what is meant by "fact"? Is a 'fact' a belief of an individual or
a community, however large, or a statement whose truth value can
somehow be determined objectively, or just a postulate for a 'possible
world', independent of anyone's acceptance of that world as "reality"
in any sense.
2) what do you mean by "count"? First, you have to be able to
distinguish individuals: when are two "facts" different? And second,
you have to tell us whether you really mean only to produce some kind
of bound _expression_ for the quantity of facts of a kind, or you want a
mechanism for enumeration of all possible facts, i.e., a mechanism for
generating facts that guarantees to generate every fact at some point
in the process.
>From a linguistic point of view, the last would imply that one could
enumerate all existing 'things' and all meaningfully distinct verb
concepts, and thus somehow generate all possible sentences, and then
have a means of eliminating those that are false or non-determinable.
Alternatively, I suppose we might have a means of generating only
provable theorems, assuming we all agreed on what the universal set of
The point of the discussion is that all of these are hard questions.
Yes, the discussion goes down many paths in trying to answer them, and
some of those paths are less rewarding than others, where "rewarding"
is indeed in the eye of the beholder (or whatever the appropriate
metaphor for this form of experience might be).
Clearly and wisely summarized, as usual, Ed.
2. The Greek letter Pi represents the
irrational number that is the ratio between a circle's circumference
By comparison with 1, this is true because mathematicians since
Pythagoras have declared it to be so. It is a fiat. Others might
argue that pi represents a voiceless labial stop, as in Pythogoras'
name, a fiat of a different community.
The fact Mr Lange might have had in mind by #2 is that π — i.e., the number 3.14159… — is the ratio in question, which is by no means a matter of convention (as, of course, you would agree).
Are these not facts?
Well, you tell me.
Indeed. Of course, in ordinary, informal parlance, all of Mr Lange's examples are facts, truths, what have you. But the relevant question in the context of formal ontology (as you suggest) is whether or not to introduce facts as a category of thing, along with, e.g., numbers, persons, NC machines, etc. In that case, there are a lot of different ways one can go depending on one's purposes, on the theoretical roles that facts are supposed to play, how closely connected facts are supposed to be to language, etc.
Are they not countable?
If you mean "bounded", I doubt it. If you mean "countable" in the
Cauchy sense -- they can be algorithmically placed in 1-to-1
correspondence with the natural numbers -- I won't hazard a guess.
Here's one way an answer might go in a theory of facts:
1. For any objects r and s, if Fact1 is about r (and only r) and Fact2 is about s (and only s) and r ≠ s, then Fact1 ≠ Fact2. (Basic axiom for facts — or perhaps a theorem in a highly articulated theory of facts)
2. For all real numbers r, there is the fact about r (and only r) that r is a real number.
3. For any distinct real numbers r and s, that r is a real number ≠ that s is a real number. (From 1 and 2)
4. There are uncountably many real numbers.
5. There are uncountably many facts. (From 3 and 4)
(One niggling point: For a set S to be countable it is not necessary that it can be algorithmically placed in 1-1 correspondence with the set N natural numbers, that is, it is not necessary that there be a recursive bijection between S and N, only that there be a bijection, recursive or not.)
Again, aside from bending the space-time continuum, or
dismissing laws of nature like thermodynamics...I fail to see the need
for relativism here...or, what am I missing? If you agree that these
are facts, then let's get pragmatic and enumerate the
properties/boundaries around the nature of a fact.
That is, I think, what this brainstorming session is trying to do.
Right, an enumeration of the basic properties of facts is exactly what a theory is.
you have some suggestion about making the session more profitable? Do
you have any idea where to start?
I will confess the uselessness of this email by simply saying: I have
no idea how to do either.