Is This Basic Proof of the Property of 0 Circular?

[AmagicalFishy]

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:

[Prove]
If a and b are real numbers, then:
ab = 0 \Leftrightarrow a = 0 or b = 0

Now recall that 'P or Q' is logically equivalent to '(not P) \Rightarrow Q', so the goal may be rewritten as 'a \neq 0 \Rightarrow b = 0. Adopting the direct strategy for proving this gives the following:

Since a \neq 0, we can divide through by a (or multiply through by 1/a) so that ab = 0 \Rightarrow 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 \Rightarrow ab = 0.

The book proves both the \Rightarrow statement and the \Leftarrow 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?

 

[DonAntonio]

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 \Rightarrow statement and the \Leftarrow 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?

No. The proof that x\cdot 0=0\,\,\,\forall x\in\mathbb R\,\, is straigthforward using the usual axioms:

x\cdot 0 =x\cdot (0+0)=x\cdot 0+x\cdot 0\Longrightarrow x\cdot 0+(-x\cdot 0)=x\cdot 0+x\cdot 0+(-x\cdot 0) \Longrightarrow 0 = x\cdot 0\,\, and we're done.

The proof of what you want is given in the extract you wrote and it's based on the existence of multiplicative inverse for any

non-zero element in \mathbb R.

 

[chiro]

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 \Rightarrow statement and the \Leftarrow 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?

Hey AmagicalFishy and welcome to the forums.

I see where your comment is headed and how it can seem pointless, but the point is to use the logic where you prove a bidirectional implication.

Yes intuitively (and remember most mathematics, if not all has an intuitive basis somewhere in it) is that if something is zero, then something must be zero.

I think to give one possible answer to your question, is that you have to use a few identities for zero, which in your argument I can see has a valid point.

The typical way of defining these things is to use some of the standard arguments that you prove when you prove that something is a vector space. In the vector space proof, you prove ten axioms and one these relates to proving things involving zero.

I don't know the formal way of proving the zero properties, but the involve proving that things don't blow up with things like distributive laws and things like that as well as division. There are a few threads here on this kind of thing where people ask why 0x = 0 and why you can't divide by zero which lends itself to why all the other associative, distributive and all that are defined.

Also the thing gets more complicated if you are working with a general group rather than just a number.

I think though, that if you want to prove the zero property, you can use the kind of things that are done in proving vector spaces, or if you want to get super anal, you can go to the very foundation of mathematics and prove it using the foundational natural number axioms which I think are based on either Peano construction or something similar to it (I know there are problems with Peano construction in terms of things to do with Godels incompleteness theorem which given you are taking logic is probably wise to look into, but I don't know the extent of the implications on proving the zero identities).

 

[HallsofIvy]

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 \Rightarrow statement and the \Leftarrow 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?

Not if the properties of 0 used are not the particular one we want to prove. If I want to prove some property of "A", I have to use the fact that it is A! And that means I have to use some properties of A- those that I have already proved.

 

[ssd]

Not if the properties of 0 used are not the particular one we want to prove. If I want to prove some property of "A", I have to use the fact that it is A! And that means I have to use some properties of A- those that I have already proved.

Excellent. Really like the explanation. Candid.