The nature of proofs...

[silkworm]

I'm currently in a linear algebra class and we are doing a massive amount of proofs, something that is foreign to me.

 

Every time I'm asked to do a proof my effort turns into a narrative argument, and my problem isn't knowing why something is true or not. I piece it together in my mind and can argue it, and if I had to teach somebody whether or not something is true I have faith in my abilities to do so, but I'm having difficulty providing any type of valid mathematical proof.

 

What's lost on me appears to be the logic behind the language of proofs and what assumptions can be made when doing them. A lot of proofs make perfect sense to me when I read them, but sometimes it appears as though the proof is only manipulating symbols instead of what the symbols mean.

 

What I'm asking for here is a clue to the nature of proofs. I'm sure the shortcoming is my own and not that of mathematics. How would you explain how to do a proof and how are valid assumptions in proofs are made?

 

Thank you in advance.

[matt grime]

I'm currently in a linear algebra class and we are doing a massive amount of proofs, something that is foreign to me.

 

It is hard to begin with, everyone finds that.

 

Every time I'm asked to do a proof my effort turns into a narrative argument, and my problem isn't knowing why something is true or not. I piece it together in my mind and can argue it, and if I had to teach somebody whether or not something is true I have faith in my abilities to do so, but I'm having difficulty providing any type of valid mathematical proof.

 

What's lost on me appears to be the logic behind the language of proofs and what assumptions can be made when doing them. A lot of proofs make perfect sense to me when I read them, but sometimes it appears as though the proof is only manipulating symbols instead of what the symbols mean.

 

well, at some level proofs are just manipulating symnbols, but the meaning of the symbol tells you what manipulations are permissible.

 

What I'm asking for here is a clue to the nature of proofs. I'm sure the shortcoming is my own and not that of mathematics.

 

I think that went withou saying.

 

How would you explain how to do a proof and how are valid assumptions in proofs are made?

 

That is difficult since there is no one way to prove something, or to write its proof out.

 

Experience is the best thing I can suggest. Just keep doing them, practising them. Try writing out some of the proofs from the book and decide what they're doing at each step.

 

Valid assumptions take one of two forms. The hypothesis specifically tells you you are allowed to assume something OR the assumption is clearly true and needs no justification.

 

Example:

 

Show that an integer is a perfect square if and only if all its prime factors occur with an even exponent.

 

So there are two things to do here (if and only if). I must show each condition implies the other.

 

Suppose that all prime factors have an even exponent, ie suppose

 

[math]n=p_1^{2e_1)\ldots p_t^{2e_2}[/math]

 

then

 

[math]n=(p_1^{e_1}\ldots p_t^{e_t})^2[/math]

 

so it is a perfect square.

 

Here I used the assumption given to me by the question. I also used the incredibly important and much over looked fact that the statement: m is an even integer is equivalent to m=2k for some integer.

 

Now, that is only one way in the implication, now I must go the other way.

 

So, suppose that n is a perfect square, ie n=m^2, and if m is the unique product

 

 

[math]m=p_1^{e_1}\ldots p_t^{e_t}[/math]

 

then n has the required properties of its prime decomposition.

 

Here I am assuming the prime number theorem, that every decomposition of an integer into the product of primes is essentially unique.

[the tree]

I'm far from sure what a proof really is.

For instance, is this a proof or just a demonstration?

 

suppose: log_ax=b and log_ay=c

or: a^b=x and a^c=y

so: xy=a^ba^c=a^b+c

or: log_axy=b+c

 

therefor: log_axy=log_ax+log_ay

[matt grime]

It is a proof, of the way logs convert mult to addition, based upon the definition of what log is.