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Flavors of Facts

Date: 10/03/2003 at 13:58:14
From: Karen
Subject: Is 1 + 1 = 2 an actual fact?

Is it a fact that 1 + 1 = 2?  I have seen your proof using the Peano 
postulate.  Is the postulate a hypothesis which is unproven, or is it 
proven, i.e., a fact?

For example, 1 + 1 = 10 in base 2.  So is the value of 1 + 1 open to
interpretation? 

I think I find some of the terminology confusing, e.g., what do we
really mean by the terms 'fact', 'premise', 'assumption', 'axiom',
'postulate', and so on? 


Date: 10/03/2003 at 16:46:46
From: Doctor Peterson
Subject: Re: Is 1 + 1 = 2 an actual fact?

Hi, Karen.

You'll have to decide for yourself what you mean by "actual fact"; 
philosophers can have trouble pinning that down!

But consider that every thought has to be based on something else; 
there has to be some starting point, since you reason ABOUT something.
So in order to say something is true, you have to believe that its
premises are true first.  That might be an observation (but how do you
know that your observations are true?); or it can be an "assumption",
something we take as true as the basis of our reasoning. 

That is how we think of math: we choose some set of axioms (or
postulates, which are the same thing) and definitions as our starting
point, the things we are thinking about.  We can choose different
axioms and come up with different mathematical systems (such as
Euclidean or non-Euclidean geometry).  But once we choose them, we
consider them to be true -- within the particular system we are
working on.

An axiom or postulate can't be proven, since there is nothing before 
it on which to build a proof; it stands at the base of the
mathematical system that is built on it.  So it is an assumption.  But 
that doesn't make it untrue; it is the truest thing there is _within 
that system_, the basis of the whole construction.  Outside of that 
system, there is really nothing to tell you whether it is true or
false.  But we are not "assuming" something about some entity outside
of the system, that might really be true or false; rather, we are
"assuming" something only in the sense of deciding what it is that our
system is about.

But your example of 1+1=10 is not really an illustration of any
problem with axioms and assumptions.  It is nothing more than
notation: the numeral 10 in binary is just a different way to WRITE
the number 2.  The fact you are stating MEANS exactly the same thing 
as 1+1=2. 

On the other hand, there is a system (modulo 2 arithmetic) in which
1+1=0.  That is no less true than 1+1=2, within its system; but the
meanings of 1, +, and = are different than in normal arithmetic.  We
are talking about different things, based on different definitions.
One is a fact about integers, the other is a fact about modulo-2
numbers.  One doesn't contradict the other; they just live in
different worlds of thought, which are built on different definitions 
and axioms.

So to answer your basic question, yes, 1+1=2 is a fact--given that 1
and 2 refer to the integers 1 and 2, and that + and = have their
normal meanings.  All the axioms and definitions on which the real
number system are based are assumed when I say that!  If you don't
make some such assumptions, then "1+1=2" has no meaning; it is just a 
string of symbols on your computer screen, and can't be said to be 
either true or false.

Here is something to read about these ideas:

  The Role of Postulates
    http://mathforum.org/library/drmath/view/62560.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 10/03/2003 at 17:45:37
From: Karen
Subject: Thank you (is 1 + 1 =  2 an actual fact)

Dear Dr. Math,

Thanks for that excellent explanation.  It has made things much
clearer in my mind. 

In your explanation you make reference to "premises" and
"assumptions".  Are these the same thing, or are there subtle
differences between the two?

Regards,
Karen


Date: 10/03/2003 at 22:31:12
From: Doctor Peterson
Subject: Re: Thank you (is 1 + 1 = 2 an actual fact)

Hi, Karen.

Well, since I was using the words in an ordinary English sense, and
not as math jargon, let's see what a dictionary (m-w.com) says about 
them, and whether I agree:

  premise

  1 a : a proposition antecedently supposed or proved as a basis of
        argument or inference; specifically: either of the first
        two propositions of a syllogism from which the conclusion
        is drawn

    b : something assumed or taken for granted: PRESUPPOSITION

  assumption

  5 a : an assuming that something is true

    b : a fact or statement (as a proposition, axiom, postulate, or
        notion) taken for granted

The (b) parts are pretty much equivalent; but as the (a) parts show,
"premise" suggests more emphasis on the logical relationship (that the
premise comes before something derived from it), while "assumption"
puts more emphasis on the fact that it is not derived from anything
else.  And I think that's about what I had in mind.  They're two
halves of the same fact, that something is assumed without proof, so
that something else can be proved.

It's nice that lexicographers spend time thinking about these
distinctions, so we have a head start!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
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