From: starpath@bard.MIT.EDU (David E. Hollingsworth)
Subject: Re: Truth
Organization: Massachvsetts Institvte of Technology
Date: Wed, 15 Apr 1992 19:24:44 GMT

	It is not at all clear that I understand what you want to know; I  
am responding to the question: "What do we know to be true?".  If this  
isn't what you want answered, then ask something else.

In article <1992Apr15.172713.5358@access.usask.ca> choy@skorpio.Usask.CA  
writes:
> 
> In the article <1992Apr15.092327.19633@athena.mit.edu> by
> David E. Hollingsworth (starpath@bard.mit.edu, NeXTMail: yummy), I get
> reply
> 
> >> > What do we know to be true?
> 
> >	Nothing without assumption, but if we don't assume a few things,  
> > we can't conclude anything either, and that's not very useful.

	I was not refering to tautologies; tautologous statements are true  
because Logic Says So, which, obviously relies on some assumptions.

> I wasn't asking "what tautologies are known?". I'm asking "what is true
> without proof (and perhaps without logic)?" (Maybe logic as we know it
> works.
	 The question "What is the set of all statements that are true  
without proof or assumption?" is answered by the statement: "The null  
set."  (Especially without logic; it doesn't make any sense to talk about  
"proof" without "logic"; you might _change_ the logic, but without logic  
at all, all one is making is assertions.)  If you are asking, "What things  
should be assumed?" then take your pick; there is no reason that you  
_have_ to assume some things and prove others using those assumptions.
	Haven't you read any Descartes?  Talk about bare-bones  
assumptions!
	Okay, okay, how about less concrete things; maybe you can prove  
simple mathematical statements:
	1 + 1 = 2 
sounds like a good starting point.  Here's what Gottlob Frege writes in  
_The Foundations of Arithmetic_, starting in the section marked, "Are  
numerical formulae provable?":
--------
	We must distinguish numerical formulae, such as 2 + 3 = 5, which  
deal with particular numbers, from general laws, which hold good for all  
whole numbers.  The former are held by some philosophers to be unprovable  
and immediately self-evident like axioms...The fact is that it conflicts  
with one of the requirements of reason, which maust be able to embrase all  
first principles in a survey.  Besides, is it really self-evident that:
	135664 + 37863 = 173527?
--------
	He goes on to talk about a problem with defining 2 to be 1 + 1, 3  
to be 2 + 1, and so on, to arrive at 2 + 2 = 4, spews for quite some time  
about whether numerical formulae are analytic or synthetic, a posteriori,  
or a priori.  He rips Mill:
--------
	Of all the whole wealth of physical facts in his apocalypse, Mill  
names for us only a solitary one, the one that holds is asserted in the  
definition of the number 3.  It consists, according to him, in this, that  
collections of objects exist, which while they impress the senses thus,
                               o   o
                                 o,
may be separated into two parts, thus:
                               o o   o.
What a mercy, then, that not everything in the world is nailed down; for  
if it were, we should not be able to bring off this separation, and 2 + 1  
would not be 3!
--------
	We can't know that things are true without proof.  We could simply  
assume them to be true (like Fermat's Last Theorem), or assume things that  
allow us to conditionally prove them to be true (like statements of the  
form: a + b = c), but there is no truth-test for statements without proof

> Henry Choy
> choy@skorpio.usask.ca

--
David E. Hollingsworth
starpath@bard.mit.edu
NeXTMail: yummy
Newsgroups: alt.atheism