|
|
Re: Proving a definition of multiplication (wrong) by induction
Posted:
Jan 30, 2013 10:47 AM
|
|
Jonathan Crabtree wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=8185797
> a = a added to a 1 time (the proposition)
>
> a does NOT equal a + a
I don't see anything of a mathematical concern here. You're
simply arguing over the English-language meaning of the phrase
"a added to a b times". Here are some examples of the statements
you're dealing with:
P(1) represents "a = 1*a"
P(2) represents "a + a = 2*a"
P(3) represents "a + a + a = 3*a"
P(4) represents "a + a + a + a = 4*a"
Each of these can be proved without mathematical induction
(in an appropriate formal setting).
We can also prove (in an appropriate formal setting) things like:
"P(1) and P(2)"
"P(1) and P(2) and P(3)"
"P(1) and P(2) and P(3) and P(4)"
This is simply because, in propositional logic, we have things like:
A,B |- (A and B)
A,B,C |- (A and B and C)
A,B,C,D |- (A and B and C and D)
That is, assuming A, B you can obtain "A and B". I say "obtain" because,
depending on the logical system you're working in, this might be an
axiom. Of course, a 1-line proof consisting of an axiom is technically
a proof of that axiom (in the sense that "proof" is understood in formal
logic), but I think there is less misunderstanding in the present context
if I say "obtain" rather than "prove".
The previous statements can be rewritten as follows:
(for all n in {1, 2})(P(n))
(for all n in {1, 2, 3})(P(n))
(for all n in {1, 2, 3, 4})(P(n))
Each of the statements just above can be proved without mathematical
induction.
To give a more complicated example, we don't need mathematical
induction to prove the following (in an appropriate formal setting,
ultrafinitistic views excepted):
(for all n in {1, 2, 3, ..., 10^10000})(P(n))
However, you do need mathematical induction to prove [*]:
[*] (for all n in {1, 2, 3, ...})(P(n))
Incidentally, being able to prove [*] (in an appropriate formal
setting) is one thing, while proving that one can't prove [*]
without mathematical induction is another thing. The "another
thing" requires a bit of formal logical training just to correctly
understand what it means to prove that something can't be proved
without mathematical induction.
Now part of the problem in proving [*] by mathematical induction
is to assign an unambiguous meaning to the statement P(n) for each
positive integer n (what you seem to be hung up on), but this is a
meta-language issue and not something that lies within the formal
system that the proofs take place in.
Dave L. Renfro
Message was edited by: Dave L. Renfro
|
|
