- #1
AmagicalFishy
- 50
- 1
Hi, folks. I'm sure this ends up being a philosophical question regarding logic—and I'm hoping someone can point me to an appropriate philosopher, logician, or better: book/paper/website. I want to say that I'm looking for Ludwig Wittgenstein (my reading regarding formal logic is fairly limited), but I'm not entirely certain.
This semester I found that I'm not as well-versed at mathematics as I should be, so I started independently reading some books. Currently I'm going through a book entitled "An Introduction to Mathematical Reasoning" which contains this proof:
The book proves both the [itex]\Rightarrow[/itex] statement and the [itex]\Leftarrow[/itex] statement—and it seems, at first, like a very simple proof. My qualm is this: If someone were to claim, "ab = 0 implies a = 0 or b = 0"—how can using the properties of 0 prove the statement? Isn't that circular, because what we want to prove in the first place is a statement of a property of 0 itself?
This semester I found that I'm not as well-versed at mathematics as I should be, so I started independently reading some books. Currently I'm going through a book entitled "An Introduction to Mathematical Reasoning" which contains this proof:
Introduction to Mathematical Reasoning said:[Prove]
If a and b are real numbers, then:
ab = 0 [itex]\Leftrightarrow[/itex] a = 0 or b = 0
Now recall that 'P or Q' is logically equivalent to '(not P) [itex]\Rightarrow[/itex] Q', so the goal may be rewritten as 'a [itex]\neq[/itex] 0 [itex]\Rightarrow[/itex] b = 0. Adopting the direct strategy for proving this gives the following:
Since a [itex]\neq[/itex] 0, we can divide through by a (or multiply through by 1/a) so that ab = 0 [itex]\Rightarrow[/itex] b = 0 as required. A formal proof could be set out as follows.
It is a basic property of 0 that 0 x b = 0 = a x 0. Therefore, a = 0 or b = 0 [itex]\Rightarrow[/itex] ab = 0.
The book proves both the [itex]\Rightarrow[/itex] statement and the [itex]\Leftarrow[/itex] statement—and it seems, at first, like a very simple proof. My qualm is this: If someone were to claim, "ab = 0 implies a = 0 or b = 0"—how can using the properties of 0 prove the statement? Isn't that circular, because what we want to prove in the first place is a statement of a property of 0 itself?